Hydromagnetic Convection Flow in a Porous Medium Bounded Between Vertical Wavy Wall and Parallel...

RESEARCH ARTICLE

Hydromagnetic Convection Flow in a Porous Medium BoundedBetween Vertical Wavy Wall and Parallel Flat Wall: AnalysisUsing Darcy–Brinkman–Forchheimer Model

R. R. Singh • Ajay Kumar Singh • Usha Singh •

Atul Kumar Singh • N. P. Singh

Received: 24 April 2013 / Revised: 18 March 2014 / Accepted: 21 March 2014 / Published online: 25 June 2014

� The National Academy of Sciences, India 2014

Abstract The present paper deals with two-dimensional

convection flow and heat transfer of a viscous, incom-

pressible, conducting fluid, through porous medium con-

fined in a vertical wavy wall and parallel flat wall under the

influence of transverse magnetic field. The non-linear

equations governing the flow are solved by linearization

technique under the assumption that the flow consists of

two parts; a mean part and a perturbed part. Darcy–

Brinkman–Forchheimer model is used to visualize the

problem. Exact solutions are obtained for the mean part

and the perturbed part is solved using long wave approxi-

mation method. The effects of various parameters, zeroth

order and total axial velocity fields, transverse velocity,

temperature distribution, shear stress and heat transfer rate

are presented graphically and discussed.

Keywords Hydromagnetic convection flow �Porous medium � Vertical wavy walls

List of Symbols

a Amplitude,

B0 Uniform magnetic field in y0-direction,

C Prescribed constant,

Cp Specific heat at constant pressure,

Da Darcy number,

F Forchheimer parameter,

g Gravitational acceleration,

Gr Buoyancy parameter,

H Distance of flat wall,

k Thermal conductivity,

K0

Permeability of the medium,

K Permeability parameter,

KT Thermal conductivity,

L Characteristic length,

M Magnetic parameter,

Nu Nusselt number,

p0

Dimensional pressure,

p Non-dimensional pressure,

Pr Prandtl number,

T0

Temperature of the fluid in the boundary layer,

T Non-dimensional temperature,

Tf

0Temperature at the surface of the flat wall,

Tw

0Temperature at the surface of wavy wall,

Tm

0Temperature of the ambient fluid,

TR Temperature ratio parameter,

u0, v0

Velocity components in x0

and y0-direction,

u, v Velocity components in x and y -direction,

x0, y0

Dimensional cartesian coordinate,

x, y Non-dimensional cartesian coordinate

Greek Symbols

e Non-dimensional amplitude,

2 Porosity of the porous medium,

b Volumetric coefficient of thermal expansion,

k0

Dimensional wave number,

k Non-dimensional wave number,

R. R. Singh (&) � A. K. Singh � U. Singh

Department of Mathematics, C. L. Jain College, Firozabad 283

203, India

e-mail: [email protected]

A. K. Singh


A. K. Singh

Department of Mathematics, V. S. S. D. College, Kanpur 208

002, India

N. P. Singh

Department of Mathematics, Rama Institute of Engineering &

Technology, Mandhana, Kanpur 209 217, India


123

Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. (July–September 2014) 84(3):409–431

DOI 10.1007/s40010-014-0153-5

c Viscosity ratio parameter,

q Density of the fluid,

r Electrical conductivity of the fluid,

leff Effective viscosity of the fluid,

l Viscosity of the fluid,

t Kinematic viscosity,

sw Skin-friction at the wavy wall (y = -1),

sf Skin-friction at the flat wall (y = 1)

Introduction

The study of heat transfer from irregular surfaces is of

importance, because irregular surfaces are often present in

engineering applications. Therefore, the study of convec-

tive flows in wavy channels is essential to observe the flow

characteristics and heat transfer processes in several heat

devices with variations in material parameters. The channel

consisting of a corrugated wall and flat wall is one of the

several devices employed for enhancing the heat transfer

efficiency of industrial processes. Surface non-uniformities

of such channels are encountered in insulating systems and

grain storage containers. The presence of roughness on the

surface disturbs the flow past surfaces and alerts the heat

transfer rate. As such, the natural convection in vertical

channels consisting of a wavy surface and a flat surface is a

model problem for the investigation of momentum and heat

transfer from roughened and flat surfaces of the wall in

order to understand effect of material parameters on flow

characteristics.

Lekoudis et al. [1] investigated incompressible boundary

layer flow of Newtonian viscous fluid over a wavy wall

analytically and observed that at low Reynolds numbers the

waviness of the wall quickly ceases as the liquid is dragged

along the wall, whereas at large Reynolds numbers the

effects of viscosity are confined to a thin layer close to the

wall. Bordner [2] presented non-linear analysis of laminar

boundary layer flow over a periodic wavy surface and

concluded that some non-linear terms in the disturbed

boundary layer equations are of first order, if the wave

amplitude and disturbance sub-layer are comparable in

magnitude. Vajravelu and Sastri [3] considered free con-

vection heat transfer in a viscous incompressible fluid

confined between a vertical wavy wall and a flat wall to

study the effects of wavy geometry, buoyancy and material

parameters. Das and Ahmed [4] extended this study con-

sidering free convection flow of a viscous incompressible

fluid between a long vertical wavy wall and parallel flat

wall under the influence of uniform transverse magnetic

field. Caponi et al. [5] obtained analytical solution of

laminar flow of an incompressible viscous fluid over a

moving wavy surface and observed significant effects of

the phase of the wavy surface on the flow field and

temperature distribution in natural convection through

porous media. Yao [6] investigated natural convection flow

of Newtonian fluids using finite-difference scheme to study

the heat transfer from isothermal vertical wavy surface.

Rao et al. [7] investigated hydromagnetic convective flow

in a vertical wavy channel with heat source/sink and dis-

cussed the effects of magnetic field, internal heat genera-

tion/absorption and amplitude wavelength ratio on the heat

transfer coefficient. Rees and Pop [8, 9] studied free con-

vection flow along a vertical wavy surface with constant

wall temperature and uniform wall flux respectively.

Hossain and Pop [10] investigated the magnetohydrody-

namic boundary layer flow as well as heat transfer from a

continuously moving wavy surface and observed that

waviness of the surface suppresses the momentum transfer

but increases the heat transfer, Alim et al. [11] studied heat

transfer from a fixed wavy vertical surface in the presence

of transverse magnetic field and discussed the effects of

waviness on flow behaviour and heat transfer rate. Reddy

et al. [12] presented theoretical analysis on convection flow

of a viscous heat generating fluid in a vertical wavy

channel considering unsteadiness due to the imposed

oscillatory flux on the flow using perturbation technique to

solve the non-linear equations governing the flow. Patidar

and Purohit [13] investigated free convection flow of a

viscous incompressible fluid in porous medium between

two long vertical wavy walls and found that the heat

transfer varies periodically along the wavy surface.

Taneja and Jain [14] examined unsteady free convection

flow of viscous fluid in the presence of temperature

dependent heat source/sink in a rotating wavy channel with

travelling thermal waves and solved the non-linear equa-

tions governing the flow using perturbation technique.

Hossain et al. [15] considered natural convection flow

along a heated vertical wavy surface in presence of tem-

perature dependent viscosity and found that the tempera-

ture dependent viscosity has significant effect on the flow

behaviour and heat transfer at the wavy surface. Molla

et al. [16] studied natural convection flow along a vertical

wavy surface and observed the effect of internal heat

generation/absorption. In their analysis, the equations

governing the flow are mapped into the domain of flat

vertical plate and then solved numerically employing the

implicit finite-difference method and Keller-box scheme.

Sarangi and Jose [17] studied unsteady flow of a viscous,

electrically conducting fluid through a porous medium

confined between two long non-conducting wavy walls

under the influence of uniform magnetic field. Tak and

Kumar [18] investigated two-dimensional hydromagnetic

free convection flow of a viscous fluid in a vertical wavy

channel with heat source considering one channel wall as

isothermal and the other as adiabatic. Yao [19] investigated

the same problem for the natural convection flow along a

410 R. R. Singh et al.

123

vertical wavy surface with complex boundary conditions.

Molla and Hossain [20] have investigated the effect of

thermal radiation on steady two-dimensional mixed con-

vection flow along a vertical wavy surface using the

appropriate similarity variables. For the analysis, the basic

equations governing the flow are transformed into conve-

nient form, and then solved numerically employing two

efficient methods, namely; Keller-box method (KBM) and

straight forward finite difference method (SFFDM).

The references pertaining to flow through porous media

are based on Darcy law. However its importance and

ramifications in process design and operation have been

recognized only during the last three/four decades. As an

improvement, Irmay [21] presented an analysis on the

theoretical derivation of Darcy and Forchheimer models

and observed that Forchheimer model is more appropriate

at low Darcy numbers. Nakayama et al. [22] studied free

convection over a non-isothermal body of arbitrary shape

in a saturated porous medium using Forchheimer equation.

Consequently, the mechanism of convective heat transfer

in a porous medium has occurred in the literature for many

processes and unit operations involving a range of geo-

metric configurations such as the Darcy-Forchheimer

model and Darcy Brinkman-Forchheimer model.

Kladias and Prasad [23] experimentally verified the

Darcy–Brinkman–Forchheimer model for natural convec-

tion in porous media. Shenoy [24] studied natural, forced

and mixed convective heat transfer in non-Newtonian

power law fluid-saturated porous medium using Darcy-

Forchheimer model. Vafai and Kim [25] presented

numerical study based on the Darcy–Brinkman–Forchhei-

mer model for the forced convection in a composite system

containing fluid and porous regions and discussed about the

limitations of the Darcy–Brinkman–Forchheimer equa-

tions. Sequentially, Knupp and Lage [26] studied general-

ized form of Forchheimer-extended Darcy flow model to

the permeability case via a variation principle. Marpu [27]

investigated the Forchheimer and Brinkman extended

Darcy flow model for natural convection in a vertical

cylindrical porous medium. Whitaker [28] presented the-

oretical derivation of the Forchheimer flow model using

modern statistical averaging technique. Nakayama [29]

considered unified treatment of Darcy-Forchheimer

boundary layer flows. Besides, Bennethum and Giorgi [30],

Giorgi [31], Lees and Yang [32], Levy et al. [33], Chen

et al. [34] and Vafai [35] have studied Darcy-Forchheimer

model for different flow situations. The papers of Srinivas

and Muthuraj [36] and Tasnim et al. [37] are also on the

same line.

Paul et al. [38] have analysed the transient behaviour of

natural convection flow in porous region bounded by two

vertical walls as a result of asymmetric heating/cooling of

the channel walls using Forchheimer-Brinkman extended

Darcy model. More recently, Singh et al. [39] extended the

work of Paul et al. [38] considering the presence of internal

heat generation/absorption under the influence of uniform

transverse magnetic field within the limitations described

by Nield and Bejan [40]. In order to simulate momentum

transfer within the porous medium, Brinkman-Forchheimer

extended Darcy model is used and convection currents

between the walls occur as a result of change in the tem-

perature of the channel walls to that of the temperature of

the flow domain. Common method of producing convec-

tive flow is used, wherein the thermal energy is supplied

externally to create temperature difference across the sys-

tem. However, heat transfer in fluid flow by natural con-

vection as a result of difference in temperature gradient of

two vertical walls is also affected by temperature ratio of

the channel walls which is of immense importance partic-

ularly in heat transfer devices in industrial processes and

aerospace technology. Therefore, the present study

addresses the free convective flow of viscous incompress-

ible fluid through porous medium confined in vertical

channel consisting of a wavy wall and a parallel flat wall. A

generalized porous media model, namely, Brinkman-

Forchheimer extended Darcy model is considered in order

to simulate momentum transfer in porous medium. Since

new emerging technological models are applicable over a

wide range of porosity, the present paper deals with flow

through porous medium using Brinkman-Forchheimer

extended Darcy model in the presence of transversely

applied uniform magnetic field. Besides, the convection

phenomenon between the walls is set by changing the

temperature of the walls to that of the fluid temperature and

a non-dimensional parameter (buoyancy force parameter)

is used to characterize the temperature of vertical walls

with respect to the fluid temperature. The present model

enhances the applicability of the models suggested by Paul

et al. [38] and Singh et al. [41]. The model can have

possible application in heat transfer devices of industrial

processes, nuclear energy technology and geothermal

systems.

Formulation of the Problem

We consider steady two-dimensional laminar free con-

vection flow of an incompressible, electrically conducting,

viscous fluid through a homogeneous porous medium

confined in an open ended vertical channel consisting of a

wavy wall and a flat wall as shown in Fig. 1. A uniform

magnetic field is applied perpendicular to the walls, i.e.

normal to the flow region. It is assumed that the magnetic

Reynolds number is small, so that the induced magnetic

field is neglected in comparison to the applied magnetic

field (Ferraro and Plumpton [42]).

Hydromagnetic Convection Flow in a Porous Medium 411

123

The wavy and flat walls are respectively maintained at

different constant temperatures Tw

0and Tf

0such that the fluid

temperature Tm

0= Tw

0. Besides, the present analysis is

based on the following assumptions:

(i) All the fluid properties are constant except influence

of density variation with temperature in buoyancy

term in the momentum equation.

(ii) The density is a linear function of temperature so that

usual Boussinesq approximation is taken into account.

(iii) The wavelength k0

of the wavy wall is large

compared with width of the channel.

(iv) In the energy equation, the dissipative effect is

negligible.

(v) The inter-particle and inertia-particle heat transfer

effects are negligible.

(vi) Darcy-Brinkman-Forchhiemer model is considered

to simulate momentum transfer in porous media,

which is applicable over a wide range of porosity

(Nield and Bejan [40]).

(vii) Transport properties of the fluid are constant and the

fluid rises in the channel driven by buoyancy forces.

(viii) Both the walls are in stationary position, as such the

boundary conditions on velocity components u0and

v0

are the no slip conditions.

