Analytical Solution of Non-linear Boundary Value Problem for the Electrohydrodynamic Flow Equation

Transcription

1 International Journal of Autoation and Control Engineering (IJACE) Volue 3 Issue, May 4 doi:.4355/ijace Analytical Solution of Non-linear Boundary Value Proble for the Electrohyodynaic Flow Equation M. Subha, V. Ananthasway, L. Rajenan 3 Departent of Matheatics, MSNPM Woen s College, Poovanthi, Sivagangai Dt., Tail Nadu, India,3 Departent of Matheatics, The Madura College (Autonoous), Madurai, Tail Nadu, India subhaasc@rediffailco; ananthu9777@rediffail.co; 3 raj_ss@rediffail.co Received 7 Sep, 3; Accepted Mar, 4; Published May, 4 4 Science and Engineering Publishing Copany Abstract The non-linear boundary value proble for the electrohyodynaic flow of a fluid in a circular cylinical conduit is discussed. This odel contains a non-linear ter related to fluid velocity. The analytical expression pertaining to the velocity of fluid is reported for all values of the rtann electric nuber and the strength of nonlinearityα. We present the analytical solution based on the Hootopy analysis ethod (HAM). The effects of the diensionless paraeters and α on velocity of fluid has been studied. These analytical results were found to be in good agreeent with the nuerical results. Keywords Electrohyodynaic Flow; Circular Cylinical Conduit; Nonlinear Boundary Value Proble; Hootopy Analysis Method; Nuerical Siulation Introduction The atheatical odel of electrohyodynaic flow of a fluid in a circular cylinical conduit is represented by a second-order non-linear ordinary differential equation [Mckee et. al (997)], Paullet et. al (999)]. The governing equation for this non-linear boundary value proble is given as follows [Khan et. al ()]: d wr () dwr () wr () + +, = r α w() r < r < () w ' () =, w () = () Here w(r) denotes the velocity of fluid, r is the radial distance fro the centre of the cylinical conduit. and α are the rtan electric nuber and easure of the strength of the non-linearity respectively. Perturbation solutions of fluid velocities for different orders of non-linearity were given by McKee et al. (997). Pallut et. al (999) obtained the solution of electrohyodynaic flow in circular cylinical conduit by solving the non-linear boundary value proble and discovered an error in perturbation and nuerical solution given in Mckee et. al (997) for large values of α. Najeeb Ala Khan et. al ( and ) introduced an analytical ethod, naely new hootopy perturbation ethod for solving a nonlinear boundary value proble in electrohyodynaic flow of a liquid in a circular cylinical conduit. McKee et al. (997). Pallut (999) obtained the solution of electrohyodynaic flow in circular cylinical conduit by solving the non-linear boundary value proble and discovered an error in perturbation and nuerical solution given in Mckee et. al (997) for large value of α. Najeeb Ala Khan et. al ( and ) introduced an analytical ethod, naely new Hootopy perturbation ethod for solving a nonlinear boundary value proble in electrohyodynaic flow of a liquid in a circular cylinical conduit. The purpose of this work is to present approxiate analytical solution of the equations () and () for all values of the relevant paraeters using the Hootopy analysis ethod (HAM), introduced by Liao (99, 995, 999, 3, 4, and ). We show that the analytical solutions are in excellent agreeent with nuerical solutions. 48

