Approximate analytical solution of MHD flow of an Oldroyd 8-constant fluid in a porous medium

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1 Approxiate analytical solution of MHD flow of an Oldroyd 8-constant fluid in a porous ediu Faisal Salah,4, Zainal Abdul Aziz*,, K.K. Viswanathan, and Dennis Ling Chuan Ching 3 UTM Centre for Industrial and Applied Matheatics & Departent of Matheatical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 83 UTM Johor Bahru, Johor, Malaysia. 3 Departent of Fundaental and Applied Sciences Universiti Teknologi PETRONAS, 375 Tronoh, Malaysia. 4 Departent of Matheatics, Faculty of Science,University of Kordofan,Elobid, Sudan. *Corresponding author e-ail: zainalabdaziz@gail.co Abstract----The steady flow in an incopressible, agnetohydrodynaic (MHD) Oldroyd 8-constant fluid in a porous ediu with the otion of an infinite plate is investigated. Using odified Darcy s law of an Oldroyd 8-constant fluid, the equations governing the flow are odelled. The resulting nonlinear boundary value proble is solved using the hootopy analysis ethod (HAM). The obtained approxiate analytical solutions clearly satisfy the governing nonlinear equations and all the iposed initial and boundary conditions. The convergence of the HAM solutions for different orders of approxiation is deonstrated. For the Newtonian case, the approxiate analytical solution via HAM is shown to be in close agreeent with the exact solution. Finally, the variations of velocity field with respect to the agnetic field, porosity and non-newtonian fluid paraeters are graphically shown and discussed. Keywords-Oldroyd 8-constant fluid, Hootopy analysis ethod, Porous ediu, Magnetohydrodynaics (MHD) I. INTRODUCTION Several fluids including butter, cosetics and toiletries, paints, lubricants, certain oils, blood, ud, jas, jellies, shapoo, soaps, soups, aralades and etc have rheological characteristics and are referred to as non- Newtonian fluids. The rheological properties of all these fluids cannot be explained by using a single constitutive relation between stress and shear rate which is quite different fro those viscous fluids [] and []. Such an understanding about non-newtonian fluids forced researchers to propose ore odels of non- Newtonian fluids. In general, the classification of non-newtonian fluid odels is given under three categories which are called the differential, the rate and the integral types [3]. In recent years there have been any analytical and nuerical studies devoted to the flows of Oldroyd-B (3-constant) fluids which is a subclass of the rate type fluids, see [-5]. Recently, Hayat et al. [6-7] carried out studies of the flow of an Oldroyd 6-constant fluid in different configurations using the hootopy analysis ethod (HAM). More recently, Wang and Ellahi [8], Ellahi et al. [9] and [], Hayat et al. [], Khan et al. [] and Sajid et al. [3] investigated the steady state flow of an Oldroyd 8-constant fluid in different configurations using hootopy analysis ethod (HAM) and soe other ethods. The Oldroyd 3-constant fluid for a unidirectional steady flow does not exhibit the non- Newtonian property. For this reason soe steady flows ay be well described by the Oldroyd 8-constant fluid, and thus with this in ind, the chosen odel in the present article is of the Oldroyd 8-constant type. The ensuing governing differential equation is non-linear. We solve the non-linear boundary value probles using the hootopy analysis ethod [4]. This ethod has already been successfully applied to probles related to non-newtonian fluids [6, 7, 8, 9, 5 and 6]. To the best of our knowledge, no investigation has been reported so far which discusses the oving flat plate MHD flow of an Oldroyd 8-constant fluid in a porous ediu, and thus the objective of the present study is to exaine this proble. Constitutive equations of the Oldroyd 8- constant fluid are used and the odified Darcy law is being utilized. The solution to the resulting proble is generated by HAM. Tables show the convergence of the HAM solutions for different orders of approxiation and for the Newtonian case the approxiate analytical solution via HAM is in close agreeent with the exact solution. Graphs are plotted in order to illustrate the variation of the velocity profile with respect to the ebedded flow paraeters. ISSN : Vol 6 No 6 Dec 4-Jan 5 53