Under the above stated configuration and assumptions,

the equations governing the flows are:

Continuity equation:

ou0

ox0þ ov0

oy0¼ 0: ð1Þ

Momentum equations:

u0ou0

ox0þ v0

ou0

oy0¼ � 1

qop0

ox0þ

leff

qo2u0

ox02þ o2u0

oy02

� �

þ gb T 0 � T0

m

� �� t

ku0 � rB2

0

qu0 � Fffiffiffi

kp u

02: ð2Þ

u0ov0

ox0þ v0

ov0

oy0¼ � 1

qop0

oy0þ

leff

qo2v0

ox02þ o2v0

oy02

� �� t

kv0

� Fffiffiffikp v

02: ð3Þ

Energy equation:

u0oT 0

ox0þ v0

oT 0

oy0¼ KT

qCp

o2T 0

ox02þ o2T 0

oy02

� �: ð4Þ

The relevant boundary conditions on velocity and

temperature are:

u0 ¼ 0; v

0 ¼ 0; T0 ¼ T

0

w at y0 ¼ �H þ a cos k

0x0

� �;

u0 ¼ 0; v

0 ¼ 0; T0 ¼ T

0

f at y0 ¼ H:

9=;ð5Þ

We introduce following non-dimensional variables and

parameters:

x ¼ x0

H; y ¼ y0

H; u ¼ Hu0

t; v ¼ Hv0

t; c ¼

leff

l;

p ¼ H2qp0

l2; T ¼ T 0 � T 0m

T 0w � T 0m; ; k ¼ k0H and, e ¼ a

H; Da

¼ k

H2:

M ¼ffiffiffiffiffirqt

rB0H magnetic parameterð Þ;Gr

¼gbH3 T 0w � T 0m

� �t2

buoyancy parameterð Þ;

Pr ¼ lCp

KT

Prandtl numberð Þ:

The symbols are defined in the nomenclature.

Introducing above mentioned non-dimensional variables

and parameters, Eqs. (1)–(4) becomes:

ou

oxþ ov

oy¼ 0: ð6Þ

uou

oxþ v

ou

oy¼ � op

oxþ c

o2u

ox2þ o2u

oy2

� �

þ GrT � 1

Dau�M2u� Fffiffiffiffiffiffi

Dap u2:

ð7Þ

Fig. 1 Physical model and coordinate system


123

uov

oxþ v

ov

oy¼�op

oyþ c

o2v

ox2þ o2v

oy2

� �� 1

Dav� Fffiffiffiffiffiffi

Dap v2: ð8Þ

Pr uoT

oxþ v

oT

oy

� �¼ o2T

ox2þ o2T

oy2: ð9Þ

The boundary conditions (5) on velocity and

temperature in non-dimensional form are:

u ¼ 0; v0 ¼ 0; T ¼ 1; y ¼ �1þ e cos kx;

u ¼ 0; v0 ¼ 0; T ¼ TR; y ¼ 1:

)ð10Þ

where TR ¼T 0

f�T 0m

T 0w�T 0m(temperature ratio parameter).

Now we proceed for the solution of Eqs. (6)–(9) under

boundary conditions (10).

Solution of the Problem

In order to obtain the solution of Eqs. (7)–(9) satisfying the

boundary conditions (10), we assume that the solution con-

sists of a mean part and a perturbed part so that the velocity,

temperature and pressure distribution can be expressed as:

u x; yð Þ ¼ u0 yð Þ þ eu1 x; yð Þ;v x; yð Þ ¼ ev1 x; yð Þ;

T x; yð Þ ¼ T0 yð Þ þ eT1 x; yð Þ;p x; yð Þ ¼ p0 yð Þ þ ep1 x; yð Þ;

9>>>=>>>;: ð11Þ

where e is the amplitude, used as perturbation parameter.

The perturbed quantities u1, v1, T1 and p1 are small com-

pared to mean quantities.

Introducing Eq. (11) in Eqs. (6)–(9) and boundary

conditions (10), we obtain:

Zeroth order equations and boundary conditions:

d2T0

dy2¼ 0: ð12Þ

cd2u0

dy2� 1

DaþM2

� �u0 þ GrT0 þ C � Fffiffiffiffiffiffi

Dap u2

0 ¼ 0:

ð13Þ

where � op0

ox¼ C.

u00 ¼ 0; u01 ¼ 0 at y ¼ �1;

u00 ¼ 0; u01 ¼ 0 at y ¼ 1:

)ð14Þ

First order equations and boundary conditions:

ou1

oxþ ov1

oy¼ 0: ð15Þ

u0

ou1

oxþ v1

ou0

oy¼ � op1

oxþ c

o2u1

ox2þ o2u1

oy2

� �þ GrT1

� 1

DaþM2

� �u1 �

2FffiffiffiffiffiffiDap u0u1: ð16Þ

u0

ov1

ox¼ � op1

oyþ c

o2v1

ox2þ o2v1

oy2

� �� 1

Dav1: ð17Þ

Pr u0

oT1

oxþ v1

oT0

oy

� �¼ o2T1

ox2þ o2T1

oy2: ð18Þ

u1 ¼� coskxdu0

dy;v1 ¼ 0;T1 ¼� coskx

dT0

dyat y¼�1;

u1 ¼ 0;v1 ¼ 0;T1 ¼ 0 at y¼ 1:

9=;ð19Þ

In arriving at the boundary conditions at the wavy wall

y = -1, the actual boundary conditions are transformed to

mean position. This is justified when e � 1, i.e., when the

amplitude (e) of the disturbance is small compared with the

disturbance of the wavelength (Lekoudis et al. [1]; Tuck

and Kouzoubov [43]).

The solution of Eq. (12) under the boundary conditions

(14) is:

T0 yð Þ ¼ TR � 1ð Þ2

yþ TR þ 1

2: ð20Þ

To solve Eq. (13), we assume:

u0 yð Þ ¼ u00 yð Þ þ Fu01 yð Þ: ð21Þ

Introducing (21) into Eq. (13), we obtain:

d2u00

dy2� A1u00 ¼ �A2y� A3: ð22Þ

d2u01

dy2� A1u01 ¼

1

cffiffiffiffiffiffiDap u2

00; ð23Þ

where A1 ¼ 1c

1DaþM2

� �; A2 ¼ Gr TR�1ð Þ

2c ; A3 ¼ Gr TRþ1ð Þþ2C

2cThe boundary conditions (14), become:

u00 ¼ 0; u01 ¼ 0 at y ¼ �1;

u00 ¼ 0; u01 ¼ 0 at y ¼ 1:

)ð24Þ

The solutions of Eqs. (22) and (23) satisfying boundary

conditions (24) are:

u00 yð Þ ¼ C1effiffiffiffiA1

py þ C2e�

ffiffiffiffiA1

py þ A2

A1

yþ A3

A1

: ð25Þ

u01 yð Þ ¼ C3effiffiffiffiA1

py þ C4e�

ffiffiffiffiA1

py þ K3e2

ffiffiffiffiA1

py þ K4e�2

ffiffiffiffiA1

py

þK5yeffiffiffiffiA1

py þ K6ye�

ffiffiffiffiA1

py þ K7 y2 � yffiffiffiffiffi

A1

p� �

effiffiffiffiA1

py

þK8 y2 þ yffiffiffiffiffiA1

p� �

e�ffiffiffiffiA1

py þ K9 y2 þ 2

A1

� �þ K10yþ K11:

ð26Þ

In order to solve the first order Eqs. (16)–(18), we

introduce the stream function w(x, y) such that:

u1 ¼ �owoy; v1 ¼

owox: ð27Þ


123

Introducing Eq. (27) into Eqs. (16)–(18), in view of the

continuity Eq. (15), the governing equations for the

perturbed flow and heat transfer after eliminating p1 are

as follows:

u0

o3wox3þ o3w

oxoy2

� �� o2u0

oy2

owox� c 2

o4wox2oy2

þ o4wox4þ o4w

oy4

� �

þ 1

DaþM2

� �o2woy2þ Gr

oT1

oyþ 1

Da

o2wox2

þ 2FffiffiffiffiffiffiDap u0

o2woy2þ ou0

oy

owoy

� �¼ 0: ð28Þ

Pr u0

oT1

oxþ ow

ox

oT0

oy

� �¼ o2T1

ox2þ o2T1

oy2: ð29Þ

In view of boundary conditions (19), we assume general

solution for w and T1 in the following form:

w x; yð Þ ¼ RealX1r¼0

wrkreikx

!¼ Real w0eikx þ w1keikx

:

ð30Þ

T1

x; yð Þ ¼ RealX1r¼0

hrkreikx

!¼ Real h0eikx þ h1keikx

ð31Þ

Introducing Eqs. (30) and (31) into Eqs. (28) and (29),

we get the following set of ordinary differential equations

for (r = 0, 1):

d2h0

dy2¼ 0: ð32Þ

cd4w0

dy4� 1

DaþM2

� �d2w0

dy2¼ Gr

dh0

dy


d2w0

dy2þ du0

dy

dw0

dy

� �: ð33Þ

cd4w1

dy4� 1

DaþM2

� �d2w1

dy2

¼ Grdh1

dyþ i u0

d2w0

dy2� w0

d2u0

dy2

� �


d2w1

dy2þ du0

dy

dw1

dy

� �:

ð34Þ

d2h1

dy2¼ iPr u0h0 þ w0

dT0

dy

� �: ð35Þ

In the light of Eqs. (30), (31) and (27), the boundary

conditions (19) corresponding to w0, h0, w1 and h1 become:

w0 ¼ 0;dw0

dy¼ du0

dy; h0 ¼ �

dT0

dyat y ¼ �1;

w0 ¼ 0;dw0

dy¼ 0; h0 ¼ 1 at y ¼ 1:

9>>=>>;

ð36:1Þ

w1 ¼ 0;dw1

dy¼ 0;h1 ¼ 0 at y¼�1;

w1 ¼ 0;dw1

dy¼ 0;h1 ¼ 0 at y¼ 1; h1ð Þy¼�1¼ 0;

dh1

dy

� �y¼1

¼ 0; w1ð Þy¼�1¼ 0;dw1

dy

� �y¼�1

¼ 0:

9>>>>>>>=>>>>>>>;ð36:2Þ

The solution of Eq. (32) satisfying corresponding

boundary condition (36.1) is:

h0 ¼ C5yþ C6: ð37Þ

Equations (33) and (34) are coupled equations and can

not be solved directly. Hence we assume:

w0 ¼ w00 þ Fw01 and w1 ¼ w10 þ Fw11: ð38Þ

Introducing Eq. (38) into Eqs. (33) and (34) we obtain:

d4w00

dy4� A1

d2w00

dy2¼ C5

cGr: ð39Þ

d4w01

dy4�A1

d2w01

dy2¼ 2


d2w00

dy2þ du00

dy

dw00

dy

� �: ð40Þ

d4w10

dy4� A1

d2w10

dy2¼ Gr

cdh1

dyþ i

cu00

d2w00

dy2� w00

d2u00

dy2

� �

ð41Þ

d4w11

dy4� A1

d2w11

dy2¼ i

cu00

d2w01

dy2þ u01

d2w00

dy2� w01

d2u00

dy2

�

�w00

d2u01

dy2

�þ 2


d2w10

dy2þ du00

dy

dw10

dy

� �:

ð42Þ

The boundary conditions corresponding to w00, w01, w10

and w11 are:

w00 ¼ 0;w01 ¼ 0;dw00

dy¼ du00

dy;dw01

dy¼ du01

dyat y ¼ �1;

w10 ¼ 0;w11 ¼ 0;dw10

dy¼ 0;

dw11

dy¼ 0; at y ¼ �1;

w00 ¼ 0;w01 ¼ 0;dw00

dy¼ 0;

dw01

dy¼ 0 at y ¼ 1;

w10 ¼ 0;w11 ¼ 0;dw10

dy¼ 0;

dw11

dy¼ 0; at y ¼ 1:

9>>>>>>>>>>>=>>>>>>>>>>>;

ð43Þ

The solutions of Eqs. (39)–(42) satisfying (43) under

corresponding boundary conditions (36.1):

w00 yð Þ ¼ C7yþ C8 þ C9effiffiffiffiA1

py þ C10e�

ffiffiffiffiA1

py � K14y2:

ð44Þ

w01 yð Þ ¼ C11yþ C12 þ C13effiffiffiffiA1

py þ C14e�

ffiffiffiffiA1

py þ K16e2

ffiffiffiffiA1

py

þ K17e�2ffiffiffiffiA1

py þ K18ye

ffiffiffiffiA1

py � K19ye�

ffiffiffiffiA1

py

þ K20y2effiffiffiffiA1

py � K21y2e�

ffiffiffiffiA1

py � K22y3 þ K23y2: ð45Þ


123

The solution of Eq. (35) introducing Eqs. (21) and (38)

satisfying conditions (43) is:

h1 ¼ C15yþ C16 þ iPr K31yþ K32ð Þe2ffiffiffiffiA1

py

h

þ K33yþ K34ð Þe�2ffiffiffiffiA1

py

þ K35y3 þ K36y3 þ K37yþ K38

� �effiffiffiffiA1

py

þ K39y3 þ K40y3 þ K41yþ K42

� �e�

ffiffiffiffiA1

py

þK43y5 þ K44y4 þ K45y3 þ K46y2�: ð46Þ

w10 ¼ C17yþ C18 þ C19effiffiffiffiA1

py þ C20e�

ffiffiffiffiA1

py

þ i K51yþ K52ð Þe2ffiffiffiffiA1

py

nþ K53yþ K54ð Þe�2

ffiffiffiffiA1

py

þ K55y4 þ K56y3 þ K57y2 þ K58y� �

effiffiffiffiA1

py

þ K59y4 þ K60y3 þ K61y2 þ K62y� �

e�ffiffiffiffiA1

py

þ K63y6 þ K64y5 þ K65y4 � K66y3 � K67y2g: ð47Þ

w11 ¼ C21yþ C22 þ C23effiffiffiffiA1

py þ C24e�

ffiffiffiffiA1

py

þ i K78e3ffiffiffiffiA1

py þ K79e�3

ffiffiffiffiA1

py

nþ K80e2

ffiffiffiffiA1

py

þ K81e�2ffiffiffiffiA1

py þ K82ye3

ffiffiffiffiA1

py þ K83ye�3

ffiffiffiffiA1

py

þ K84ye2ffiffiffiffiA1

py þ K85ye�2

ffiffiffiffiA1

py þ K86ye

ffiffiffiffiA1

py

þ K87ye�ffiffiffiffiA1

py þ K88y2e2

ffiffiffiffiA1

py þ K89y2e�2

ffiffiffiffiA1

py


py þ K91y2e�

ffiffiffiffiA1

py þ K92y3e2

ffiffiffiffiA1

py

þ K93y3e�2ffiffiffiffiA1

py þ K94y3e

ffiffiffiffiA1

py þ K95y3e�

ffiffiffiffiA1

py

þ K96y4e2ffiffiffiffiA1

py þ K97y4e�2

ffiffiffiffiA1

py þ K98y4e

ffiffiffiffiA1

py

þ K99y4e�ffiffiffiffiA1

py þ K100y5e

ffiffiffiffiA1

py þ K101y5e�

ffiffiffiffiA1

py


py þ K103y6e�

ffiffiffiffiA1

py � K104y7 � K105y6

� K106y5 þ K107y4þK108y3 þ K109y2�:

ð48Þ

Based on these solutions, the first order quantities w1

and h1 can be expressed in the following form:

w1 ¼ wr þ iwið Þ ¼ w10 þ kw11: ð49Þh1 ¼ hr þ ihið Þ ¼ h10 þ kh11; ð50Þ

where the suffix r denotes the real part and i denotes the

imaginary part.

