Applied Mathematics, 3, 4, 864-875 http://dx.doi.org/.436/am.3.469 Published Online June 3 (http://.scirp.org/journal/am) Soret-Dufour Effects on the MHD Flo and Heat Transfer of Microrotation Fluid over a Nonlinear Stretching Plate in the Presence of Suction Md Abdullah Al Mahbub, Nasrin Jahan Nasu, Shomi Aktar, Zillur Rahman Department of Mathematics, Comilla University, Comilla, Bangladesh Department of Accounting and Information System, University of Chittagong, Chittagong, Bangladesh Email: dipmahbub3@yahoo.com Received March 8, 3; revised April 9, 3; accepted May 4, 3 Copyright 3 Md Abdullah Al Mahbub et al. This is an open access article distributed under the Creative Commons Attribution License, hich permits unrestricted use, distribution, and reproduction in any medium, provided the original ork is properly cited. ABSTRACT In this ork, the Micropolar fluid flo and heat and mass transfer past a horizontal nonlinear stretching sheet through porous medium is studied including the Soret-Dufour effect in the presence of suction. A uniform magnetic field is applied transversely to the direction of the flo. The governing differential equations of the problem have been transformed into a system of non-dimensional differential equations hich are solved numerically by Nachtsheim-Sigert iteration technique along ith the sixth order Runge-Kutta integration scheme. The velocity, microrotation, temperature and concentration profiles are presented for different parameters. The present problem finds significant applications in hydromagnetic control of conducting polymeric sheets, magnetic materials processing, etc. Keyords: Heat Transfer; Micropolar Fluid; Porous Media; Stretching Sheet; Soret Number; Dufour Number. Introduction The natural convection processes involving the combined mechanism of heat and mass transfer are encountered in many natural and industrial transport processes such as hot rolling, ire draing, spinning of filaments, metal extrusion, crystal groing, continuous casting, glass fiber production, paper production, and polymer processing, etc. Ostrach [] the initiator of the study of convection flo, made a technical note on the similarity solution of transient free convection flo past a semi infinite vertical plate by an integral method. Goody [] considered a neutral fluid. Sakiadis [3] analyzed the boundary layer flo over a solid surface moving ith a constant velocity. This boundary layer flo situation is quite different from the classical Blasiuss problem of boundary flo over a semi-infinite flat plate due to entrainment of ambient fluid. Micropolar fluids, distinctly non-netonian in nature, are referred to those that contain micro-constituents belonging to a class of complex fluids ith nonsymmetrical stress tensor. These fluids respond to micro-rotational motions and spin inertia, and therefore can support couple stress and distributed body torque hich are not achievable ith the classical Navier-Stokes equations or the viscoelastic flo models. The Micropolar fluid models are developed to make an analysis of the flo characteristics of physiological fluids (blood containing corpuscles), colloidal suspensions, paints, liquid crystal suspensions, concentrated silica particle suspensions, oils containing very fine suspensions, industrial contaminants containing toxic chemicals, lubricants, organic/inorganic hybrid nano-composites and clay hich are fabricated by melt intercalation etc. Eringen [4] first designed the study on micropolar fluid making an analysis on the theory of micropolar fluids hich provided a mathematical model for non-netonian behavior. Crane [5] noted that usually the sheet is assumed to be inextensible, but situations may arise in the polymer industry in hich it is necessary to deal ith a stretching plastic sheet. For examples, materials manufactured by aerodynamic extrusion processes and heat-treated materials traveling beteen a feed roll and a ind-up roll or on a conveyor belt possess the characteristics of a moving continuous stretching surface. Moreover lots of metallurgical processes occupy the system of cooling of continuous strips or filaments by draing them through a quiescent fluid and that in the process of draing, these strips are sometimes stretched. Copyright 3 SciRes.