2 International Journal of Autoation and Control Engineering (IJACE) Volue 3 Issue, May 4 Solution of boundary value proble using Hootopy analysis ethod (HAM) 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 (99, 995, 999, 3, 4, 7, and ), Doairry (8), Tan et. al (7), Abbasbandy (8), Cheng et. al (8), yat et. al (8 and ), Jafari et. al (9)]. 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 proved to be the ost effective ethod for obtaining analytical solutions to highly non-linear differential equations. Previous applications of HAM have ainly focused on non-linear differential equations in which the nonlinearity is a polynoial in ters of the unknown function and its derivatives. As seen in (), the nonlinearity present in electrohyodynaic flow takes the for of a rational function, and thus, poses a greater challenge with respect to finding approxiate solutions analytically. Our results show that even in this case, HAM yields excellent results. Liao (99, 995, 999, 3, 4, 7, and ) 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 syste of differential equations were solved. The Hootopy analysis ethod is a good technique coparing to another perturbation ethod. Hootopy perturbation ethod [Chowdhuri et. al (7), Eswari et. al (), Ghori et. al (7), Coyle et. al (986), Ozis et. al (7), Maideen et. al (3) and Ananthasway et. al ( and 3)] 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 way to control and adjust the convergence region and rate of approxiation series, when necessary. Briefly speaking, this 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 non-linear proble by choosing different sets of base functions. The Hootopy analysis ethod contains the auxiliary paraeter h, which provides us with a siple way to adjust and control the convergence region of solution series. Using this ethod, we can obtain the following solution to () and () (see Appendix B). The approxiate analytical solution of the equations () and () using HAM is given by cosh( ) r wr () = cosh( ) 4 α + 3cosh( ) α + + cosh ( ) 6cos( ) α cosh( ) + α tanh( ) 6cosh ( ) 4 h α + (3) cosh( r) 3cosh( ) cosh( ) α + cosh ( ) α cosh( r) + 6cosh ( ) α r sinh( r) r cosh( ) 6cosh( ) Nuerical Siulation In order to find the accuracy of our analytical ethod, the non-linear differential eqns. () and () are also solved by nuerical ethods. The function bvp4c in Matlab/Scilab software which is a function of solving boundary value probles (BVPs) for ordinary differential equations are used to solve these equations nuerically. Our analytical result is copared with nuerical solution and it gives a good agreeent (See Figures: (a)-(e) and (a)-(d)). The Matlab/Scilab progra is also given in Appendix D. Discussions The ain interest in this section is to investigate the effects of rtann electric nuber and the strength of nonlinearity α on the velocity eerging in the electrohyodynaics flows by using Hootopy analysis ethod. Figures (a)-(e) shows the radial 49

3 International Journal of Autoation and Control Engineering (IJACE) Volue 3 Issue, May 4 distance r versus the fluid velocity w (r). Fro these figures, it is evident that, when the rtann electric nuber increases, the fluid velocity w(r) also increases for the fixed value of α. Also we note that when α increases, the fluid velocity w(r) decreases for all values of the rtann electric nuber. Figures (a)-(d) infer that the radial distance r versus the fluid velocity w (r). Fro these figures we notice that when α increases, the fluid velocity w (r) decreases for the fixed values of the rtann electric nuber. Also we conclude that when the rtann electric nuber increases, the fluid velocity w(r) also increases for the various values of the strength of non-linearityα. The efficiency of HAM is very uch depending on choosing auxiliary paraeter h. (c) (d) (a) (e) (b) FIGURE FLUID VELOCITY wr () VERSUS THE DIMENSIONLESS RADIAL DISTANCE r. THE FLUID VELOCITY WERE COMPUTED USING EQN. (3) FOR VARIOUS VALUES OF THE DIMENSIONLESS PARAMETERS α AND WHEN (A) α =.5 (B) α =.5 (C) α = (D) α = (E) α = 5 AND h =.58. 5

4 International Journal of Autoation and Control Engineering (IJACE) Volue 3 Issue, May 4 (d) (a) FIGURE THE FLUID VELOCITY w (r) VERSUS THE DIMENSIONLESS RADIAL DISTANCE r. THE FLUID VELOCITY WERE COMPUTED USING EQN. (3) FOR VARIOUS VALUES OF THE DIMENSIONLESS PARAMETERS α AND WHEN (A) =. 5 (B) = (C) = 5 (D) = AND h =.58. (b) FIGURE 3 THE h CURVE TO INDICATE THE CONVERGENCE REGION FOR w (.), WHEN α =.5, H a = 5 (c) FIGURE 4 THE h CURVE TO INDICATE THE CONVERGENCE ' REGION FOR w (.), WHEN α =.5, H a = 5 5