Considering only the real part, the expressions for first

order velocity and first order temperature become:

u1 ¼ edwi

dxksin kxð Þ � dwr

dycos kxð Þ

� �: ð51Þ

v1 ¼ e �kwrsin kxð Þ � k2wicos kxð Þ� �

: ð52Þ

T1 ¼ e hrcos kxð Þ � ksin kxð Þhið Þ: ð53Þ

The entire solution for the velocity components and

temperature distribution is the summation of the mean part

and the perturbed part.

Shear Stress

The shearing stress (sxy) at any point in the fluid in non-

dimensional form is given by:

sxy ¼ou

oyþ ov

ox¼ du0

dyþ eeikx du1

oyþ eikeikxv1 yð Þ: ð54Þ

Therefore, the shear stress (sw) at the wavy wall y =

-1 ? e cos (kx) and the shear stress (sf) at the parallel flat

wall y = 1 in non-dimensional form can be expressed as:

sw ¼du0

dy

� �y¼�1

þeRe eikx d2u0

dy2�1ð Þ þ d2w1

dy2

� �� : ð55Þ

and

sf ¼du0

dy

� �y¼1

þeRe eikx d2wdy2

1ð Þ� �

: ð56Þ

Heat Transfer Rate

The heat transfer rate in terms of Nusselt number (Nu) in

non-dimensional form is given by:

Nu ¼ oT

oy¼ dT0

dyþ eRe eikx dT1

dy

� �: ð57Þ

Hence, the Nusselt number (Nuw) at the wavy wall and

the Nusselt number (Nuf) at the flat wall in non-

dimensional form take the form:

Nuw ¼dh0

dy

� �y¼�1

þeRe eikx dhdy�1ð Þ

� �: ð58Þ

Nuf ¼dh0

dy

� �y¼1

þeRe eikx dhdy

1ð Þ� �

: ð59Þ

Results and Discussion

Consideration of the Darcy–Brinkman–Forchheimer model

gives rise to an additional empirical parameter (F), which

is known as Forchheimer parameter (inertia constant). The

numerical value of the Forchheimer parameter (F) depends

upon the porosity of the porous medium (2). If the porous

medium consists of concrete (ordinary mixes) or concrete

(bituminous) then2 lies in the range 0.02 B 2 B 0.07

(Neild and Bejan, [40]). When 2 = 0.02, the value of F is

found as 0.01 approximately. As such F � 1, which is

used as perturbation parameter. The solutions for axial

velocity (u), transverse velocity (v) and temperature dis-

tribution (T) are numerically evaluated for several sets of

values of parameters. The solutions for the velocity fields

and temperature distribution show that the flow is governed

by the Darcy number (Da), Grashof number (Gr),


123

kinematic viscosity ratio (c), magnetic parameter (M),

Forchheimer parameter (F), Prandtl number (Pr) and wall

temperature ratio parameter (TR). In order to analyse the

effect of different parameters on the behaviour of transport

phenomenon, several sets of the values of parameters are

used for all the graphs except the varying one. Also, the

shear stress at the wavy wall, sw; shear stress at the flat

wall, sf; the Nusselt number at the wavy wall, Nuw and the

Nusselt number at the flat wall, Nuf, are calculated

numerically and presented graphically. Equations (2) and

(3) show that we distinguish between the fluid viscosity (l)

and the effective viscosity (leff) used in the Brinkman term

following computer simulation study of Martys et al. [44]

and experimental investigation by Gilver and Altobelli

[45]. These studies demonstrate that there are situations,

particularly in aerospace technology, when it becomes

important to distinguish between these two coefficients.

The experimental study of Gilver and Altobelli [45], dis-

closes the fact that the ratio of effective viscosity (leff) to

the fluid velocity (l) lies in the range 5.1 \ leff/lf \ 10.9.

The Forchheimer parameter (inertia constant) F, appearing

in Eq. (1) is evaluated from the following empirical for-

mula suggested by Ergun [46].

F ¼ 1:75ffiffiffiffiffiffiffiffiffiffiffiffiffiffi150 23p

The temperature ratio parameter (TR) is important due to

its fundamental effects on transport processes between the

vertical wavy wall and the parallel flat wall. This parameter

essentially fixes the orientation of the ambient fluid

temperature (Tm

0) with respect to the temperature (Tw

0) of

the vertical wavy wall and the temperature (Tf

0) of the parallel

flat wall with the consideration T0

w [ T0

m. The case TR \ 0

implies T0f \T

0m, whereas the case TR [ 0 implies T

0f [ T

0m.

The case TR = 0 is a symmetric wall heating case, i.e. the

flat wall temperature (Tf

0) is equal to the temperature of the

ambient fluid (Tm

0). The case TR = 1 implies that the wavy

wall and flat wall are maintained at equal temperature,

whereas TR = -1 implies that the average of the

temperatures of the wavy wall (Tw

0) and the flat wall (Tf

0) is

equal to the fluid temperature (Tm

0). In the analysis, the

amplitude parameter (e) and kx are fixed as 0.02 and 0.78

respectively for all the computations. For practical

applications in nuclear technology and geothermal

systems, one important case; namely, the cooling (Gr [ 0)

of the wavy and flat wall is considered. The values of Prandtl

number (Pr) are chosen to be 0.71, 1.0 and 7.0, which

correspond to air, electrolyte solutions and water

respectively at 20 �C and one atmosphere pressure. These

are important fluids generally used in energy and aerospace

technologies (Rosa, [47]; Blums, [48]). The numerical

values of the remaining parameters are chosen either

arbitrarily or following Malashetty et al. [49]. The effects

of material parameters on the axial velocity (u), zeroth order

velocity (u0), transverse velocity (v), temperature

distribution (T), shear-stress at the wavy wall (sw); shear

stress at the flat wall (sf); rate of heat transfer (Nusselt

number) at the wavy wall (Nuw) and the parallel flat wall

(Nuf) are presented graphically and discussed.

Figure 2a illustrates the effect of Grashof number (Gr)

and the magnetic parameter (M) on the zeroth order axial

velocity (u0).The effect of Grashof number (Gr) and the

magnetic parameter (M) on the total axial velocity (u) is

shown in Fig. 2b. Physically, an increase in the numerical

value of Grashof number (Gr) implies an increase in the

buoyancy force, which supports the zeroth order axial

velocity (u0) as well as the total axial velocity (u). As such,

an increase in Grashof number (Gr) increases the flow.

Also, TR = 0.5 indicates that the temperature of the flat

wall (Tf

0) is less than the temperature of the wavy wall,

which implies an enhanced velocity in the vicinity of the

wavy wall. The physics behind the decrease in the zeroth

order axial velocity and the total axial velocity due to

increase in the magnetic field lies in the fact that the pre-

sence of the transversely applied magnetic field produces a

retarding force (Lorentz force) similar to the drag force,

which reduces the axial velocities u0 and u. Mathemati-

cally, the hydromagnetic drag embodied in the term -

M2u in the Eq. (7) retards the zeroth order axial velocity as

well as the total axial velocity. Due to this property of the

magnetic field, it is used as an important controlling device

in nuclear energy systems heat transfer, where momentum

development can be reduced by enhancing the magnetic

field.

(a) (b)

Fig. 2 Effect of Grashof number (Gr) and magnetic field (M) on

a zeroth order axial velocity (u0) and b total axial velocity (u) at

Da = 10-5, k = 0.05, F = 0.01, Pr = 1.0, [ = 0.02, c = 6.0,

TR = 0.5 and kx = 0.78


123

Figure 3a illustrates the effect of the temperature ratio

parameter (TR) and the magnetic field parameter (M) on the

zeroth order axial velocity (u0) for the case TR \ 0, in the

range -1 B TR \ 0. The effect of the temperature ratio

parameter (TR) and the magnetic field (M) on the axial

velocity (u) is shown in Fig. 3b. The physics behind this

phenomenon lies in the fact that TR \ 0 implies that the

temperature of the wavy wall (Tw

0) is greater than the

temperature of the flat wall (Tf

0) and less than the temper-

ature of the ambient fluid (Tm

0), i.e. the inequality

Tw

0[ Tm

0[ Tf

0holds. As such the fluid is gaining heat from

the wavy wall only so that enhanced velocity is noted in the

vicinity of the wavy wall. In particular, when TR = -1, the

average of the temperatures of the wavy wall and the flat

wall is equal to the fluid temperature Tm

0so that the velocity

up to half of the channel width from the wavy wall

decreases, which remains positive, whereas reverse flow is

noted in the remaining half of the channel width towards

the flat wall.

Figure 4a shows the influence of the temperature ratio

parameter (TR) and the magnetic parameter (M) on the

zeroth order axial velocity (u0) for the case TR C 0, in the

range 0 B TR B 1. The effect of the temperature ratio

parameter (TR) and the magnetic field (M) on the axial

velocity (u) is illustrated in Fig. 4b. Figures 3a, b and 4a, b

show that the effect of wall temperature ratio (TR) on the

zeroth order velocity (u0) and the total axial velocity (u) is

to increase both the velocities for the cases TR = 0 and

TR = 1, whereas for TR = -1, the velocity increases in

half of the channel width from the wavy wall and decreases

at the flat wall, i.e. flow reversal is noted at the right flat

wall. In addition, it is noted that the magnitude of velocity

is optimum for TR = 1 and minimum for TR = -1. When

TR = 0, the velocity profiles of the zeroth order velocity

and the total axial velocity are observed more enhanced in

the locality of the wavy wall in comparison with the flat

wall because the fluid is gaining heat from the wavy wall

only.

Figure 5a shows the effect of Darcy number (Da) and

the magnetic field (M) on the zeroth order axial velocity

(u0). The effect of Darcy number (Da) and the magnetic

field (M) on the total axial velocity (u) is shown in Fig. 5b.

Increase in the Darcy number (Da) physically implies less

solid particles in the porous matrix so that the resistance

opposing the flow decreases, i.e. the bulk porous medium

resistance is lowered as Darcy number increases. As such,

the momentum development in the flow region is increased,

which enhances the velocity.

Figure 6a illustrates the effect of viscosity ratio (c) and

the magnetic field (M) on the zeroth order velocity (u0).

The effect of viscosity ratio parameter (c) and the magnetic

parameter (M) on total axial velocity is represented in

Fig. 6b. Physically, increase in viscosity ratio implies,

decrease in the viscousness of the ambient fluid, so that the

resistance between the flow layers decreases, As such, the

zeroth order axial velocity as well as total axial velocity

increase.

Figure 7a represents the effect of Forchheimer param-

eter (F) and the magnetic field (M) on the zeroth order axial

velocity (u0). The effect of Forchheimer parameter (F) and

(a) (b)

Fig. 3 Effect of temperature ratio (TR \ 0) on a zeroth order axial

velocity (u0) and b total axial velocity (u) at Gr = 4.0, Da = 10-5,

k = 0.05, F = 0.01, Pr = 1.0,[ = 0.02, c = 6.0 and kx = 0.78

(a) (b)

Fig. 4 Effect of temperature ratio (TR C 0) on a zeroth order axial

velocity (u0) and b total axial velocity (u) at Gr = 4.0, Da = 10-5,

k = 0.05, F = 0.01, Pr = 1.0, [ = 0.02, c = 6.0 and kx = 0.78


123

the magnetic parameter (M) on total axial velocity is rep-

resented in Fig. 7b. The reason behind this phenomena lies

in the fact that increase in Forchheimer parameter

(F) implies presence of less solid particles in the porous

matrix so that the resistance of bulk porous medium is

lowered, which increases momentum development of the

flow regime, thereby enhancing the axial velocity.

Figure 8a shows the effect of Prandtl number (Pr) and

the magnetic field (M) on the zeroth order velocity (u0).

The effects of Prandtl number (Pr) and the magnetic

parameter (M) on total axial velocity are shown in Fig. 8b.

Mathematically, Prandtl number (Pr) defines the ratio of

momentum diffusivity to thermal diffusivity, consequently

lower axial velocity is observed with increase in Prandtl

number. A distinct decrease in velocity with increase in M,

confirms the property of magnetism, which is used as an

important controlling mechanism in nuclear energy sys-

tems and heat transfer, where momentum development can

be reduced in axial flow regime by enhancing the magnetic

field.

Figure 9a represents profiles of the transverse velocity

(v) for different values of the Grashof number (Gr) and the

magnetic field (M). The effect of viscosity ratio parameter

(c) and magnetic field (M) on the transverse velocity (v) is

shown in Fig. 9b. The transverse velocity (v) is normal to

the axial velocity (u), as such the effects of the Grashof

number (Gr), the viscosity ratio parameter (c) and the

magnetic parameter (M) are reverse to that of axial flow.

Figure 10a shows the effect of Prandtl number (Pr) and

the magnetic field (M) on the transverse velocity (v). The

effect of temperature ratio parameter (TR) and the magnetic

field (M) on the transverse velocity (v) is shown in

Fig. 10b. It may be noted that water (Pr = 7.0) enhances

the transverse velocity (v) more effectively in the negative

direction in comparison with air (Pr = 0.71) and electro-

lyte solutions (Pr = 1.0).