M. A. A. MAHBUB ET AL. 865 An important matter is that the final product depends to a great extent on the rate of cooling. By draing such strips in an electrically conducting fluid subjected to a magnetic field, the rate of cooling can be controlled and a final product of desired characteristics can be achieved. The study of heat and mass transfer is necessary for determining the quality of the final product. Sparro [6] explained a parameter named Rosseland approximation to describe the radiation heat flux in the energy equation in his book. The boundary layer models for steady or unsteady micropolar fluids in various geometries (stationary or moving surface, linear or nonlinear stretching surface etc.) ith/or ithout heat transfer considering various flo conditions (no slip or slip, suction/injection at the surface) and thermal boundary conditions (constant/variable surface temperature or heat flux) have extensively been studied by numerous researchers [7-6]. Moreover, the thermal-diffusion (Soret) effect, for instance, has been utilized for isotope separation, and in mixtures beteen gases ith very light molecular eight (Hz, He) and of medium molecular eight (Nz, air) the diffusion-thermo (Dufour) effect as found to be of a considerable magnitude such that it cannot be ignored, described by Eckert and Drake [7] in their book. Recently plenty of investigators [8-] are getting interest ork on Soret-Dufour effects. From the above referenced ork and the numerous possible industrial applications of the problem, it is of paramount interest in this study in order to clarify the parametric behavior of magneto-hydrodynamic flo of free convection of a micropolar fluid over a nonlinear stretching sheet in the presence of dynamic effects of suction, thermal-diffusion and diffusion-thermo.. Mathematical Model We consider the isothermal, steady, laminar, hydromagnetic free convection flo of an incompressible micropolar fluid floing past a nonlinear stretching sheet coinciding ith the plane y, the flo being confined in the region y. The flo under consideration is also subjected to a strong transverse magnetic field B ith a constant intensity along the y-axis. To equal and opposite forces are introduced along the x-axis so that the surface is stretched keeping the origin fixed. The flo configurations and the coordinate system are shon in Figure. We assume that the velocity of a point on a sheet is proportional to its distance from the slit. We assume that all the fluid properties are isotropic and constant. Under the usual boundary layer and Boussinesq approximations, the governing equations for the problem under consideration can be ritten as follos: Figure. Flo configuration and coordinate system. u v x y u u u v x y S S u B S N g T T u u y y kp () N N S N S u u v N x y j y j y T T u v x y S c c k y c c T u u DmK T C p y p s p C C C C DK T u v Dm x y T m T y m y In Equation () the Darcian porous drag force term is S u defined by the term,, hich is linear in k p terms of the translational velocity, u. With S, the micropolar effects disappears and this term reduces to the u conventional Netonian Darcy drag force i.e.. k The micro-rotation component, N, is coupled to the linear momentum Equation () via the angular velocity gradient S N term,. Very strong coupling exists beteen the y translational velocity components, u and v, in Equation N (3) via the convective acceleration terms, u and x () (3) (4) (5) p Copyright 3 SciRes.
866 M. A. A. MAHBUB ET AL. N v. Furthermore, there is a second coupling term link- y ing the angular velocity ith the x-direction velocity gra- S u dient, in Equation (3), N. The microrota- j y tion viscosity (or spin-gradient viscosity) S is defined S by S j (Rahman [5]). We note that in the S u viscous shear diffusion term,, the Neto- y nian kinematic viscosity is no supplemented by the Eringen micropolar vortex viscosity, S. In the present ork, e assume that the micro-inertia per unit mass j is a constant. Also, positive or negative n indicate the acceleration and deceleration of the sheet from the extruded slit respectively. Here uv, are the fluid velocity components in the x-, y-directions respectively, N is the microrotation, T is the temperature, is the kinematic viscosity, is the fluid density, is the electric conductivity, g is the acceleration due to gravity, is the volumetric coefficient of thermal expansion, B is the uniform magnetic field strength, k p is the Darcy permeability of porous medium, j is the Microinertia per unit mass, is the thermal conductivity of the fluid, c p the specific heat at constant pressure, D m is the chemical molecular diffusivity, K T is the Thermophoretic constant, T m is the Mean fluid temperature and c s is the Concentration susceptibility. The appropriate boundary conditions suggested by the physical conditions are: ) on the plate surface at y : n u u U Bx, vv x, N S, y (6) T TT Ax, C CC Dx ) matching ith the quiescent free stream as y : uu, N, T T, C C (7) here the subscripts and refer to the all and boundary layer edge, respectively. The relationship beteen the microrotation function N and the surface shear u is chosen for investigating the effect of different y surface conditions for the microrotation of the micropolar fluid elements. The conditions are generally of importance in micropolar boundary layer analysis. When microrotation parameter S, e obtain N hich represents no-spin condition i.e. the microelements in a concentrated particle flo-close to the all are not able to rotate (Rahman [5]). Finally AD, and are the constants... Similarity Solutions The partial differential Equations () to (5) are transformed into non-dimensional form by mean of folloing dimensionless variables n B n n y x, u Bx f, n n n v B x f f, n (8) 3 n B n T T N B x h,, T T C C C C Implementing Equation (8) into Equations () to (5) produces the folloing ordinary differential equations: f ff M f n Da nf h Gr n n (9) 3 n h h f hf hf () n n f f Pr n Ec f f Du n Da () Scf Sc f ScSr () n and corresponding boundary conditions are reduce to: f f, f, h Sf,, at (3) f, h,, as here the primes denote differentiation ith respect to S (non-dimensional y-coordinate) and is the vortex g T T x viscosity parameter, Gr is the local U Bx Grashof number, M is the local magnetic U B k parameter and B is the magnetic field, p Da x x Copyright 3 SciRes.
