Analytical Expression for the Hydrodynamic Fluid Flow through a Porous Medium

Transcription

1 Analytical Expression for the Hydronaic Fluid Flow through a Porous Mediu * V.Ananthasway 1, S.Ua Maheswari 1 Departent of Matheatics, The Madura College (Autonoous), Maduri, Tail Nadu, India M. Phil., Matheatics, The Madura College (Autonoous), Maduri, Tail Nadu, India 1* ananthu9777@rediffail.co; uashanuga199@gail.co Abstract In this research article the effect of variable viscosity on the teporal developent of sall disturbances in a pressure-driven fluid flow through a channel saturated with porous ediu is investigated. The approxiate analytical solution of the second order boundary value proble for the diensionless velocity is derived by using the Hootopy analysis ethod. This ethod can be easily extended to solve the non-liner initial and boundary value probles in physical, cheical in engineering and sciences. Keywords Variable Viscosity; Porous Mediu; Hydronaic Flow; Boundary Value Proble; Hootopy Analysis Method Introduction In the last few years, studies related to hydronaic stability of a oving viscous fluid through a channel filled with saturated porous ediu played a key role in transport process, petrocheical engineering and geo-physical flows. It was because of that the stu provided useful inforation on the sequence of fluids fro laina to turbulent flows. Turbulent flow has been used in soe real life application. For exaple, in arterial blood flow with ultiple stenosis, shipping over deep seas and in aviation industry and any ore. Currently, several work has been done in this area of research for exaple, Makinde [3] reported the linear stability of hydroagnetic plane-poiseuille flow at high Reynolds nubers by using the ultideck asyptotic approach. Makinde and Mhone [7] investigated the teporal developent of sall disturbances in agneto hydronaic Jeffery Hael flows through a convergent-divergent channel. Furtherore Makinde [9] exained the teporal developent of sall disturbances in a pressure-driven fluid flow through a channel filled with a saturated porous ediu by using the Brinkan flow odel. In all the above entioned studies, the fluid viscosity has been studied constant. Viscosity is a very sensitive fluid property that varies with teperature, pressure or both in soe cases. Therefore, as suggested in [Liao, (199 & 1995)] the effect of stenosis can be captured in the odel by taking the artery as a porous ediu. Motivated by the results in [Protter (1984), Liao., (199 & 1995) ], the specific objective of this paper is to investigate the effects of variations in viscosity and porous pereability on the flow stability which has not been accounted for in the previous odels in literature. Matheatical Forulations of the Proble Consider the flow of a variable viscous, incopressible fluid through a channel filled with saturated porous aterials. The naic viscosity of the fluid is assued to vary with the channel width. If we eployed a Cartesian coordinate syste such that the x-axis corresponds to the flow direction and the y-axis is noral to it, then in -diensions, the flow governing equations can be written as: u v x y (1) International Journal of Autoation and Control Engineering, Vol. 4, No. October /15/ DEStech Publications, Inc. doi:1.1783/ijace.15.4.