Figure 11 shows the profiles of the temperature distri-

bution (T) the channel for different values of the temper-

ature ratio parameter (TR). The curve demonstrates that the

temperature remain constant throughout the channel and is

(a) (b)

Fig. 5 Effect of Darcy number (Da) on a zeroth order axial velocity

(u0) and b total axial velocity (u) at Gr = 4.0, TR = 0.5, k = 0.05,

F = 0.01, Pr = 1.0, [ = 0.02, c = 6.0 and kx = 0.78

(a) (b)

Fig. 6 Effect of viscosity ratio (c) on a zeroth order axial velocity

(u0) and b total axial velocity (u) at Gr = 4.0, TR = 0.5, k = 0.05,

F = 0.01, Pr = 1.0, [ = 0.02, Da = 10-5 and kx = 0.78

(a) (b)

Fig. 7 Effect of Forchheimer parameter (F) on a zeroth order axial

velocity (u0) and b total axial velocity (u) at Gr = 4.0, TR = 0.5,

k = 0.05, c = 6.0, Pr = 1.0, [ = 0.02, Da = 10-5 and kx = 0.78


123

independent of the distance y, whereas the curve for

TR = -1, is a straight line which makes an angle of 45� to

the y-axis. This explains the fact that the temperature

diminishes uniformly with distance. Also, for the case

TR = 0, the curve is also a straight line.

(a) (b)

Fig. 8 Effect of Prandtl number (Pr) on the a zeroth order axial

velocity (u0) and b total axial velocity (u) at Gr = 4.0, TR = 0.5,

k = 0.05, c = 6.0, F = 0.01, [ = 0.02, Da = 10-5 and kx = 0.78

(a) (b)

Fig. 9 Profiles of transverse velocity (v) for different values of

a Grashof number (Gr) and magnetic field (M), b viscosity ratio (c)

and magnetic field (M) at TR = 0.5, k = 0.05, F = 0.01, Pr = 1.0,

[ = 0.02, Da = 10-5 and kx = 0.78

(a) (b)

Fig. 10 Profiles of transverse velocity (v) for different values of

a Prandtl number (Pr) and magnetic field (M), b temperature ratio

parameter (TR) and magnetic field (M) at Da = 10-5, k = 0.05,

F = 0.01, c = 6.0, [ = 0.02, Gr = 4.0 and kx = 0.78

Fig. 11 Profiles of temperature (T) for different values of the

temperature ratio parameter (TR) at Da = 10-5, k = 0.05, F = 0.01,

Pr = 1.0, M = 0.5, c = 6.0, [ = 0.02, Gr = 4.0 and kx = 0.78


123

Figure 12 shows the profiles of temperature distribution

(T) versus y for different values of the Prandtl number (Pr).

Higher Pr value fluids transfer heat less effectively com-

pared with lower Pr value fluids. Consequently, lower tem-

perature is observed with increase in the Prandtl number.

Besides, two curves (M = 0 and M = 0.5) for each Pr value

expose the fact that the presence of magnetic field increases

the temperature due to Brownian moment.

Figure 13 shows shear stress (sw) at the wavy wall and

shear stress (sf) at the flat wall versus Grashof number (Gr)

for different values of the magnetic parameter (M). It is

noted that the shear stress (sw) at the wavy wall increases

with increase in Grashof number (Gr) and the magnetic

parameter (M) for the assigned values of the material

parameters.

Figure 14 shows the curve for Nusselt number (Nuw, Nuf)

versus Grashof number (Gr) for different values of the

magnetic parameter (M). It is observed that the effect of the

Grashof number (Gr) is to decrease the Nusselt number

(Nuw) at the wavy wall and to increase the Nusselt number

(Nuf) at the flat wall. Also, as the magnetic parameter

(M) increases, the Nusselt number at the wavy wall (Nuw) as

well as the Nusselt number at the flat wall (Nuf) decreases.

Fig. 12 Profiles of temperature (T) for different values of Prandtl

number (Pr) at Da = 10-5, k = 0.05, F = 0.01, TR = 0, M = 0.5,

c = 6.0, [ = 0.02, Gr = 4.0 and kx = 0.78

Fig. 13 Shear stress versus Grashof number (Gr) for different values

of magnetic field (M) at Da = 10-5, k = 0.05, F = 0.01, Pr = 1.0,

TR = 0.5, c = 6.0, [ = 0.02 and kx = 0.78

Fig. 14 Nusselt number versus Grashof number (Gr) for different

values of magnetic field (M) at Da = 10-5, k = 0.05 F = 0.01,

Pr = 1.0, TR = 0.5, c = 6.0, [ = 0.02 and kx = 0.78


123

Figure 15 represents the graphs of the present study and

that of steady case of Paul et al. [38]. It is observed that the

graphs of the present study are in agreement under the

same physical situation and the boundary conditions.

However, the curves shown in the figure differ due to the

change of the physical situation of the channel and the

boundary conditions.

Conclusion

From the present study, the following conclusions may be

drawn:

(i) The effect of increasing Grashof number (Gr) is to

increase the zeroth order axial velocity (u0).

(ii) The effect of the magnetic parameter (M) is to

decrease the zeroth order axial velocity and total axial

velocity.

(iii) An increase in the numerical value of the temper-

ature ratio TR \ 0 in the range -1 B TR \ 0

decreases the zeroth order axial velocity (u0) and

the total axial velocity (u), whereas an increase in TR

increases these velocities and the reverse flow exists.

For the case TR [ 0 an increase in TR increases u0 as

well as u in the range 1 C TR C 0. When TR = 0, the

enhanced velocity is noted near the wavy wall and

when TR = 1, the velocity increases symmetrically.

(iv) The magnitude of velocity is optimum for TR = 1

and minimum TR = -1. When TR = -1, the reverse

flow is noted at the right flat wall.

(v) The effect Darcy number (Da) is to increase in the

zeroth order velocity (u0) as well as in total axial

velocity (u).

(vi) The zeroth order velocity (u0) and total axial velocity

(u) increase with increase in the viscosity ratio

parameter (c).

(vii) An increase in the Grashof number (Gr) decreases

the transverse velocity (v), whereas reverse effect is

noted for magnetic field.

(viii) An increase in the Prandtl number (Pr) or the

magnetic parameter (M) increases the magnitude of

the transverse velocity (v).

(ix) For TR = 1, the temperature field is constant and is

independent of the distance parallel to the y-axis;

whereas for TR = -1, the temperature decreases at

constant rate towards the y-axis.

(x) The temperature increases with increase in the

magnitude of Prandtl number.

(xi) The shear stress (sf) at the flat wall decreases whereas

the shear stress (sw) at the wavy wall increases as

Grashof number (Gr) increases.

(xii) The effect of Grashof number (Gr) is to decrease the

Nusselt number (Nuw) at the wavy wall and to

increase the Nusselt number (Nuf) at the flat wall.

Appendix

C1 ¼K2e�

ffiffiffiffiA1

p� K1e

ffiffiffiffiA1

p

e2ffiffiffiffiA1

p� e�2

ffiffiffiffiA1

p ; C2 ¼K1e�

ffiffiffiffiA1

p� K2e

ffiffiffiffiA1

p

e2ffiffiffiffiA1

p� e�2

ffiffiffiffiA1

p

C3 ¼K13e�

ffiffiffiffiA1

p� K12e

ffiffiffiffiA1

p

e2ffiffiffiffiA1

p� e�2

ffiffiffiffiA1

p ; C4 ¼K12e�

ffiffiffiffiA1

p� K13e

ffiffiffiffiA1

p

e2ffiffiffiffiA1

p� e�2

ffiffiffiffiA1

p ;

C5 ¼1þ TR

4; C6 ¼

3� TR

4;

C7 ¼1

a5

C10 � C9ð Þ; C8 ¼ �K14 � a1C9 � a2C10;

C9 ¼a5K15 a4 � a2ð Þ þ 4a7K14

a6 a4 � a2ð Þ þ a7 a3 � a1ð Þ

C10 ¼4a6K14 � a5 a3 � a1ð ÞK15

a6 a4 � a2ð Þ þ a7 a3 � a1ð Þ ; C11 ¼K26 � K24

2

� effiffiffiffiA1

p� e�

ffiffiffiffiA1

p

2

!C13 � C14ð Þ;

C12 ¼ K25 � K24 � a1C13 � a2C14; C13 ¼a9K30 � a4 � a2ð ÞK29

a9 a3 � a1ð Þ � a8 a4 � a2ð Þ

C14 ¼a3 � a1ð ÞK29 � a8K30

a9 a3 � a1ð Þ � a8 a4 � a2ð Þ ;C15 ¼ iK47;C16 ¼ iK48;

Fig. 15 Comparison of the effects of the present problem and the

temperature ratio parameter (TR) of the problem of Paul et al. (2006)

[38] at Da = 10-5, k = 0.05, Gr = 4.0, M = 0.5, F = 0.01,

Pr = 1.0, c = 6.0, [ = 0.02 and kx = 0.78


123

C17 ¼ iK76;C18 ¼ iK77;C19 ¼ iK74;C20 ¼ iK75;

C21 ¼ iK118;C22 ¼ iK119;C23 ¼ iK116;C24 ¼ iK117;

K1 ¼A3 þ A2

A1

; K2 ¼A3 � A2

A1

; K3 ¼C2

1

3cA1

ffiffiffiffiffiffiDap ;

K4 ¼C2

2

3cA1

ffiffiffiffiffiffiDap ; K5 ¼

A3C1

cA1

ffiffiffiffiffiffiffiffiffiffiffiA1Dap ; K6 ¼

�A3C2

cA1

ffiffiffiffiffiffiffiffiffiffiffiA1Dap ;

K7 ¼A2C1

2cA1


�A2C2

2cA1


�A22

cA31


K10 ¼�2A2A3

cA31


� 2C1C2A21 þ A2

3

� �cA2

1


K12 ¼ K3e2ffiffiffiffiA1

pþ K4e�2

ffiffiffiffiA1

pþ K5e

ffiffiffiffiA1

p

þ K6e�ffiffiffiffiA1

pþ K7 1� 1ffiffiffiffiffi

A1

p� �

effiffiffiffiA1

p

þ K8 1þ 1ffiffiffiffiffiA1

p� �

e�ffiffiffiffiA1

pþ K9 1þ 2

A1

� �þ K10 þ K11;

K13 ¼ K3e�2ffiffiffiffiA1

pþ K4e2

ffiffiffiffiA1

p� K5e�

ffiffiffiffiA1

p

� K6effiffiffiffiA1

pþ K7 1þ 1ffiffiffiffiffi

A1

p� �

e�ffiffiffiffiA1

p

þ K8 1� 1ffiffiffiffiffiA1

p� �

effiffiffiffiA1

pþ K9 1þ 2

A1

� �� K10 þ K11;

K14 ¼GrC5

2cA1

; K15 ¼ C1

ffiffiffiffiffiA1

pe�

ffiffiffiffiA1

p� C2

ffiffiffiffiffiA1

peffiffiffiffiA1

pþ A2

A1

� 2K14;

K16 ¼C1C9

3cA1


C2C10

3cA1


K18 ¼1

cA1

ffiffiffiffiffiffiffiffiffiffiffiA1Dap A3C9 � 2K14C1 þ

A1C1C7 þ A2C9ffiffiffiffiffiA1

p� ��

� 5

2ffiffiffiffiffiA1

p A2C9 � 2ffiffiffiffiffiA1

pK14C1

� � ;

K19 ¼1

cA1

ffiffiffiffiffiffiffiffiffiffiffiA1Dap A3C10 � 2K14C2 �

A1C2C7 þ A2C10ffiffiffiffiffiA1

p� ��

þ 5

2ffiffiffiffiffiA1

p A2C10 þ 2ffiffiffiffiffiA1

pK14C2

� � ;

K20 ¼1

2cA1

ffiffiffiffiffiffiffiffiffiffiffiA1Dap A2C9 � 2

ffiffiffiffiffiA1

pK14C1

� �;

K21 ¼1

2cA1

ffiffiffiffiffiffiffiffiffiffiffiA1Dap A2C10 þ 2

ffiffiffiffiffiA1

pK14C2

� �;

K22 ¼ �4K14A2

3cA21

ffiffiffiffiffiffiDap ;K23 ¼ �

1

cA21

ffiffiffiffiffiffiDap A2C7 � 2K14A3ð Þ

K24 ¼ K16e2ffiffiffiffiA1

pþ K17e�2

ffiffiffiffiA1

pþ K18e

ffiffiffiffiA1

p� K19e�

ffiffiffiffiA1

p

þ K20effiffiffiffiA1

p� K21e�

ffiffiffiffiA1

p� K22 þ K23;

K25 ¼ 2K16

ffiffiffiffiffiA1

pe2ffiffiffiffiA1

p� 2K17

ffiffiffiffiffiA1

pe�2

ffiffiffiffiA1

pþ K18

ffiffiffiffiffiA1

peffiffiffiffiA1

p


p� K19e�

ffiffiffiffiA1

pþ K19

ffiffiffiffiffiA1

pe�

ffiffiffiffiA1

pþ K20

ffiffiffiffiffiA1

peffiffiffiffiA1

p

þ 2K20effiffiffiffiA1

p� 2K21e�

ffiffiffiffiA1

pþ K21

ffiffiffiffiffiA1

pe�ffiffiffiffiA1

p� 3K22 þ 2K23;

K26 ¼ K16e�2ffiffiffiffiA1

pþ K17e2

ffiffiffiffiA1

p� K18e�

ffiffiffiffiA1

pþ K19e

ffiffiffiffiA1

p


p� K20e

ffiffiffiffiA1

pþ K22 þ K23;

K27 ¼ C3

ffiffiffiffiffiA1

pe�

ffiffiffiffiA1

p� C4

ffiffiffiffiffiA1

peffiffiffiffiA1

pþ 2K3

ffiffiffiffiffiA1

pe�

ffiffiffiffiA1

p

� 2K4

ffiffiffiffiffiA1

pe2ffiffiffiffiA1

pþ K5e�

ffiffiffiffiA1

p1�

ffiffiffiffiffiA1

p� �


p1þ

ffiffiffiffiffiA1

p� �� K7e�

ffiffiffiffiA1

p1�

ffiffiffiffiffiA1

pþ 1ffiffiffiffiffi

A1

p� �

� K8effiffiffiffiA1

p1þ

ffiffiffiffiffiA1

p� 1ffiffiffiffiffi

A1

p� �

� 2K9 þ K10;