M. A. A. MAHBUB ET AL. 867 ju is the Darcy number, is the micro-inertia x c density parameter, p Pr is the Prandtl number, U Ec is the Eckert number, c T T p DmKT C C Du c c T T s p is the Schmidt number, is the Dufour number, DmKT T T Sr T C C m n number and f vxb x is the suction parameter. n.. Skin Friction, Rate of Heat and Mass Transfer Sc D m is the Soret The parameters of engineering interest for the present problem are the skin friction coefficient c f, plate couple stress M, local Nusselt number Nu and Sherood number Sh hich indicate physically the all shear stress, couple stress, the rate of heat transfer and the local surface mass flux respectively. The dimensionless skin-friction coefficient, Couple stress, Nusselt number and Sherood number for impulsively started plate are given by n Cf f (4) U Re S j N M n h (5) U y y Nu Sh x T T T y x C C C y y y Re n Re n (6) (7) here Re is the Reynolds number. And hence the values proportional to the skin-friction coefficient, couple stress, Nusselt number and Sherood number are f, h, and respectively. 3. Numerical Computation The numerical solutions of the non-linear differential Equations (9) to () under the boundary conditions (3) have been performed by applying a shooting method namely Nachtsheim and Sigert [] iteration technique (guessing the missing values) along ith sixth order Runge-Kutta iteration scheme. We have chosen a step size. to satisfy the convergence criterion of 6 in all cases. The value of has been found to each iteration loop by. The maximum value of to each group of parameters, n, M, Gr, Da,,Pr, Ec, S, Sc,, Du, Sr and f has been determined hen the values of the unknon boundary conditions at not change to successful loop ith error less than 6. In order to verify the effects of the step size, e have run the code for our model ith three different step sizes as Δη =., Δη =.5 and Δη =., and in each case e have found excellent agreement among them shon in Figures -5....9.8.7 f'.6 Curves.5.4.3.. -.5 -.6 3 4 5. -. -. Figure. Distribution of velocity profiles for Δη. h Curves -.3 -.4 3 4 5 Figure 3. Distribution of microrotation profiles for Δη. Copyright 3 SciRes.
