Mathematical Study on MHD Squeeze Flow between Two Parallel Disks with Suction or Injection via HAM and HPM and Its Applications

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1 International Journal of Engineering Trends and Technology (IJETT) Volue-45 Nuber -March 7 Matheatical Study on MHD Squeeze Flow between Two Parallel Disks with Suction or Injection via HAM and HPM and Its Applications Anil Kuar #, S P Agrawal # Departent of Applied Sciences (Matheatics) Chandigarh Engineering College, Landran Mohali Punjab India # Departent of Civil Engineering, Sai Nath University Ranchi, Jharkhand India Abstract-In this paper, we are considering the proble of agneto-hydrodynaic MHD squeeze flow of an electrically conducting fluid between two infinite, parallel disks are investigated. The analytical ethod called Hootopy Analysis Method (HAM) and Hootopy Perturbation Method (HPM) are used to copute an approxiation for the solution of nonlinear differential equations governing on the proble. The results of the entioned ethods are copared with a type of nuerical analysis as Boundary Value Proble ethod. Keywords Magneto-hydrodynaic Hootopy Perturbation Method, Squeeze flow, Teprature, nonlinear differential equations, incopressible flow. I. INTRODUCTION The application of a MHD fluid in lubrication prevents the adverse ipact of teperature on the fluid viscosity when the syste operates under boundary conditions. The proble considered is of general interest in the theory of agnetohydrodynaic lubrication and other related applications. In particular, the results of the present investigation are directly applicable to the hydrodynaics of high teperature bearings lubricated with liquid etals. A nuber of theoretical and experiental investigations into agneto-hydrodynaic effects in lubrication have been reported. These include aong other works of Hughes and Elco [], Kuza et al. [] and Krieger et al. [3]. Most scientific probles such as twodiensional viscous flow between slowly expanding or contracting walls with weak pereability and other fluid echanic probles are inherently nonlinear. In ost cases, these probles do not adit analytical solution, so these equations should be solved using special techniques. In recent years, uch attention has been devoted to the newly developed ethods to construct an analytic solution of equation such as the ethod included the Perturbation techniques. Perturbation techniques are too strongly dependent upon the so-called sall paraeters [4]. Other different ethods have introduced to solve nonlinear equations such as the δ-expansion ethod [5], Adoian s decoposition ethod [6], Hootopy Perturbation Method (HPM) [7 ], Variational Iteration Method (VIM) [ 4], Hootopy analysis ethod [5-8], Optial Hootopy Asyptopic Method (OHAM)[9,] and optial Hootopy Perturbation Method (OHPM)[]. In this letter, analytical solutions of nonlinear equations arising of agnetohydrodynaic MHD squeeze flow of an electrically conducting fluid between two infinite, parallel disks have been studied by the two analytical ethods. These ethods called Hootopy Analysis Method (HAM), Hootopy Perturbation Method (HPM) do not have sall paraeters. Obtaining the analytical solution of the odels and coparing with the nuerical result reveal the capability, effectiveness and convenience of HAM and HPM. These ethods give successive approxiations of high accuracy solution. Kuar et al. [3]investigated a finite difference technique for reliable MHD steady flow through channels pereable boundaries.kuaret al [4] investigated MHD free convective fluctuating flow through a porous effect with variable pereability Paraeter.Kuar et al. [5] investigated atheatical analysis of MHD on lainar ixed convection of newtonian fluid between vertical parallel plates channel. Kuar et al. [6] investigated a Crank-Nicholson schee to transient MHD free convective flow through seiinfinite vertical porous plate with constant suction and teperature dependent heat source. II. MATHEMATICAL MODEL In the present investigation, consider an axisyetric incopressible flow between two parallel infinite disks, which at tie t, are space a distance H at apart and a agnetic field ISSN: Page 7