2 II. MATHEMATICAL FORMULATION A Cartesian coordinate syste is chosen by considering an infinite plate at y. An incopressible fluid occupying the porous space conducts electrically by exerting a unifor agnetic field B, and is applied noral to the plate. The electric field is not taken into consideration and the agnetic Reynolds nuber is assued to be sall so that the induced agnetic field does not need to be accounted for. The Lorentz force J under these conditions is equal to B V. Here J is the current density, V is the velocity field, B is the electrical conductivity of fluid. The governing equations are divv, () V ( V. ) V p div B t S V R, () in which is the fluid density, p is the pressure, and R is Darcy s resistance. The extra stress tensor S for an Oldroyd 8-constant fluid satisfies T Ip S, DS DA 7 S SA A S A trs tr 4 tr, Dt SA I = A A Dt A I (3) T is the Cauchy stress tensor, S the extra stress tensor, I the identity tensor, L the velocity gradient, D A L+ L T the first Rivlin Eriksen tensor, the dynaic viscosity of fluid and is defined by Dt DS ds T d LS SL, in which indicates the aterial derivative. Dt dt dt For steady unidirectional flow, the extra stress tensor and the velocity are denoted respectively, as Sxx y Sxy y Sxz y S y S yx y S yy y S yz y, y S y S y S y zx zy zz Here uy ( ) is the velocity in the x direction. uy ( ) V. (4) According to M. Khan et al. [], the Darcy resistance in Oldroyd 8-constant fluid satisfies the following expression R du dy k u x , is the porosity and k is the pereability of the porous ediu. With the help of Eq. (4), the continuity equation () is satisfied identically and the x-coponent of the oentu equation () along with Eqs. (3) and (5) for steady flow in the absence of the pressure gradient, and the non-linear odel equation is obtained as du dy (5) ISSN : Vol 6 No 6 Dec 4-Jan 5 54

3 d u 3 du du d u dy dy dy dy dy k dy dy du du du N u N B. The last fifth, non-linear, ter in equation (6) represents the porous effects in the odel equation. In order to solve this equation, the proficient ethod HAM is resorted, the corresponding results with the porous effects are taken into consideration, and the influence of pertinent paraeters on the fluid otion is graphically underlined. III. FLOW ANALYSIS The steady flow of an incopressible, electrically conducting Oldroyd 8-constant fluid occupying the space y is considered. The fluid is bounded by an infinite non conducting rigid plate at y. A unifor agnetic field B is applied noral to the plate. The flow is aintained due to the sudden otion of the plate, and there is no flow far away fro the plate. The governing nonlinear differential equation is ODE (6) and the boundary conditions are u U for y, u as y, U is the constant plate velocity. Introducing the following diensionless variables 4 4 U u N U U y, f, M,,, (8) U U K ku is the kineatic viscosity. The proble (6) with conditions (7) now becoe d f 3 df df d f d d d d df df df M f d K d d with f f Consider and. () u r, t IV. ESSENTIAL IDEAS OF HAM, () which is a nonlinear equation in a general for, indicates a nonlinear operator,, (6) (7) (9) u r t iplies a unknown function. By using hootopy analysis ethod, the zeroth-order deforation equation is constructed as r, t; u r, t ћr, t r, t;, () ISSN : Vol 6 No 6 Dec 4-Jan 5 55

4 u r,t denotes an initial guess of the exact solution u r, t, an auxiliary linear operator, ћ an auxiliary paraeter, and, r, t an auxiliary function, is an ebedding paraeter. It should be noted, that the auxiliary paraeter ћ which is being featured in HAM can be chosen freely. Obviously, when, Eq. () holds for r, t; u r, t, r, t; ur, t respectively. Then as long as increases fro to, the solution rt,; u r,t to the exact solution u r, t. By using Taylor theore, Liao [7] has expanded rt,; r, t; r, t; u r, t u r, t varies fro the initial guess in a power series of as follows (3)! rt,;. (4) The convergence of the series (3) depends upon the auxiliary function rt, auxiliary linear operator and initial guess u convergence at, and one has,,, u r t u r t u r t,, auxiliary paraeter ћ, r t. If these are selected properly, the series (3) is Based on Eq. (4), the governing equation can be derived fro the zeroth-order deforation equation () and we can define the vector u ( r, t) u r, t, u r, t,, u ( r, t). n n Differentiating -ties of zeroth-order deforation equation () with respect to and dividing the by! and also setting, the result will be so-called th-order deforation equation,,,,, u r t u r t ћ r t u r t (6),, u, r, t u r, t! (5) (7) (8) THEOREM 3. (Liao [7]): The series (5) is convergent to the exact solution of () as long as the series is convergent. V. SOLUTION OF THE PROBLEM Now to solve the non-linear ordinary differential equation (9) subject to boundary conditions (), the ethod HAM is applied to give an approxiate, uniforly valid and analytic solution. For the HAM solution, the initial guess function is chosen to be in the for f (9) e and f f () ISSN : Vol 6 No 6 Dec 4-Jan 5 56

