Analytical Expression for the Hydrodynamic Fluid Flow through a Porous Medium by Shirley Wang

Analytical Expression for the Hydrodynamic Fluid Flow through a Porous Medium V.Ananthaswamy1, S.Uma Maheswari2

Department of Mathematics, The Madura College (Autonomous), Maduri, Tamil Nadu, India

M. Phil., Mathematics, The Madura College (Autonomous), Maduri, Tamil Nadu, India

ananthu9777@rediffmail.com; 2 umashanmugam1992@gmail.com

Abstract

In this research article the effect of variable viscosity on the temporal development of small disturbances in a pressure-driven fluid flow through a channel saturated with porous medium is investigated. The approximate analytical solution of the second order boundary value problem for the dimensionless velocity is derived by using the Homotopy analysis method. This method can be easily extended to solve the non-liner initial and boundary value problems in physical, chemical in engineering and sciences. Keywords

Variable Viscosity; Porous Medium; Hydrodynamic Flow; Boundary Value Problem; Homotopy Analysis Method Introduction In the last few years, studies related to hydrodynamic stability of a moving viscous fluid through a channel filled with saturated porous medium played a key role in transport process, petrochemical engineering and geo-physical flows. It was because of that the study provided useful information on the sequence of fluids from lamina to turbulent flows. Turbulent flow has been used in some real life application. For example, in arterial blood flow with multiple stenosis, shipping over deep seas and in aviation industry and many more. Currently, several work has been done in this area of research for example, Makinde [2003] reported the linear stability of hydromagnetic plane-Poiseuille flow at high Reynolds numbers by using the multideck asymptotic approach. Makinde and Mhone [2007] investigated the temporal development of small disturbances in magneto hydrodynamic Jeffery–Hamel flows through a convergent-divergent channel. Furthermore Makinde [2009] examined the temporal development of small disturbances in a pressure-driven fluid flow through a channel filled with a saturated porous medium by using the Brinkman flow model. In all the above mentioned studies, the fluid viscosity has been studied constant. Viscosity is a very sensitive fluid property that varies with temperature, pressure or both in some cases. Therefore, as suggested in [Liao, (1992 & 1995)] the effect of stenosis can be captured in the model by taking the artery as a porous medium. Motivated by the results in [Protter (1984), Liao., (1992 & 1995) ], the specific objective of this paper is to investigate the effects of variations in viscosity and porous permeability on the flow stability which has not been accounted for in the previous models in literature. Mathematical Formulations of the Problem Consider the flow of a variable viscous, incompressible fluid through a channel filled with saturated porous materials. The dynamic viscosity of the fluid is assumed to vary with the channel width. If we employed a Cartesian coordinate system such that the x-axis corresponds to the flow direction and the y-axis is normal to it, then in 2-dimensions, the flow governing equations can be written as: u v  0 x y

(1)

V.Ananthaswamy, S.Uma Maheswari

u  u   u   p   2   u   u   v          t  x   y   x   Re x  x 

(2)

 1     u v     u            Re y    y x    Re Da  v  v   v   p   2   v   u   v          t  x   y   y   Re y  y 

(3)

 1     u v     v            Re x    y x    Re Da 

Where the following dimensionless variables and parameters have been used in (1)-(3) u

v t' u' v' x' y' ,v  ,t  0 , x  , y  , v0 v0 h h h

p

p' K dp ,   e  (1 y ) , Da  2 , G  Re dx  v02 h 2

(4)

In (4), v0 is the characteristic fluid velocity, h is the half-width of the channel that represents the tube radius, K is the porous permeability,  is the fluid density,  is the dynamic viscosity, Da is the Darcy parameter, Re is the









flow Reynolds number, p, p ' is the dimensionless and dimensional fluid pressure, u, u ' is the dimensionless





and dimensional fluid velocity, x, x , y, y

 are the dimensionless and dimensional Cartesian coordinates, 

the viscosity variation parameter. The basic flow equation is given by, d 2u u du   1 y 2    2 y  Ge 0 2 Da dy dy