5 International Journal of Autoation and Control Engineering (IJACE) Volue 3 Issue, May 4 TABLE COMPARISON OF ANALYTICAL SOLUTION (HAM) AND THE NUMERICAL RESULTS FOR VARIOUS VALUES OF α AND. α Analytical solution (3) Nuerical solution Error % TABLE COMPARISON OF ANALYTICAL SOLUTION (HAM) AND THE NUMERICAL RESULTS FOR VARIOUS VALUES OF α AND. α Analytical solution (3) Nuerica l solution Error % Conclusion The non-linear boundary value proble for the electrohyodynaic flow of a fluid has been solved analytically and nuerically. Analytical expressions of the fluid velocity can be derived by using the Hootopy analysis ethod (HAM). The priary result of this work is siple and approxiate expressions of the fluid velocity for all values of the diensionless paraeters and α.. This analytical result will be useful to analyze the behavior of the electrohyodynaic flow of a fluid in an ion-ag configuration in a circular cylinical conduit. This ethod is an extreely siple and it is also a proising ethod to solve other non-linear equations. ACKNOWLEDGEMENTS This work was supported by the University Grants Coission (F. No /(SR)), New Delhi, India. The authors are thankful to Shri. S. Natanagopal, Secretary, Madua College Board, Madurai and Dr. R. Murali, The Principal for their encourageent. Appendix A: Basic Concept of Hootopy Analysis Method (HAM) Consider the following differential equation: Nut [ ( )] = (A.) Where Ν is a nonlinear operator, t denotes an independent variable, u(t) 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 () constructed the so-called zero-order deforation equation as: ( p) L[ φ( t; p) u ( t)] = phh ( t) N[ φ( t; p)] (A.) where p [,] is the ebedding paraeter, h is a nonzero auxiliary paraeter, H(t) is an auxiliary function, L an auxiliary linear operator, u () t is an initial guess of u(t), φ (: t p) is an unknown function. It is iportant to note that one has great freedo to choose auxiliary unknowns in HAM. Obviously, when p = and p =, it holds: φ ( t;) = u ( t) and φ ( t;) = ut ( ) (A.3) respectively. Thus, as p increases fro to, the solution φ(; t p) varies fro the initial guess u () t to the solution u (t). Expanding φ (; t p) in Taylor series with respect to p, we have: where φ(; t p) = u () t + u () t p (A.4) + = φ(; 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 5

6 International Journal of Autoation and Control Engineering (IJACE) Volue 3 Issue, May 4 so properly chosen, the series (A.4) converges at p = then we have: ut () = u() t + u () t (A.6) + = 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: where And Applying L[ u χ u ] = hh () t R ( u ) (A.7) N[ φ( t; p)] R ( u ) = ( )! p L,, χ =, >. on both sides of equation (A7), we get χ (A.8) (A.9) u () t = u () t + hl [ H () t R ( u )] (A) In this way, it is easily to obtain u for, at order, we have M th M ut () = u () t (A.) = When M +, we get an accurate approxiation of the original equation (A.). For the convergence of the above ethod we refer the reader to Liao [9]. If equation (A.) adits unique solution, then this ethod will produce the unique solution. Appendix B: Solution of the Boundary Value Proble Using HAM In this Appendix, we indicate how the eqn.(3) in this paper is derived. When is sall then the α wr () eqn. () reduces to d wr () dwr () + + ( wr () α wr ()) = (B.) r We construct the Hootopy as follows d w ( p) w + d w dw = hp + + w α w r The approxiate solution of the eqn.(b.) is, (B.) w = w + pw+ p w +... (B.3) The initial approxiations are as follows ' w () = and w () = (B.4) Substituting the eqn. (B.3) into an eqn. (B.) we have d ( w + pw+ p w +...) ( p) (( w + pw+ p w +...) + d ( w + pw+ p w +...) d( w + pw+ p w +...) = hp + + r ( w + pw + p w +...) α ( w + pw+ p w +...) (B.5) Coparing the coefficients of like powers of p in the eqn. (B.5) we get d w p : w + = (B.6) d w d w : ( + ) p w h dw + ( h + ) w h + h αw = x (B.7) Solving the eqns.(b.6) and (B.7) and using the boundary conditions the eqn.(b.4) we can obtain the following results: cosh( ) r w () r = cosh( ) (B.8) 4 α + 3cosh( ) α + + cosh ( ) 6cos( ) α cosh( ) + α tanh( ) 6cosh ( ) 4 w () r h α + = (B.9) cosh( r) 3cosh( ) cosh( ) α + cosh ( ) α cosh( r) + 6cosh ( ) α r sinh( r) r cosh( ) 6cosh( ) 53

