_______________________________________________________________________

*Corresponding Author: sabirali@cuisahiwal.edu.pk; Cell No: +92333-5532785 1

ANALYSIS OF SECOND-GRADE FLUID FLOW IN POROUS CHANNEL

WITH CATTANEO-CHRISTOV AND GENERALIZED FICK’S

THEORIES

Sami Ullah Khana, Nasir Ali

b, S. A. Shehzad

a,* and M.N. Bashir

a

aDepartment of Mathematics, COMSATS University Islamabad, Sahiwal 57000, Pakistan

bDepartment of Mathematics and Statistics, International Islamic University, Islamabad 44000,

Pakistan

Abstract: This attempt executes the combined heat mass transport features in steady MHD

viscoelastic fluid flow through stretching walls of channel. The channel walls are considered to

be porous. The analysis of heat transport is made by help Cattaneo-Christov heat diffusion

formula while generalized Fick’s theory is developed for study of mass transport. The system of

partial differential expressions is changed into set of ordinary differential by introducing suitable

variables. The homotopic scheme is introduced for solving the resultant equations and then

validity of results are verified by various graphs. An extensive analysis has been performed for

the influence of involved constraints on liquid velocity, concentration and temperature profiles. It

is noted that the normal component of velocity decreases by increasing Reynolds number while

retards for increasing viscoelastic constraint. Both temperature and concentration profiles

enhanced by increasing combined parameter and Reynolds number. The presence of thermal

relaxation number and concentration relaxation number declines both temperature and

concentration profiles, respectively.

Keywords: Cattaneo-Christov model; Fick’s law; viscoelastic fluid; porous medium; stretching

walls channel

2

1. Introduction

The phenomenon of heat transportation is produced by thermal energy movement from one

object to an-other object due to difference in temperature. Such objects can be solid, a liquid and

solid or gas, or within a gas or liquid. The interest in heat transfer phenomena is increasing

substantially due to its countless industrial and technological applications. The wide-ranging

applications of heat transfer can be seen in many industrial and engineering processes. The most

valuable applications are fuel cells, energy production and cooling process in various atomic

devices. To predict heat transfer analysis, Fourier’s [1] developed one of most effective models

to predict heat transport analysis. The unique feature of this fluid model is that it is more capable

to envisage heat transfer analysis in macroscopic systems. Further, the mathematical modeling

based on Fourier’s model results the parabolic type form of energy expression. Cattaneo [2]

presented the generalization of this theory by addition of relaxation time phenomenon which

transformed the expression of energy in hyperbolic form. This formula enables the transport of

heat by mean of thermal waves with confined speed. The theory of Cattaneo [2] was extended by

Christov [3] in which he changed the time derivative with Oldroyd-B upper convected

derivative. Straughan [4] utilized this law to elaborate the thermal convection aspects in flow of

horizontal layer viscous liquid. Han et al. [5] observed that presence of thermal relaxation

constraints which suppressed the thermal thickness of boundary-layer. Khan et al. [6] computed

numerical solution by using shooting technique to examine the heat transportation characteristics

based on Cattaneo-Christov expressions. Later on, Meraj et al. [7] employed this model of heat

diffusion to elaborate the heat transport effects in the flow of Jeffrey fluid with variable

conductivity.

3

In several medicinal and engineering processes, the involvement of stretched

magnetohydrodynamic (MHD) flows in electrical conducting materials have various applications

in industry of metal-working, nuclear reactors, plasma, thermal insulators, modern metallurgical,

oil exploration, extraction of geothermal energy and MHD generators. MHD flows through an

arteries in the several physiological processes has keen interest of researchers and scientists. To

control the fluid flows, mixing of samples, heat transfer rate and biological transportation applied

magnetic field is used. Many authors proved an extensive analysis regarding flow of various

fluids models in presence of applied magnetic field [8-18].

The phenomenon of porous medium has many applications in different engineering processes

include petroleum technology, solar collectors, porous insulation, drying processes, geothermal

energy, geophysics, oil recovery, packed beds etc. Fluid flows in porous medium mainly depend

on the differential expression that developed by macroscopic motion of liquid. Darcy [19]

observed in his experimental study that applied pressure-gradient and flow velocity are linearly

proportional for the case of unidirectional flow under uniform medium. This Darcy’s theory is

very effective role in different applications of biomedical engineering like biological tissues [20].

