HYDROMAGNETIC BOUNDARY LAYER MICROPOLAR FLUID FLOW OVER A STRETCHING SURFACE EMBEDDED IN A NON-DARCIAN POROUS MEDIUM WITH RADIATION

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1 HYDROMAGNETIC BOUNDARY LAYER MICROPOLAR FLUID FLOW OVER A STRETCHING SURFACE EMBEDDED IN A NON-DARCIAN POROUS MEDIUM WITH RADIATION MOSTAFA A. A. MAHMOUD, MAHMOUD ABD-ELATY MAHMOUD, AND SHIMAA E. WAHEED Received 3 January 2006; Accepted 6 July 2006 We ave studied te effects of radiation on te boundary layer flow and eat transfer of an electrically conducting micropolar fluid over a continuously moving stretcing surface embedded in a non-darcian porous medium wit a uniform magnetic field. Te transformed coupled nonlinear ordinary differential equations are solved numerically. Te velocity, te angular velocity, and te temperature are sown grapically. Te numerical values of te skin friction coefficient, te wall couple stress, and te wall eat transfer rate are computed and discussed for various values of parameters. Copyrigt 2006 Mostafa A. A. Mamoud et al. Tis is an open access article distributed under te Creative Commons Attribution License, wic permits unrestricted use, distribution, and reproduction in any medium, provided te original work is properly cited.. Introduction Eringen [7] introduced te concept of micropolar fluid in an attempt to explain te beavior of a certain fluid containing polymeric additives and naturally occurring fluids suc as te penomenon of te flow of colloidal fluids, real fluid wit suspensions, liquid crystals, and animal blood. Te teory of termomicropolar fluids as been developed by Eringen [8], takinginto account te effect of microelementsoffluids on bot te kinematics and conduction of eat. Micropolar fluid teory as been used to describe in detail te effect of dirt in journal bearing, see [2,, 4, 22]. Te review articles by Ariman et al. [4, 5] describe some of te various applications wic ave been explored. Boundary layer on continuous surface is an important type of flow occurring in a number of tecnical problems. Examples may be found in continuous casting, glass fiber production, metal extrusion, ot rolling, textiles, and wire drawing (see [3, 20]). Sakiadis [7] initiated te teoretical study of boundary layer on a continuous semi-infinite seet moving steadily troug an oterwise quiescent fluid environment, wereas its eat transfer aspect was studied by Tsou et al. [23]. Karwe and Jaluria [9] carried out a numerical study of te transport arising due to te movement of a continuous eated body. Matematical Problems in Engineering Volume 2006, Article ID 39392, Pages 0 DOI 0.55/MPE/2006/39392

2 2 MHD boundary layer micropolar fluid in a porous medium Te boundary layer flow of a micropolar fluid past a semi-infinite plate as been studied by Peddieson and Mcnitt [3] wereas a similarity solution for boundary layer flow near stagnation point was presented by Ebert [6]. Te boundary layer flow of micropolar fluids past a semi-infinite plate was studied by Amadi [], taking into account te gyration vector normal to te xy-plane and te microinertia effects. Flow and eat transfer of a micropolar fluid past a continuously moving plate are studied by Takar and Soundalgekar [8, 2]. By drawing te continuous strips troug a quiescent electrically conducting fluid subject to a magnetic field, te rate of cooling can be controlled and final product of desired caracteristics can be acieved. Kelson and Farrell [0] studied micropolar flow over a porous stretcing seet wit strong suction or injection. Flow and eat transfer troug porous media ave several practical engineering applications suc as transpiration cooling, packed bed cemical reactors, geotermal systems, crude oil extraction, ground water ydrology, and building termal insulation. We know tat te radiation effect is important under many nonisotermal situations. If te entire system involving te polymer extrusion process is placed in a termally controlled environment, ten radiation could become important. Te knowledge of radiation eat transfer in te system can peraps lead to a desired product wit sougt caracteristic. Recently, te effects of radiation on te flow and eat transfer of a micropolar fluid past a continuously moving plate ave been studied by many autors, see [2, 5]. Raptis [6] studied te boundary layer flow of a micropolar fluid troug non-darcian porous medium. Te problem of ydromagnetic boundary layer micropolar fluid flow over a continuously moving stretcing surface troug a fluid saturated porous medium wit radiation is terefore an important one. It is now proposed to study te flow and eat transfer of an electrically conducting micropolar fluid on a continuously moving plate embedded in a non-darcian porous medium in te presence of a uniform magnetic field and radiation. 2. Matematical formulation Consider a steady, two-dimensional laminar flow of an incompressible, electrically conducting micropolar fluid over a continuously moving stretcing surface embedded in a non-darcian porous medium wic issues from a tin slit. Te x-axis is taken along te stretcing surface in te direction of te motion and y-axis is perpendicular to it. We assume tat te velocity is proportional to its distance from te slit. A uniform magnetic field B 0 is imposed along y-axis. Ten under te usual boundary layer approximations, te flow and eat transfer of a micropolar fluid in porous medium wit te non-darcian effects included are governed by te following equations [6]. (i) Te equation of continuity is (ii) Te equation of momentum is u x + v = 0. (2.) y u u x + v u y = ν 2 u y 2 + k N y νϕ k u cϕu2 σb2 0 u. (2.2) ρ