7 International Journal of Autoation and Control Engineering (IJACE) Volue 3 Issue, May 4 According to the HAM, we can conclude that w= li wr ( ) = w + w (B.) p After putting the eqns.(b.8) and (B.9) into an eqn. (B.), we obtain the solution in the text the eqn. (3). Appendix C: Deterining the region of h for validity The analytical solution should converge. It should be noted that the auxiliary paraeter h controls the convergence and accuracy of the solution series. The analytical solution represented by eqn.(3) contains the auxiliary paraeter h, which gives the convergence region and rate of approxiation for the Hootopy analysis ethod. In order to define region such that the solution series is independent of h, a ultiple of h curves are plotted. The region where the fluid velocities w (r) and w '( r) versus h is a horizontal line known as the convergence region for the corresponding function. The coon region aong w (r) and its derivatives are known as the over all convergence region. To study the influence of h on the convergence of solution, h - curves of w (.) and w '(.) are plotted in Fig. (3) and (4) respectively for α =.5, = 5. These figures clearly indicate that the valid region of h is about (-.65 to -.35). Siilarly we can find the value of the convergence control paraeter h for different values of the constant paraeters. Appendix D : Matlab/Scilab progra to find the nuerical solution of non-linear equations () and (): function = ; x =linspace(,); t=linspace(,); sol = pdepe(,@pdex4pde,@pdex4ic,@pdex4bc,x,t); u = sol(:,:,); figure plot(x,u(end,:)) title('u(x,t)') xlabel('distance x') ylabel('u(x,)') % function [c,f,s] = pdex4pde(x,t,u,dudx) c = ; f = DuDx; alpha=; ^=; F =^*(-((u())/(-(alpha)*u()))); s = F; % function u = pdex4ic(x); %create a initial conditions u = ; % function[pl,ql,pr,qr]=pdex4bc(xl,ul,xr,ur,t) %create a boundary conditions pl = ; ql = ; pr = ur()-; qr = ; Appendix E : Noenclature Sybol Meaning Diensionless distance fro the center of r cylinical conduit w (r) Diensionless fluid velocity α Measure of the strength of nonlinearity rtann Electric nuber REFERENCES Abbasbandy S., Soliton solutions for the FitzhughNaguo equation with the Hootopy analysis ethod, Appl Math Model. 3 (8): Ananthasway V., and Rajenan L., Analytical solutions of soe two-point non linear elliptic-boundary value probles, Applied Matheatics. 3 (): Ananthasway V., and Rajenan L., Analytical solution of Non-isotheral diffusion-reaction processes and effectiveness factors, ISRN Physical cheistry, Hindawi publishing corporation. 3 (): -4. Cheng J., Liao SJ., Mohapatra RN., and Vajravelu K., Series solutions of nano boundary layer flows by eans of the Hootopy analysis ethod. J Math Anal Appl. 343 (8): Chowdhury M. S. H., shi I., Solutions of Tie- Dependent Eden-Fowler Type Equations by Hootopy Perturbation Method. Phys. Lett. A. 368 (7): Coyle J. M., Flaherty J. E., and Ludwig R., On the Stability of Mesh Equidistribution Strategies for Tie-Dependent Partial Differential Equations. J. Coput.Phys.6 (986): Doairry G., Bararnia H., An Approxiation of Analytical Solution of Soe Non-Linear Heat Transfer Equations: A 54

8 International Journal of Autoation and Control Engineering (IJACE) Volue 3 Issue, May 4 Survey by using Hootopy Analysis Method.Adv. Studies Theor. Phys. (8): Eswari A., Rajenan L., Analytical Solution of Steady State Current at a Microdisk Biosensor. J. Electroanal. Che. 64 (): Adv. Studies Theor. Phys. (8): Ghori Q. K., Ahed M., and Siddiqui A. M., Application of Hootopy Perturbation Method to Squeezing Flow of a Newtonian Fluid. Int. J. Nonlinear. Sci. Nuer. Siulat 8 () (7): yat T., and Abbas Z., Heat transfer analysis on MHD flow of a second grade fluid in a channel with porous ediu,chaos Solitons Fractals. 38 (8): yat T., Naz R., and Sajid M., On the Hootopy solution for Poiseuille flow of a fourth grade Fluid, Coun Nonlinear Sci Nuer Siul, 5 (): Jafari H., Chun C., and Saeidy S. M., Analytical Solution for Non-Linear Gas Dynaic using Hootopy Analysis Method. Appl. Math. 4 (9): Khan N., Ara A., and Jail M., An approach for solving the Riccati equation with fractional orders, Coputers and Matheatics with applications. 6 (): Khan N., Ara A., Jail M.,and Khan N.U., On efficient et ethod for syste of fractional differential equations, Advances in Differential equations. (): Article ID Khan N., Jail M., MahaoodA., and Ara A., Approxiate solution for the electro hyodynaic flow in a circular cylinical conduit, ISRN Coputation Matheatics, Hindawi publishing corporation. : (): -5. Liao S. J., The Proposed Hootopy Analysis Technique for the Solution of Non-Linear Probles, Ph.D. Thesis, Shanghai Jiao Tong University, 99. Liao S. J., An Approxiate Solution Technique Which does not Depend upon Sall Paraeters: A Special Exaple. Int.J. Non-Linear Mech. 3 (995): Liao S. J., Beyond Perturbation Introduction to the Hootopy Analysis Method st Edn., Boca Raton 336, Chapan and ll, CRC press, 3. Liao S. J., On the Hootopy Analysis Method for Non- Linear Probles. Appl. Math. Coput. 47 (4): Liao S. J., An Optial Hootopy Analysis Approach for Strongly Non-Linear Differential Equations. Coun. Nonlinear Sci. Nuer. Siulat.5 (): 3-6. Liao S. J., The Hootopy Analysis ethod in Non-Linear Differential Equations, Springer and Higher Education press,. Liao S.J., An explicit totally analytic approxiation of Blasius viscous flow probles, Int J Nonlinear Mech. 34 (999): Liao S.J., On the analytic solution of agnetohyodynaic flows non-newtonian fluids over a stretching sheet, J Fluid Mech. 488 (3):89. Liao S.J., A new branch of boundary layer flows over a pereable stretching plate, Int J Nonlinear Mech. 4 (7): 9 3. Madden N., and Stynes M., A Uniforly Convergent erical Method for a Coupled Syste of Two Singularly Perturbed Linear Reaction-Difusion Probles. IMA J. Nuer.Anal.3 (4) (3): Mckee S., Watson R., Cuinato J.A., Caldwell J., and Chen M.S., Calculation of electrohyodynaic flow in a circular cylinical conduit, Zeitschrift fur Angewandt Matheatik und Mechanik. 77(6) (997): Ozis T., and Yildiri A., Relation of a Non-Linear Oscillator with Discontinuities.Int. J. Nonlinear. Sci. Nuer. Siulat.8 () (7): Paullet J.E., On the solutions of electrohyodynaic flow in a circular cylinical conduit, Z.Angew Math Mech 79 (999): Tan Y., Xu H., and Liao S.J., Explicit series solution of travelling waves with a front of Fisher Equation, Chaos Solitons Fractals. 3 (7): 46 7 Dr. V. Ananthasway received his M.Sc. Matheatics degree fro The Madura College (Autonoous), Madurai-65, Tail Nadu, India during. He has received his M.Phil degree in Matheatics fro Madurai Kaaraj University, Madurai, Tail Nadu, India during. He has received his Ph.D., degree (Under the guidance of Dr. L. Rajenan, Assistant Professor, Departent of Matheatics, The Madura College, Tail Nadu, India) fro Madurai Kaaraj University, Madurai, Tail Nadu, India, during October 3. He has 3 years and 6 onths experiences of teaching for Engineering 55