K28 ¼ 2K16

ffiffiffiffiffiA1

pe�2

ffiffiffiffiA1

p� 2K17

ffiffiffiffiffiA1

pe2ffiffiffiffiA1

p


p1�

ffiffiffiffiffiA1

p� �� K19e

ffiffiffiffiA1

p1þ

ffiffiffiffiffiA1

p� �


p ffiffiffiffiffiA1

p� 2

� �þ K21e

ffiffiffiffiA1

p ffiffiffiffiffiA1

pþ 2

� �

� 3K22 � 2K23;

K29 ¼ K26 � K24 � 2K27 þ 2K28;K30 ¼ K24 � 2K25 � K26;

K31 ¼FK3C5

4A1

; K32 ¼2FK3a12 þ TR � 1ð ÞFK16

8A1

; K33 ¼FK4C5

4A1

;

K34 ¼2FK4a13 þ TR � 1ð ÞFK17

8A1

; K35 ¼FK7C5

A1

;

K36 ¼2FK5C5 þ 2FK7a18 þ TR � 1ð ÞFK20

2A1

;

K38 ¼C1a10 þ FC3a10 þ FK5a15 þ FK7a20

A1

þ TR � 1ð Þ2A1

C9 þ FC13 �2FK18ffiffiffiffiffi

A1

p þ 6FK20

A1

� �;

K39 ¼FK8C5

A1

;

K40 ¼2FK6C5 þ 2FK8a21 þ TR � 1ð ÞFK21

2A1

þ TR � 1ð Þ2A1

FK19 þ4FK21ffiffiffiffiffi

A1

p�

;

K41 ¼2C2C5 þ 2FC4C5 þ 2FK6a16 þ 2FK8a22

2A1

þ TR � 1ð Þ2A1

C9 þ FC13 �2FK18ffiffiffiffiffi

A1

p þ 6FK20

A1

� �;

K42 ¼C2a11 þ FC4a11 þ FK6a17 þ FK8a23

A1

þ TR � 1ð Þ2A1

C10 þ FC14 þ2FK19ffiffiffiffiffi

A1

p þ 6FK21

A1

� ;

K43 ¼2FK9C5 þ TR � 1ð ÞFK22

40;

K44 ¼A2C5 þ A1 FK9C6 þ FK10C5ð Þ

12A1

þ TR � 1ð Þ24

K14 þ FK23ð Þ;


123

K45¼A3C5þA2C6þ2FK9C5þA1 FK10C6þFK11C5ð Þ

6A1

þ TR�1ð Þ12

C7þFC11ð Þ;

K46¼A3C6þ2FK9C6þA1FK11C6

2A1

þ TR�1ð Þ4

C8þFC12ð Þ;

K47¼Pr

2K49�K50ð Þ; K48 ¼�

Pr

2K49þK50ð Þ;

K49 ¼ �K31 þ K32ð Þe�2ffiffiffiffiA1

pþ �K33 þ K34ð Þe2

ffiffiffiffiA1

p

þ �K35 þ K36 � K37ð þK38Þe�ffiffiffiffiA1

p

þ �K39 þ K40 � K41 þ K42ð ÞeffiffiffiffiA1

p

� K43 þ K44 � K45

þ K46;

K50 ¼ K31 þ K32ð Þe2ffiffiffiffiA1

pþ K33 þ K34ð Þe�2

ffiffiffiffiA1

p

þ K35 þ K36 þ K37ð þK38ÞeffiffiffiffiA1

p

þ K39 þ K40 þ K41 þ K42ð Þe�ffiffiffiffiA1

pþ K43 þ K44

þ K45 þ K46;

K51 ¼PrGrK31

6cA1

ffiffiffiffiffiA1

p ; K52 ¼PrGra32

12cA21

; K53 ¼ �PrGrK33

6cA1

ffiffiffiffiffiA1

p ;

K54 ¼PrGra33

12cA21

; K55 ¼PrGrK35

8cA1

; K56 ¼3PrGra34 þ A1K14C1

6cA1

ffiffiffiffiffiA1

p ;

K57 ¼2PrGra35 � 5

ffiffiffiffiffiA1

pK14C1 þ a45

4cA1

ffiffiffiffiffiA1

p ;

K58 ¼2PrGra36 þ 2a43 þ 17K14C1

4cA1

ffiffiffiffiffiA1

p � 5a45

4cA21

;

K59 ¼PrGrK39

8cA1

; K60 ¼3PrGra37 � A1K14C2

6cA1

ffiffiffiffiffiA1

p ;

K61 ¼2PrGra38 � 5

ffiffiffiffiffiA1

pK14C2 � a46

4cA1

ffiffiffiffiffiA1

p ;

K62 ¼2PrGra39 � 2a44 � 17K14C2

4cA1

ffiffiffiffiffiA1

p � 5a46

4cA21

;

K63 ¼ �PrGrK43

6cA1

; K64 ¼ �PrGrK44

5cA1

;

K65 ¼ �PrGra40

cA1

; K66 ¼3PrGra41 þ K14A2

3cA1

;

K67 ¼2PrGra42 þ 2K14A3 þ GrK47

2cA1

;

K68 ¼ K52 � K51ð Þe�2ffiffiffiffiA1

pþ K54 � K53ð Þe2

ffiffiffiffiA1

p

þ K55 � K56 þ K57 � K58ð Þe�2ffiffiffiffiA1

p

þ K59 � K60 þ K61 � K62ð ÞeffiffiffiffiA1

p

þ K63 � K64 þ K65 þ K66 � K67;



ffiffiffiffiA1

p

þ K55 þ K56 þ K57 þ K58ð Þe2ffiffiffiffiA1

p

þ K59 þ K60 þ K61 þ K62ð Þe�ffiffiffiffiA1

p

þ K63 þ K64 þ K65 � K66 � K67;

K70 ¼ 2ffiffiffiffiffiA1

pK52 � K51ð Þ þ K51

h ie�2

ffiffiffiffiA1

p

þ K53 � 2ffiffiffiffiffiA1

pK54 � K53ð Þ

h ie2ffiffiffiffiA1

p

þ �4K55 þ 3K56 � 2K57 þ K58ð Þ½þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA1 K55 � K56 þ K57 � K58ð Þ

p�e�

ffiffiffiffiA1

p

þ �4K59 þ 3K60 � 2K61 þ K62ð Þ½�

ffiffiffiffiffiA1

pK59 � K60 þ K61 � K62ð Þ�e

ffiffiffiffiA1

p

� 6K63 þ 5K64 � 4K65 � 3K66 � 2K67;


pK51 þ K52ð Þ þ K51

h ie2ffiffiffiffiA1

p

þ �2ffiffiffiffiffiA1

pK53 þ K54ð Þ þ K53

h ie�2

ffiffiffiffiA1

p

þ 4K55 þ 3K56 þ 2K57 þ K58ð Þ½þ

ffiffiffiffiffiA1

pK55 þ K56 þ K57 þ K58ð Þ�e

ffiffiffiffiA1

p

þ 4K59 þ 3K60 þ 2K61 þ K62ð Þ½�

ffiffiffiffiffiA1

pK59 þ K60 þ K61 þ K62ð Þ�e�

ffiffiffiffiA1

p

þ 6K63 þ 5K64 þ 4K65 � 3K66 � 2K67

K72 ¼ K68 � K69 þ 2K70;K73 ¼ K69 � 2K71 � K68;

K74 ¼a9K73 � a4 � a2ð ÞK72

a9 a3 � a1ð Þ � a8 a4 � a2ð Þ ;

K75 ¼a3 � a1ð ÞK72 � a8K73

a9 a3 � a1ð Þ � a8 a4 � a2ð Þ ;

K76 ¼K68 � K69ð Þ

2� e

ffiffiffiffiA1

p� e�

ffiffiffiffiA1

p

2

!K74 � K75ð Þ;

K77 ¼ �K69 þ K71 � a1K74 � a2K75;

K78 ¼1

cb1 þ a47 � b38 � b64 þ

2ffiffiffiffiffiffiDap a85 þ a109ð Þ

� ;

K79 ¼1

cb2 þ a48 � b39 � b65 þ


� ;

K80 ¼1

cb3 þ a49 � b40 � b66 þ


� ;

K81 ¼1

cb4 þ a50 � b41 � b67 þ


� ;

K82 ¼C1K51

6cA1



123

K83 ¼C2K53

6cA1


K84 ¼1

cb7 þ a51 � b42 � b68 þ


� ;

K85 ¼1

cb8 þ a52 � b43 � b69 þ


� ;

K86 ¼1

cb5 þ a53 � b44 � b70 þ


� ;

K87 ¼1

cb6 � a54 þ b45 þ b71 þ

2ffiffiffiffiffiffiDap �a92 þ a116ð Þ

� ;

K88 ¼1

cb11 þ a55 � b46 � b72 þ


� ;

K89 ¼1

cb12 þ a56 � b47 � b73 þ


� ;

K90 ¼1

cb9 þ a57 � b48 � b74 þ


� ;

K91 ¼1

cb10 � a58 � b49 þ b75 þ


� ;

K92 ¼2

cffiffiffiffiffiffiDap a97 þ a121ð Þ; K93 ¼

2

cffiffiffiffiffiffiDap a98 þ a122ð Þ;

K94 ¼1

cb13 þ a59 � b50 � b76 þ


� ;

K95 ¼1

cb14 þ a60 � b51 � b77 þ


� ;

K96 ¼C1K55

3cA1


K97 ¼C2K59

3cA1


K98 ¼1

cb15 � b52 � b78 þ


� ;

K99 ¼1

cb16 � b53 � b79 þ


� ;

K100 ¼1

c�b54 þ


� ;

K101 ¼1

c�b55 þ


� ;

K102 ¼A2K55 þ 6

ffiffiffiffiffiA1

pC1K63

6cA1

ffiffiffiffiffiffiffiffiffiffiffiA1Dap

� �;

K103 ¼6ffiffiffiffiffiA1

pC2K63 � A2K59

6cA1

ffiffiffiffiffiffiffiffiffiffiffiA1Dap

� �;

K104 ¼12A2K63

7cA21


b103 þ b128ð Þ15cA1


K106 ¼2 a105 þ a129ð Þ

cffiffiffiffiffiffiDap ;

K107 ¼1

cb19 þ a61 � b56 � b80 �


� ;

K108 ¼1

cb18 þ a62 � b57 � b81 �


� ;

K109 ¼1

cb17 þ a63 � b58 � b82 �


� ;

K110 ¼ K78 � K82ð Þe�3ffiffiffiffiA1

pþ K79 � K83ð Þe3

ffiffiffiffiA1

p

þ K80 � K84ð þ K88

� K92 þ K96Þe�2ffiffiffiffiA1

p

þ K81 � K85ð þ K89 � K93 þ K97Þe2ffiffiffiffiA1

p

þ �K86 þ K90ð � K94 þ K98 � K100 þ K102Þe�ffiffiffiffiA1

p

þ �K87 þ K91ð � K95 þ K99 � K101 þ K102ÞeffiffiffiffiA1

p

þ K104 � K105 þ K106 þ K107 � K108 þ K109;



ffiffiffiffiA1

p

þ K80 þ K84ð þ K88

þ K92 þ K96Þe2ffiffiffiffiA1

p

þ K81 þ K85ð þ K89 þ K93 þ K97Þe�2ffiffiffiffiA1

p

þ K86 þ K90ð þ K94 þ K98 þ K100 þ K102ÞeffiffiffiffiA1

p

þ K87 þ K91ð þ K95 þ K99 þ K101 þ K103Þe�ffiffiffiffiA1

p

� K104 � K105 � K106 þ K107 þ K108 þ K109;


pK78 � 3

ffiffiffiffiffiA1

pK82 þ K82

� �e�3

ffiffiffiffiA1

p


pK79 þ 3

ffiffiffiffiffiA1

pK83 þ K83

� �e3ffiffiffiffiA1

p

þ 2ffiffiffiffiffiA1

pK80�

�2ffiffiffiffiffiA1

pK84 þ 2

ffiffiffiffiffiA1

pK88

� 2ffiffiffiffiffiA1

pK92 þ 2

ffiffiffiffiffiA1

pK96 þ K84 � 2K88 þ 3K92 � 4K96

�e�2

ffiffiffiffiA1

p


pK81

�þ 2

ffiffiffiffiffiA1

pK85 � 2

ffiffiffiffiffiA1

pK89 þ 2

ffiffiffiffiffiA1

pK93


pK97þK85 � 2K89 þ 3K93 � 4K97Þe2

ffiffiffiffiA1

p

þ �ffiffiffiffiffiA1

pK86

�þ

ffiffiffiffiffiA1

pK90 �

ffiffiffiffiffiA1

pK94 þ

ffiffiffiffiffiA1

pK98

�ffiffiffiffiffiA1

pK100 þ

ffiffiffiffiffiA1

pK102 þ K86 � 2K90

þ 3K94 � 4K98 þ 5K100 � 6K102Þe�ffiffiffiffiA1

p

þffiffiffiffiffiA1

pK87 �

ffiffiffiffiffiA1

pK91

�þ

ffiffiffiffiffiA1

pK95 �

ffiffiffiffiffiA1

pK99

þffiffiffiffiffiA1

pK101 �

ffiffiffiffiffiA1

pK103 þ K87 � 2K91

þ 3K95 � 4K99 þ 5K101 � 6K103ÞeffiffiffiffiA1

p

� 7K104 þ 6K105 � 5K106 � 4K107

þ 3K108 � 2K109;


