Chemical Reaction and Viscous Dissipation Effects on an Unsteady MHD Flow of Heat and Mass Transfer along a Porous Flat Plate

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1 International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 2 Issue 2 December 24 PP ISSN X (Print) & ISSN (Online) Chemical Reaction and Viscous Dissipation Effects on an Unstead MHD Flow of Heat and Mass Transfer along a Porous Flat Plate L. Anand Babu 2 Department of Mathematics O.U. Campus Hderabad Telangana India Venkat Redd.V Department of Mathematics Kamala Nehru Poltechnic Hderabad Telangana India venkatreddiv@gmail.com Narsimlu.G 3 Department of Mathematics C.B.I.T Hderabad Telangana India. dr.narsimlu@gmail.com Abstract: The influence of chemical reaction and viscous dissipation on an unstead MHD flow and heat transfer along a porous flat plate with mass transfer is investigated. The transformed governing equations in the present stud were solved numericall b using Galerkin Finite element method. The effects of viscous dissipation chemical reaction over primar velocit u secondar velocit w heat transfer and mass transfer are analzed. Effects of magnetic parameter M Prandtl number Pr Schmidt number Sc Modified Grashof number Gr for heat transfer and Modified Grashof number Gc for mass transfer on the velocit components and temperature is also examined. Kewords: MHD viscous dissipation chemical reaction finite element method.. INTRODUCTON The combined heat and mass transfer problems with chemical reaction are of importance in man processes and therefore have received considerable amount of attention in recent ears. In processes such as dring evaporation at the surface of a water bod energ transfer in a wet cooling tower and flow in a desert cooler the heat and mass transfer occurs simultaneousl. Chemical reaction can be codified as either homogeneous or heterogeneous processes. A homogeneous reaction is one that occurs uniforml through a given phase; in contrast a heterogeneous reaction takes place in a restricted region or within the boundar of a phase. A reaction is said to be the first order reaction if the rate of reaction is directl proportional to the concentration itself. In man chemical engineering processes a chemical reaction between foreign mass and fluid occur. Das et.al. [] Studied the effect of mass transfer on flow past an impulsivel started infinite vertical plate with constant heat flux and chemical reaction. Muthukumarswam and GanesanP [23] discussed flow past an impulsivel started infinite plate with constant heat and mass flux and effect of chemical reaction on moving isothermal surface with suction. In the recent ears theoretical stud of MHD channel flows has been a subject of great interest due to its widespread applications in designed cooling sstems with liquid metals petroleum industr purification of crude oil polmer technolog centrifugal separation of matter from liquid MHD generators pumps accelerators and flow meters. In an ionized gas where the densit is low and/or the magnetic field is ver strong the conductivit normal to the magnetic field is reduced to the free spiraling of electrons and ions about the magnetic lines of force before suffering collision; also a current is induced in a direction normal to both the electric and magnetic fields. The phenomenon well known in the literature is called the Hall Effect. The stud of magnetohdrodnamic flows with Hall currents has important engineering applications in problems of magnetohdrodnamic generators and of Hall accelerators as well as in flight ARC Page 977

2 Venkat Redd. V et al. magnetohdrodnamics. Sato [4] has studied the effect of Hall currents on the stead hdromagnetic flow between two parallel plates. Various workers Sherman and Sutton [5] Hossain [6] Ram [7] Pop [8] studied the Hall effects. This effect can be taken into account within the range of magnetohdrodnamical approximation. Katagiri [9] studied the stead incompressible boundar laer flow past a semi-infinite flat plate in a transverse magnetic field at small magnetic Renolds number considering with the effect of Hall current. Agarwal et.al [] gave the solution of flow and heat transfer of a micro polar fluid over a stretching sheet using finite element technique. Sri Ramulu et.al [] studied the Effect of Hall Current on MHD Flow and Heat Transfer along a Porous Flat Plate with Mass Transfer. Hossain and Rashid [2] investigated the effect of Hall current on the unstead free-convection flow of a viscous incompressible fluid with mass transfer along a vertical porous plate subjected to a time dependent transpiration velocit when the constant magnetic field is applied normal to the flow. And an analsis of heat transfer taking dissipation function into account in boundar laer flow of a hdromagnetic fluid over a stretching sheet in the presence of uniform transverse magnetic field has been given b Lodha and Tak [3] Gupta [4] studied the Hall current effects in the stead hdromagnetic flow of an electricall conducting fluid past an infinite porous flat plate. Sattar and Maleque [5] studied the effect of viscous dissipation as well as joule heating on two-dimensional stead MHD free-convection and mass transfer flow with Hall current of an electricall conducting viscous incompressible fluid past an accelerated infinite vertical porous plate with wall temperature and concentration. Recentl Bala Siddulu Malga et.al [6] studied the effects of viscous dissipation on unstead free convection and mass transfer boundar laer flow past an accelerated infinite vertical porous flat plate with suction. In this paper we investigated Hall current viscous dissipation and chemical reaction of an unstead free-convection MHD flow of an electrical conducting viscous incompressible fluid with mass transfer along an infinite vertical porous plate. We emploed Galerkin finite element method to solve the governing partial differential equations. The effects of various important phsical parameters on the flow and heat mass transfer are analzed. 2. MATHEMATICAL FORMULATION An unstead free-convection flow of an electricall conducting viscous incompressible fluid with mass transfer along an infinite vertical porous plate has been considered. The flow is assumed to be in direction which is taken along the plate in upward direction. The axis is taken to be normal to the direction of the plate. At time the temperature and the species concentration at the plate are raised to ( ) and ( ) and thereafter maintained uniform. The level of species concentration is assumed to be ver low and hence species thermal diffusion and diffusion thermal energ effects can be neglected. A magnetic field of uniform strength is assumed to be applied transversel to the porous plate. The magnetic Renolds number of the flow is taken to be small enough so that the induced magnetic field can be neglected. The equation of conservation of electric change gives =constant where.it is also assumed that the plate is non-conducting. This implies at the plate and hence zero everwhere. When the strength of magnetic field is ver large the generalized Ohm s law in the absence of electric field takes the form where V is the velocit vector; the electric conductivit; the magnetic permeabilit; the electron frequenc; the electron collision time; e the electron charge; the number densit of the electron; and the electron pressure. Under the assumption that the electron pressure (for weakl ionized gas) the thermo-electric pressure and ion-ship are negligible equation () becomes () (2) International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 978

