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UNSTEADY MHD BOUNDARY LAYER FLOW OF AN INCOMPRESSIBLE MICROPOLAR FLUID OVER A STRETCHING SHEET K.Govardhan1 N.KISHAN2 1Department of mathematics, CVR College of Engineering, R.R.Dist,A.P.India. 2Department of mathematics, Osmania University, Hyderabad 500007, A.P, India, E-mail: govardhan_kmtm@yahoo.co.in ABSTRACT: Aim of the paper is to investigate the MHD effects on the unsteady boundary layer flow of an incompressible micropolar fluid over a stretching sheet when the sheet is stretched in its own plane. The stretching velocity is assumed to vary linearly with the distance along the sheet. Two equal and opposite forces are impulsively applied along x axis so that the sheet is stretched, keeping the origin fixed in a micropolar fluid. The governing non-linear equations and their associated boundary conditions are first cast into dimensionless form by a local non-similarity transformation. The resulting equations are solved numerically using the Adams- Predictor Corrector method for the whole transient from the initial state to final steady- state flow. Numerical results are obtained and a representative set is diaplaced graphically to illustrate the influence of the various physical parameters on the velocity profiles, microrotation profiles as well as the Skin friction coefficient for various values of the material parameter K. It is found that there is a smooth transition from the small- time solution to the large- time solution. Results for the local skin friction coefficient are presented in table as well as in graph. Key words: Unsteady flow, micro polar fluid, stretching surface, skin friction, MHD. NOMENCLATURE u, v velocity components j micro-inertia x, y cartesian co-ordinates N micro rotation dynamic viscosity K matériel parameter density of the fluid B0 applied magnetic field Spin gradient M magnetic field parameter k vortex viscocity electrical conductivity n constant 1. INTRODUCTION The fluid dynamics over a stretching surface is important in considerd various problems in micropolar fluids. Nazar et.al extrusion process. The production of sheeting material arises (2004) studied the stagnation point flow of a non- Newtonian in a number of industrial manufacturing process and includes micropolar fluids with zero vertical velocity at the surface or both metal and polymer sheets. Examples are numerous and heat generation. Rajeshwari and Nath (1992) studied they include the cooling of an infinite metallic plate in a unsteady flow over a stretching surface in a rotating fluid, cooling bath, the boundary layer along material handling Noor (1992) investigated Heat transfer from a stretching conveyers, the aerodynamic extrusion of plastic sheets, the sheet. boundary layer along a liquid film in condensation process, Guram and Smith (1980) investigated the stagnation flows paper production, glass blowing, metal spinning, and drawing of micropolar fluids with strong and weak interactions. They plastic films, to name just a few. The quality of the final obtained numerical results using a fourth order Runge – product depends on the rate of heat transfer at the stretching Kutta method. Gorla (1983) obtained numerical results by a surface. A comprehensive review of micropolar fluids Runge – Kutta method for the micropolar boundary layer mechanically has been presented by Ariman et al (1973). flow at a stagnation point on a moving wall. Heat transfer Since the pioneering study by Crane (1970) who presented an over a stretching surface with variable surface heat flux in analytical solution for the steady two – dimensional micropolar fluids and MHD stagnation point flow towards a stretching of a surface in a quiescent fluid, many authors stretching vertical sheet in a micropolar fluid is studied by have considered various aspects of this problem and obtained Ishak et al. (2008). Recently Nazar et al (2008) studied the similar solutions. Some mathematical results were presented unsteady boundary layer flow of an incompressible by many authors, and a good number of references can be