123


pK78 þ 3

ffiffiffiffiffiA1

pK82 þ K82

� �e3ffiffiffiffiA1

pþ �3

ffiffiffiffiffiA1

pK79

�


pK83 þ K83

�e�3

ffiffiffiffiA1

pþ 2

ffiffiffiffiffiA1

pK80þ

�2ffiffiffiffiffiA1

pK84 þ 2

ffiffiffiffiffiA1

pK88


pK92 þ 2

ffiffiffiffiffiA1

pK96 þ K84 þ 2K88 þ 3K92 þ 4K96

�e2ffiffiffiffiA1

p


pK81

�� 2

ffiffiffiffiffiA1

pK85 � 2

ffiffiffiffiffiA1

pK89 � 2

ffiffiffiffiffiA1

pK93 � 2

ffiffiffiffiffiA1

pK97

þK85 þ 2K89 þ 3K93 þ 4K97Þe�2ffiffiffiffiA1

pþ

ffiffiffiffiffiA1

pK86 þ

ffiffiffiffiffiA1

pK90

�

þffiffiffiffiffiA1

pK94 þ

ffiffiffiffiffiA1

pK98 þ

ffiffiffiffiffiA1

pK100 þ

ffiffiffiffiffiA1

pK102 þ K86 þ 2K90

þ 3K94 þ 4K98 þ 5K100 þ 6K102ÞeffiffiffiffiA1

pþ �

ffiffiffiffiffiA1

pK87

�

�ffiffiffiffiffiA1

pK91 �

ffiffiffiffiffiA1

pK95 �

ffiffiffiffiffiA1

pK99 �

ffiffiffiffiffiA1

pK101 �

ffiffiffiffiffiA1

pK103

þK87 þ 2K91 þ 3K95 þ 4K99 þ 5K101 þ 6K103Þe�ffiffiffiffiA1

p

� 7K104 � 6K105 � 5K106 þ 4K107 þ 3K108 þ 2K109;

K114 ¼ K110 � K111 þ 2K112;

K115 ¼ K111 � 2K113 � K110;

K116 ¼a9K115 � a4 � a2ð ÞK114

a9 a3 � a1ð Þ � a8 a4 � a1ð Þ ;

K117 ¼a3 � a1ð ÞK114 � a8K115

a9 a3 � a1ð Þ � a8 a4 � a2ð Þ ;

K118 ¼1

2K110 � K111ð Þ � 1

2effiffiffiffiA1

p� e�

ffiffiffiffiA1

p� �K116 � K117ð Þ;

K119 ¼ �a1K116 � a2K117 þ K113 � K111;

a1 ¼ effiffiffiffiA1

p1�

ffiffiffiffiffiA1

p� �; a2 ¼ e�

ffiffiffiffiA1

p1þ

ffiffiffiffiffiA1

p� �;

a3 ¼ e�ffiffiffiffiA1

pþ

ffiffiffiffiffiA1

peffiffiffiffiA1

p;

a4 ¼ effiffiffiffiA1

p�

ffiffiffiffiffiA1

pe�

ffiffiffiffiA1

p; a5 ¼

2

effiffiffiffiA1

p� e�

ffiffiffiffiA1

p ;

a6 ¼ a5

ffiffiffiffiffiA1

pe�

ffiffiffiffiA1

p� 1;

a7 ¼ a5

ffiffiffiffiffiA1

peffiffiffiffiA1

p� 1; a8 ¼ e

ffiffiffiffiA1

p� e�

ffiffiffiffiA1

p1þ 2

ffiffiffiffiffiA1

p� �;

a9 ¼ e�ffiffiffiffiA1

p� e

ffiffiffiffiA1

p1� 2

ffiffiffiffiffiA1

p� �; a10 ¼ C6 �

2C5ffiffiffiffiffiA1

p ;

a11 ¼ C6 þ2C5ffiffiffiffiffi

A1

p ;

a12 ¼ C6 �C5ffiffiffiffiffiA1

p ; a13 ¼ C6 þC5ffiffiffiffiffiA1

p ; a14 ¼ C6 �4C5ffiffiffiffiffi

A1

p ;

a15 ¼6C5 � 2

ffiffiffiffiffiA1

pC6

A1

; a16 ¼ C6 þ4C5ffiffiffiffiffi

A1

p ;

a17 ¼6C5 þ 2

ffiffiffiffiffiA1

pC6

A1

;

a18 ¼ C6 �7C5ffiffiffiffiffi

A1

p ; a19 ¼22C5

A1

� 5C6ffiffiffiffiffiA1

p ;

a20 ¼8C�6

A1

� 30C5

A1

ffiffiffiffiffiA1

p ;

a21 ¼ C6 þ7C5ffiffiffiffiffi

A1

p ; a22 ¼22C5

A1

þ 5C6ffiffiffiffiffiA1

p ;

a23 ¼8C6

A1

þ 30C5

A1

ffiffiffiffiffiA1

p ;

a24 ¼ K31 þ 2ffiffiffiffiffiA1

pK32;

a25 ¼ K33 � 2ffiffiffiffiffiA1

pK34;

a26 ¼ 3K35 þffiffiffiffiffiA1

pK36;

a27 ¼ 2K36 þ 2ffiffiffiffiffiA1

pK37; a28 ¼ K37 þ K38;

a29 ¼ 3K39 �ffiffiffiffiffiA1

pK40;

a30 ¼ 2K40 �ffiffiffiffiffiA1

pK41; a31 ¼ K41 þ K42;

a32 ¼ a24 �14

3K31

a33 ¼ a25 �14

3K33; a34 ¼

1

3a26 �

15

3K35

� �;

a35 ¼1

2a27 þ

51K35

2ffiffiffiffiffiA1

p � 5a26ffiffiffiffiffiA1

p� �

;

a36 ¼ a28 �5a27

2ffiffiffiffiffiA1

p þ 17a26

2A1

� 147K35

4A1

;

a37 ¼ �1

3a29 �

15

2K39

� �;

a38 ¼ �1

2a30 �

51K39

2ffiffiffiffiffiA1

p þ 5a29ffiffiffiffiffiA1

p� �

;

a39 ¼ � a31 �5a30

2ffiffiffiffiffiA1

p þ 17a29

2A1

� 147K39

4A1

� �;

a40 ¼1

4K45 þ

20

A1

K43

� �;

a41 ¼1

3K46 þ

12

A1

K44

� �;

a42 ¼3A1K45 þ 60K43

A21

� �;

a43 ¼ A3C9 � 2K14C1 � A1C1C8;

a44 ¼ A3C10 � 2K14C2 � A1C2C8;

a45 ¼ A2C9 � A1C1C7; a46 ¼ A2C10 � A1C2C7;

a47 ¼b20

72A21

;

a48 ¼b21

72A21

; a49 ¼3ffiffiffiffiffiA1

pb22 � 7b26 þ 3

ffiffiffiffiffiA1

pb30b37

� �36A2

1

ffiffiffiffiffiA1

p ;

a50 ¼3ffiffiffiffiffiA1

pb23 þ 7b27 þ 3

ffiffiffiffiffiA1

pb31b37

� �36A2

1

ffiffiffiffiffiA1

p ;


pb26 � 14b30

� �36A2

1

ffiffiffiffiffiA1

p ;


123


pb27 þ 14b31

� �36A2

1

ffiffiffiffiffiA1

p ; a53

¼2A1b24 � 5

ffiffiffiffiffiA1

pb28 þ 17b32þ

� �4A2

1

ffiffiffiffiffiA1

p ;

a54 ¼2A1b25 þ 5

ffiffiffiffiffiA1

pb29 þ 17b33

� �4A2

1

ffiffiffiffiffiA1

p ; a55 ¼b30

12A21

;

a56 ¼b31

12A21

;

a57 ¼ffiffiffiffiffiA1

pb28 � 5b32

� �4A2

1

; a58 ¼ffiffiffiffiffiA1

pb29 þ 5b33

� �4A2

1

;

a59 ¼b32

6A1

ffiffiffiffiffiA1

p ;

a60 ¼ �b33

6A1

ffiffiffiffiffiA1

p ; a61 ¼ �b34

12A1

; a62 ¼ �b35

6A1

;

a63 ¼ �2b34 þ A1b36ð Þ

2A21

; a64 ¼ 4A1C9K3;

a65 ¼ 4A1C10K4;

a66 ¼ 4A1C8K3 þ A1C3C9 þ 2ffiffiffiffiffiA1

pC9K5;

a67 ¼ 4A1C8K4 þ A1C4C10 � 2ffiffiffiffiffiA1

pC10K6;

a68 ¼ 4A1C10K3 þ A1C3C8 þ 2ffiffiffiffiffiA1

pC8K6 þ 2C9K9;

a69 ¼ 4A1C9K4 þ A1C4C8 � 2ffiffiffiffiffiA1

pC8K6 þ 2C10K9;

a70 ¼ 4A1C7K3 þffiffiffiffiffiA1

pC9K5 þ 3

ffiffiffiffiffiA1

pC9K7; a71

¼ 4A1C7K4 þffiffiffiffiffiA1

pC10K6 � 3

ffiffiffiffiffiA1

pC10K8;

a72 ¼ A1C3C7 þ 2ffiffiffiffiffiA1

pC7K5 þ

ffiffiffiffiffiA1

pC8K5 þ 3

ffiffiffiffiffiA1

pC8K7;

a73 ¼ A1C4C7 � 2ffiffiffiffiffiA1

pC7K6 þ

ffiffiffiffiffiA1

pC8K6 � 3

ffiffiffiffiffiA1

pC8K8;

a74 ¼ A1C9K7 � 4A1K3K14; a75 ¼ A1C10K8 þ 4A1K4K14;

a76 ¼ A1C8K7 � A1C3K14 þffiffiffiffiffiA1

pC7K5 þ 3

ffiffiffiffiffiA1

pC7K7


pK5K14;

a77 ¼ �A1C8K8 � A1C4K14 þffiffiffiffiffiA1

pC7K6 þ 2

ffiffiffiffiffiA1

pK6K14;

a78 ¼ A1C7K7 �ffiffiffiffiffiA1

pK5K14 � 3

ffiffiffiffiffiA1

pK7K14; a79

¼ A1C7K8 þffiffiffiffiffiA1

pK6K14 � 3

ffiffiffiffiffiA1

pK8K14;

a80 ¼ �A1K7K14; a81 ¼ A1K8K14; a82

¼ A1C9K8 � A1C10K7 þ 2K9K14;

a83 ¼ffiffiffiffiffiA1

pC9K6 � 3

ffiffiffiffiffiA1

pC9K8 þ

ffiffiffiffiffiA1

pC10K5


pC10K7 þ 2C7K9;

a84 ¼ A1C4C9 þ A1C3C10 þ 2C8K9 þ 2ffiffiffiffiffiA1

pC9K5


pC9K6;

a85 ¼b83

72A21

� 17C1K51

216A1

ffiffiffiffiffiA1

p ; a86 ¼b84

72A21

þ 17C2K53

216A1

ffiffiffiffiffiA1

p ;

a87 ¼b85

12A21

� 7b89

36A21

ffiffiffiffiffiA1

p þ b37b93

12A21

� 239b97

108A31

ffiffiffiffiffiA1

p

þ 6529C1K55

648A31

;

a88 ¼b86

12A21

þ 7b90

36A21

ffiffiffiffiffiA1

p þ b37b93

12A21

þ 239b97

108A31

ffiffiffiffiffiA1

p

þ 6529C2K59

648A31

;

a89 ¼b89

12A21

� 7b93

18A21

ffiffiffiffiffiA1

p þ 127b97

72A31

� 239C1K55

27A21

ffiffiffiffiffiA1

p ;

a90 ¼b90

12A21

þ 7b94

18A21

ffiffiffiffiffiA1

p þ 127b98

72A31

þ 239C2K59

27A21

ffiffiffiffiffiA1

p ;

a91 ¼b87

2A1

ffiffiffiffiffiA1

p � 5b91

4A21

þ 17b95

4A21

ffiffiffiffiffiA1

p � 147b99

8A31

þ 387b101

4A31

ffiffiffiffiffiA1

p

� 4815A2K55

8A41

;

a92 ¼b88

2A1

ffiffiffiffiffiA1

p þ 5b92

4A21

þ 17b96

4A21

ffiffiffiffiffiA1

p þ 147b101

8A31

þ 387b102

4A31

ffiffiffiffiffiA1

p

þ 4815A2K59

8A41

;

a93 ¼b93

12A21

� 7b97

12A21

ffiffiffiffiffiA1

p þ 127C1K55

36A21

;

a94 ¼b94

12A21

þ 7b98

12A21

ffiffiffiffiffiA1

p þ 127C2K59

36A21

;

a95 ¼b91

4A1

ffiffiffiffiffiA1

p � 5b95

4A21

þ 51b99

8A21

ffiffiffiffiffiA1

p � 147b101

4A31

þ 1935A2K55

8A31

ffiffiffiffiffiA1

p ;

a96 ¼b92

4A1

ffiffiffiffiffiA1

p þ 5b96

4A21

þ 51b100

8A21

ffiffiffiffiffiA1

p þ 147b102

4A31

þ 1935A2K59

8A31

ffiffiffiffiffiA1

p ;

a97 ¼b97

12A21

� 7C1K55

9A1

ffiffiffiffiffiA1

p ; a98 ¼b98

12A21

þ 7C2K59

9A1

ffiffiffiffiffiA1

p ;

a99 ¼b95

6A1

ffiffiffiffiffiA1

p � 5b99

4A21

þ 17b101

2A21

ffiffiffiffiffiA1

p � 245A2K55

4A31

;

a100 ¼b96

6A1

ffiffiffiffiffiA1

p þ 5b100

4A21

þ 17b102

2A21

ffiffiffiffiffiA1

p þ 245A2K59

4A31

;

a101 ¼b99

8A1

ffiffiffiffiffiA1

p � 5b101

4A21

þ 85A2K55

4A21

ffiffiffiffiffiA1

p ;

a102 ¼b100

8A1

ffiffiffiffiffiA1

p þ 5b102

4A21

þ 85A2K59

4A21

ffiffiffiffiffiA1

p ;

a103 ¼b101

10A1

ffiffiffiffiffiA1

p � 5A2K55

4A21

; a104 ¼b102

10A1

ffiffiffiffiffiA1

p þ 5A2K59

4A21

;