(5)

du  0   0, u 1  0 dy

(6)

the boundary conditions are,

To obtain the solution of the non-linear equation (5), we assume 0 <  <<2. Approximate Analytical Solution of the Initial Value Problem Using the Homotopy Analysis Method (HAM) Homotopy analysis method (HAM) is a non-perturbative analytical method for obtaining series solutions to nonlinear equations and has been successfully applied to numerous problems in science and engineering [Liao (1992, 1995, 1999, 2003, 2004, 2007, 2010 & 2012), Ananthaaswamy et. al., (2013), Saravanakumar et. al., (2013), Subha et. al., (2014) ]. In comparison with other perturbative and non-perturbative analytical methods, HAM offers the ability to adjust and control the convergence of a solution via the so-called convergence-control parameter. Because of this, HAM has been proved to be the most effective method for obtaining analytical solutions to highly nonlinear differential equations. Previous applications of HAM have mainly focused on non-linear differential equations in which the non-linearity is a polynomial in terms of the unknown function and its derivatives. Liao [1992, 1995, 1999, 2003, 2004, 2007, 2010 & 2012] proposed a powerful analytical method for non-linear problems, namely the Homotopy analysis method. This method provides an analytical solution in terms of an infinite power series. However, there is a practical need to evaluate this solution and to obtain numerical values from the infinite power series. In order to investigate the accuracy of the Homotopy analysis method (HAM) solution with a finite number of terms, the systems of differential equations were solved. The Homotopy analysis method is a good technique comparing to another perturbation. Homotopy perturbation method is a special case of Homotopy analysis method. Different from all reported perturbation and non-perturbative techniques, the Homotopy analysis method itself provides us with a convenient

Analytical Expression for the Hydrodynamic Fluid Flow through a Porous Medium

way to control and adjust the convergence region and rate of approximation series, when necessary. Briefly speaking, the Homotopy analysis method has the following advantages: It is valid even if a given non-linear problem does not contain any small/large parameter at all; it can be employed to efficiently approximate a nonlinear problem by choosing different sets of base functions. The Homotopy analysis method contains the auxiliary parameter, which provides us with a simple way to adjust and control the convergence region of solution series. Using this method, the approximate analytical solutions of the eqn. (6) (see Appendix B) are shown as follows:    Ge    y 2     2a cosh ky       b 2 k             a   e 2 ky  1   yc  2sinh ky    3     4k 2  k         Ge     c      2   sinh ky    4   2sinh ky    k  k    u  y             Ge  1  e  Ky   2c  e  Ky  1    4     k2  k         a   2 Ky 2  1  a  y  cosh ky        4k 2  e      2     a y   cy   sinh ky    2          k    k     







Where k 2 





 



(7)

1 Da

Where a, b and c are defined by the following:  1   Ge  G  e  2G  e   a    2  k2 k4   2cosh ky   k  Ge    2G  e   b   2   4  k   k  2   4G  e   c   G e 2 k  





(8)

(9) (10)

Results and Discussion

FIGURE 1. THE DIMENSIONLESS CARTESIAN COORDINATE y VERSES THE DIMENSIONLESS VELOCITY u  y  . THE VARIATION OF THE DIMENSIONLESS VELOCITY u  y  ARE COMPUTED BY USING THE EQN. (7) IN SOME FIXED PARAMETER VALUES  , G AND THE DIFFERENT VALUES OF Da WHEN y  1....1.

V.Ananthaswamy, S.Uma Maheswari

Figures (1)-(10) represent the dimensionless Cartesian coordinate y versus the dimensionless fluid velocity u  y  . The variation of the dimensionless fluid velocity u  y  is obtained by using the eqn.7. From Figs. (1)-(3) it is clear that when the darcy parameter Da increases, the dimensionless fluid velocity u  y  is also increases in some fixed values of the viscosity variation parameter  and the body acceleration parameter G . From Figs. (4)-(7), it is observed that when the viscosity variation parameter  decreases, the dimensionless fluid velocity u  y  increases in some fixed values of darcy parameter Da and the body acceleration parameter G. From Figs. (8)-(10) it is noted that when the body acceleration parameter G increases, the dimensionless fluid velocity u  y  also increases in some fixed values of darcy parameter Da and the viscosity variation parameter  .