3 Mostafa A. A. Mamoud et al. 3 (iii) Te equation of angular momentum is (vi) Te equation of energy is (v) Te boundary conditions are G 2 N u 2N = 0. (2.3) y2 y u T x + v T y = K 2 T ρc p y 2 q r ρc p y. (2.4) y = 0:u = ax, v = 0, T = T w, N = 0, y : u 0, T T, N 0, (2.5) were ν = (μ + S)/ρ is te apparent kinematic viscosity, μ is te coefficient of dynamic viscosity, S is a constant caracteristic of te fluid, N is te microrotation component, k = S/ρ(> 0) is te coupling constant, G (> 0) is te microrotation constant, ρ is te fluid density, u and v are te components of velocity along x and y directions, respectively, ϕ is te porosity, k is te permeability of te porous medium, c is Forceimer s inertia coefficient, T is te temperature of te fluid in te boundary layer, T is te temperature of te fluid far away from te plate, T w is te temperature of te plate, K is te termal conductivity, C p is te specific eat at constant pressure, σ is te electrical conductivity, B 0 is an external magnetic field, and q r is te radiative eat flux. We now introduce te following transformations: ( ) a /2 η = y, ψ = (aν) /2 xf(η), ν ( a 3 ) /2 N = xg(η), θ = T T, ν T w T u = ψ y, v = ψ x. (2.6) Using te Rosselant approximation [9], we ave ( q r = 4σ ) 0 T 4 3k 0 y, (2.7) were σ 0 is te Stefan-Boltzmann constant and k 0 istemeanabsorptioncoefficient. Substituting expressions in (2.6)-(2.7)into(2.) (2.5), we ave f + ff + Lg ( Da + R ) f ( + α) f 2 = 0, (2.8) Gg (2g + f ) = 0, (2.9) [ 3Fθ +3FP r fθ +4P r ( + rθ) 3 θ +3r( + rθ) 2 θ 2 ] = 0, (2.0)

4 4 MHD boundary layer micropolar fluid in a porous medium were L = k /ν denotes te coupling constant parameter; Da = ϕν/ka denotes te inverse Darcy number; R = (σ 0 B0)/ρa 2 denotes te magnetic parameter; α = cϕx denotes te inertia coefficient parameter; G = G a/ν denotes te microrotation parameter; P r = (νρc p )/k denotes te Prandtl number; F = (ρc p k 0 ν)/(4σ 0 T ) 3 denotes te radiation parameter; and r = (T w T )/T is te relative difference between te temperature of te surface and te temperature far away from te surface. Te corresponding boundary conditions are f (0) = 0, f (0) =, θ(0) =, g(0) = 0, f ( ) = 0, θ( ) = 0, g( ) = 0. (2.) In te above equations, a prime denotes differentiation wit respect to η. In te case of L = 0andα = 0, (2.8) togeter wit te boundary conditions f (0) = 0, f (0) =, and f ( ) = 0 as an exact solution in te form f (η) = ( ) e ( +Da +R)η. (2.2) +D a + R Te sear stress at te surface of te plate is given by [9], τ w = (μ + S) du dy + SN. (2.3) y=0 Te skin-friction coefficient is given by ( ) 2τ c f = ρu 2 = 2 R ex f (0), (2.4) y=0 were R ex = Ux/ν is te local Reynolds number. Te couple stress at te wall is given by te following: ( ) N m w = G y y=0 ( ) G U = R ex x 2 g (0). (2.5) From te temperature field, we can now study te rate of eat transfer. It is given by ( ) T q w = K. (2.6) y y=0 From (2.6), (2.5) isreducedto q w = K ( T w T ) u 0 2νx θ (0). (2.7) Te numerical values of f (0), θ (0), and g (0) are displayed in Table 3..