micropolar fluid over a stretching sheet. They solved found in the papers by Magyari and Keller (1999,2000). numerically using Keller-box method. Sriramulu et.al (2001) studied steady flow and heat transfer Viscous dissipation effects were consider on mhd nonlinear of a viscous incompressible fluid through porous medium flow and heat transfer past a stretching porous surface over a stretching sheet. embeded in a porous medium under a transverse magnetic On the other hand, it is well known that the theory of field is studied Anjalidevi and ganga(2010). Sharma and micropolar fluids has generated a lot of interest and many singh(2009) is investigated the effect of variable thermal flow problems have been studied. The theory of micropolar conductivity and heat source/sink on flow of a viscous fluids was originally developed by Eringen (1964,1966) and incompressible electrically conducting fluid in the presence has now been applied in the investigation of various fluids. of uniform transverse magnetic fluid and variable free stream The theory takes into account the microscopic effects arising near a stagnation point on a non conducting stretching sheet. from the local structure and micro-motions of the fluid The purpose of the present paper is to study the elements and provides the basis for a mathematical model for magneto hydrodynamic effects on the unsteady boundary non-newtonian fluids which can be used to analysis the layer flow of an incompressible micropolar fluid over a behavoiur of exotic lubricants, polymers, liquid crystals, stretching sheet when the sheet is stretched in its own plane. animal bloods and colloidal or suspension solutions, etc. A numerical solution is obtained for the governing Since introduced by Eringen many researchers have momentum using the Adams predictor-corrector method. j is microinertia per unit mass, is spin gradient viscosity and k is vortex viscosity. Further, n is a constant and 2.MATHEMATICAL ANALYSIS 0 n 1. The case n 0 , which indicates N=0 at the Consider the flow of an incompressible micropolar wall represents concentrated particle flows in which the microelements close to the wall surface are unable to rotate. fluid in the region y 0 driven by a plane surface located at This case is also known as the strong concentration of y 0 with a fixed end at x 0 . It is assumed that the microelements.Thecase n 1/ 2 indicates the vanishing of surface is stretched in the x direction such that the anti – symmetric part of the stress tensor and denotes weak x component of the velocity varies linearly along it, concentration of microelements. The case n 1 is used for i.e. u w ( x) cx , where c is an arbitrary constant and the modeling of turbulent boundary layer flows. We shall c 0 . The simplified two - dimensional equations consider here both cases of n 0 and n 1/ 2 . governing the flow are the equations of the continuity, Introducing the new variables as momentum equations under the influence of externally cv 1 / 2 1 / 2 xf , , N c / v 1 / 2 1 / 2 cxg , , c / v 1 / 2 1 / 2 y, 1 e , ct , imposed transverse magnetic field in the boundary layer (5) steady laminar and incompressible micropolar fluids are; Where is the stream function defined in the usual way as u v 0 ,(1) u and v , and identically satisfy (1). x y y x Substituting variables (5) in to equations (2) and (3) gives u u u k 2u k N B 20 u v u ,(2) t x y y 2 y 1 K f 1 2 f f f f 2 Mf ' (6) N N N 2N u f j t u v k 2 N ,(3) Kg 1 x y y 2 y K 1 Subject to the initial and boundary conditions 1 2 g 1 2 g 2 g fg f g g (7) t 0 : u v N 0 , for any x, y , K 2 g f 1 u t 0 : v 0, u u w ( x) cx, N n , at y 0 , K= k is the material parameter. Here and j are y Where (4) k K u 0 , , as y , N 0 assumed to be given by j 1 j 2 2 Where u and v are the velocity components along the v x and y axes, respectively, t is time, N is the and j , respectively. The boundary conditions equation c microrotation or angular velocity whose direction of rotation (4) becomes is in the xy plane, is dynamic viscosity, is density, f ,0 0, f ,0 1, g ,0 n f ,0 , K 1 g f g f g K 2 g f 0 (16) 