868 M. A. A. MAHBUB ET AL..9.8.7.6.5.4.3.. Curves 3 4 5 Figure 4. Distribution of temperature profiles for Δη..9.8.7.6.5.4.3.. Curves 3 4 5 Figure 5. Distribution of concentration profiles for Δη. 4. Results and Discussions For the purpose of discussing the results of the flo field represented in the Figure, the numerical calculations are presented in the form of non-dimensional velocity, microrotation, temperature and concentration profiles. The values of buoyancy parameter Gr is taken to be both positive to represent cooling of the plate. The parameters are chosen arbitrarily here Pr =.7 corresponds physically to air at C, Pr =. corresponds to the electrolyte solution such as salt ater and Pr = 7. corresponds to ater, and Sc.,.6 and. corresponds to hydrogen, ater vapor and methanol respectively at 5 C and atmosphere. The values of Dufour and Soret numbers are chosen in such a ay their production is constant provided that the meat temperature T m is kept constant as ell. Due to free convection problem positive large values of Gr is chosen. The value of M.5, Da., Sc.5, Pr.7 and.5. Hoever, numerical computations have been carried out for different values of the vortex viscosity parameter, surface nonlinearity parameter n, Eckert number Ec, constant parameter, Dufour number Du, Soret number Sr and suction parameter f. The numerical results for the velocity, microrotation, temperature and concentration profiles are displayed in Figures 6-33. Figure 6 shos the effect of vortex viscosity parameter (.,.5,,.5 are chosen) on the velocity profiles. From here e see that velocity profiles decrease ith the increase of. Figure 7 demonstrates the effect of on the microrotation profiles. From this figure it is seen that microrotation increases very evidently ith the increase of the vortex viscosity parameter. It is also understood that as the vortex viscosity increases the rotation of the micropolar constituents gets induced in most part of the boundary layer here kinematic viscosity dominates the flo. From Figure 8 it is found that the temperature profiles increase for the increase of. The effect of vortex viscosity parameter on the concentration profile is not so noteorthy displayed in Figure 9. The effects of the surface nonlinearity constant n are characterized in the Figures -3. At the beginning the velocity profiles decrease ith the increase of the value of nn,, 3, 4 but far aay from the plate they increase after.76 displayed in Figure. Figure expresses that the microrotation profiles at the beginning increase extensively but at a distance from the plate they overlap and start to decrease very sloly. Figures and 3 enlighten the temperature and the concentration profiles for the increasing influence of the parameter n respectively. Figures 4-7 exhibit the velocity, microrotation, temperature and concentration profiles for the different values of the Eckert number Ec (.3,.,.5 and.). Figure 4 demonstrates that the effect of the Ec on velocity profiles very significant. We observe that velocity increases rapidly ith increasing the value of Ec. From the Figure 5 e notice that microrotation profiles decreases ith the increase of the value of Ec. Figure 6 presents the increasing effect of Ec on the temperature profiles. The concentration profiles decrease ith the increase of the value of Ec illustrated in Figure 7. Figures 8- represent the influence of the constant parameter for the values,,, 5. All the profiles except microrotation profiles decrease ith the increase of. The effects of are very significant smooth on the distributions. The microrotation profiles increase ith the increase of the value of. It is observed from the Figure that ith the increase Copyright 3 SciRes.
M. A. A. MAHBUB ET AL. 869.9.9.8.8.7.7 f'.6.5.4.,.5,.,.5.6.5.4.,.5,.,.5.3.3.... 3 4 5 Figure 6. Distribution of velocity profiles for. 3 4 5 Figure 9. Distribution of concentration profiles for..6.5..9 h.4.3.,.5,.,.5 f'.8.7.6.5 n=,, 3, 4..4.3.. 3 4 5 Figure 7. Distribution of microrotation profiles for.. 3 4 5 Figure. Distribution of velocity profiles for n...9.8.7.6.5.4.3.,.5,.,.5 h -. -. -.3 -.4 n=,, 3, 4. -.5. -.6 3 4 5 Figure 8. Distribution of temperature profiles for. 3 4 5 Figure. Distribution of microrotation profiles for n. Copyright 3 SciRes.
87 M. A. A. MAHBUB ET AL...9.8 -..7.6.5 n=,, 3, 4 h -.4 -.6 -.8 Ec=.3,.,.5,..4 -.3.. 3 4 5 Figure. Distribution of temperature profiles for n. -. -.4 -.6 4 Figure 5. Distribution of microrotation profiles for Ec..9.8.7.6.5 n=,, 3, 4.8.6 Ec=.3,.,.5,..4.3.4... 3 4 5 Figure 3. Distribution of concentration profiles for n..8 4 Figure 6. Distribution of temperature profiles for Ec..6.4.8 f'..8 Ec=.3,.,.5,..6.4 Ec=.3,.,.5,..6.4.. 4 Figure 4. Distribution of velocity profiles for Ec. 4 6 Figure 7. Distribution of concentration profiles for Ec. Copyright 3 SciRes.