2 68 V.Ananthasway, S.Ua Maheswari u u u p u u v t x y x Re x x 1 u v u Re y y x Re Da v v v p v u v t x y y Re y y 1 u v v Re x y x Re Da () (3) Where the following diensionless variables and paraeters have been used in (1)-(3) ' ' ' ' ' ' vt (1 y ) v h u v x y p K dp u, v, t, x, y, p, e, Da, G Re v v h h h dx (4) In (4), v is the characteristic fluid velocity, h is the half-width of the channel that represents the tube radius, K is the porous pereability, is the fluid density, is the naic viscosity, Da is the Darcy paraeter, Re is the flow Reynolds nuber, pp, ' is the diensionless and diensional fluid pressure, uu, ' is the diensionless ' ' and diensional fluid velocity,,,, the viscosity variation paraeter. The basic flow equation is given by, the boundary conditions are, x x y y are the diensionless and diensional Cartesian coordinates, is To obtain the solution of the non-linear equation (5), we assue < <<. d u u du 1y y Ge (5) Da du, u 1 (6) Approxiate Analytical Solution of the Initial Value Proble Using the Hootopy Analysis Method (HAM) Hootopy analysis ethod (HAM) is a non-perturbative analytical ethod for obtaining series solutions to nonlinear equations and has been successfully applied to nuerous probles in science and engineering [Liao (199, 1995, 1999, 3, 4, 7, 1 & 1), Ananthaasway et. al., (13), Saravanakuar et. al., (13), Subha et. al., (14) ]. In coparison with other perturbative and non-perturbative analytical ethods, HAM offers the ability to adjust and control the convergence of a solution via the so-called convergence-control paraeter. Because of this, HAM has been proved to be the ost effective ethod for obtaining analytical solutions to highly nonlinear differential equations. Previous applications of HAM have ainly focused on non-linear differential equations in which the non-linearity is a polynoial in ters of the unknown function and its derivatives. Liao [199, 1995, 1999, 3, 4, 7, 1 & 1] proposed a powerful analytical ethod for non-linear probles, naely the Hootopy analysis ethod. This ethod provides an analytical solution in ters of an infinite power series. However, there is a practical need to evaluate this solution and to obtain nuerical values fro the infinite power series. In order to investigate the accuracy of the Hootopy analysis ethod (HAM) solution with a finite nuber of ters, the systes of differential equations were solved. The Hootopy analysis ethod is a good technique coparing to another perturbation. Hootopy perturbation ethod is a special case of Hootopy analysis ethod. Different fro all reported perturbation and non-perturbative techniques, the Hootopy analysis ethod itself provides us with a convenient

3 Analytical Expression for the Hydronaic Fluid Flow through a Porous Mediu 69 way to control and adjust the convergence region and rate of approxiation series, when necessary. Briefly speaking, the Hootopy analysis ethod has the following advantages: It is valid even if a given non-linear proble does not contain any sall/large paraeter at all; it can be eployed to efficiently approxiate a nonlinear proble by choosing different sets of base functions. The Hootopy analysis ethod contains the auxiliary paraeter, which provides us with a siple way to adjust and control the convergence region of solution series. Using this ethod, the approxiate analytical solutions of the eqn. (6) (see Appendix B) are shown as follows: u y Ge y a cosh ky b k a ky yc e 1 sinh ky 3 4k k Ge c 4 k k Ge 4 k k a 4k a y cy sinh ky k k sinh ky sinh ky Ky c Ky 1 e e 1 Ky e 1 a y cosh ky (7) Where k 1 Da Where a, b and c are defined by the following: 1 Ge Ge Ge a 4 cosh ky k k k Ge Ge b 4 k k 4G e c k Ge (8) (9) (1) Results and Discussion u y. THE VARIATION OF u y ARE COMPUTED BY USING THE EQN. (7) IN SOME FIXED PARAMETER VALUES, G AND FIGURE 1. THE DIMENSIONLESS CARTESIAN COORDINATE y THE DIMENSIONLESS VELOCITY THE DIFFERENT VALUES OF Da WHEN y

4 7 V.Ananthasway, S.Ua Maheswari u y. u y is obtained by using the eqn.7. Fro Figs. (1)-(3) it is clear u y is also increases in soe fixed Figures (1)-(1) represent the diensionless Cartesian coordinate y versus the diensionless fluid velocity The variation of the diensionless fluid velocity that when the darcy paraeter Da increases, the diensionless fluid velocity values of the viscosity variation paraeter and the bo acceleration paraeter G. Fro Figs. (4)-(7), it is observed that when the viscosity variation paraeter decreases, the diensionless fluid velocity u y increases in soe fixed values of darcy paraeter Da and the bo acceleration paraeter G. Fro Figs. (8)-(1) it is noted that when the bo acceleration paraeter G increases, the diensionless fluid velocity soe fixed values of darcy paraeter Da and the viscosity variation paraeter. u y also increases in FIGURE. THE DIMENSIONLESS CARTESIAN COORDINATE y u y. THE VARIATION u y ARE COMPUTED BY USING THE EQN. (7) IN SOME FIXED PARAMETER VALUES, G AND THE DIFFERENT VALUES OF Da WHEN y FIGURE 3. THE DIMENSIONLESS CARTESIAN COORDINATE y u y. THE VARIATION u y ARE COMPUTED BY USING THE EQN. (7) IN SOME FIXED PARAMETER VALUES, G AND THE DIFFERENT VALUES OF Da WHEN y FIGURE 4. THE DIMENSIONLESS CARTESIAN COORDINATE y u y. THE VARIATION u y ARE COMPUTED BY USING THE EQN. (7) IN SOME FIXED PARAMETER VALUES Da, G AND THE DIFFERENT VALUES OF WHEN y FIGURE 5. THE DIMENSIONLESS CARTESIAN COORDINATE y u y. THE VARIATION u y ARE COMPUTED BY USING THE EQN. (7) IN SOME FIXED PARAMETER VALUES Da, G AND THE DIFFERENT VALUES OF WHEN y