2 f , 0, g , 0. (8) Subject to the boundary conditions (14). The physical quantity of interest in this problem is the skin friction coefficient C f , which is defined as 3.METHODOFSOLUTION To solve the equations (6) and (7), we have convert into a system of five first order equations, we have at , w Cf , (9) y1 f , y 4 g , y1 y 2 , y 2 y3 uw 2 / 2 w is the skin friction, given by 1 y2 y2 1 2 y3 Where y3 y1 y3 y 2 2 M * y2 ky5 u / 1 k w k kN . (10) y y 0 , y 4 y 5 and Using variables (5) in (9) and (10), we obtain y4 y 5 1 1 1 y4 y5 y5 2 2 y1 y5 y2 y4 , C f Re x1/2 1/2 1 1 n K f ,0 . (11) k Further, we can obtain some particular cases of this problem. / 1 2 A. Early Unsteady Flow Early unsteady flow is obtained by solving these equations For early unsteady flow 0 1, we have 0 , so with 0 . For 0 , the above equations reflect a fully implicit scheme with respect to . In both cases, assuming (6) and (7) reduce in the leading order approximation to y 3 ,0 and y 5 ,0 , the above system is 1 K f f Kg 0, (12) solved up to m ax . 2 To solve and by Newton –Raphson method. We need K 1 y 2 y 4 y 2 y 4 1 g g g 0, (13) , , and at m ax , these quantities are 2 2 2 obtained by solutions and the boundary conditions (8) become y1 y 2 , y 2 y3 , f 0 0, f 0 1, g 0 n f 0 , y2 1 1 y3 (14) f 0, g 0. y3 2 / 1 k y1Y3 Y1 y3 2 y2Y2 M * y2 ky5 B. Final steady- state Flow For this case, 1 and (6) and (7) take the following forms: y 4 y5 and 1 K f f f f Mf Kg 0 (15) 2 y4 1 state flow 1 is represented in the figures 5&6 for the 1 1 y4 y5 k y5 2 2 / 1 cases n 0 and n 1/ 2 , respectively. These figures show y Y Y y y Y Y y 2 that the velocity profiles corresponding to increasing of 0 1 1 5 1 5 2 1 2 1 approach the final steady profile Once corresponding to 1 . It has seen that there is a smooth with y1 0 y2 0 0, y3 0 0, y4 0 n , transition from small time solution 0 to large time y 5 0 0 and another time with solution 1 . The effect of magnetic parameter M on velocity y1 0 y 2 0 0, y 2 0 0, y 3 0 0, y 4 1 0, y 5 0 distribution f ( ) is shown in figure 7. The magnetic field 1. parameter m, effect is shown in fig 7 for the velocity distribution f ( ) of final steady flow 1 with K 1 This procedure converging in about three iterations giving correct values of and . The system of Ordinary and n 0 . The velocity distribution of final steady state differential equation is solved by Adams predictor- corrector flow ( 1) for various M values with K 1, n 0 is methods of fourth order. Accuracy is ensured by solving with shown in figure 7. It is obvious that existence of Magnetic different , m ax , . field M decelaretes the velocity profiles. 4.DISSCUSSION OF THE Figures.8-13 represents microrotation distribution for various values of K , n for steady state and unsteady state RESULTS flow. The microrotation distribution of final steady state flow The transformed equations (6) and (7) satisfying the 1 is increases with the increase of the material boundary conditions (8) were solved numerically using the parameter K is observed from Figure.8 when n 0 .The Adams predictor-corrector method for several values of the microrotation distribution of final steady state flow 1 material parameter K . Numerical results for Skin friction with n 1/ 2 is shown in fig.9, from which the coefficients, the velocity distribution and microrotation microrotation decreases as K increases in the vicinity of the plate where as it increases as one moves away from it. Fig. distribution are shown graphically. 