5 as the auxiliary linear operator satisfying Ce, C e () C and C are arbitrary constants. It is very significant that one has great freedo to choose auxiliary objects in HAM in accordance to the rule of its solution expression. Moreover, any value of the initial operator ay be selected which yields rapid convergence in the given doain. In the present proble, the flow doain is sei-infinite and therefore having the boundary condition at infinity, an initial operator of the for () which satisfies Eq. () is chosen. If q, is the ebedding paraeter and ћ is the auxiliary nonzero paraeter then we have: Zeroth-order deforation qf, q f qћ f, q, () f, q, f, q, (3) f, q 4 d f, q df d f df d f 3 d d d d d 4 4 df df df df, q f, q M f. d d K d d Mth-order deforation proble f f ћ, (5) f f,, (6) k d f df df d f d K k i d d d K k ki i M f 3 M fi k i j df df df df d f d d d d d K k ki ij jl l M fl (7) k i j l,,,. MATHEMATICA is used to solve the set of linear equations (5) with conditions (6). It is found that the series solution for is given by f e e e 9 e 9 e K e 3 Ke 3 Ke Me 6 Me ћ e 7 Kћe 7 Kћe 4 Kћ e Kћe... (9) The approxiate analytical solution given by (9) contains the auxiliary paraeter ћ, which influences the convergence region and rate of approxiation for the HAM solution. In Fig. the ћ - curve is plotted for f when 5, M K at 3 th -order approxiation. Table shows the convergence of the HAM solutions for different orders of approxiation. This iplies that the 3 rd order is the appropriate order of approxiation to be chosen. Table (4) (8) ISSN : Vol 6 No 6 Dec 4-Jan 5 57

6 Convergence of the HAM solutions for different orders of approxiation when M 3and K Table Order of approxiation f () f '() f ''() Fig. clearly elucidates that the range for the adissible values of ћ is.5 ћ.. These calculations indicate that the series solution as given in Eq. (9) converges in the whole region of when ћ.. K M..5 f5 f 5 f Fig.. ћ -curve f K M M M M Fig.. Velocity profile for MHD paraeter M. ISSN : Vol 6 No 6 Dec 4-Jan 5 58

7 ..8.6 M K K K f.4 K Fig.3. Velocity profile for porous paraeter when K.. MK f Fig.4. Velocity profile for paraeter. f K M Fig.5. Velocity profile for paraeter. ISSN : Vol 6 No 6 Dec 4-Jan 5 59

8 VI. RESULT AND DISCUSSION In this section, the graphical illustrations of the velocity profiles as shown in Figures -5 are described. These graphs have been deterined for the MHD flow of the Oldroyd 8 constant fluid in a porous ediu over a oving flat plate. The eerging paraeters here are () M is the agnetic field paraeter, () K is the porous paraeter, (3) & are the aterial constant paraeters. In order to illustrate the role of these paraeters on the velocity profile of f ( ), the Figs. - 5 have been displayed. Fig. is prepared to see the effects of applied agnetic field (Hartan nuber) M on the velocity profile. Keeping,, K, ћ fixed and varying M, it is noted that the velocity profile decreases by increasing the agnetic field paraeter M. Clearly, we observe that with increasing the values of M, the velocity profile of f ( ) decreases, in fact this is because of the effects of the transverse agnetic field on the electrically conducting fluid which gives rise to a resistive type Lorentz force which tends to slow down the otion of the fluid. Very interesting situation is observed fro the results in Fig. 3. By increasing the porous paraeter K whiles the other paraeters M,,, ћ are fixed, would lead to a decrease in the velocity profile. This is physically due to the fact that by increasing the values of K the ediu would appear to be soe less porous and thus this increases the friction forces, and would then reduce the flow of the fluid. Fig. 4 shows the effects of the aterial constant paraeter on the velocity profile when M, K,, ћ are fixed. It is of interest to notice that by increasing the paraeter, this would lead to an increase in the velocity profile (this is uch related to decrease in the boundary layer thickness). This is in fact true since by increasing the values of would then reduce the friction forces, and, thus, assists the flow of the fluid considerably; and hence the fluid oves with greater velocity. Fig. 5 shows the effects of the other aterial constant paraeter on the velocity profile when M, K,, ћ are fixed. It is worth noticing that by increasing the paraeter would lead to a decrease in the velocity profile (this is uch related to increase in the boundary layer thickness). This is because of the fact that increasing the values of would increase the friction forces, and, thus, slow down the otion of the fluid. Table shows that for the Newtonian case, the approxiate analytical solution via HAM is in close agreeent with the exact solution. Thus this verifies the for of the HAM solution for this proble. Table Illustrating the variation of exact solution (Faisal et al. [5]) f when M., K and i.e. Newtonian case using HAM and z Table f '( z) HAM Exact [5] ISSN : Vol 6 No 6 Dec 4-Jan 5 5