9 International Journal of Autoation and Control Engineering (IJACE) Volue 3 Issue, May 4 College, Arts College and Deeed University and 3 years of research experiences. At present he is working as Assistant Professor in Matheatics, The Madura College (Autonoous), Madurai-65 fro 8 onwards. He has published ore than 6 articles in peer-reviewed National and International Journals and counicated 4 research articles in National and International Journals. Currently he has one inor research project fro UGC. 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. Also, he has participated and presented research papers in National and International Conferences. Mrs. M. Subha received her M.Sc., degree () and M.Phil., degree (3) in Matheatics fro The Madura College and E.M.G. Yadhava Woen s College, Madurai, Tailnadu, India. At present, She is working as Assistant Professor in the Departent of Matheatics, Madurai Sivakasi Nadars Pioneer Meenakshi Woen s College, Sivagangai District, Tail Nadu, India. Also, she is doing her Ph.D entitled Asyptotic Methods for Solving Initial and Boundary Value Probles at Madurai Kaaraj University, Madurai under the guidance of Dr. L. Rajenan, Assistant Professor, Departent of Matheatics, The Madura College, Madurai. Her present research interest include: Matheatical odelling, Analytical solution of syste of nonlinear reaction diffusion processes in biosensor, Hootopy analysis ethod, Hootopy perturbation and nuerical ethods. She has published papers in International Journals and papers have been counicated in International Journals. Also, She has participated and presented research papers in International and National Conferences. Dr. L. Rajenan received his M.Sc. in Matheatics in 98 fro Presidency College, Chennai, TN, India. He obtained his Ph.D. in Applied Matheatics fro Alagappa University, Karaikudi, TN, India during. At present, he is an Assistant Professor in Matheatics at The Masdura College(Autonoous), Madurai, TN, India. Before this position (986 7), he was working as a Post Graduate Assistant in Matheatics at SMSV Higher Secondary School, Karaikudi, TN, India. He has years teaching experience and 5 years research experience. He has authored and coauthor over 3 research publications including about 8 scholarly articles in peer-reviewed journals. He visited institute fur Organische Cheie, Universitry at Tubingen, D- 776 Tubingen, Gerany in year 3 under INSA and DFG Postdoctoral Research Fellowship. Currently he has three research projects fro DST, CSIR and UGC. His current research interests include atheatical and coputational odeling of electrocheical biosensor. 56