FIGURE 2. THE DIMENSIONLESS CARTESIAN COORDINATE y VERSES THE DIMENSIONLESS VELOCITY u  y  . THE VARIATION OF THE DIMENSIONLESS VELOCITY u  y  ARE COMPUTED BY USING THE EQN. (7) IN SOME FIXED PARAMETER VALUES  , G AND THE DIFFERENT VALUES OF Da WHEN y  1....1.

FIGURE 4. THE DIMENSIONLESS CARTESIAN COORDINATE y

FIGURE 3. THE DIMENSIONLESS CARTESIAN COORDINATE y VERSES THE DIMENSIONLESS VELOCITY u  y  . THE VARIATION OF THE DIMENSIONLESS VELOCITY u  y  ARE COMPUTED BY USING THE EQN. (7) IN SOME FIXED PARAMETER VALUES  , G AND THE DIFFERENT VALUES OF Da WHEN y  1....1.

FIGURE 5. THE DIMENSIONLESS CARTESIAN COORDINATE y

VERSES THE DIMENSIONLESS VELOCITY u  y  . THE VARIATION VERSES THE DIMENSIONLESS VELOCITY u  y  . THE VARIATION OF THE DIMENSIONLESS VELOCITY u  y  ARE COMPUTED BY

OF THE DIMENSIONLESS VELOCITY u  y  ARE COMPUTED BY

USING THE EQN. (7) IN SOME FIXED PARAMETER VALUES Da , G AND THE DIFFERENT VALUES OF  WHEN y  1....1.

Analytical Expression for the Hydrodynamic Fluid Flow through a Porous Medium

FIGURE 6. THE DIMENSIONLESS CARTESIAN COORDINATE y

FIGURE 7. THE DIMENSIONLESS CARTESIAN COORDINATE y

VERSES THE DIMENSIONLESS VELOCITY u  y  . THE VARIATION VERSES THE DIMENSIONLESS VELOCITY u  y  . THE VARIATION OF THE DIMENSIONLESS VELOCITY u  y  ARE COMPUTED BY

OF THE DIMENSIONLESS VELOCITY u  y  ARE COMPUTED BY

USING THE EQN. (7) IN SOME FIXED PARAMETER VALUES Da , G AND THE DIFFERENT VALUES OF  WHEN y  1....1.

FIGURE 8. THE DIMENSIONLESS CARTESIAN COORDINATE y

FIGURE 9. THE DIMENSIONLESS CARTESIAN COORDINATE y

USING THE EQN. (7) IN SOME FIXED PARAMETER VALUES Da , 

VERSES THE DIMENSIONLESS VELOCITY u  y  . THE VARIATION VERSES THE DIMENSIONLESS VELOCITY u  y  . THE VARIATION OF THE DIMENSIONLESS VELOCITY u  y  ARE COMPUTED BY OF THE DIMENSIONLESS VELOCITY u  y  ARE COMPUTED BY AND THE DIFFERENT VALUES OF G WHEN y  1....1.

AND THE DIFFERENT VALUES OF G WHEN y  1....1.

FIGURE 10. THE DIMENSIONLESS CARTESIAN COORDINATE y VERSES THE DIMENSIONLESS VELOCITY u  y  . THE VARIATION OF THE DIMENSIONLESS VELOCITY u  y  ARE COMPUTED BY USING THE EQN. (7) IN SOME FIXED PARAMETER VALUES Da ,  AND THE DIFFERENT VALUES OF G WHEN y  1....1.