Attia [21] computed asymptotic solutions of viscous liquid flow through insulated disk by

utilizing Darcy model. In another attempt, Attia et al. [22] presented the flow characteristics to

time varying non-Newtonian liquid over rotating insulated disk in the application of porous

medium. Siddiq et al. [23] examined the Darcy-Forchheimer porous theory for convectively

heating nanofluid flow. Some novel numerical computations regarding flow of alumina

nanoparticles through a permeable porous enclosure was by Sheikholeslami et al. [24].

In past few decades, a good deal attention has be devoted to analysis and mathematical modeling

of flow between stretching boundaries because of its scientific applications such as extrusion

4

process in plastic and metal industries, artificial fibers, metal spinning, glass blowing, drawing

plastic films and metal industries. This is why many authors studied this phenomenon in various

fluids models with different physical properties. Ashraf and Batool [25] developed the algorithm

of shooting method to find the numerical solution of buoyancy driven flow of micropolar fluid in

a disk. Turkyilmazoglu [26] computed the numerical expressions of flow of viscous fluid

between radially stretchable rotating disk by employing spectral numerical integration scheme.

Hayat et al. [27] implemented homotopy analysis method to discuss steady-state flow of

Newtonian fluid generated due to stretchable rotating disk.

The flow problem associated with pulsating motion of walls of channel gained the attentions of

investigators in recent years. The magnetohydrodynamic channel flow is one of great topic of

interest for engineers because of its theoretical and practical applications such as space vehicle

reentry, accelerators, astrophysical flows solar power technology. The flow causes by stretching

walls channel was extensively studied for viscoelastic fluid model via analytically [28]. Misra et

al. [29] captured the novel features of viscous fluid model by considering MHD flow in channel

with pulsating walls by using finite difference method. The investigations of electrically

conducing flow of non-Newtonian fluid through a stretching wall channel were shown

numerically et al. Misra et al. [30]. They considered the blood as non-Newtonian fluid in their

investigation. In another contribution, Misra et al. [31] explored the effects of induced magnet

field in flow of second grade occupied between walls of channel. They also utilized the heat

transfer phenomenon by using well Fourier law of heat conduction. Raftari and Vajravelu [32]

showed that series solution obtained by homotopy analysis method has an excellent agreement

with [31]. Abbasi et al. [33] presented slip flow of electrically conducting Maxwell fluid in

stretchable wall porous channel.

5

Due to immense industrial applications, the aim of this study is to examine combined mass

transport analysis of electrically-conducting viscoelastic fluid in a porous stretching walls

channel by using Cattaneo-Christov heat diffusion. The results of this investigation may be of

interest to fluid dynamicity and for physiologists in their investigations on blood flow in arteries

under chemical reaction. The solution of modeled problem is computed via homotopy analysis

method. The physical properties of various flow features parameters are explored with various

graphs.

2. Flow Analysis

We assume the steady-state situation of laminar, incompressible second grade fluid flow

confined by planes .y a It is assumed that the fluids particles flows due to solely motion of

stretching walls. The magnetic force of strength 0B is imposed perpendicular to the walls of

channel (see Fig. 1). It is mention beforehand that dominant of induced magnetic force can be

neglected as we consider low magnetic Reynolds number assumptions. Let wT represents the

constant temperature of the walls. The develop equations of the flow dynamics in the channel

can be considered as [24]:

0,u v

x y

(1)

22 3 3 2 22 0

02 2 3 2 2.

