ENTROPY GENERATION ANALYSIS FOR SECOND ORDER …raops.org.in/epapers/dec19_4.pdfexplicitly presented...

Journal of Rajasthan Academy of Physical Sciences

ISSN : 0972-6306; URL : http://raops.org.in

Vol.18, No.3&4, July-December, 2019, 155-168

ENTROPY GENERATION ANALYSIS FOR SECOND ORDER

FLUID FLOW OVER A NON ISOTHERMAL STRETCHING

SURFACE

Paresh Vyas

1 and Kusum Yadav

2

Department of Mathematics, University of Rajasthan, Jaipur, 302004, India

E-mail: [email protected],

[email protected]

Abstract: The paper presents entropy analysis for radiative dissipative second order fluid

flow over a stretching sheet subjected to power-law temperature in a saturated porous

medium. Similarity transformations are used to convert the governing partial differential

equations to ordinary differential equations (ODE). The BVP is solved numerically to get

the velocity distribution, temperature distribution and entropy generation. The quantities

of interest are portrayed graphically and discussed.

Keywords: Entropy, stretching sheet, porous, second order fluid, Radiation, Dissipation

1. Introduction

Entropy generation analysis for non Newtonian fluid flow over a stretching sheet is

peculiarly interesting owing its vast applications in biological sciences, geophysics &

mining industries and chemical & plastic processing industries. For long, Non-Newtonian

behaviour has been treated for heat transfer aspects only due to industrial applications.

Recently it has been realised that an entropy generation analysis would be more insight

giving in designing optimal device and to understand the phenomenon.

There has been long history of studies in Boundary layer flow on a moving continuous

solid surfaces pioneered first by Sakiadis [28, 29] and later on experimentally verified by

Tsou et al. [31]. Gupta and Gupta [12] investigated heat transfer from an isothermal

stretching sheet subject to suction or blowing. The case of exponentially stretching sheet

in viscous fluid was studied by Crane [10]. All these studies were restricted to Newtonian

fluids. Bujurke et al. [6] studied the flow and heat transfer of second order fluid over

stretching sheet. Vajravelu and Rollins [32] devised an analytical solution for heat

transfer characteristics in viscoelastic second order fluid over a stretching sheet taking

frictional heating and internal heat generation into account. Akkayya and Talla [2]

explicitly presented the analysis of boundary layer flow and heat transfer of visco–elastic

fluid through porous medium over an exponentially stretching sheet with non-uniform

heat source employing Homotopy Analysis Method (HAM). Nataraja et al. [22] obtained

the closed-form solutions for the flow of a second-order fluid over a stretching surface

having power-law temperature. Pillai et al. [24] studied heat transfer in a visco-elastic

http://raops.org.in/mailto:[email protected]:[email protected]

156 Paresh Vyas and Kusum Yadav

boundary layer flow through a porous medium. Heydari and Taleghani [14] analysed

boundary layer in MHD flow of non-Newtonian viscoelastic fluid on stretching sheet.

The flow of a power-law fluid due to a linearly stretching sheet and heat transfer

characteristics using variable thermal conductivity in the presence of a non-uniform heat

source/sink was studied by Abel et al. [1]. Zaib et al. [40] obtained Dual similarity

solutions for velocity and temperature of a non-Newtonian Casson fluid and heat transfer

due to an exponentially permeable shrinking sheet with viscous dissipation. Khan and

Rahman [17] investigated flow and heat transfer of a modified second order fluid over

non-linearly stretching sheet. Vijaya Laxmi and Shankar [33] proposed radiative

boundary layer flow and heat transfer of Nanofluid over a nonlinear stretching sheet with

slip conditions and suction. Imtiaz et al. [16] explored the flow of Jeffrey fluid due to a

curved stretching sheet.

Second law of thermodynamics is a pertinent tool to address inherent thermodynamic

irreversibility. It underscores the processes that involves transfer or conversion of energy

are irreversible. Bejan [3-5] in his innovative works shown that entropy generation

minimization can be actualised by selecting parameters’ vlues that have bearing on

entropy generation. In thermofluidic configurations entropy production is caused by heat

transfer, fluid friction, imposed magnetic field, and radiative heat transfer.