V.Ananthaswamy, S.Uma Maheswari

Conclusion The second order boundary value problem for hydrodynamic viscous fluid flow through a porous medium has been solved analytically. The graphical representations of the dimensionless fluid velocity u  y  are obtained by varying the dimensionless parameters namely darcy parameter Da , the viscosity variation parameter  and the body acceleration parameter G . This analytical result will be used to analyze the behavior of arterial blood flow with multiple stenosis. This method is an extremely simple and it is also a promising method to solve other nonlinear equations. ACKNOWLEDGEMENT

Researchers express their gratitude to the Secretary Shri. S. Natanagopal, Madura College Board, Madurai, Dr. K. M. Rajasekaran, The Principal and Dr. S. Muthukumar, Head of the Department, Department of Mathematics, The Madura College, Madurai, Tamilnadu, India for their constant support and encouragement. REFERENCES

[1] Ananthaswamy V., Eswari A., and Rajendran L., “Nonlinear reactiondiffusion process in a thin membrane and Homotopy analysis method”, International Journal of Automation and Control Engineering, 2(1), 10-18, (2013). [2] Ananthaswamy, S.P Ganesan, and L. Rajendran, “Approximate analytical solution of non-linear reaction-diffusion equation in microwave heating model in a slab: Homotopy analysis method”, International Journal of Mathematical Archive, 4(7), 178-189, (2013). [3] El-Sayed M., “Pulsatile flow of blood through a stenosed porous medium under periodic body acceleration, Applied Mathematics and Computation, 138 (2003) 479-488. [4] Liao S. J., “The proposed Homotopy analysis technique for the solution of nonlinear problems”, Ph.D. Thesis, Shanghai Jiao Tong University, 1992. Liao S. J., “An Approximate Solution Technique Which does not Depend upon Small Parameters: A Special Example”. Int.J. Non-Linear Mech. 30 (1995), 371-380. [5] Liao S. J., Beyond Perturbation Introduction to the [6] Homotopy Analysis Method 1st Edn., Boca Raton 336, Chapman and Hall, CRC press, 2003. [7] Liao S. J., “On the Homotopy Analysis Method for Non-Linear Problems”. Appl. Math. Comput. 147, (2004), 499-513. [8] Liao S. J., “An Optimal Homotopy Analysis Approach for Strongly Non-Linear Differential Equations”.

Commun.

Nonlinear Sci. Numer. Simulat.15, (2010), 2003- 2016. [9] Liao S. J., The Homotopy Analysis method in Non-Linear Differential Equations, Springer and Higher Education press, 2012. [10] Liao S. J., ”An explicit totally analytic approximation of Blasius viscous flow problems”, International Journal of Nonlinear Mech. 34 (1999), 759–78. [11] Liao S. J., “On the analytic solution of magneto-hydrodynamic flows non-Newtonian fluids over a stretching sheet”, J Fluid Mech. 488, (2003), 189–212. [12] Liao S. J., “A new branch of boundary layer flows over a permeable stretching plate”, Int J Nonlinear Mech., 42, (2007), 19– 30. [13] Makinde O. D., “Magneto-hydrodynamic stability of plane-poiseuille flow using Multideck asymptotic technique”, Mathematical and Computer Modelling, 37, (2003), 251-259. [14] Makinde O. D., Mhone Y., “Temporal stability of small disturbances in MHD Jeffery–Hamelows”, Computers and Mathematics with Applcations, 53, (2007), 128-136. [15] Makinde O. D., “On the Chebyshev collocation spectral approach to stability of fluid flow in a porous medium”, International Journal for Numerical Methods in Fluids, 59, (2009), 791-799. [16] Makinde O.D., “Computation hemodynamics analysis in large blood vessels: Effect of hematocrit variation on the flow