Bu v u u u u u u uu v k u v u u

x y y x y y x y y x y k

(2)

here u and v reflects velocity component along coordinate axis x and y , respectively, v

signifies the kinematic viscosity, the electric intensity, 0B the strength of applied magnetic

6

field, the fluid density, 0k represents the elasticity parameter, is porosity parameter and

k is permeability parameter. The imposed boundary conditions are [24]:

, 0, at ,

0, 0, at 0.

u bx v y a

uv y

y

(3)

Let us introduce dimensionless quantities [21]:

, , .y

u bxf v abfa

(4)

Using above dimensionless variables, Eqs. (2)-(3) transformed as

2 2

2 Re 0,f K f f f ff f ff Mf

(5)

0 0, 1 0, 1 1, 0 0, f f f f (6)

in which, 1 /K b is viscoelastic parameter, 0M B abk

is the combined magnetic

and porosity parameter (combined parameter) and 2Re /a b is the Reynolds number.

2.1 Heat and Mass Transfer Analysis

The generalized expressions of Fourier’s and Fick’s theories are [5,6]:

,E k Tt

qq V q V q q V (7)

,C BD Ct

JJ V J V J J V (8)

where q and J denotes the heat and mass fluxes, respectively, BD stands for molecular

diffusivity of the species, E and C are the thermal and concentration relaxation times,

respectively. It is note-worthy to point-out that for 0E C in Eqs. (7) and (8), the

7

expressions for Classical Fourier’s and Fick’s laws retained. Thus the energy equation and

presence of thermal radiation and concentration equation are defined as

2

2,E E

T T Tu v

x y y

(9)

2

2,C C B

C C Cu v D

x y y

(10)

where represents the liquid thermal diffusivity and E and C can be expresses as

2 2 22 2

2 22 ,E

u T v T T v T u T T Tu v uv u v u v

x x y y x y x y y x x y

(11)

2 2 22 2

2 22 ,C

u C v C C v C u C C Cu v uv u v u v

x x y y x y x y y x x y

(12)

We impose following boundary conditions for governing energy and concentration equations

, at ,

0, 0, at 0.

w wT T C C y a

T Cy

y y

(13)

The governing equations are made dimensionless by introducing following dimensionless

variable

, ,w w

T C

T C (14)

21Re Re 0,

PrTf ff f (15)

21Re Re 0,Cf ff f

Sc (16)

where Pr / reflects the Prandtl number, T Eb stands for thermal relaxation constraint,

/ BSc D denotes the Schmidt number and C Cb reflects the concentration relaxation

8

constraint.

The dimensionless boundary conditions are

'(0) 0, (1) 1, '(0) 0, (1) 1. (17)

3. Homotopy Analysis Method

The mathematical modeling of many engineering problems involves the differential equations of

highly nonlinear nature. Such equations set a challenge for engineers to compute either analytical

or numerical solution. A famous analytical method, homotopy analysis method (HAM) is a

power full technique to derive the series solution of differential expressions without restriction of

large or small constraint. This useful method offers great freedom to adopt and control the region

of convergence. The benefit of this method over various numerical techniques is that of rounding

off errors free which take place by discretization process. The computations of this method do

not require any large time and computer memory. This approach was originally proposed by

Liao [34]. After that, distinct researchers adopted this scheme to solve the governed differential

equations [35-41]. In present section, we introduce one of the most powerful techniques to solve

the system of Eqs. (5) and (15)-(16). We used homotopy analysis scheme to discuss the

analytical solution for all values of given parameters. To proceed, we suggest following initial

approximations for given flow problem

3 2 2 2

0 0 0

1 1 1( ) , ( ) , ( ) .

2 2 2f (18)

The differential operators are

3 2 2

3 2 2, , ,f£ £ £

(19)

which satisfying

9

2 3

1 2 3 4 0,f£ A A A A (20)

2

5 6 0,£ A A (21)

2

7 8 0,£ A A (22)

whereiA ( 1,2,...8)i denotes constants.

4. Convergence of Solution

The series solution obtained in above section is strictly reliable on auxiliary constraints

,fh h and .h The appropriate values of these parameters are necessary for convergence of

HAM solution. Graphs of these auxiliary parameters

have been plotted at 16th order of

approximations for the selection of suitable range of these parameters. In Fig. 2 it has been

analyzed admissible values for 0.8 0.2fh , 1.5 0.3 and 1.5 0.3.h The role

of convergence in the given region is quit necessary for guaranteed the solution. It is commonly

noted that the values of involved parameters are not properly selected, the comparison of

homotopy analysis method with other exact or numerical methods becomes quite difficult and

time consuming. A suitable selection of these parameters guaranteed the solution for various

sundry parameters [42]. In order to validate our solution, present results are compared with the

already available numerical values reported by Hayat et al. [28]. It is seen that our results has an

excellent agreement with these reported results.