There has been a great surge of studies in entropy generation aspects in fluid flow

systems. These investigations were carried first primarily in idealized flow

configurations. The second law characteristics of heat transfer and fluid flow for forced

convection inside channel with circular cross-section and channel made of two parallel

plates was analyzed by Mahmud and Fraser [20]. Chauhan and Olkha [7] examined

entropy generation on non-Newtonian fluid flow in annular pipe with naturally permeable

boundaries. Chauhan and Rastogi [8] studied entropy generation in flow through a porous

medium over a stretching sheet in the presence of a magnetic field. Srivastava et al.[30]

analysed entropy generation analysis for oscillatory flow in vertical channel filled with

porous medium. Vyas and his coworkers [34,35,36,37,38.39] examined entropy

generation in variety of flow configurations. Makinde and Aziz [21] performed analytical

and numerical analysis of the second law of thermodynamics for plane Poiseuille flow

with asymmetric convective heat transfer taking variable fluid viscosity. M.H.Yazdi et al.

[19] presented the first and second law analysis of power-law non-Newtonian flow over

embedded open parallel micro channels within micro patterned permeable continuous

moving surface are examined at prescribed surface temperature. Chauhan and kumar [9]

studied heat transfer and entropy generation for compressible fluid flow in a channel

partially filled with a porous medium. Hooshmand et al. [15] numerically investigated the

effects of magnetic field strength, thermal radiation, Joule heating, and viscous heating

on a forced convective flow of a non-Newtonian, incompressible power law fluid in an

axisymmetric stretching sheet with variable temperature wall. Rashidi et al.[25,26]

studied the analysis of entropy generation for Stagnation point flow in a porous medium

over a permeable stretching surface and Rashidi et al. [26] also investigated entropy

generation in an MHD flow over a rotating porous disk with variable physical properties.

Entropy Generation Analysis for... 157

Rashidi et al. [27] also studied convective flow of a third grade non-Newtonian fluid due

to a linearly stretching sheet subject to a magnetic field.

Dehsara et al. [11] discussed entropy analysis for MHD flow over a non-linear inclined

transparent plate in a porous medium taking solar radiation into account. Lin and Chen

[18] analysed irreversibility of heat transfer for a Non-Newtonian power-law flow with

power-law temperature. They exploited differential transform method (DTM) with

shooting method to solve the non-Newtonian power-law flow problem. Habibi Matin et

al. [13] studied numerically entropy analysis in mixed convection MHD flow of

nanofluid over a non-linear stretching sheet taking viscous dissipation and variable

magnetic field into account using the Keller-Box scheme. Noghrehabadi et al. [23]

presented entropy analysis for Nano fluid flow over a stretching sheet in the presence of

heat generation/ absorption and partial slip.

The central objective of this paper is to analyse the entropy generation in second order

fluid over a stretching sheet embedded in a porous medium. The governing equations are

solved numerically using Runge Kutta method along with shooting technique. The results

obtained for the pertinent parameters are depicted graphically.

2. Mathematical Formulation

We consider a 2-D boundary layer flow of a viscous, incompressible, electrically

conducting second order fluid past a stretching sheet placed at the bottom of the porous

medium. A Cartesian coordinate system is chosen wherein the x-axis is along the sheet

and the y-axis is taken normal to it. Two equal and opposite forces are applied along the

sheet so that the position of the origin is unaltered. The stretching velocity varies linearly

with the distance from the origin. A magnetic field of uniform strength B is applied along

y-axis. Further, it is assumed that both the fluid and the porous medium are in thermal

equilibrium. The Rooseland approximation is followed to describe the heat flux in the

energy equation. The governing equation and the associated boundary condition for the

set up are as follows:

u v0

x y

(1)

2 2 2 3 2

2 2 2 3

o

u u d u u u v u νu σB uu v ν k u v

x y y x y y y y K ρ

(2)

2 22 2r

p 2

o

T T T q µuρC u v κ σB u

x x y y K

(3)

and the associated boundary conditions are


Tβ

w w

xy=0: u u =cx,v 0, T

L=T T

(4)

u 0, T Ty : (5)

where (u, v) are the velocity components along (x, y) directions respectively, uw is the

stretching velocity, c > 0 is a given stretching parameter, T is the temperature, Tw is the

surface temperature, T∞ is a uniform temperature of the ambient fluid, Ko is the

permeability of the porous medium, σ is the electrical conductivity, ρ is the density, ν =

µ/ρ is the kinematic viscosity, A is constant, L is the characteristic length, βT is a

constant, κ is the thermal conductivity, B is the applied magnetic field, k is a positive

parameter associated with viscoelastic fluid, Cp is the specific heat at constant pressure.