5 Analytical Expression for the Hydronaic Fluid Flow through a Porous Mediu 71 FIGURE 6. THE DIMENSIONLESS CARTESIAN COORDINATE y u y. THE VARIATION u y ARE COMPUTED BY USING THE EQN. (7) IN SOME FIXED PARAMETER VALUES Da, G AND THE DIFFERENT VALUES OF WHEN y FIGURE 7. THE DIMENSIONLESS CARTESIAN COORDINATE y u y. THE VARIATION u y ARE COMPUTED BY USING THE EQN. (7) IN SOME FIXED PARAMETER VALUES Da, G AND THE DIFFERENT VALUES OF WHEN y FIGURE 8. THE DIMENSIONLESS CARTESIAN COORDINATE y u y. THE VARIATION u y ARE COMPUTED BY USING THE EQN. (7) IN SOME FIXED PARAMETER VALUES Da, AND THE DIFFERENT VALUES OF G WHEN y FIGURE 9. THE DIMENSIONLESS CARTESIAN COORDINATE y u y. THE VARIATION u y ARE COMPUTED BY USING THE EQN. (7) IN SOME FIXED PARAMETER VALUES Da, AND THE DIFFERENT VALUES OF G WHEN y u y. THE VARIATION OF u y ARE COMPUTED BY USING THE EQN. (7) IN SOME FIXED PARAMETER VALUES Da, AND FIGURE 1. THE DIMENSIONLESS CARTESIAN COORDINATE y THE DIMENSIONLESS VELOCITY THE DIFFERENT VALUES OF G WHEN y

6 7 V.Ananthasway, S.Ua Maheswari Conclusion The second order boundary value proble for hydronaic viscous fluid flow through a porous ediu has been solved analytically. The graphical representations of the diensionless fluid velocity u y are obtained by varying the diensionless paraeters naely darcy paraeter Da, the viscosity variation paraeter and the bo acceleration paraeter G. This analytical result will be used to analyze the behavior of arterial blood flow with ultiple stenosis. This ethod is an extreely siple and it is also a proising ethod 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. Muthukuar, Head of the Departent, Departent of Matheatics, The Madura College, Madurai, Tailnadu, India for their constant support and encourageent. REFERENCES [1] Ananthasway V., Eswari A., and Rajendran L., Nonlinear reactiondiffusion process in a thin ebrane and Hootopy analysis ethod, International Journal of Autoation and Control Engineering, (1), 1-18, (13). [] Ananthasway, S.P Ganesan, and L. Rajendran, Approxiate analytical solution of non-linear reaction-diffusion equation in icrowave heating odel in a slab: Hootopy analysis ethod, International Journal of Matheatical Archive, 4(7), , (13). [3] El-Sayed M., Pulsatile flow of blood through a stenosed porous ediu under periodic bo acceleration, Applied Matheatics and Coputation, 138 (3) [4] Liao S. J., The proposed Hootopy analysis technique for the solution of nonlinear probles, Ph.D. Thesis, Shanghai Jiao Tong University, 199. Liao S. J., An Approxiate Solution Technique Which does not Depend upon Sall Paraeters: A Special Exaple. Int.J. Non-Linear Mech. 3 (1995), [5] Liao S. J., Beyond Perturbation Introduction to the [6] Hootopy Analysis Method 1st Edn., Boca Raton 336, Chapan and Hall, CRC press, 3. [7] Liao S. J., On the Hootopy Analysis Method for Non-Linear Probles. Appl. Math. Coput. 147, (4), [8] Liao S. J., An Optial Hootopy Analysis Approach for Strongly Non-Linear Differential Equations. Coun. Nonlinear Sci. Nuer. Siulat.15, (1), [9] Liao S. J., The Hootopy Analysis ethod in Non-Linear Differential Equations, Springer and Higher Education press, 1. [1] Liao S. J., An explicit totally analytic approxiation of Blasius viscous flow probles, International Journal of Nonlinear Mech. 34 (1999), [11] Liao S. J., On the analytic solution of agneto-hydronaic flows non-newtonian fluids over a stretching sheet, J Fluid Mech. 488, (3), [1] Liao S. J., A new branch of boundary layer flows over a pereable stretching plate, Int J Nonlinear Mech., 4, (7), [13] Makinde O. D., Magneto-hydronaic stability of plane-poiseuille flow using Multideck asyptotic technique, Matheatical and Coputer Modelling, 37, (3), [14] Makinde O. D., Mhone Y., Teporal stability of sall disturbances in MHD Jeffery Haelows, Coputers and Matheatics with Applcations, 53, (7), [15] Makinde O. D., On the Chebyshev collocation spectral approach to stability of fluid flow in a porous ediu, International Journal for Nuerical Methods in Fluids, 59, (9), [16] Makinde O.D., Coputation heonaics analysis in large blood vessels: Effect of heatocrit variation on the flow