123

a105 ¼30A2K63

A31

þ b104

20A1

; a106 ¼b103

A21

þ b105

12A1

;

a107 ¼600A2K63

A41

þ b104

A21

þ b106

6A1

;

a108 ¼12b103

A31

þ b105

A21

þ b107

2A1

;

a109 ¼b108

72A21

� 17C1K51

432A21

ffiffiffiffiffiA1

p ; a110 ¼b109

72A21

þ 17C2K53

432A21

ffiffiffiffiffiA1

p ;

a111 ¼b110

12A21

� 7b114

36A21

ffiffiffiffiffiA1

p þ b37b118

12A21

� 239b122

108A31

ffiffiffiffiffiA1

p

þ 6529C1K55

648A31

;

a112 ¼b111

12A21

þ 7b115

36A21

ffiffiffiffiffiA1

p þ b37b119

12A21

þ 239b123

108A31

ffiffiffiffiffiA1

p

þ 6529C2K59

648A31

;

a113 ¼b114

12A21

� 7b118

18A21

ffiffiffiffiffiA1

p þ 127b122

72A31

� 239C1K55

27A21

ffiffiffiffiffiA1

p ;

a114 ¼b115

12A21

þ 7b119

18A21

ffiffiffiffiffiA1

p þ 127b123

72A31

þ 239C2K59

27A21

ffiffiffiffiffiA1

p ;

a115 ¼b112

2A1

ffiffiffiffiffiA1

p þ 5b116

4A21

þ 17b121

4A21

ffiffiffiffiffiA1

p � 147b124

8A31

þ 387b126

4A31

ffiffiffiffiffiA1

p

� 14445C1K63

4A31

ffiffiffiffiffiA1

p ;

a116 ¼b113

2A1

ffiffiffiffiffiA1

p � 5b117

4A21

� 17b121

4A21

ffiffiffiffiffiA1

p þ 147b125

8A31

þ 387b127

4A31

ffiffiffiffiffiA1

p

þ 14445C2K63

4A31

ffiffiffiffiffiA1

p ;

a117 ¼b118

12A21

� 7b122

12A21

ffiffiffiffiffiA1

p þ 127C1K55

36A21

;

a118 ¼b119

12A21

þ 7b123

12A21

ffiffiffiffiffiA1

p þ 127C2K59

36A21

;

a119 ¼ �b116

4A1

ffiffiffiffiffiA1

p � 5b120

4A21

þ 51b124

8A21

ffiffiffiffiffiA1

p � 147b126

4A31

þ 5805C1K63

4A31

;

a120 ¼ �b117

4A1

ffiffiffiffiffiA1

p � 5b121

4A21

þ 51b125

8A21

ffiffiffiffiffiA1

p þ 147b127

4A31

þ 5805C2K63

4A31

;

a121 ¼b122

12A21

� 7C1K55

9A1

ffiffiffiffiffiA1

p ; a122 ¼b123

12A21

þ 7C2K59

9A1

ffiffiffiffiffiA1

p ;

a123 ¼b120

6A1

ffiffiffiffiffiA1

p � 5b124

4A21

þ 17b126

2A21

ffiffiffiffiffiA1

p � 735C1K63

2A21

ffiffiffiffiffiA1

p ;

a124 ¼ �b121

6A1

ffiffiffiffiffiA1

p þ 5b125

4A21

þ 17b127

2A21

ffiffiffiffiffiA1

p þ 735C2K63

2A21

ffiffiffiffiffiA1

p ;

a125 ¼b124

8A1

ffiffiffiffiffiA1

p � 5b126

4A21

þ 255C1K63

4A21

;

a126 ¼b125

8A1

ffiffiffiffiffiA1

p þ 5b127

4A21

þ 255C2K63

4A21

;

a127 ¼b126 � 75C1K63ð Þ

10A1

ffiffiffiffiffiA1

p ; a128 ¼b127 þ 75C2K63ð Þ

10A1

ffiffiffiffiffiA1

p ;

a129 ¼6A2K63

A31

þ b129

20A1

; a130 ¼12b128 þ A1b130ð Þ

12A1

;

a131 ¼120A2K63

A41

þ b129

A21

þ b131

6A1

;

a132 ¼12b128

A31

þ b130

A21

þ b132

2A1

;

b1 ¼C1K16

18A1

; b2 ¼C2K17

18A1

;

b3 ¼1

12A21

A1C1C13 þ 2ffiffiffiffiffiA1

pC1K18 þ 2C1K20 þ 4A3K16

� �

� 7

36A21

ffiffiffiffiffiA1

p 4ffiffiffiffiffiA1

pC1K20 þ A1C1K18 þ 4A2K16

� �

þ 127C1K20

216A21

;

b4 ¼1

12A21


pC2K19 � 2C2K21 þ 4A3K17

� �

þ 7

36A21

ffiffiffiffiffiA1

p 4ffiffiffiffiffiA1

pC2K21 � A1C2K19 þ 4A2K17

� �� 127C2K21

216A21

;

b5 ¼1

2A1

ffiffiffiffiffiA1

p 2C1K23 þ 4C2A1K16 þ A3C13 þ2A3K18ffiffiffiffiffi

A1

p þ 2A3K20

A1

� �

� 5

4A21

�6K22C1 þ A3C13 þ2A3K18ffiffiffiffiffi

A1

p þ 2A3K20

A1

þ A3K18

� �

þ 17

4A21

ffiffiffiffiffiA1

p A2K18 þ4A2K20ffiffiffiffiffi

A1

p þ A3K20

� �� 147A2K20

8A31

;

b6 ¼ �1

2A1

ffiffiffiffiffiA1

p 2C2K23 þ 4C1A1K17 þ A3C14 þ2A3K19ffiffiffiffiffi

A1

p � 2A3K21

A1

� �

� 5

4A21

�6K22C2 þ A2C14 þ2A2K19ffiffiffiffiffi

A1

p � 2A2K21

A1

þ 4A3K21ffiffiffiffiffiA1

p � A3K19

� �

� 17

4A21

ffiffiffiffiffiA1

p �A2K19 þ4A2K21ffiffiffiffiffi

A1

p � A3K21

� �þ 147A2K21

8A31

;

b7 ¼1

12A21

4C1K20

ffiffiffiffiffiA1

pþ A1C1K18 þ 4A2K16

� �

� 7C1K20

18A1

ffiffiffiffiffiA1

p ;

b8 ¼1

12A21

4C2K21

ffiffiffiffiffiA1

p� A1C2K19 þ 4A2K17

� �

� 7C2K21

18A1

ffiffiffiffiffiA1

p ;


123

b9 ¼1

4A1

ffiffiffiffiffiA1

p �6K22C1 þ A3C13 þ2A3K18ffiffiffiffiffi

A1

p�

þ 2A3K20

A1


p þ A3K18

� 5

4A21

A2K18 þ4A2K20ffiffiffiffiffi

A1

p þ A3K20

� �þ 51A2K20

8A21

ffiffiffiffiffiA1

p ;

b10 ¼ �1

4A1

ffiffiffiffiffiA1

p �6K22C2 þ A2C14 þ2A2K19ffiffiffiffiffi

A1

p�

� 2A2K21

A1


p � A3K19

þ 5

4A21

A2K19 �4A2K21ffiffiffiffiffi

A1

p þ A3K21

� �þ 51A2K22

8A21

ffiffiffiffiffiA1

p ;

b11 ¼C1K20

12A1

; b12 ¼ �C2K21

12A1

;

b13 ¼1

6A1

ffiffiffiffiffiA1

p A2K18 þ4A2K20ffiffiffiffiffi

A1

p þ A3K20

� �� 5A2K20

4A21

;

b14 ¼1

6A1

ffiffiffiffiffiA1

p A2K19 �4A2K21ffiffiffiffiffi

A1

p þ A3K21

� �þ 5A2K21

4A21

;

b15 ¼A2K20

8A1

ffiffiffiffiffiA1

p ; b16 ¼A2K21

8A1

ffiffiffiffiffiA1

p ;

b17 ¼1

A21

A1C1K21 � A1C2K20 þ6A2K22ffiffiffiffiffi

A1

p�

� 1

2A1


phC1K19

� 2C1K21 þ A1C2C13 þ 2ffiffiffiffiffiA1

pC2K18

þ 2C2K20þ2A3K23

A1

;

b18 ¼ �1

6A1

4ffiffiffiffiffiA1

pC1K21 � A1C1K19 þ A1C2K18

h


pC2K20 þ

2A2K23

A1

;

b19 ¼1

12A1

A1C1K21 � A1C2K20 þ6A2K22

A1

� ;

b20 ¼ A1K3C9;

b21 ¼ A1K4C10; b22 ¼ A1C3C9 � 2K14K3;

b23 ¼ A1C4C10 � 2K14K4;

b24 ¼ 2K9C9 þ A1K11C9 þ A1K3C10 � 2K14C3;

b25 ¼ A1K4C9 þ 2K9C10 þ A1K11C10 � 2K14C4;

b26 ¼ A1K5C9 �ffiffiffiffiffiA1

pK7C9; b27 ¼ A1K6C10 þ

ffiffiffiffiffiA1

pK8C10;

b28 ¼ A1K10C9 � 2K14K5 þ2K14K7ffiffiffiffiffi

A1

p ;

b29 ¼ A1K10C10 � 2K14K6 �2K14K8ffiffiffiffiffi

A1

p ;

b30 ¼ A1K7C9; b31 ¼ A1K8C10; b32 ¼ A1K9C9 � 2K14K7;

b33 ¼ A1K9C10 � 2K14K8;

b34 ¼ A1K8C9 þ A1K7C10 � 2K14C9;

b35 ¼ A1K6C9 þ A1K5C10 þffiffiffiffiffiA1

pK8C9 �

ffiffiffiffiffiA1

pK7C10;

b36 ¼ A1C4C9 þ A1C3C10 �4K14K9

A1

� 2K14K11;

b37 ¼127

18A1

; b38 ¼a64

72A21

; b39 ¼a65

72A21

;

b40 ¼3ffiffiffiffiffiA1

pa66 � 7a70 þ 3

ffiffiffiffiffiA1

pb37a74

36A21

ffiffiffiffiffiA1

p ;


pa67 þ 7a71 � 3

ffiffiffiffiffiA1

pb37a75

36A21

ffiffiffiffiffiA1

p ;


pa70 � 14a74

36A21

ffiffiffiffiffiA1

p ;


pa71 � 14a75

36A21

ffiffiffiffiffiA1

p ;

b44 ¼4A2

1a68 � 10A1

ffiffiffiffiffiA1

pa72 þ 34A1a76 � 147

ffiffiffiffiffiA1

pa78 � 774a80

8A31

ffiffiffiffiffiA1

p ;

b45 ¼4A2

1a69 þ 10A1

ffiffiffiffiffiA1

pa73 þ 34A1a77 � 147

ffiffiffiffiffiA1

pa79 þ 774a81

8A31

ffiffiffiffiffiA1

p ;

b46 ¼a74

12A21

; b47 ¼ �a75

12A21

;

b48 ¼2A1

ffiffiffiffiffiA1

pa72 � 10A1a76 þ 51

ffiffiffiffiffiA1

pa78 þ 294a80

8A31

;

b49 ¼ �2A1

ffiffiffiffiffiA1

pa73 þ 10A1a77 � 51

ffiffiffiffiffiA1

pa79 þ 294a81

� �8A3

1

;


pa76 � 15a78 � 102a80

12A21

;

b51 ¼�2

ffiffiffiffiffiA1

pa77 þ 15a79 � 102a81

12A21

;

b52 ¼ffiffiffiffiffiA1

pa78 þ 10a80

8A21

; b53 ¼ffiffiffiffiffiA1

pa79 � 10a81

8A21

;

b54 ¼ �a80

10A1

ffiffiffiffiffiA1

p ;


123

b55 ¼ �a81

10A1

ffiffiffiffiffiA1

p ; b56 ¼a82

12A1

; b57 ¼ �a83

6A1

;

b58 ¼2a82 � A1a84

2A21

;

b59 ¼ A1C1C12 þ A1C2K6; b60 ¼ A1C2C12 þ A1C1K17;

b61 ¼ A1C2K20 � A1C1K21; b62 ¼ A1C2K18 � A1C1K19;

b63 ¼ A1C1C14 þ A1C2C13; b64 ¼C1K16

72A1

; b65 ¼C2K17

72A1

;


pC1C13 � 7C1K18 þ 3

ffiffiffiffiffiA1

pC1K20b37

36A1

ffiffiffiffiffiA1

p ;


pC2C14 � 7C2K19 � 3

ffiffiffiffiffiA1

pC2K21b37

36A1

ffiffiffiffiffiA1

p ;


pC1K18 � 14C1K20

36A1

ffiffiffiffiffiA1

p ;


pC2K19 þ 14C2K21

36A1

ffiffiffiffiffiA1

p ;


pb59 � 10A1C1C11 þ 34

ffiffiffiffiffiA1

pC1K23 þ 147C1K22

8A21

;


pb60 þ 10A1C2C11 þ 34

ffiffiffiffiffiA1

pC2K23 � 147C2K22

8A21

;

b72 ¼C1K21

12A1

; b73 ¼ �C2K21

12A1

;

b74 ¼2A1C1C11 � 5

ffiffiffiffiffiA1

pC1K23 � 51C1K22

8A1

ffiffiffiffiffiA1

p ;

b75 ¼2A1C2C11 þ 5

ffiffiffiffiffiA1

pC2K23 � 51C2K22

8A1

ffiffiffiffiffiA1

p ;


pC1K23 þ 15C1K22

12A1

;

b77 ¼15C2K22 � 2

ffiffiffiffiffiA1

pC2K23

12A1

; b78 ¼ �C1K22

8ffiffiffiffiffiA1

p ;

b79 ¼C2K22

8ffiffiffiffiffiA1

p ;

b80 ¼ �b61

12A1

; b81 ¼ �b62

6A1

; b82 ¼ �2b61 þ A1b63ð Þ

2A21

;

b83 ¼ 4ffiffiffiffiffiA1

pC1K51 þ 4A1C1K52;

b84 ¼ 4A1C2K54 � 4

ffiffiffiffiffiA1

pC

2K53;


pC1K58 þ 2C1K57 þ A1C1K74 þ

4A3K51ffiffiffiffiffiA1

p

þ 4A3K52

b86 ¼ 2C2K61 � 2ffiffiffiffiffiA1

pC2K62 þ A1C2K75 þ 4A3K54 �


p� �

;