5 Mostafa A. A. Mamoud et al. 5 Table 3.. Values of f (0), g (0), and θ (0) wit L = 0., G = 2, α =, and r = 0.3. D a R P r F f (0) g (0) θ (0) Table 3.2. Comparison between analytical and numerical values of f (0) for various values of D a and R wit L = 0andα = 0. D a R Analytical Numerical Solutions and discussion Te system of coupled nonlinear ordinary differential equation (2.8) (2.0) togeter wit te boundary conditions (2.) is solved numerically by using te fourt-order Runge-kutta metod along wit te sooting tecnique. In order to assess te accuracy of te present numerical metod, we ave compared our numerical results obtained for te skin-friction coefficient taking L = 0andα = 0in(2.8) wit tose obtained analytically. Te analytical and numerical values of f (0)forvariousvaluesofDa and R are tabulated in Table 3.2. Te numerical values of f (0) are in good agreement wit te obtained analytical values. We ave considered in some detail te influence of te pysical parameters Da, R, P r, and F on te velocity, microrotation, and temperature distributions wic are sown in Figures Figures3. and 3.2 sow te velocity, angular velocity, and temperature profilesforvariousvaluesoftemagneticparameterr, respectively. Application of a transverse magnetic field normal to te flow direction gives rise to a resistive drag-like force acting in a direction opposite to tat of flow. Tis as a tendency to reduce bot te fluid velocity and angular velocity and increase te fluid temperature. Tis is indicative from te decreases in te fluid velocity f, te angular velocity g and increases in te temperature θ as sown in Figures 3. and 3.2, respectively.

6 6 MHD boundary layer micropolar fluid in a porous medium D a = 2 F = f R = R = 2 g R = 2 R = 2 3 Figure 3.. Velocity and microrotation distributions for various values of R. 4 D a = 2 F = q 0.4 R = R = Figure 3.2. Temperature distributions for various values of R. Figures 3.3 and 3.4 display te influence of te inverse Darcy number Da on te velocity and te temperature profiles, respectively. It is obvious tat te presence of porous medium causes iger restriction to te fluid, wic reduces bot te velocity and te

7 Mostafa A. A. Mamoud et al. 7 R = F = f D a = 0.5 D a = 2 g 2 3 D a = 0.5 D a = 2 Figure 3.3. Velocity and microrotation distributions for various values of D a. 4 R = F = q 0.4 D a = 2 D a = Figure 3.4. Temperature distributions for various values of Da. angular velocity and enanced te temperature. Figures 3.5 and 3.6 depect te influence of te Prandtl number P r and te radiation parameter F on te temperature distributions, respectively. It is observed tat te temperature at fixed values of η decreases

8 8 MHD boundary layer micropolar fluid in a porous medium D a = 2 F = R = q 0.4 P r = Figure 3.5. Temperature distributions for various values of P r. D a = 2 R = q 0.4 F = F = Figure 3.6. Temperature distributions for various values of F. wit an increase in te Prandtl number P r as sown in Figure 3.5. Tis is in agreement wit te pysical fact tat te termal boundary tickness decreases wit te increase of P r. Figure 3.6 displays te variation of temperature for different values of te radiation

9 Mostafa A. A. Mamoud et al. 9 parameter F. We observe tat te temperature decreases as te radiation parameter F increase. Table 3. displays numerical results for te skin friction coefficient, wall couple stress, and te wall eat transfer rate for various values of Da, R, P r,andf. Weobservefrom tis table tat te skin friction coefficient and wall couple stress increase wit te increase of R or Da and te wall eat transfer rate decreases wit te increase of R or Da. Tis is because, as mentioned before, increases in R or Da cause respective decreases in te velocity, te angular velocity, and te temperature, respectively. Tis results in increasing and decreasing te slopes of velocity and temperature, respectively. Tis as te direct effect of increasing te skin-friction coefficient, te wall couple stress and decreasing te rate of wall eat transfer as sown in Table 3.. Also te wall eat transfer rate increases wit te increase of P r. Furtermore, te negative values of te wall temperature gradient, for all values of te parameters, are indicative of te pysical fact tat te eat flows from te surface to te ambient fluid. 4. Conclusions Te problem of ydromagnetic boundary layer flow and eat transfer of an electrically conducting micropolar fluid on a continuously moving stretcing surface embedded in a non-darcian porous medium in te presence of radiation was investigated. Te resulting partial differential equations, wic describe te problem, are transformed into ordinary differential equations by using similarity transformations. Numerical evaluations were performed and grapical results were obtained. It was found tat bot te velocity and te angular velocity are decreased as eiter te inverse Darcy number or te magnetic parameter was increased. Also it is observed tat te temperature decreased as te Prandtl number or te radiation parameter increased. Analysis of te tables sows tat te skin-fricition coefficient is found to increase wit te increase of magnetic parameter or of te inverse Darcy number. Also, we found tat te wall eat transfer rate increased as te Prandtl number or te radiation parameter increased, wile it decreased as te magnetic parameter or te inverse Darcy number increased. References [] G. Amadi, Self-similar solution of imcompressible micropolar boundary layer flow over a semiinfinite plate, International Engineering Science 4 (976), no. 7, [2] S. J. AllenandK. A. Kline, Lubrication teory for micropolar fluids, Applied Mecanics: Transactions of te ASME 38 (97), no. 3, [3] T.Altan,S.O,andH.Gegel,Metal Forming Fundamentals and Applications,AmericanSociety of Metals, Oio, 979. [4] T. Ariman, M. A. Turk, and N. D. Sylvester, Microcontinuum fluid mecanics a review, International Engineering Science (973), no. 8, [5], Applications of microcontinuum fluid mecanics, International Engineering Science 2 (974), no. 4, [6] F. Ebert, A similarity solution for te boundary layer flow of a polar fluid, Te Cemical Engineering Journal 5 (973), no., [7] A. C. Eringen, Teory of micropolar fluids, Matematics and Mecanics 6 (966), 8.