The radiative heat flux qr is described as follows

4

r

4β Tq

3δ y

(6)

where β and δ are the Stephen-Boltzmann constant and the mean absorption coefficient

respectively. Temperature difference within the flow is assumed to be sufficiently small

so that we can express T4 as a linear function of temperature T, using a truncated Taylor

series about the free stream temperature T∞ to yield

4 3 4T 4T T 3T (7)

3. Solution

Introducing the stream function ѱ such that

u , vy x

0 w

x T Tf ( ),

u L T TRe

where 0 1xx y kc

, y ,u cL,k , Re yL L L

.

The equations (1)-(5) are transformed to

2 2 211 1

ff f k 2f f f ff M fR K

fe

(8)

2 2 2Br 1

1 N θ Prfθ M f x 0Re K

(9)


subject to the boundary conditions

0: f 0, f 1, 1

(10)

: f 0, 0

(11)

Here prime denotes differentiation with respect to η and Re, Pr, K, M. Br and N are the

Reynolds number, Prandtl number, permeability parameter, Magnetic field parameter,

Brinkman number and radiation parameter respectively defined as Re = uoL /ν, 2 32

2 0 0P

w 0

K c uC 16 TBPr ,M ,K ,Br , N

c (T T ) 3

,.

The equation (8) subject to boundary conditions given in equation (11) admits the

solution of the form

1

1

m1 1f (1 e ), m , 0 k 1m 1 k

The energy equation (9) is obtained in the following form

2 2 2Br 1(1 N) Prf M x f 0Re K

(12)

Constrained by the end conditions are 0 1, 0

(13)

3.1 Solution Technique:

The non dimensional boundary value problem (BVP) described by (12)-(13) has been solved

by Runge-Kutta integration scheme together with shooting technique. In order to employ

shooting method, the BVP is converted first to a system of initial value problems. In the

present case, the BVP is reduced to following system of initial value problems (IVP):

f p (14)

q (15)

2 2 21 Br 1q Prf M x f

1 N Re K

(16)

Together with initial conditions

0f (0) 1, q(0) a (17)

The above system of initial value problems is solved in the solution space with a proper

guess value a0 (say) for the unknown quantity 0 such that the end condition θ (1) = 0 is satisfied. If the end condition is not satisfied for this guess value then the system of


IVP’s is solved with another guess value a1. If the end conditions still not satisfied with

suitable error tolerance, then new refined guess value is generated using secant method.

The process continues until the end condition is satisfied to the prescribed accuracy. In

order to have confidence in numerical code of the problem in hand, a step size of

∆η=0.001 was found to be satisfactory for prescribed error tolerance of magnitude10-7

after a rigourous grid independence exercise.

4. Second Law Analysis

The local volumetric rate of entropy generation SG for the present problem is given as follows

2 22 2 2 2

G 2

0

k T 16 T B u u uS

T x 3 y T T K T y

(18)

In order to define the dimensionless entropy generation rate we prescribe the

characteristic entropy generation rate SG0 and the entropy generation number Ns

respectively as follows

2

w GG0 s2 2

G0

Re(T T ) SS and N

T L S

(19)

Consequently the entropy generation number Ns is found to be

2 2 2 2 2

s 1 2

Br 1N (1 N) M x f Br x N N

Re K

(20)

where

2 2 2 2 2

1 2

Br 1N (1 N) , N M x f Br x

Re K

(21)

Obviously the terms N1 and N2 stand for heat transfer irreversibility and dissipative

irreversibility respectively.