2 International Journal of Engineering Trends and Technology (IJETT) Volue-45 Nuber -March 7 B at proportional to is applied perpendicularly to the disks [6, ]. The upper disk z H at is oving with velocity at H a t towards the stationary lower disk at z. The axial coordinate is denoted by z and the radial coordinate by r. With the axial and radial velocities denoted by w and u, respectively, we introduce the following quantities: r H B u f, w f, B at at at z, r r, t t H at The equation of continuity is satisfied and the oentu equations are reduced to: f ( ) S f ( ) 3 f ( ) f ( ) f ( ) M f ( ) () p H Where p () t has been used, S r r B and M that denotes density, denotes kinetic viscosity and denotes fluid electrical conductivity. The boundary conditions are given by: f () A, f (), f (), f () (3) Where, A is the constant paraeter such that A corresponds to suction and A to injection. III. APPLICATION OF HOMOTOPY ANALYSIS METHOD For HAM solutions, we choose the initial guess and auxiliary linear operator in the following for: 3 3 f A 3 A A, L( f ) f, (5) (4) 3 L( c y c y c3 y c4), (6) 6 Where ci ( i,,3, 4) are constant. Let P, denotes the ebedding paraeter and indicates non zero auxiliary paraeters. () Zeroth order deforation equations ( P) L F( ; p) f ( ) p H( ) N F( ; p) (7) F(; p) A; F(; p), F(; p), F(; p) (8) 4 3 M 4 3 d F( ; p) d F( ; p) d F( ; p) d F( ; p) N[ F( ; p)] S 3 d d d d 3 d F( ; p) df( ; p) d F( ; p) F( ; p) 3 d d d S For p and p we have F( ;) f ( ) F( ;) f ( ) () When p increases fro to then F( ; p) varies fro f ( ) to f ( ). By Taylor's theore and using Eq. (9), F( ; p) can be expanded in a power series of p as follows: ( F( ; p)) () F( ; p) f( ) f( ) p, f( )! p p In which is chosen in such a way that this series is convergent at p, therefore we have in according to Eq. () that f ( ) f ( ) f ( ), () th order deforation equations L f ( ) f ( ) H( ) R ( ) (3) F(; p) ; F(; p), F(; p), F(; p) (4) 3 k k k R f S f f M f S f f (5) Now we deterine the convergence of the result, the differential equation, and the auxiliary function according to the solution expression. So, let us assue: H( ) (6) We have found the answer by the aple analytic solution device. The first deforation of the solution are presented below : (9) ISSN: Page 8

3 International Journal of Engineering Trends and Technology (IJETT) Volue-45 Nuber -March 7 f A S S A A S A S A M S 3 A M SA S S A S M S A M A S A S M S A M A (7) The solutions f ( ) were too long to be entioned here, therefore they are shown graphically. IV. APPLICATION OF HOMOTOPY PERTURBATION METHOD In this section, we eploy HPM to solve Eq. ( ) subject to boundary conditions Eq.(3).We can construct Hootopy function of Eq. () as described in []: H f, p P f g y p f ( ) S f ( ) 3 f ( ) f ( ) f ( ), M f ( ), (8) Where p, is an ebedding paraeter. For p and p we have:,,, f f f f (9) Solving Esq. () (4) with boundary conditions, we have (for exaple S., M, A ): 3 f.5, (5) f , (6) 9 f (7) f (8) The solution of this equation, when p, will be as follows: 3 i f Li p fi (9) i p Note that when p increases fro to, f, p varies fro f to f substituting: n i i i f f p f p f p f, g ().By Fro equation (33) and rearranging the result based on powers of p-ters, we get: P : f ( ) () f() A; f, f(), f () P : f S f Sf f 3S f M f f () ; f, f (), f () P : f S f f 3S f Sf f S f M f f () ; f, f (), f () P : f 3S f M f Sf f Sf f 3 3 Sf f S f f () ; f, f (), f () (4) () (3) Fig. The - validity for S., M 5 and different value of A. Fig. The - validity for S.3, A and different value of M. As pointed out by Liao, the convergence and rate of approxiation for the HAM solution strongly depends on the value of auxiliary paraeter. The ISSN: Page 9

4 International Journal of Engineering Trends and Technology (IJETT) Volue-45 Nuber -March 7 auxiliary paraeter provides us with a convenient way to adjust and control the convergency. The range of for convergency is obtained according to figs. and. For S., M 5 and 7 A 7 the ranges.4.5, for S.3, A and M 5 the ranges.5.3, give suitable value of for convergency. Then,.9 is a suitable value for ranges which is used for the solution. Fig 4. The coparison between the HAM, HPM and nuerical solutions for f( ), f ( ) when S., M, A. Fig 3. The coparison between the HAM, HPM. and nuerical solutions for f( ), f ( ) when S., M 5, A. Fig 6. The coparison between the HAM, HPM and nuerical solutions for f( ), f ( ) when S, M 3, A. Table The results of HAM, HPM and Nuerical ethods for f when S.4, M, A HAM HPM NUM Error of HAM Error of HPM ISSN: Page 3