Analytical Expression for the Hydrodynamic Fluid Flow through a Porous Medium

stability”, Poster Presentation.IMA Design in Biological Systems, University of Minnesota, (2008), 21 -25. [17] Protter M. H., Weinberger H. F., “Maximum Principles in Differential Equations”, Springer, New York (1984). [18] Rathod V. P., Tanveer S., “Pulsatile flow of couple stress fluid through a porous medium with periodic body acceleration and magnetic field”, Bull.Malays. Math. Sci. Soc., 32(2) (2009) 245-259. [19] Samuel O., Adesanya J., and Falade A., “Hydrodyna-mic stability analysis for variable viscous fluid flow through a porous medium”, International journal of differential equation and application, 13(4), (2014) , 219-230. [20] Subha M., Ananthaswamy V., and Rajendran L., “Analytical solution of non-linear boundary value problem for the electrohydrodynamic flow equation”, International Jouranal of Automation and Control Engineering, 3(2), 48-56, (2014). [21] Saravanakumar K., Ananthaswamy V., Subha M., and Rajendran L., “Analytical Solution of nonlinear boundary value problem for in efficiency of convective straight Fins with temperature-dependent thermal conductivity”, ISRN Thermodynamics, Article ID 282481, 1-18, (2013). APPENDIX A:

Basic Concept of the Homotopy Analysis Method (HAM) (Liao , 1992, 1995, 1999, 2003, 2004, 2007, 2010 & 2012] Consider the following differential equations

N[u(t )]  0

(A.1)

where N is a nonlinear operator, t denotes an independent variable, u (t ) is an unknown function. For simplicity, we ignore all boundary or initial conditions, which can be treated in the similar way. By means of generalizing the conventional Homotopy method, Liao [21-29] constructed the so-called zero-order deformation equation as: (1  p) L[ (t; p)  u0 (t )]  phH (t ) N[ (t; p)]

where

(A.2)

p  0,1 is the embedding parameter, h  0 is a nonzero auxiliary parameter, H (t )  0 is an auxiliary

u (t ) ,  (t : p) is an unknown function. It is

function, L an auxiliary linear operator, u0 (t ) is an initial guess of

important, that one has great freedom to choose auxiliary unknowns in HAM. Obviously, when p  0 and p  1 , it holds:

 (t;0)  u0 (t ) and  (t;1)  u(t )

(A.3)

respectively. Thus, as p increases from 0 to 1, the solution  (t; p) varies from the initial guess u0 (t ) to the solution u (t ) . Expanding  (t; p) in Taylor series with respect to p , we have: 

 (t; p)  u0 (t )   um (t ) p m m 1

(A.4)

where um (t ) 

1  m (t; p) m! p m

p 0

(A.5)

If the auxiliary linear operator, the initial guess, the auxiliary parameter h, and the auxiliary function are so properly chosen, the series eqn.(A.4) converges at p  1 then we have: 

u (t )  u0 (t )   um (t ) .

(A.6)

m 1

Differentiating (A.2) for m times with respect to the embedding parameter p , and then setting p  0 and finally dividing them by m !, we will have the so-called m  th order deformation equation as: 

L[um  mum1 ]  hH (t )m ( u m1 )

(A.7)

V.Ananthaswamy, S.Uma Maheswari

where 

m ( u m1 ) 

1  m1 N [ (t; p)] (m  1)! p m1

(A.8)

and 0, m  1, 1, m  1.

m  

(A.9)

Applying L1 on both side of the eqn. (A.7), we get 

um (t )  mum1 (t )  hL1[ H (t )m (um1 )]

(A.10)

In this way, it is easily to obtain um for m  1, at M th order, we have M

u (t )   um (t )

(A.11)

m 0

When M   , we get an accurate approximation of the original eqn.(A.1). For the convergence of the above method we refer the reader to Liao [17]. If the eqn.(A.1) admits unique solution, then this method will produce the unique solution. APPENDIX: B

Solution of the Boundary Value Problem Eqn. (5) Using the Homotopy Analysis Method In this appendix we indicate the eqn. (6) is derived in this paper. To find the solution of the eqn. (5) , we construct a Homotopy as follows:  d 2u

1  p  

 dy

 uk 2  Ge



  1 y 2

  

(B.1)

 d 2u du   1 y 2    hp  2  uk 2  2 y   Ge  dy  dy 

The analytical solution of equation (B.1) is, u  u0  pu1  p 2u2  p3u3  ......

