International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:15 No:01 11

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1 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:5 No: An Analytical Solution for MD Micro Polar Fluid Transport Phenoena over a Stretching Pereable Sheet A. Majidian a, M. Parvizi Oran b,g, A. Aani c, M. Golipour d,. Borzouei Bazgir e, G. Doairry f a Mechanical Engineering Departent, Islaic Azad University, Sari Branch, Sari, Iran b Mechanical Engineering Departent, Noshirvani University of Technology, Babol, Iran c Faculty of Industrial Design Engineering, Delft University of Technology, Delft, Netherlands d Mechanical Engineering Departent, Islaic Azad University, Sari Branch, Sari, Iran e Mechanical Engineering Departent, Islaic Azad University, borujerd Branch, Borujerd, Iran f Mechanical Engineering Departent, Noshirvani University of Technology, Babol, Iran g Corresponding author: Shariati St, Babol, Iran, ojtaba.parvizioran@gail.co Abstract-- In this paper, an analytical solution for twodiensional boundary layer of an electrically conducting icro polar fluid spreading over pereable flat plate is aied. The effect of Magno-ydrodynaic (MD), viscous dissipation and internal heat generation has been considered. A set of siilarity paraeters is eployed to convert the governing partial differential equations into ordinary differential equations.nonlinear ODE which is obtained by siilarity solution has been solved through, a powerful analytical ethod, called ootopy Analysis Method (AM). The convergence of the obtained series solutions is explicitly studied. The obtained analytical solution in coparison with the nuerical ones represents a favorable accuracy. The Results are presented graphically and in tabular for and the different physical aspects of the proble have been discussed. Index Ter ootopy Analysis Method (AM), MD Flow, Micropolar Fluid, Joule eating Effect, Analytical Solution, Non- Linear ODE, Siilarity Solution, Variable Electrical Conductivity. I. INTRODUCTION The basic continuu theory for icropolar fluid was originally introduced and forulated by Eringen [-]. This theory take into consideration the icroscopic effects resulting fro the local structure and icro otions of the fluid eleents and has been eployed to study a nuber of various flow situations such as the flow of low concentration suspensions, liquid crystals, blood, colloidal fluids, ferroliquids, etc, for which the classical Navier-Stokes theory is inadequate. During the recent decades, the dynaics of icropolar fluids have been a popular area of research and a significant aount of research papers dealing with icropolar fluid flow over a flat plate were reported. Lukaszewicz [] and Eringen [4] presented Extensive reviews of atheatical aspects of the icro-polar fluid flow theory and its applications in their book. A study on the boundary layer flow of icropolar fluids past a sei-infinite plate was perfored by Ahadi [5] taking into account the gyration vector noral to the xy-plane and the icro-inertia effects. Soundalgekar and Takhar [6] investigated the flow and heat transfer past a continuously oving plate in a icropolar fluid. Chen and Char [7] have studied the suction and injection on a linearly oving plate subject to unifor wall teperature and heat flux. Nuerical solution was obtained for a nonisotheral stretching sheet by assanien and Gorla [8], including suction and injection effects. The sae proble solution using a ethod of successive approxiations has been reported by ady [9].Mohaadein and Gorla [] studied the heat transfer characteristics of a lainar boundary layer of a icropolar fluid over a linearly stretching sheet with prescribed unifor surface teperature or prescribed wall heat flux and viscous dissipation and internal heat generation. Abo-Eldahab and Ghonai [] investigated convective heat transfer in an electrically conducting icropolar fluid at a stretching surface with unifor free strea. Aissa and Modaadein [] studied joule heating effects on a icropolar fluid past a stretching sheet with variable electric conductivity. Odda and Farhan [] studied the effects of variable viscosity and variable theral conductivity on heat transfer to a icro-polar fluid fro a non-isotheral stretching sheet with suction and blowing. Eldabe and Ouaf [4] solved the proble of heat and ass transfer in a hydro agnetic flow of a icropolar fluid past a stretching surface with ohic heating and viscous dissipation using the Chebyshev finite difference ethod. Mahoud [5] considered theral radiation effects on MD flow of a icropolar fluid over a stretching surface with variable theral conductivity. Aouadi [6] reported a nuerical study for icropolar flow over a stretching sheet. Patil and Kulkarni [7] studied the effects of cheical reaction on free convective flow of a polar fluid through a porous ediu in the presence of internal heat generation. Recently Rahan et al. [8 ] have investigated soe new aspects of icro-polar fluids and their applications. Because of the nonlinear nature of icro-polar fluids, solving the related equations is generally ore difficult to obtain. This is not only true for analytical solutions but also true for nuerical solutions. In tradition, perturbation ethods are widely applied to give analytic approxiations of non February 5 IJENS I J E N S