10 0 MHD boundary layer micropolar fluid in a porous medium [8], Teory of termomicro fluids, Matematical Analysis and Applications 38 (972), no. 9, [9] M.V.KarweandY.Jaluria,Termal transport from a eated moving surface, Heat Transfer: Transactions of te ASME 08 (986), no. 4, [0] N. A. Kelson and T. W. Farrell, Micropolar flow over a porous stretcing seet wit strong suction or injection, International Communications in Heat and Mass Transfer 28 (200), no. 4, [] M. M. Konsari, On te self-excited wirl orbits of a journal in a sleeve bearing lubricated wit micropolar fluids, Acta Mecanica 8 (990), no. 3-4, [2] Y. J. Kim and A. G. Fedorov, Transient mixed radiative convection flow of a micropolar fluid past a moving, semi-infinite vertical porous plate, International Heat and Mass Transfer 46 (2003), no. 0, [3] J. Peddieson and R. P. Mcnitt, Boundary layer teory for a micropolar fluid, Recent Advances in Engineering Science 5 (970), [4] J. Prakas and P. Sina, Lubrication teory for micropolar fluids and its application to a journal bearing, International Engineering Science 3 (975), no. 3, [5] A. Raptis, Flow of a micropolar fluid past a continuously moving plate by te presence of radiation, International Heat and Mass Transfer 4 (998), no. 8, [6], Boundary layer flow of a micropolar fluid troug a porous medium, JournalofPorous Media 3 (2000), no., [7] B. C. Sakiadis, Boundary-layer beavior on continuous solid surfaces: I. Boundary-layer equations for two-dimensional and axisymmetric flow, AICE Journal 7 (962), no., [8] V. M. Soundalgekar and H. S. Takar, Flow of micropolar fluid past a continuously moving plate, International Engineering Science 2 (983), no. 8, [9] E. M. Sparrow and R. D. Cess, Radiation Heat Transfer, Hemispere, Wasington, DC, 978. [20] Z. Tadmor and I. Klein, Engineering Principles of Plasticating Extrusion, Polymer Science and Van nostrand Reinold, New York, 970. [2] H. S. Takar and V. M. Soundalgekar, Flow and eat transfer of a micropolar fluid past a continuously moving porous plate, International Engineering Science 23 (985), no. 2, [22] N. Tipei, Lubrication wit micropolar liquids and its application to sort bearings,journaloflubrication Tecnology: Transactions of te ASME 0 (979), no. 3, [23] F. K. Tsou, E. M. Sparrow, and R. J. Goldstein, Flow and eat transfer in te boundary layer on a continuous moving surface, International Heat and Mass Transfer 0 (967), no. 2, Mostafa A. A. Mamoud: Department of Matematics, Faculty of Science, Bena University, Bena 358, Egypt address: mostafabdelameed@yaoo.com Mamoud Abd-elaty Mamoud: Department of Matematics, Faculty of Science, Bena University, Bena 358, Egypt address: m abdelaty2000@yaoo.com.uk Simaa E. Waeed: Department of Matematics, Faculty of Science, Bena University, Bena 358, Egypt address: simaa ezat@yaoo.com

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