7 Analytical Expression for the Hydronaic Fluid Flow through a Porous Mediu 73 stability, Poster Presentation.IMA Design in Biological Systes, University of Minnesota, (8), 1-5. [17] Protter M. H., Weinberger H. F., Maxiu Principles in Differential Equations, Springer, New York (1984). [18] Rathod V. P., Tanveer S., Pulsatile flow of couple stress fluid through a porous ediu with periodic bo acceleration and agnetic field, Bull.Malays. Math. Sci. Soc., 3() (9) [19] Sauel O., Adesanya J., and Falade A., Hydrona-ic stability analysis for variable viscous fluid flow through a porous ediu, International journal of differential equation and application, 13(4), (14), [] Subha M., Ananthasway V., and Rajendran L., Analytical solution of non-linear boundary value proble for the electrohydronaic flow equation, International Jouranal of Autoation and Control Engineering, 3(), 48-56, (14). [1] Saravanakuar K., Ananthasway V., Subha M., and Rajendran L., Analytical Solution of nonlinear boundary value proble for in efficiency of convective straight Fins with teperature-dependent theral conductivity, ISRN Theronaics, Article ID 8481, 1-18, (13). APPENDIX A: Basic Concept of the Hootopy Analysis Method (HAM) (Liao, 199, 1995, 1999, 3, 4, 7, 1 & 1] Consider the following differential equations N[ u( t)] (A.1) where N is a nonlinear operator, t denotes an independent variable, ut () is an unknown function. For siplicity, we ignore all boundary or initial conditions, which can be treated in the siilar way. By eans of generalizing the conventional Hootopy ethod, Liao [1-9] constructed the so-called zero-order deforation equation as: where p,1 is the ebedding paraeter, (1 p) L[ ( t; p) u ( t)] phh ( t) N[ ( t; p)] (A.) h is a nonzero auxiliary paraeter, Ht ( ) is an auxiliary function, L an auxiliary linear operator, u () t is an initial guess of ut (), ( t: p) is an unknown function. It is iportant, that one has great freedo to choose auxiliary unknowns in HAM. Obviously, when p and p 1, it holds: ( t;) u ( t) and ( t;1) u( t) (A.3) respectively. Thus, as p increases fro to 1, the solution ( t; p) varies fro the initial guess u () t to the solution ut (). Expanding ( t; p) in Taylor series with respect to p, we have: where ( t; p) u ( t) u ( t) p (A.4) 1 1 ( t; p) u() t! p p (A.5) If the auxiliary linear operator, the initial guess, the auxiliary paraeter h, and the auxiliary function are so properly chosen, the series eqn.(a.4) converges at p 1 then we have: u( t) u ( t) u ( t). (A.6) 1 Differentiating (A.) for ties with respect to the ebedding paraeter p, and then setting p and finally dividing the by!, we will have the so-called th order deforation equation as: L[ u u ] hh ( t) ( u ) (A.7) 1 1