2 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:5 No: linear probles. Since there are soe liitations with the coon perturbation ethod, and also because the basis of the coon perturbation ethod is upon the existence of a sall paraeter, developing the ethod for different applications is very difficult. Recently a powerful, easy to- use analytical technique developed by Liao [-5] naely the ootopy Analysis Method (AM) has been successfully applied to several non-linear flow fields. [6 ]. Motivated by all these studies, we conteplate to apply AM to find the analytical solution of three-couple nonlinear ordinary differential equations arising fro siilarity solution of ass, angular oentu and energy equations of MD icropolar fluid over a stretching sheet. The series solution in ters of the recurrence forulae is firstly coputed and then its convergence is properly discussed. Then Results have been copared with nuerical solutions which are solved by Maple software using fourth fifth order Runge Kutta (RKF45) ethod [4-5]. Finally, the graphs are plotted and discussed for the variation of the agnetic paraeter ( Mn ), suction paraeter ( f w ), Eckert nuber ( E ), icrorotaion paraeter ( ) and Prandtl nuber ( Pr ) on the velocity of the fluid, teperature distribution and angular velocity of icrostructures. Different values of physical paraeters are tabulated and discussed nuerically and graphically. II. FORMULATION OF TE PROBLEM Let us consider an incopressible lainar two-diensional MD icropolar fluid pasting over a pereable plane surface. The flow is assued to be in the x-direction, which is taken along the plate in the forward direction and y-axis is noral to it. The flow configuration and the coordinate syste are shown in the Fig.. Fig.. D diagra of MD icro polar flow over pereable stretching sheet. A variable agnetic field is applied in the y-direction that is noral to flow direction. No external electric field is applied. Moreover, the agnetic Reynolds nuber is so sall that the agnetic field induced by the oving fluid is negligible with respect to the external agnetic field. The electrical conductivity is assued to have the for: u () Where is a constant. For the flow under study, it is relevant to assue that the applied agnetic field strength has the for [6]: B ( x ) / Bx () B is a constant. With usual boundary conditions, the governing equation for the oentu, teperature and angular velocity field within the boundary layer are given by: Continuity equation: u v x y Moentu equation: u v K u K N B ( ) x y y x u v u y Angular oentu equation: N N N K N u v ( N ) x y j y j y Energy equation: ( ) B ( ) p y p p T T k T K u u v u x y C C y C x Where u and v are the velocity coponent along the x and y axes respectively, is kineatic viscosity, N is angular velocity, K is vortex viscosity, is fluid density, is spin gradient viscosity, j is the icro inertia per unit ass and B( x ) and are variable agnetic field and the electrical conductivity, respectively. In the energy equation, T is teperature, k is the theral conductivity and C is the specific heat at constant pressure. The appropriate boundary conditions for the above proble are given by: At: N T y : u Cx, v vw, N s, k qw E x y y And As: y : u, N, T T (8) Where q w is the rate of heat flux, C and E are a positive constant, T is a constant teperature of the abient fluid at a large distance fro the sheet and is icrorotation paraeter. When icrorotaion paraeter, s we obtain N ( x,) which represents no-spin condition i.e. the icroeleents in a p () (4) (5) (6) (7) February 5 IJENS I J E N S