10 represents the microrotation distribution of fully To validate our method we have compared the Skin friction developed unsteady flow for n 0 and K 1 coefficients C f Re1/ 2 x values with Rosilinda nazar (2008) is for 0 1 . It is noticed that the microrotation distribution shown in Table, there is very good agreement between the as parabolic distribution and increases with the increase of . results when we solved fully unsteady boundary layer Fig. 11 shows that microrotation distribution of early equations and final steady state equations. unsteady flow 0 1 for various K values Though computations have been carried out for various with n 1/ 2 . The microrotation distribution decreases as values of the n and the material parameter K are presented. K increases near the plate but reverse phenomena is The velocity distribution of initial flow ( 0 ) and observed as one moves away from the plate. The microrotation distribution of fully developed unsteady flow unsteady flow (0 1) for various values of K with when K 1 and n 1/ 2 and final steady state flow n 0 and n 1/ 2 is shown graphically in figures 1& 2 1 is shown in fig.12. The microrotation distribution respectively. From both the figures it is observed that the increases near the plate while, the reverse happens far away velocity boundary layer thickness increases with the the plate with the increase of is observed. The magnetic increasing values of K , for both the cases n 0 and n 1/ 2 . The figures 3&4 represent for final steady field effect on the microrotation distribution is ploted in state flow 1 for the cases n 0 and n 1/ 2 , figure 13. It can be seen from the figure that the magnetic field effect occelerates the microrotation distribution near the respectively. It is observed from the figures that the velocity plate, where as it decelerates the microrotation distribution increases with the increase of K . The velocity distribution of far away from the plate.The magnetic field effect is more on fully developed unsteady flow 0 1 and final steady microrotation distribution when far away from the plate. C f Re x with for various impulsive motion, the skin friction coefficients as large 1/ 2 The Skin friction coefficient magnitude (absolute value) for small time ( 0 or values of K is drawn when n 0 in fig.14 and n 1/ 2 in fig.15. It can be seen that the values of C f Re1/ 2 x decreases as K increases. 0) after the start of the motion, and decreases CONCLUSION. monotonically and reaches the steady state value at 1 It is clear from the figs that the microrotation effects are . more pronounced for n 1/ 2 when compared to those of n 0 . The microrotation profile for n 0 is different as compared to n 1/ 2 it has a parabolic distribution The magnetic field effects is to decelerates the velocity distribution f ( ) . The microrotation distribution of final when n 0 , where as it has continuously decreasing when n 1/ 2 . steady flow 1 is to increase near the plate, where as it decreases far away from the plate with the effect of magnetic The values of the Skin friction coefficients for the final field is observed. steady flow are shown in Table1. It is noticed that due to TABLE.1 VALUES OF THE SKIN FRICTION COEFFICIENT C f Re1/ 2 x FOR VARIOUS VALUES OF K AND n WHEN 1 . K /n 0 1/2 0 -1.0043 -1.0043 1 -1.3952 -1.24005 2 -1.6635 -1.4532 4 -2.0092 -1.8105 ACKNOWLEDGEMENT I thank you for referees, their valuable suggestions for making my paper more lucid REFERENCESS Anjalidevi, S.P and Ganga, B (2010). Dissipation effects on MHD nonlinear flow and heat Magyari E. and Keller B. (2000). Exact transfer past a porous medium with prescribed Solutions for Self – Similar Boundary – Layer heat flux. Journal of Applied Fluid Mechanics Flows induced by Permeable Stretching Vol 13, No 1(5),1-6. Surfaces. European Journal of Mechanics B – fluids Vol 19, 109-122. Ariman T, Turk M A and Sylvester N.D,(1973). Microcontinum fluid mechanics- a review. Int j Nazar, R, Amin, N, Filip, D and Pop I(2004). Eng sci Vol 11, 905-930. Stretching point flow of a Micropolar Fluid towards a stretching sheet. Int . J. non-Linear Crane, L.J (1970). Flow Past a stretching Plane. Mech Vol 39, 1227-1235. Journal of Applied Mathematics and Physics (ZAMP) Vol 21, 645-647. Noor, A (1992). Heat transfer from a stretching sheet. Int J Heat and Mass Transfer.Vol 4, 1128- Eringen, A. C (1964).Simple micropolar fluids. 