5. Results and Discussion

After computing the explicit analytic solution discussed in previous section, the aim of this

section is to present the inner sight of the study against various flow parameters. A graphical

analysis for distinct controlling governing parameters for velocity, temperature and concentration

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profiles are presented by sketching Figs. 3–17. The graphical representation of viscoelastic

parameter K on normal velocity component f is shown in Fig. 3. The values of K are

ranging from 0 to 5.5. It is observed that the magnitude of velocity enhanced for larger values of

K. Fig. 4 which is proportional to mass flow rate, shows that back mass flow rate decreases when

the M increases. Since the combined parameter is combination of both Hartmann number and

porosity parameter which strictly reduce the magnitude of the velocity of fluids particles. Fig. 5

depicts the influence of Reynolds number Re in the velocity component .f The magnitude of

component of velocity f decreases in the whole region by increasing Reynolds number Re .

Fig. 6 exhibits that when M and Re are constants and K is varying, the back flow velocity in

the center line decreases but the region of back flow increases. Here two different observations

are observed within the region of channel. The flow near the central line of channel increases

smoothly while decay in flow is observed near the walls of channel.

The variation of velocity f against for distinct values of M is depicted in Fig. 7. It is evident

from this figure that horizontal component of velocity f shows increasing behavior near the

walls of the channel but shows opposite effects near the central line of channel. By increasing the

magnetic field strength, back flow velocity in the center line decreases but the region of back

flow increases. The influence of Re ranging from 0 to 12 on f are sketched in Fig. 8. The

movement of stretching walls results a back flow near the central line of channel. It is observed

that velocity near the walls of channel decreases up to 0.5 . However, opposite effects are

observed far away from the center of channel. As expected a reversed trend for velocity is

observed near central line of channel is opposite.

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The outcomes of different values of Prandtl number Pr on are demonstrated in Fig. 9. In Fig.

9(a), variation of Prandtl number Pr 0.0,1.4,2.4,4.4 is observed by taking 2.5T . The

temperature profile showing decaying behavior because of low thermal diffusivity. Fig. 8(b)

explains the variation of same parameter for 5.5.T Again similar trend is observed, however,

the rate of heat transfer is smaller as compared to Fig. 9(a). Thus the proper choice of T can be

more useful for increasing or decreasing the fluid temperature. Fig. 10 manifested the superiority

of combined parameter M on . .The flow field develops a Lorentz force, acts like frictional

force and thus increase the temperature of fluid. In Fig. 11, the temperature of fluid decrease in

the whole domain by increasing viscoelastic parameter .K In Fig. 12 the influence of Reynolds

number Re is depicted on . A rise in temperature is observed for dominant values of Re.

Moreover the intensifying values of Re strengthen the boundary layer thickness. Fig. 13 reflects

the influence of relaxation time constant T on temperature profile . It is scrutinized that the

temperature profiles with related thermal boundary layer thickness decayed for increased T in

whole domain. Fig. 14 shows the behavior of concentration profile for viscoelastic parameter.

The concentration field decreases by increasing viscoelastic parameter. Contrary to Fig. 14, the

concentration profile is found to be increase with increasing M (Fig. 15). Fig. 16 elucidated the

outcomes of Re on concentration distribution. It is noted that concentration profile enhanced by

increasing Re. Finally Fig. 17 presents the effects of relaxation time constant C on

concentration profile. It is seen that concentration profile shows a decaying behavior by

increasing .C

6. Concluding Remarks

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MHD steady-state flow of second-grade fluid in a stretching walls channel in presence of porous

medium has been considered. The partial differential system which govern the flow is first

transformed to a forth order nonlinear differential equation and then solved by employing HAM.

This method can be useful in other analytical methods that use the series solution.