3 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:5 No: concentrated particle flow-close to the wall are not able to rotate as was stated by Jena and Mathur [7]. The case corresponding to s.5 results in the vanishing of the antisyetric part of the stress tensor and represents weak concentrations. The case corresponding to s. is the representative of turbulent boundary layer flows (see Peddison and McNitt [8]). In order to obtain local siilarity solution of the proble, following transforations are introduced: / / ( x, y ) C xf ( ), [ C / ] y / C (9) N C x ( ) And D / x T T ( ) ( ) G ( ) () K C L B K Mn,, j k L C C B, E, Pr Cj D CC k P p () Where ( x, y) is the strea function that satisfies the continuity Eq., is the siilarity variable, Mn is the agnetic paraeter,c and D are equation constants, and B are diensionless aterial paraeter, E is Eckert nuber, is vortex viscosity paraeter, Pr is Prandtl nuber and L is a characteristic length. Applying the transforation of variables fro Eqs 8-, the governing Eqs. -6 are transfored to a syste of diensionless nonlinear ordinary differential equations: ' ( ) F ''' ' FF '' MnF () '' B ( F '') F ' F ' () G '' FG ' GF ' E ( Mn ) F ' E ( ) F '' Pr () vw F () f w, F '(), (), G '() ( C ) and F '( ), ( ), G( ) III. EXACT ANALYTICAL SOLUTION (4) (5) In this section we eploy the hootopy analysis ethod to solve Eqs. -.For that we select: F ( ) ( f w ) e (6) ( ) (7) G ( ) e (8) As an initial approxiations of F ( ), ( ) and G ( ).the corresponding linear operators are: L ( F ) F ''' F '' (9) L ( ) '' () L ( '' ' () Which satisfy: L [ C C C e ] L [ C e C e ], L [ C e C e ] Where C, C to C 7 are constants. We construct the zeroth order deforation probles as: ( ) [ (, ) ( )] [ (, ), (, )] () q L F q F q N F q q () F(, q) fw, F '(, q), F '(, q) (4) ( q) L [ (, q) ( )] q N [ F(, q), (, q)] (5) (, q), (, q) (6) ( q) L [ G (, q) G ( )] (7) q N [ F (, q), (, q), G (, q)] G '(, q), G(, q) (8) Where, and are non-zero auxiliary paraeters F(, q), (, q), G(, q) are apping functions, q [,] is the ebedding paraeter and the non-linear differential operators N, N and N are defined by: F (, q) (, q) N [ F (, q)] ( ) F (, q) F (, q) F (, q) Mn( ) (, q) N [ (, q)] B ( (, q) F (, q) F (, q) (, q) ) ( (, q) ) F (, q) G(, q) N [ G (, q)] Pr F (, q) F (, q ) E ( )( ) G (, q)( ) G (, q) F (, q) F (, q) E ( Mn )( ) Obviously for q and q we have: F(,) F ( ), F(,) F( ) (9) () () () G(,) G ( ), G(,) G( ) () (,) ( ), (,) ( ) (4) As q increases fro to, F(, q), (, q) and G(, q) vary fro, initial values to the exact solution February 5 IJENS I J E N S

4 F''() International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:5 No: 4 F(, q), (, q) and G(, q).due to Taylor s theore and Eqs.-4, we can express that: F (, q) F ( ) F ( ) q (5) (, q) ( ) ( ) q (6) G (, q) G( ) G ( ) q (7) F(, q) F ( )! q q (, q) ( )! q q G(, q) G ( )! q q (8) Differentiating the zeroth order deforation Eqs ties with respect to q, then dividing by!, and finally setting q,we get the following th-order deforation probles for. L[ F ( ) F ( )] R ( ) (9) L[ G ( ) G ( )] G ( ) (4) L [ ( ) ( )] ( ) ' ' (4) F () F () F ( ) (4) G ' () ( ) (4) () G( ) (44) ''' ' ( ) ( ) R F k '' ' ' F k Fk MnF k Fk (45) '' '' R ( ) B ( F ) k ' ' [ F k k F k k ] (46), (48), F ( ) F ( ) F ( ) C C C e (49) C e C e (5) ( ) ( ) ( ) 4 5 G G G C e C e (5) ( ) ( ) ( ) 6 7 Where C to C 7 are constant that can be obtain by applying the boundary condition in Eq. (9-4). We used Maple to solve above equations.the first few order of solution are given in Appendix A. Substituting the expressions for F, and G in higher order deforation equations, the final results are explicit expression for desired series solutions: N F( ) li F ( ) (5) N N ( ) li ( ) (5) N N G( ) li G ( ) (54) N N ust be sufficiently large. In practice, we decided to stop the calculations at N (at th- order), having realized that a sufficiently sall tolerance has been et. IV. CONVERGANCE OF ANALYTICAL SOLUTION As it can be seen fro appendix A, The analytical solution contains the auxiliary paraeter which deterines the convergence and rate of the AM solution approxiation as pointed out by Liao [4]. The -curves are plotted to see the range of adissible values for the paraeters, and th order appr. th order appr h Fig.. curve of F ''() for two different orders of approxiation when.5, Mn., E.5, fw.7 and Pr.7. Fro Fig.-4, under converged conditions, a horizontal line segent that is appeared in -curves can be considered as the valid region. As long as we choose paraeter in this region we can be sure that the series converges to its unique value. In our case study, according to Fig., the acceptable range of auxiliary paraeter is.5.5 and in Fig., the acceptable range of auxiliary paraeter is..75.also, it is easy to discover that..8 is the valid region of (Fig. 4) February 5 IJENS I J E N S