b87 ¼2A3K58ffiffiffiffiffi

A1

p þ 2A3K57

A1

þ A3K74 � 2C1K67 þ 4C2

ffiffiffiffiffiA1

pK51

þ 4C2A1K52;

b88 ¼ 4A1C1K54 � 4C1

ffiffiffiffiffiA1

pK53 þ

2A3K61

A1

� 2A3K62ffiffiffiffiffiA1

p

þ A3K75 � 2C2K67;

b89 ¼ A1C1K58 þ 4ffiffiffiffiffiA1

pC1K57 þ 6C1K56 þ


p

þ 4A2K52 þ 4A3K51;

b90 ¼ A1C2K62 � 4ffiffiffiffiffiA1

pC2K61 þ 6C2K60 þ 4A2K54


p þ 4A3K53;

b91 ¼ 4A1C2K51 þ2A2K58ffiffiffiffiffi

A1

p�

þ 2A2K57

A1

þ A2K74 þ A3K58


p þ 6A3K56

A1

� 6C1K66

�;

b92 ¼ 4A1C1K53 �2A2K62ffiffiffiffiffi

A1

p�

þ 2A2K61

A1

þ A2K75 þ A3K62


p þ 6A3K60

A1

� 6C2K66

�;


pC1K56 þ 12C1K55 þ 4A2K51;


pC2K60 � 12C2K59 þ 4A2K53;

b95 ¼ 12C1K65 þ A2K58 þ4A2K57ffiffiffiffiffi

A1

p þ 6A2K56

A1

þ A3K57 þ6A3K56ffiffiffiffiffi

A1

p þ 12A3K55

A1

;

b96 ¼ 12C2K65 þ A2K62 �4A2K61ffiffiffiffiffi

A1

p þ 6A2K60

A1

þ A3K61


p � 12A3K59

A1

;


pC1K55;


pC2K59;

b99 ¼ 20C1K64 þ A2K57 þ6A2K56ffiffiffiffiffi

A1

p þ 12A2K55

A1

þ A3K56


p ;

b100 ¼ 20C2K64 þ A2K61 �6A2K60ffiffiffiffiffi

A1

p � 12A2K59

A1

þ A3K60


p ;


123

b101 ¼ A3K55 þ 30C1K63 þ A2K56 þ8A2K55ffiffiffiffiffi

A1

p ;

b102 ¼ A3K59 þ 30C2K63 þ A2K60 �8A2K59ffiffiffiffiffi

A1

p ;

b103 ¼20A2K64

A1

þ A1C1K59 þ30A3K63

A1

þ A1C2K55;


pC1K59 þ A1C2K56 þ 8

ffiffiffiffiffiA1

pC2K55

þ 12A2K65

A1

þ 20A3K64

A1

;


pC1K60 � 12C1K59 þ A1C2K57

�


pC2K56 þ 12C2K55 �

6A2K66

A1

þ 12A3K65

A1

�;


pC1K61 þ 6C1K60 þ A1C2K58

�


pC2K57 þ 6C2K56 �

2A2K67

A1

� 6A3K66

A1

�;

b107 ¼ 2C1K61 � 2ffiffiffiffiffiA1

pC1K62 þ A1C1K75

�þ 2

ffiffiffiffiffiA1

pC2K58

þ2C2K57 þ A1C2K74 �2A3K67

A1

�;

b108 ¼ 2A1C1K52 þffiffiffiffiffiA1

pC1K52; b109

¼ 2A1C1K54 �ffiffiffiffiffiA1

pC2K53;

b110 ¼ A1C1K74 þffiffiffiffiffiA1

pC1K58 þ


p þ A2K51

A1

;

b111 ¼ A1C2K75 �ffiffiffiffiffiA1

pC2K62 �


p þ A2K53

A1

;


pC1K76 � 2A1C2K52 �

ffiffiffiffiffiA1

pC2K51 þ

A2K74ffiffiffiffiffiA1

p

þ A2K58

A1

;


pC2K76 þ 2A1C1K54 �

ffiffiffiffiffiA1

pC1K53 þ


p

� A2K62

A1

;


pC1K56 þ


p ;


pC2K61 �


p ;

b116 ¼ 2A1C2K51 þ 2ffiffiffiffiffiA1

pC1K67 �


p � 2A2K57

A1

;


pC2K67 � 2A1C1K53 þ

2A2K61

A1

� A2K62ffiffiffiffiffiA1

p ;


pC1K56;


pC2K60;

b120 ¼A2K57ffiffiffiffiffi

A1

p þ 3A2K56

A1

� 3K66

ffiffiffiffiffiA1

pC1;

b121 ¼3A2K60

A1

� A2K61ffiffiffiffiffiA1

p þ 3ffiffiffiffiffiA1

pC2K66;


pC1K55; b123

¼ A1C2K60 � 4ffiffiffiffiffiA1

pC2K59;


pC1K65 þ


p þ 4A2K55

A1

;


pC2K65 þ


p � 4A2K59

A1

;


pC1K64 þ


p ;


pC2K64 þ


p ;

b128 ¼5A2K64

A1

� A1C1K59 � A1C2K55;


pC1K59 � A1C1K60 � A1C2K56 � 4

ffiffiffiffiffiA1

pC2K55

þ 4A2K65

A1

;


pC1K60 � A1C1K61 � A1C2K57 � 3

ffiffiffiffiffiA1

pC2K56

� 3A2K66

A1

;


pC1K61 � A1C1K62 � A1C2K58 � 2

ffiffiffiffiffiA1

pC2K57

� 2A2K67

A1

;


pC1K62 � A1C1K75 �

ffiffiffiffiffiA1

pC2K58 � A1C2K74

þ A2K74

A1

;

References

1. Lekoudis SG, Nayfeh AH, Saric WS (1976) Compressible

boundary layers over wavy walls. Phys Fluid 19:514–519

2. Bordner GL (1978) Non-linear analysis of laminar boundary

layer flow over a periodic wavy surface. Phys Fluid 21:

1471–1474

3. Vajravelu K, Sastri KS (1978) Free convective heat transfer in a

viscous incompressible fluid confined between a long vertical

wavy wall and a parallel flat wall. J Fluid Mech 86:365–383


123

4. Das UN, Ahmed N (1992) Free convective MHD flow and heat

transfer in a viscous incompressible fluid confined between a long

vertical wavy wall and parallel flat wall. Ind J Pure Appl Math

23:295–304

5. Caponi EA, Fornberg B, Knight DD, Mclean JW, Saffman PG,

Yuen HC (1982) Calculations of laminar viscous flow over a

moving wavy surface. J Fluid Mech 124:347–362

6. Yao LS (1983) Natural convection along a vertical wavy surface.

ASME J Heat Transf 105:465–468

7. Rao DRVP, Krishna DV, Sivaprasad R (1987) MHD convection

flow in a vertical wavy channel with temperature dependent heat

source. Pro Ind Natn Sci Acad 53:63–74

8. Rees DAS, Pop I (1994) A note on free convection along a

vertical wavy surface in a porous medium. ASME J Heat Transf

116:505–508

9. Rees DAS, Pop I (1995) Free convection induced by a vertical

wavy surface with uniform heat flux. ASME J Heat Transf

117:547–550

10. Hossain MA, Pop I (1996) Magnetohydrodynamic boundary

layer flow and heat transfer on a continuously moving wavy

surface. Arch Mech 48:813–823

11. Alim KCA, Hossain MA, Rees DAS (1997) Magnetohydrody-

namic free convection flow along a vertical wavy surface. Appl

Mech Eng 1:555–556

12. Reddy YP, Rao DRVP, Krishna DV (1997) Unsteady mixed

convection flow in a channel of varying gap under oscillatory

flux. Ind J Pure Appl Math 28:1053–1056

13. Patidar RP, Purohit GN (1998) Free convection flow of a viscous

incompressible fluid in a porous medium between two long ver-

tical wavy walls. Ind J Math 40:67–86

14. Taneja R, Jain NC (2001) Non-linear analysis of free convection

flow in a rotating wavy channel with temperature dependent heat

source. J Energy Heat Mass Transf 23:473–481

15. Hossain MA, Kabir S, Rees DAS (2002) Natural convection of

fluid with temperature dependent viscosity from heated vertical

wavy surface. ZAMP 53:48–52

16. Molla MM, Hossain MA, Yao LS (2004) Natural convection flow

along a vertical wavy surface with uniform surface temperature in

presence of heat generation/absorption. Int J Thermal Sci 43:157–163

17. Sarangi KC, Jose CB (2004) Unsteady MHD free convective flow

and mass transfer through porous medium with constant suction

and constant heat flux. J Ind Acad Math 26:115–126

18. Tak SS, Kumar H (2005) Natural convection MHD flow and heat

transfer in viscous incompressible fluid confined between two

vertical wavy walls. Acta Ciencia Indica 31:1089–1102

19. Yao LS (2006) Natural convection along a vertical complex wavy

surface. Int J Heat Mass Transf 49:281–286

20. Molla MM, Hossain MA (2007) Radiation effect on mixed

convection laminar flow along a vertical wavy surface. Int J

Thermal Sci 40:926–935

21. Irmay S (1958) On the theoretical derivation of Darcy and

Forchheimer formulas. EOS Trans AGU 39:702–707

22. Nakayama A, Kokudai T, Koyama H (1990) Forchheimer free

convection over a non-isothermal body of arbitrary shape in a

saturated porous medium. ASME J Heat Transf 112:511–515

23. Kladias N, Prasad V (1991) Experimental verification of Darcy–Brinkman–Forchheimer model for natural convection in porous

media. AIAA J Thermophys Heat Transf 5:560–576

24. Shenoy AV (1993) Darcy-Forchheimer natural, forced and mixed

convection heat transfer in non-Newtonian power law fluid-sat-

urated porous medium. Transp Porous Media 11:219–241

25. Vafai K, Kim SJ (1995) On the limitations of the Brinkman-

Forchheimer Darcy equation. Int J Heat Fluid Flow 16:11–15

26. Knupp PM, Lage JL (1995) Generalization of the Forchheimer-

extended Darcy flow model to the tensor permeability case via a

variational principle. J Fluid Mech 299:97–104

27. Marpu DR (1995) Forchheimer and Brinkman extended Darcy

flow model on natural convection in a vertical cylindrical porous

medium. Acta Mech 109:41–48

28. Whitaker S (1996) The Forchheimer equation: a theoretical

development. Transp Porous Media 25:27–61

29. Nakayama A (1998) A unified treatment of Darcy-Forchheimer

boundary layer flows. In: Ingham DB, Pop I (eds) Transport

phenomena in porous media, Elsevier

30. Bennethum LS, Giorgi T (1997) Generalized Forchheimer

equation for two-phase flow based on hybrid mixture theory.

Transport Porous Media 26:261–275

31. Giorgi T (1997) Derivation of the Forchheimer law via matched

asymptotic expansions. Transport Porous Media 29:191–206

32. Lees SL, Yang JH (1997) Modelling of Darcy-Forchheimer drag

for fluid flow across a bank of circular cylinders. Int J Heat Mass

Transf 40:2123–2132

33. Levy A, Levi-Hevroni D, Sorek S, Ben-Dor G (1999) Derivation

of Forchheimer terms and their verification by applications to

waves propagated in porous media. Int J Multiphase Flows 25:

683–704

34. Chen Z, Lyons SL, Qin G (2001) Derivation of the Forchheimer

equation via homogenization. Transport Porous Media 44:

325–335

35. Vafai K (2005) Hand book of porous media, 2nd edn. Taylor and

Francis, New York

36. Srinivas S, Muthuraj R (2010) MHD flow with slip effects and

temperature dependent heat source in a vertical wavy porous

plate. Chem Eng Commun 197:1387–1403

37. Tasnim SH, Mahmud S, Fraser RA, Pop I (2011) Brinkman-

Forchheimer modeling for porous media thermo-. Int J Heat Mass

Transf 54:3811–3821

38. Paul T, Singh AK, Mishra AK (2006) Transient free convection

flow in a porous region bounded by two vertical walls heated/

cooled asymmetrically. J Energy Heat Mass Transf 28:193–207

39. Singh AK, Kumar R, Singh U, Singh NP (2011) Unsteady

hydromagnetic convective flow in a vertical channel using

Darcy–Brinkman–Forchheimer extended model with heat gen-

eration/absorption: analysis with asymmetric heating/cooling of

the channel walls. Int J Heat Mass Transf 54:5633–5642

40. Nield DA, Bejan A (2006) Convection in Porous Media.

Springer-Verleg, New York

41. Singh AK, Agnihotri P, Singh NP, Singh AK (2011) Transient

and non-Darcien effects on natural convection flow in a vertical

channel partially filled with porous medium: analysis with

Forchheimer-Brinkman extended Darcy model. Int J Heat Mass

Transf 54:1111–1120

42. Ferraro VCA, Plumpton C (1961) An Introduction to Magneto-

Fluid Mechanics. Oxford University Press, London

43. Tuck EO, Kouzoubov A (1995) A Laminar Roughness Boundary.

J Fluid Mech 300:59–70

44. Martys N, Bentz DP, Garboczi EJ (1994) Computer simulation

study of the effective viscosity in Brinkman’s equation. Phys

Fluids 6:1434–1439

45. Gilver RC, Altobelli SA (1994) A determination of the effective

viscosity for Brinkman-Forchheimer flow model. J Fluid Mech

258:355–370

46. Ergun S (1952) Fluid flow through pocket columns. Chem Engng

Prog 48:89–94

47. Rosa RJ (1968) Magneto-hydrodynamic Energy Conversion.

McGraw-Hill, New York

48. Blums E (1987) Heat and Mass Transfer in MHD Flows. World

Sciences, Singapore

49. Malashetty MS, Umavati JC, Prathap Kumar J (2005) Flow and

heat transfer in an inclined channel containing a fluid layer

sandwiched between two porous layers. J Porous Media 8:

443–453


123

Hydromagnetic Convection Flow in a Porous Medium Bounded Between Vertical Wavy Wall and Parallel...

Documents

Transcript of Hydromagnetic Convection Flow in a Porous Medium Bounded Between Vertical Wavy Wall and Parallel...