8 74 V.Ananthasway, S.Ua Maheswari where 1 1 N[ ( t; p)] ( u 1) 1 ( 1)! p (A.8) and, 1, 1, 1. (A.9) Applying 1 L on both side of the eqn. (A.7), we get In this way, it is easily to obtain u for 1, at u ( t) u ( t) hl [ H( t) ( u )] (A.1) th M order, we have M u( t) u ( t) (A.11) When M, we get an accurate approxiation of the original eqn.(a.1). For the convergence of the above ethod we refer the reader to Liao [17]. If the eqn.(a.1) adits unique solution, then this ethod will produce the unique solution. APPENDIX: B Solution of the Boundary Value Proble Eqn. (5) Using the Hootopy 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 Hootopy as follows: du 1 p uk Ge 1y d u du hp uk y Ge 1y (B.1) The analytical solution of equation (B.1) is, 3 u u pu1 p u p u3... (B.) substituting the eqn. (B.) in (B.1) we get, 1 p d u pu1 p u... 1y u pu1 p u... k Ge d u pu 1 p u... hp u pu p u k Ge d y u pu1 p u... 1y 1... (B.3) Coparing the coefficient likes powers of p in eqn. (B.3) we get, du : 1 p u k Ge y (B.4)

9 Analytical Expression for the Hydronaic Fluid Flow through a Porous Mediu 75 1 du 1 d u du p : u1k h 1 yh h Ge y h u k (B.5) The initial approxiations are as follows: u i du u 1,. (B.6) dui,, where i 1,,3,... (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 e u y a cosh ky b k ky yc e 1 sinh ky a 3 4k k Ge c 4 k k Ge u1 y h e e 4 k k a 4k a y cy sinh ky k k sinh ky sinh ky ky c ky 1 1 ky e 1 a y cosh ky (B.8) (B.9) According to the HAM, we conclude that u li u( y) u u (B.1) After putting the eqns. (B.8) and (B.9) into an eqn. (B.1) we obtain the solutions in the text eqn. (7). p1 1 APPENDIX C: NOMENCLATURE Sybol Meaning Da Darcy paraeter. Viscosity variation paraeter. y The diensionless Cartesian coordinates G The bo acceleration paraeter v The characteristic fluid velocity h The half-width of the channel that represent the tube radius K The porous pereability The fluid density The naic viscosity Re The flow Reynolds nuber, pp, The diensionless and diensional fluid pressure, ', ',, ' uu The diensionless and diensional fluid velocity x x y y The diensionless and diensional Cartesian coordinates

10 76 V.Ananthasway, S.Ua Maheswari AUTHOR INTRODUCTION Dr. V. Ananthasway received his M.Sc. Matheatics degree fro The Madura College (Autonoous), Madurai-6511, Tail Nadu, India during the year. He has received his M.Phil degree in Matheatics fro Madurai Kaaraj University, Madurai, Tail Nadu, India during the year. He has received his Ph.D., degree (Under the guidance of Dr. L. Rajendran, Assistant Professor, Departent of Matheatics, The Madura College, Tail Nadu, India) fro Madurai Kaaraj University, Madurai, Tail Nadu, India, during the year October 13. He has 15 years & 6 onths of teaching experiences for Engineering Colleges, Arts & Science Colleges and Deeed University. He has 4 years of research experiences. At present he is working as Assistant Professor in Matheatics, The Madura College (Autonoous), Madurai-65 11, Tail Nadu, India fro 8 to till date. He has published ore than 45 research articles in peer-reviewed National and International Journals and counicated 7 research articles in National and International Journals. He has guided ore than 8 M. Phil., Scholars and presently guiding 7 M.Phil., scholars. Currently he has Reviewer/Editorial Board Meber/Advisory Board Meber/Editor in 377 reputed National and International Journals and including this journal also. He has copleted one inor research project of Rs. 6,/- sanctioned by UGC in the duration of 18 onths. His present research interest includes: Matheatical odeling based on differential equations and asyptotic approxiations, Analysis of syste of non-linear reaction diffusion equations in physical, cheical and biological sciences, Nuerical Analysis, Matheatical Biology, Matheatical and Coputational Modeling, Matheatical Modeling for Ecological systes. Also, he has participated and presented 8 research papers in National and International Conferences.