5 G'() F' '() International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:5 No: th order appr. th order appr. variables. As entioned, we can adjust to get ost accurate solution. In this proble we have wide acceptable range values of. Therefore we can take it fro this region. As you can see, the table shows a good agreeent between nuerical and analytical solution and give us AM as a reliable analytical approach with high degree of accuracy in solution of nonlinear flow field proble. The influence of the agnetic paraeter Mn on diensionless velocity F ( ), icro rotation ( ) and the teperature profile G ( ) are depicted in Figs Fig.. curve of '() h for two different orders of approxiation when.5, Mn., E.5, fw.7 and Pr Mn=. Mn=. Mn=. Mn= th order appr. th order appr h Fig. 4. curve of G '() for two different orders of approxiation when.5, Mn., E.5, fw.7 V. RESULTS AND DISCUSSION and Pr.7. The present work generalized the proble of joule heating effects on a boundary layer of a MD icropolar fluid over a non-isotheral pereable stretching sheet with variable electric conductivity. The nonlinear ordinary differential equations - subject to the boundary conditions 4-5 have been solved using an analytical ethod known as AM. In order to assess the accuracy of the solution, we defined relative errors as: AM sol NUM sol RE. (55) NUM sol We have copared our results with nuerical solution that has been perfored in Maple software.as presented in Table I, the results are obtained for the wall value of teperature G (), icrorotaion gradient '() and velocity gradient F ''(), respectively while f w and Mn are considered as Fig. 5.Diensionless velocity profile variation for different values of agnetic paraeter, Mn when., E.5, fw.7 and Pr.7. The presence of a transverse agnetic field in an electrically conducting fluid gives rise to a drag-like effect called the Lorentz force. This force tends to slow down the otion of the fluid in the boundary layer and reduce the oentu boundary layer thicknesses. On the contrary, it increases the theral boundary layer thickness. As it can be seen, the effect of agnetic paraeter on angular velocity profile is not considerable. Figs8- exhibit the behavior of flow fields diensionless profiles for different values of suction/injection paraeter, f w. Fro these figures, it can be concluded that a wall suction flow. ( f w ) tends to decrease all of the fluid velocity, angular velocity, and teperature as well as their boundary-layer thicknesses, revealing the usual fact that suction akes stable the boundary layer growth. On the other hand, injection of fluid at the plate surface ( f w ) causes the exact opposite effect naely increases in the fluid velocity, icro rotation and teperature boundary-layer thicknesses February 5 IJENS I J E N S

6 G F' International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:5 No: 6 TABLE I COMPARISON BETWEEN AM SOLUTION AND MAPLE NUMERICAL SOLUTION FOR.,. AND.5 f w Mn -.7 T -ORDER APPROXIMATION WEN.5, E.5, Pr.7 AND.5. F ''() '() G () AM NM R.E AM NM R.E AM NM R.E WIT Mn=. Mn=. Mn=. Mn=. except in the iediate vicinity of the plate surface. Fig..shows the velocity profiles for different values of vortex viscosity paraeter. Fro here we observe that as increases boundary layer thicknesses increases too...8 fw=-.7 fw=. fw= Fig. 6. Diensionless icrorotaion profile variation for different values of agnetic paraeter, Mn when., E.5, fw.7 and Pr.7.4. Mn=. Mn=. Mn=. Mn= Fig. 8. Diensionless velocity distribution for various values of suction/injection paraeter, f w when Mn.,.5 Pr Fig. 7. Diensionless teperature profile variation for different values of agnetic paraeter, Mn when., E.5, fw.7 and Pr.7. It is also seen fro Fig. 9 that increent of f w decreases the icro rotation everywhere within the boundary layer February 5 IJENS I J E N S