M. A. A. MAHBUB ET AL. 87.3...9.8 f'.7,,, 5.6.5.4.3.. 4 Figure 8. Distribution of velocity profiles for... -. -. h -.3,,, 5 -.4 -.5 -.6 -.7 -.8 -.9 4 Figure 9. Distribution of microrotation profiles for..9.8.7.6.5.4.3..,,, 5 4 Figure. Distribution of temperature profiles for..9.8.7.6.5.4.3..,,, 5 4 Figure. Distribution of concentration profiles for. f'.4.3...9.8.7.6.5.4.3.. Du =.5,,, 3 4 Figure. Distribution of velocity profiles for Du. of the value of Du the velocity profiles occur higher. The effect of Du on the microrotation profiles is insignificant illustrated in Figure 3. From Figure 4, it is noticed that Du has remarkable effect on temperature profiles; quantitatively hen. Du increases from.5 to and there is 3.8% increase in the temperature value, hereas the corresponding increase is 9.5%, hen Du increases from to 3. The Dufour number has a falling effect on the concentration field shon in Figure 5. Quantitatively hen. and Du increases from.5 to, there is 5.56% decrease in the concentration value, hereas the corresponding decrease is 6.67% hen Du increases from to 3. Figures 6-9 display the effects of the Soret number Sr on the velocity, microrotation, temperature and concentration profiles respectively. It is observed that Sr has very negligible effect on the velocity, microrotation and temperature profiles. Figure 9 reveals that the Soret Copyright 3 SciRes.
87 M. A. A. MAHBUB ET AL. h.3.. -. -. -.3 -.4 -.5 -.6 -.7 -.8 -.9 - -. Du =.5,,, 3 -. 4 6 Figure 3. Distribution of microrotation profiles for Du. f'..9.8.7.6.5.4.3.. Sr=.5,,, 3 4 6 Figure 6. Distribution of velocity profiles for Sr..9.8..7 -..6.5.4 Du =.5,,, 3 h -. -.3 Sr=.5,,, 3.3 -.4. -.5. 4 Figure 4. Distribution of temperature profiles for Du..9.8.7 -.6 4 6 8 Figure 7. Distribution of microrotation profiles for Sr..9.8.7.6.5 Du =.5,,, 3.6.5 Sr=.5,,, 3.4.4.3.3.... 4 Figure 5. Distribution of concentration profiles for Du. 4 Figure 8. Distribution of temperature profiles for Sr. Copyright 3 SciRes.
M. A. A. MAHBUB ET AL. 873 number Sr influences the concentration profiles to a great extent. Quantitatively hen. and Du increases from.5 to, there is 3.8% decrease in the concentration value, hereas the corresponding decrease is 3.7% hen Du increases from to 3. Figure 3 displays that the suction parameter f has strong effect on the velocity profiles. With the increase of the value of f the velocity profiles decrease. Elaborately hen. and Du decreases from to.5, there is 76.79% decrease in the concentration value, hereas the corresponding decrease is.47% hen Du increases from to 3. It is observed that, hen suction f increases, the microrotation increase monotonically seen in Figure 3. These Figures 3 and 33 indicate that temperature as ell as concentration profiles decrease ith the increase of suction velocity or mass transfer parameter frequently. h. -. -.4 -.6 f=,.5,, 3 4 Figure 3. Distribution of microrotation profiles for f..9.8.9.8.7.6.5 Sr=.5,,, 3.7.6.5 f=,.5,, 3.4.4.3.3.... 4 Figure 9. Distribution of concentration profiles for Sr. 4 Figure 3. Distribution of temperature profiles for f...9.8.8 f' f=,.5,, 3.6.7.6.5 f=,.5,, 3.4.4.3... 4 Figure 3. Distribution of velocity profiles for f. 4 Figure 33. Distribution of concentration profiles for f. Copyright 3 SciRes.