A decrease in the normal component of velocity is investigated by increasing combined

parameter and Reynolds number while it shows opposite trend by increasing viscoelastic

parameter.

By increasing combined parameter and Reynolds number, the fluid temperature in whole

domain increases and demotion in profile is observed by increasing viscoelastic

parameter, Prandtl number and thermal relaxation number.

It is executed that the larger values of thermal relaxation number deflate temperature

profile.

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[40]. Turkyilmazoglu, M. “Convergence accelerating in the homotopy analysis method: A new

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Dr. Sami Ullah Khan is working as an Assistant Professor at Department of Mathematics,

COMSATS University Islamabad, Sahiwal Pakistan. He has completed his Ph.D in applied

mathematics from International Islamic University Islamabad, Pakistan in 2016. He has

published more than 30 international impact factor research papers. His research interests include

Newtonian and non-Newtonian fluids flow, flow through porous media, heat and mass transfer

and magnetohydrodynamics flows.

Dr. Nasir Ali is an Associate Professor in Department of Mathematics, International Islamic

University Islamabad, Pakistan. He earned his Ph. D. in applied mathematics from Quaid-I-

Azam University Islamabad, Pakistan in 2008. He has published more than 150 international

impact factor research papers with high impact factor. His research interests include Newtonian

and non-Newtonian fluids flow, gliding motion, heat and mass transfer, magnetohydrodynamics

and fluid dynamics of Peristaltic flows.

Dr. Sabir Ali Shehzad is working as an Assistant Professor at Department of Mathematics,

COMSATS University Islamabad, Sahiwal Pakistan. He has completed his Ph.D in Fluid

18

Mechanics from Quaid-I-Azam University, Islamabad, Pakistan in 2014. His main research

interests are Newtonian and non-Newtonian fluids, nanofluids and heat and mass transfer

analysis. He has published 230 research articles in various high reputed international journals.

Dr Bahsir is working as an Assistant Professor at COMSATS University Islamabad, Sahiwal

campus. He completed his PhD from University of Engineering and Technology, Lahore in

2017. He has published several research articles. His main areas of research are computational

methods, numerical analysis and heat transfer.

19

Figure Captions

Fig. 1: Geometry of problem.

Fig. 2(a-c): h-curve for (a) velocity (b) temperature and (c) concentration profile.

Fig. 3: Graph of K on .f

Fig. 4: Graph of M on .f

Fig. 5: Graph of Re on .f

Fig. 6: Graph of K on .f

Fig. 7: Graph of M on .f

Fig. 8: Graph of Re on .f

Fig. 9(a,b): Graph of Pr on for (a) 2.5T (b) 5.5T

Fig. 10: Graph of M on

Fig. 11: Graph of K on

Fig. 12: Graph of Re on

Fig. 13: Graph of T on

Fig. 14: Graph of K on .

Fig. 15: Graph of M on .

Fig. 16: Graph of Re on .

Fig. 17: Graph of C on .

Table Caption

Table 1: Comparison of present results with Hayat et al. [28] when 6.0.

20

Fig. 1: Geometry of problem.

Fig. 2(a-c): h-curve for (a) velocity (b) temperature and (c) concentration profile.

21

Fig. 3: Graph of K on .f

Fig. 4: Graph of M on .f

Fig. 5: Graph of Re on .f

Fig. 6: Graph of K on .f

22

Fig. 7: Graph of M on .f

Fig. 8: Graph of Re on .f

Fig. 9(a,b): Graph of Pr on for (a) 2.5T (b) 5.5T

23

Fig. 10: Graph of M on

Fig. 11: Graph of K on

Fig. 12: Graph of Re on

Fig. 13: Graph of T on

24

Fig. 14: Graph of K on .

Fig. 15: Graph of M on .

Fig. 16: Graph of Re on .

Fig. 17: Graph of C on .

25

Table 1: Comparison of present results with Hayat et al. [28] when 6.0.

K Re Hayat et al. [28] Present results

0.0

0.4

0.6

10 7.7958

15.4094

22.1710

7.79583

15.40952

22.17100

0.6 0.0

5.0

15.0

17.7101

18.2930

19.4301

17.71010

18.29329

19.43020