5 International Journal of Engineering Trends and Technology (IJETT) Volue-45 Nuber -March Table The results of HAM, HPM and Nuerical ethods for f when S.4, M, A HAM HPM NUM Error of HAM Error of HPM In this study, the proble of agneto-hydrodynaic MHD squeeze flow of an electrically conducting fluid between two infinite, parallel disks was analyzed using HAM and HPM. By the drawing of -D fig. 3 to 6, of the nuerical solution, HPM and HAM solutions for f( y ) and f ( y) with different values of SM, and A, we see that the Hootopy Analysis Method and Hootopy Perturbation Method are ore accurate than NUM. According to fig. 3 to 6 and tables and these ethods provide highly accurate analytic solutions for nonlinear probles in coparison with the nuerical solution. The coparison of the ethod reveals that the approxiations obtained by HAM and HPM converge to the exact solution quite fast. Also, the auxiliary paraeter provides us with a convenient way to adjust and control the convergence and its rate for the solutions series. Finally, it has been attepted to show the capabilities and wide-range applications of the HAM and HPM in coparison with the nuerical solution of nonlinear equations. 7. CONCLUSION In this paper, the proble of agneto-hydrodynaic MHD squeeze flow of an electrically conducting fluid between two infinite, parallel disks was analyzed by using HAM and HPM. Furtherore, the obtained solutions by HAM and HPM are copared with nuerical solutions. The results deonstrate that HAM and HPM are very effective and siple. ISSN: Page 3

6 International Journal of Engineering Trends and Technology (IJETT) Volue-45 Nuber -March 7 They offer superior accuracy in coparison with the NUM. Also, it is found that these ethods are powerful atheatical tools and that they can be applied to a large class of linear and nonlinear probles arising in different fields of science and engineering specially soe heat transfer equations. Acknowledgent Author is grateful to Chandigarh Engineering college Landran Mohali, India for providing facilities and encourageent to coplete this work. Also, the corresponding authors are thankful to the learned referees for their fruitful suggestions for iproving the presentation of this work. The author would like to thanks St. Naita Varshney, Ridansh and Siddika for providing eotional happiness during this research work. REFERENCES [] Hughes, W. F., Elco, R. A., Magnetohydrodynaic lubrication flow between parallel rotating disks, Journal of Fluid Mechanics, 3 (96),, pp. 3. [] kuza, D. C., Maki, E. 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Aziz, Approxiate Analysis of MHD Squeeze Flow between Two Parallel Disks with Suction or Injection by Hootopy Perturbation Method, journal of Matheatical Probles in Engineering, doi:.55/9/6396. [3] Anil Kuar, R. K. Saket, C L Varshney and Sajjan Lal : Finite difference technique for reliable MHD steady flow through channels pereable boundaries, International Journal of Bioedical Engineering and Technology (IJBET) UK, Vol. 4() pp -,. [4] Anil Kuar, CL Varshney and Sajjan Lal: MHD free convective fluctuating flow through a porous effect with variable pereability Paraeter, International Journal of Engineering, Iran, Volue 3-3&4 - Transactions A: Basics, ISSN Noveber, pp [5] Anil Kuar and S P Agrawal (5): Matheatical Analysis of MHD on Lainar Mixed Convection of Newtonian Fluid Between Vertical Parallel Plates Channel,, 9 th INDIACo; 5 nd International Conference on Coputing for Sustainable Global Developent, -3 March 5, IEEE Bharti Vidyapeeth New Delhi, ISSN ; ISBN pp [6] Anil Kuar, CL Varshney and Sajjan Lal : Crank-Nicholson Schee to Transient MHD Free Convective Flow through Sei-Infinite Vertical Porous Plate with Constant Suction and Teperature Dependent Heat Source, Proceedings International Conference on Advances in Coputing and Artificial Intelligence (ACAI ) pp 86-9,,USA. ISSN: Page 3