874 M. A. A. MAHBUB ET AL. Finally, Finally, the effects of various parameters on the skin friction C f, couple stress M, local Nusselt number Nu and local Sherood number Sh are shon in the Tables -. 5. Conclusions In the present paper, Soret Dufour effect on the boundary layer flo and heat transfer of microrotation fluid over a nonlinear stretching plate in the presence of suction has been studied. The governing momentum and energy equations ere transformed to a set of non linear ordinary differential equations by employing the appropriate similarity transformations and solve numerically for various combinations of problem parameters. The effects of the vortex viscosity parameter, surface nonlinearity parameter n, Eckert number Ec, constant parameter, Dufour number Du, Soret number Sr and suction parameter f are investigated through the use of graphs. Table. C f, M, Nu and Sh for different values of the parameters and n. Parameters C f M Nu Sh =..76654.6785573.98754.4347764.5.94367.74868.8969659.4495..475.865383.865346.4443.5.758.843893.837798.394873 n =..6567.659867.65.4556 3..39368.758.894876.65374 4..8769.97969.857597.7768 Table. C f, M, Nu and Sh for different values of the parameter Ec,, Du, Sr and f. Parameters C f M Nu Sh Ec =..377933.89.875.869.5.879.78963.7748798.7767. 3.46737 3.49488.57758.98357 =..8534.739.75796.685..54758.4357.978664.86748 5..938.88778.894767.4954 Du =.57844.396758.5834.39687.934.7744343.65564.379 3.436488.4699.7946.54848 Sr =.7836.945495.434.884983.94364.58.58953.6 3.959358.37859.8467657.7597 f =.438645.7336.69.997766.6648.559658.53565.35838 3.894534.9339.48893.564 From the present study the folloing conclusions are made: ) The effect of vortex viscosity parameter on velocity and microrotation is prominent; ) Nonlinearity of the stretching surface n is effective on the boundary layer flo; 3) Effect of Eckert number Ec is uniform; 4) Boundary layer groth can be controlled by using constant parameter ; 5) The Dufour Du effect is significant; 6) The Soret number Sr plays a role on concentration; 7) The effect of suction parameter f is dominating on the velocity, microrotation, temperature and concentration profiles. So, using suction boundary layer groth can be stabilized. REFERENCES [] S. Ostrach, An Analysis of Laminar Free-Convection Flo and Heat Transfer about a Flat Plate Parallel to the Direction of the Generating Body Force, Technical Note, naca Report, Washington DC, 95. http://naca.central.cranfield.ac.uk/report.php?nid=5 [] R. M. Goody, The Influence of Radiative Transfer on Cellular Convection, Journal of Fluid Mechanics, Vol., No. 4, 956, pp. 44-435. doi:.7/s5663 [3] B. C. Sakiadis, Boundary-Layer Behavior on Continuous Solid Surfaces: I. Boundary-Layer Equations for To-Dimensional and Axisymmetric Flo, AIChE Journal, Vol. 7, No., 96, pp. 6-8. doi:./aic.6978 [4] A. C. Eringen, Theory of Micropolar Fluids, Journal of Mathematics and Mechanics, Vol. 6, No., 966, pp. -8. [5] L. J. Crane, Flo past a Stretching Sheet, Zeitschrift für Angeandte Mathematik und Physik (ZP), Vol., No. 4, 97, pp. 645-647. doi:.7/bf587695 [6] E. M. Sparro, Radiation Heat Transfer, Augmented Edition, Hemisphere Publishing Corp., Washington DC, 978. [7] O. Aydin and I. Pop, Natural Convection from a Discrete Heater in Enclosures Filled ith a Micropolar Fluid, International Journal of Engineering Science, Vol. 43, No. 9-, 5, pp. 49-48. doi:.6/j.ijengsci.5.6.5 [8] M.-I. Char and C.-L. Chang, Effect of Wall Conduction on Natural Convection Flo of Micropolar Fluids along a Flat Plate, International Journal of Heat and Mass Transfer, Vol. 4, No. 5, 997, pp. 364-365. doi:.6/s7-93(97)6-9 [9] H. A. M. El-Arabay, Effect of Suction/Injection on the Flo of a Micropolar Fluid past a Continuously Moving Plate in the Presence of Radiation, International Journal of Heat and Mass Transfer, Vol. 46, No. 8, 3, pp. 47-477. doi:.6/s7-93()3-4 Copyright 3 SciRes.
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