(B.2)

substituting the eqn. (B.2) in (B.1) we get,  d2 2  2 u0  pu1  p u2  ..... dy 1  p      1 y   u0  pu1  p 2 u2  ..... k 2  Ge    d2  2  2 u0  pu1  p u2  .....   dy     1 y   2 2  hp   u0  pu1  p u2  ..... k  Ge     2 y  d u  pu  p 2u  .....  0 1 2   dy  

















     

(B.3)





Comparing the coefficient likes powers of p in eqn. (B.3) we get, p0 :

d 2u0 dy 2



 u0 k 2  Ge  1   y 2



(B.4)

Analytical Expression for the Hydrodynamic Fluid Flow through a Porous Medium

 d 2u 0   du  2  u k  h  1   2 yh    0     1 2 2   dy  dy   dy 

d 2u1

p1 :



  h  1 Ge



1   y    h  1 u k 2

(B.5)

The initial approximations are as follows: u 1  0, ui  0   0,

du  0   0. dy

(B.6)

dui  0   0, where i  1, 2,3,..... dy

(B.7)

Solving the eqns. (B.4) and (B.5) and using the initial condition eqn. (B.7), we obtain the following results:  G  y 2 e   u0  y    2a cosh ky    b k2  

(B.8)

 a   2 ky   yc   2  e  1   3   2sinh ky   k   4k     Ge     c     2   sinh ky    4   2sinh ky   k    k       Ge   2c   ky   ky u1  y   h    2  1  e   4  e  1  k  k       a   2 ky   1  a  y 2 cosh ky    2  e   4k     a y    cy 2       sinh ky    2   k    k  











 





(B.9)



According to the HAM, we conclude that u  lim u( y)  u0  u1 p 1

After putting the eqns. (B.8) and (B.9) into an eqn. (B.10) we obtain the solutions in the text eqn. (7). APPENDIX C: NOMENCLATURE

Symbol Da

Meaning Darcy parameter. Viscosity variation parameter. The dimensionless Cartesian coordinates The body acceleration parameter The characteristic fluid velocity

 y

G v0

The half-width of the channel that represent the tube radius The porous permeability The fluid density The dynamic viscosity The flow Reynolds number The dimensionless and dimensional fluid pressure

The dimensionless and dimensional fluid velocity

h K

 

 p, p   u, u   x, x ,  y, y  '

The dimensionless and dimensional Cartesian coordinates

(B.10)

V.Ananthaswamy, S.Uma Maheswari

AUTHOR INTRODUCTION

Dr. V. Ananthaswamy received his M.Sc. Mathematics degree from The Madura College (Autonomous), Madurai-625011, Tamil Nadu, India during the year 2000. He has received his M.Phil degree in Mathematics from Madurai Kamaraj University, Madurai, Tamil Nadu, India during the year 2002. He has received his Ph.D., degree (Under the guidance of Dr. L. Rajendran, Assistant Professor, Department of Mathematics, The Madura College, Tamil Nadu, India) from Madurai Kamaraj University, Madurai, Tamil Nadu, India, during the year October 2013. He has 15 years & 6 months of teaching experiences for Engineering Colleges, Arts & Science Colleges and Deemed University. He has 4 years of research experiences. At present he is working as Assistant Professor in Mathematics, The Madura College (Autonomous), Madurai-625 011, Tamil Nadu, India from 2008 to till date. He has published more than 45 research articles in peer-reviewed National and International Journals and communicated 7 research articles in National and International Journals. He has guided more than 8 M. Phil., Scholars and presently guiding 7 M.Phil., scholars. Currently he has Reviewer/Editorial Board Member/Advisory Board Member/Editor in 377 reputed National and International Journals and including this journal also. He has completed one minor research project of Rs. 60,000/sanctioned by UGC in the duration of 18 months. His present research interest includes: Mathematical modeling based on differential equations and asymptotic approximations, Analysis of system of non-linear reaction diffusion equations in physical, chemical and biological sciences, Numerical Analysis, Mathematical Biology, Mathematical and Computational Modeling, Mathematical Modeling for Ecological systems. Also, he has participated and presented 8 research papers in National and International Conferences.