1131. Int. J. Engng. Sci.Vol 2, 205-217. Rajeshwari V and Nath G,(1992). Unsteady Eringen, A. C (1966). Theory of micropolar flow over a stretching surface in a rotating fluid. fluids. J. Math. Mech Vol 16, 1-18. Int j Eng sci Vol 30, No 6, 747-756. Gorla, R. S. R (1983). Micropolar boundary Roslinda Nazr, Anuar Ishak, and Ioan layer flow at stagnation point on a moving wall. Pop,(2008).Unsteady Boundary Layer Flow Int. J. Engng. Sci Vol 21, 25-33. Over a Stretching Sheet in a Micropolar Fluid. International Journal of Mathematical, Physical Guram G. S. and Smith, A. C. (1980).Stagnation and Engineering Sciences Vol 2(3), 161-165. flows of micropolar fluids with strong and weak interaction. Comp. Maths. With Appls Vol 6,1980 P.213-233. Sharma, P.R and Singh, G (2009).Effect of variable thermal conductivity and heat source/ Ishak, A. Nazar, R. and Pop, I. (2008). Heat sink on MHD flow near a stagnation point on transfer over a stretching surface with Variable linearly stretching sheet.Journal of Applied surface heat flux in micropolar fluids. Phys. Lett: Fluid Mechanics Vol 2, No 1(3), 13-21. A Vol 372, 559- 561. Sriramulu, A, Kishan, N and Anadarao, J Magyari E. and Keller B. (1999). Heat and (2001). Steady flow and heat transfer of a Mass Transfer in the Boundary Layers on an viscous incompressible fluid flow through Exponentially Stretching Continuous Surface. porous medium over a stretching sheet. Journal Journal of Physics D: Applied Physics Vol 32, of Energy, Heat and Mass Transfer Vol 23, 483- 577-586. 495. 1 0.9 0.8 0.7 0.6 0.5 K= 0, 1, 2, 3 0.4 f 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 Fig.1. Velocity distribution of initial flow ( 0) and early unsteady flow ( 0 1) for various K with n 0 1 0.9 0.8 0.7 0.6 K= 0, 1, 2, 4 f 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 5 6 7 8 9 10 Fig.2.Velocity distribution of initial flow ( 0) and early unsteady flow ( 0 1) for various K with n 1/ 2 1 0.9 0.8 0.7 0.6 0.5 K= 0, 1, 2, 4 0.4 f 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 Fig.3. Velocity distribution of final steady- state flow ( 1) for various K with n 0 1 0.9 0.8 0.7 0.6 0.5 K= 0, 1, 2, 3 f 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 Fig.4. Velocity distribution of final steady-state flow ( 1) for various K with n 1/ 2 1 — Unsteady-state flow 0 1 --- Final steady- state flow 1 0.9 0.8 0.7 0.6 0,0.2,0.4,0.6,0.8,1 f , 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 Fig.5. Velocity distribution of fully developed unsteady flow for K=1 when n 0 1 0.9 — Unsteady-state flow 0 1 0.8 ….. Final steady – state flow 1 0.7 0.6 0, 0.2, 0.4, 0.6, 0.8,1 f , 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 Fig.6. Velocity distribution of fully developed unsteady flow for K=1 when n 1/ 2 1.2 1 0.8 f ( ) M 0,5,10 0.6 0.4 0.2 0 0 2 4 6 8 10 12 Fig.7.Velocity distribution of final steady-state flow ( 1 ) for various M with n 0 and K=1 0.14 0.12 K=1, 2, 3 0.1 0.08 0.06 g 0.04 0.02 0 0 1 2 3 4 5 6 7 8 9 10 Fig.8.Micro rotation distribution of final steady – state flow ( 1) for various K when n 0 0.45 0.4 0.35 0.3 0.25 0.2 K=1, 2, 3 g 0.15 0.1 0.05 0 0 1 2 3 4 5 6 7 8 9 10 Fig.9. Micro rotation distribution of final steady- state flow ( 1) for various K when n 1/ 2 0.1 Unsteady state flow ( 0 1) 0.09 Final steady-state flow ( 1) 0.08 0.07 0.1, 0.2, 0.4, 0.6, 0.8,1 0.06 g , 0.05 0.04 0.03 0.02 0.01 0 0 1 2 3 4 5 6 7 8 9 10 Fig.10.Micro rotation distribution of fully developed unsteady flow for n 0 and K=1 0.3 0.25 0.2 g 0.15 K=1, 2, 4 0.1 0.05 0 0 1 2 3 4 5 6 7 8 9 10 Fig.11. Micro rotation distribution of early unsteady flow ( 0 1) for various K with n 1/ 2 — Unsteady- state flow 0 1 0.45 0.4 ….. Final steady-state flow 1 0.35 0.3 0.25 g , 0,0.2,0.4,0.6,0.8,1 0.2 0.15 0.1 0.05 0 0 1 2 3 4 5 6 7 8 9 10 Fig.12.Microrotation distribution of fully developed unsteady flow for n 1/ 2 and K 1 0.09 0.08 M 0,5,10 0.07 0.06 g ( ) 0.05 0.04 0.03 0.02 0.01 0 0 2 4 6 8 10 12 Fig.13. Micro rotation distribution of final steady-state flow ( 1 ) for various M when K=1, n 0 . 0 0.2 0.4 0.6 0.8 1 1 -1 -3 C f Re1/ 2 x -5 K= 0, 1, 2, 4 Final steady- state flow -7 -9 -11 Fig.14. Variation with of the skin friction coefficient for various K with n 0 0 0.2 0.4 0.6 0.8 1 1 -1 -3 1/ 2 C f Re x -5 K= 0, 1, 2, 4 Final steady- state flow -7 -9 -11 Fig.15. Variation with of the skin friction coefficient for various K with n 1/ 2