9 VII. CONCLUDING REMARKS In this paper, we have odelled the governing equation for steady and agnetohydrodynaic flow of an Oldroyd 8-constant fluid over a plate in a porous ediu and deterined the corresponding velocity field. The HAM is used to solve the governing non-linear second-order differential equation. The obtained solution indicates the effects of aterial constant paraeters, porous paraeter and the agnetic field. It is shown that an increase in the applied agnetic field leads to a decrease in the velocity due to the resistive Lorentz force. Fro the presented analysis, the ain observations are described as follows: i. The behaviours of and on the velocity f based on the graphs are quantitatively opposite in nature. ii. The flow characteristics subject to M and K on the velocity coponents based on the graphs leads to the sae effects. iii. The results corresponding to viscous fluid can be obtained by choosing. iv. The convergence of the HAM solutions for different orders of approxiation is deonstrated. v. For the Newtonian case, the approxiate analytical solution via HAM is shown to be in close agreeent with the exact solution. ACKNOWLEDGMENT This research is partially funded by UTM-CIAM Flagship Project Vote No. PY/3/474 and MOE FRGS Vote No. R.J F354, Research Manageent Centre (RMC), Universiti Teknologi Malaysia, Malaysia. Also Dr. Faisal is thankful to UTM for the postdoctoral fellowship. REFERENCES [] C. Fetecau, T. Hayat, and M. Khan, "Unsteady flow of an Oldroyd-B fluid induced by the ipulsive otion of a plate between two side walls perpendicular to the plate," Acta Mechanica, vol. 98(-), pp. -33, 8. [] D. Vieru, C. Fetecau, and C. Fetecau, "Flow of a generalized Oldroyd-B fluid due to a constantly accelerating plate," Applied Matheatics and Coputation, vol. (), pp , 8. [3] C. Fetecau, Sharat C. Prasad, and K. R. Rajagopal, "A note on the flow induced by a constantly accelerating plate in an Oldroyd-B fluid." Applied atheatical odelling, vol. 3(4), pp , 7. [4] M. Husain, T. Hayat, C. Fetecau, and S. Asghar, "On accelerated flows of an Oldroyd-B fluid in a porous ediu." Nonlinear Analysis: Real World Applications, vol. 9(4), pp , 8. [5] S. Faisal, Z.A. Aziz, S. A. Norsarahaida, and L. C. C. Dennis, "Exact Solution For Rayleigh Stokes Probles of An Oldroyd-B Fluid in a Porous Mediu and Rotating Frae." Journal of Porous Media, vol. 5(), pp. 9-98,. [6] T. Hayat, M. Khan, and S. Asghar, "Magnetohydrodynaic flow of an Oldroyd 6-constant fluid." Applied Matheatics and Coputation, vol. 55(), pp , 4. [7] T. Hayat, M. Khan, and M. Ayub, "Couette and Poiseuille flows of an Oldroyd 6-constant fluid with agnetic field." Journal of atheatical analysis and applications, vol. 98(), pp. 5-44, 4. [8] Y. Wang, T. Hayat, and K. Hutter, "On non-linear agnetohydrodynaic probles of an Oldroyd 6-constant fluid." International Journal of Non-Linear Mechanics, vol. 4(), pp , 5. [9] R. Ellahi, T. Hayat, T. Javed, and S. Asghar, "On the analytic solution of nonlinear flow proble involving Oldroyd 8-constant fluid." Matheatical and Coputer Modelling, vol. 48(7), pp. 9-, 8. [] R. Ellahi, T. Hayat, F. M. Mahoed, and A. Zeeshan, "Exact solutions for flows of an Oldroyd 8-constant fluid with nonlinear slip conditions." Counications in Nonlinear Science and Nuerical Siulation, vol. 5(), pp. 3-33,. [] T. Hayat, M. Khan, and M. Ayub, "The effect of the slip condition on flows of an Oldroyd 6-constant fluid." Journal of Coputational and Applied Matheatics, vol. (), pp. 4-43, 7. [] M. Khan, T. Hayat, and Y. Wang, "Slip effects on shearing flows in a porous ediu." Acta Mechanica Sinica, vol. 4(), pp. 5-59, 8. [3] M. Sajid, A. M. Siddiqui, and T. Hayat, "Wire coating analysis using MHD Oldroyd 8-constant fluid." International Journal of Engineering Science, vol. 45(), pp , 7. [4] S. J. Liao, Beyond Perturbation: Introduction to Hootopy Analysis Method, Boca Raton: Chapan & Hall. 3. [5] S. Abbasbandy, "The application of hootopy analysis ethod to solve a generalized Hirota Satsua coupled KdV equation." Physics Letters A, vol. 36(6), pp , 7. [6] Z.A. Aziz, M. Nazari, S. Faisal, and L. C. C. Dennis, "Constant Accelerated Flow for a Third-Grade Fluid in a Porous Mediu and a Rotating Frae with the Hootopy Analysis Method." Matheatical Probles in Engineering, vol.,. [7] S. J. Liao, Beyond Perturbation: Introduction to the Hootopy Analysis Method, USA: Chapan and Hall. 4. ISSN : Vol 6 No 6 Dec 4-Jan 5 5