The Bejan Number Be is defined as follows

1

1 2

HTI NBe

HTI FFI N N

(22)

5. Results and Discussions

In order to have the plots for entropy generation number Ns we require the distributions

for velocity, temperature and their gradients. The Runge- Kutta method together with

shooting technique has been used to solve the BVP to yield temperature and its gradients

for different sets of parameter values. Thus numerically computed values for the velocity,

temperature and their gradientsare readily exploited to compute entropy generation and


Bejan number depicted in figures 1-18. Figure 1 shows the effect of Brinkman number on

non dimensional temperature θ(η).We observe that the thickness of thermal boundary

layer registers an increase with an increment in values of Brinkman number. This goes

fine with reasoning that larger values of Brinkman number indicate more frictional

heating in the system causing a rise of temperature θ(η). The effect of magnetic field has

been depicted in Figure 2, which shows that θ(η) increases with increasing values of

Hartmann number M. Figure 3 demonstrates variation of θ(η) for different values of

Prandtl number Pr. We find that θ(η) decreases with an increase in Pr. The effect of

radiation parameter N on θ(η) has been shown in figure 4, which demonstrates that the

temperature θ(η) registers a considerable increase with an increase in radiation parameter

N. Figure 5 exhibits variations in θ(η) for different values of Reynolds number Re. It

clearly shows that increase in Re results in decay of θ(η). Figure 6 shows the impact of

permeability parameter K on the temperature. We find that the temperature decreases

with the increase in permeability parameter K.

Figure 7-18 depict variation in entropy generation number and in Bejan number for varying

values of pertinent parameters. The plots for Bejan number show that there is a rise in

Bejan number in a region adjacent to the surface and after attaining a peak at different

spatial distances the Be decreases as we move farther away from the surface. Figure 7

demonstrates that the effect of Brinkman number over entropy generation number. The

figure reveals that the entropy increases with an increase in Brinkman number. Figure 8

shows that Bejan number decreases throughout the solution space with an increase in the

values of Br. Figure 9 and Figure 10 display the effect of permeability parameter K on

entropy generation number and Bejan number respectively. It is revealed that with an

increase in values of K there is a decay in entropy where as the Bejan number increases

with increasing K in the region adjacent to the surface though it decreases subsequently

with increasing values of K. Figure 11 and Figure 12 depict the effects of magnetic field

parameter M on entropy and Bejan number. Figure 11 shows that with an increase in

magnetic parameter M, the entropy increases. The Bejan number decreases with increasing

values of M near the surface till a certain spatial distance and then increases. Figure 13

exhibits the effect of radiation parameter N on entropy generation number. It shows that

with increasing values of N, the entropy decays near the surface and then the trend is

reversed beyond a certain spatial distance. Figure 14 depicts the decrement of Bejan

number Be with an increment in the values of radiation parameter N. Figure 15 and figure

16 depict the effect of ω on the entropy generation number and Bejan number respectively.

We find that the entropy increases with an increase in ω whereas the Bejan Number

decreases with the increasing values of ω. Figure 17 and figure 18 demonstrate the effect of

Reynolds number Re on entropy generation number and the Bejan number respectively. It

is shown that the entropy decreases with an increase in Reynolds number whereas Bejan

number increases near the surface and then the trend is reversed.

6. Conclusion

It is found that the parameters entering into the problem have qualitative and quantitative

bearing on entropy generation number and the Bejan number. The information can be

utilized for entropy minimization.


Fig 1 Temperature Profiles for varying

values of Brinkman number Br


values of radiation parameter N


values of Hartmann number M


values of Reynolds number Re


values of Prandtl number Pr


values of permeability parameter K


Fig 7 Entropy generation number for

varying values of Brinkman number Br

Fig 10 Bejan number for varying values of

permeability parameter K


Brinkman number Br


varying values of Hartmann number M

Fig 9 Entropy generation number for varying

values of permeability parameter K


Hartmann number M



varying values of Radiation paramrter N


characteristic ratio ω


Radiation parameter N


varying values of Reynolds number Re


varying values of characteristic ratio ω


Reynolds number Re

Acknowledgements : The authors are thankful to the Referee for valuable comments and

suggestions.


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ENTROPY GENERATION ANALYSIS FOR SECOND ORDER …raops.org.in/epapers/dec19_4.pdfexplicitly presented...

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