7 G G F' International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:5 No: 7..5 fw=-.7 fw=. fw= Fig. 9. Diensionless icrorotaion distribution for different values of suction/injection paraeter, f w when Mn.,.5, E.5 and Pr Fig.. Diensionless velocity profile for different values of vortex viscosity paraeter, when Mn., E.5, fw.7 and Pr fw= -.7 fw=. fw = Fig.. Diensionless teperature profile for different values of suction/injection paraeter, f w when Mn.,.5 Pr.7. The effect of on diensionless angular velocity (icro rotation) of the icroeleents has been shown in the Fig.. We observe that as increases, the axiu of the angular velocity and the location of the axiu value increases and shifts away fro the surface of the plate slightly. Fro Fig. also it is understood that increent of has no significant effect on theral boundary layer. In the Fig.4, the diensionless teperature for various values of the Prandtl nuber is plotted. As Prandtl increase, the teperature decreases. This is consistent with the fact the higher Prandtl Nuber iplies ore viscous fluid which tends to reduce the theral boundary layer along the plate. This yields a reduction in the fluid teperature. Also an increasing effect for Eckert nuber over teperature profile has been found through Fig Fig.. Diensionless icro rotation variation for different values of vortex viscosity paraeter, when Mn., E.5, fw.7 and Pr Fig.. Diensionless teperature profile for various values of vortex viscosity paraeter when Mn., E.5, fw.7 and Pr February 5 IJENS I J E N S

8 G G International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:5 No: 8.5 Pr=. Pr=. Pr=.4 Pr=.7 ore viscous fluid which tends to reduce the theral boundary layer. Also it is evident that the teperature increases with an increasing Eckert nuber. It is hoped that the present work will serve as a vehicle for understanding ore coplex probles involving the various physical effects investigated in Micropolar proble..5 APPENDIX A Results are given for.5, Mn., E.5, f.7, Pr.7. w Fig. 4. Diensionless teperature profile for various values of Prandtl nuber, Pr when Mn., E.5,. and fw E=. E=. E=.5 E= Fig. 5. Diensionless teperature profile for various values of Eckert nuber, E when Mn., Pr.7,. fw.7 VI. CONCLUSION A two-diensional, lainar icro fluid flow over a stretching sheet is studied considering variable electric conductivity, viscous dissipation and joule heating effect in the presence of a agnetic field. The set of governing equations and the boundary condition were reduced to ordinary differential equations and a series solution is obtained by applying AM. Effects of agnetic paraeter, the suction/injection paraeter, vortex viscosity paraeter Prandtl nuber and Eckert nuber on velocity, icro rotation and teperature profile have been analyzed and discussed in detail. The results indicate that the velocity boundary layer thickness decreases as the agnetic paraeter coefficient increases. But it increases teperature profile thickness. The effect of agnet paraeter over the icro rotation is found insignificant. Results reveal that a wall suction flow leads to decrease in velocity, icro rotation and teperature boundary layer thicknesses while an opposite effect is observed fro injection of fluid at the plate surface. It is observed that increase in Prandtl nuber iplies ( ).8.8 F e e..8 ( ) e e ( )..4.5 e.8 G e e ( ) e.774 e.85 e F e e G ( ).647 ( ) e.5 ( ) e..75 e.5 ( ) e.78 e.495 e.8.4 ( ).494 ( ) e.57 ( ) e.59.5 (.5.5) e. ( ) e.5 e.8976 e.454 e e p REFERENCES [] A. C. Eringen, Theory of Micropolar, Fluid Journal of Matheatical Analysis and Applications, 966, 6, -8. [] A. C. Eringen, Theory of Thero Micropolar Fluids. Journal of Matheatical Analysis and Applications, 97, 8, [] G. Lukaszewicz, Micropolar Fluids. Theory and Applications, Modeling and Siulation in Science, Engineering and Technology, Birkh User, Boston, 999. [4] A. C. Eringen, Micro Continuu Field Theories II. Fluent Media, Springer, New York,. [5] G. Ahadi, Self-Siilar Solution of Incopressible Micropolar Boundary Layer Flow over a Sei- Infinite Plate. International Journal of Engineering Science 976; 4, [6] V. M. Soundalgekar,. S. Takhar, Flow of a Micropolar Fluid on a continuous Moving Plate, International Journal of Engineering Science, 98; : [7] C. K. Chen, N. I. Char, eat Transfer of a Continuous Stretching Surface with Suction or Blowing, Journal of Matheatical Analysis and Applications, 988;, February 5 IJENS I J E N S

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Mathur, Siilarity Solutions For Lainar Free Convection Flow of a Thero Micropolar Fluid Past a Non-isotheral Vertical Flat Plate, International Journal of Engineering Science, 98, 9, [8] J. Peddison, R. P. McNitt, Boundary Layer Theory For Micropolar Fluid, Recent Advance in. Engineering Science, 97, 5, February 5 IJENS I J E N S