3 Chemical Reaction and Viscous Dissipation Effects on an Unstead MHD Flow of Heat and Mass Transfer along a Porous Flat Plate where u is the x component of V; w the z component of V and m (= ) the Hall parameter. Within this frame work the equations which govern the flow under the usual Boussinesq approximation are: Continuit: (3) Linear momentum: Angular momentum: (5) (4) Energ: (6) Diffusion: (7) Where g is the acceleration due to gravit; the volumetric coefficient of thermal expansion; the coefficient of volume expansion with species concentration; T the temperature of the fluid within the boundar laer; the species concentration; are respectivel densit viscosit kinematic viscosit thermal conductivit specific heat at constant pressure; and D the chemical molecular diffusivit Chemical reaction parameter. The initial and boundar conditions of the problem are (8) The non-dimensional quantities introduced in the equations (3) to (7) are (9) where is the reference velocit. Therefore the governing equation in the dimensionless form are obtained as () () (2) (3) (4) where International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 979

4 Venkat Redd. V et al. and the boundar conditions (equation (8)) in the non-dimensional form are (5) From equation () it is seen that is either constant or a function of time t. Similarl solutions of equations () to (4) with the boundar conditions equation (5) exist onl if where λ is a non-dimensional transpiration parameter. For suction and for blowing λ <. From equation (6) it can be observed that the assumption is valid onl for small values of time variable. 3. METHOD OF SOLUTION B appling the Galerkin finite element method for equation () over a tpical two-noded linear element is (6) (7) (8) The element equation given b (9) (2) where Here the prime and dot denote differentiation with respect to and t. we obtain We write the element equation for the elements and. Assembling these element equations we get International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 98

5 Chemical Reaction and Viscous Dissipation Effects on an Unstead MHD Flow of Heat and Mass Transfer along a Porous Flat Plate (2) Now put row corresponding to the node i to zero from equation (2) the difference schemes with is (22) Using the Cranck-Nicolson method to the equation (22) we obtain: (23) Similarl the equations (2) (3) (4) are becoming as follows: (24) (25) (26) The initial and boundar conditions (5) reduce to (27) Here where k h is mesh sizes along direction and time direction respectivel. Index i refers to space and j refers to time. The mesh sstem consists of h=. for velocit profiles and International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 98

6 Venkat Redd. V et al. concentration profiles and k=.5 has been considered for computations. In equation (23)-(26) taking i=() n and using initial and boundar conditions (5) the following sstem of equation are obtained. where s are matrices of order n and are column matrices having n- components. The solution of above sstem of equations are obtained using Thomas algorithm for velocit temperature and concentration. Also numerical solutions for these equations are obtained b MATLAB-Program Galerkin finite element method. The same MATLAB-Program was run with slightl changed values of h and k and no significant change was observed in the values of 4. RESULTS AND DISCUSSION To get the phsical insight of the problem the results are discussed through graphs. The numerical results are obtained for the velocit profiles of main (primar) flow and cross (secondar) flow and the temperature profiles concentration profiles have been shown graphicall for different flow parameters such as magnetic field parameter M Hall current m Eckert number Ec Chemical reaction parameter Modified Grashof number Gr for heat transfer Modified Grashof number Gc for mass transfer Schmidt number Sc transpiration parameter and time t. The primar velocit profiles are shown in fig. (a) fig.2 (a) and fig.3 (a). The Hall current effect is to increases the primar velocit component u and this is because of the fact that Hall Effect in general reduces the level of Lorentz forces effect on the fluid whose tendenc is to suppress the flow. The primar velocit flow is decreases with the increase of magnetic field parameter M and the transpiration parameter λ. It is also observed that the primar velocit u increases with the time t. It can be seen that the effect of Gr and Gc is to increase the primar velocit profiles u which is due to with an increase in buoanc force due to temperature differences the flow is accelerated in the boundar laer. Here the positive of Gr (>) correspond to cooling of the plate b natural convection. The effect of Prandtl number Pr is reduces velocit profiles the reason is that the smaller values of Pr are equivalent to increasing the thermal conductivities and therefore heat is able to diffuse awa from the heated surface more rapidl than for higher values of Pr. The influence of Schmidt number Sc is to reduce the velocit profiles. It is also noticed that the influence of Eckert number Ec on the velocit profiles is increases the primar velocit profiles u and designates the ratio of kinetic energ of the flow into the boundar laer enthalp differences. It embodies the conversion of kinetic energ to internal energ b work done against the viscous fluid stresses. Hence greater viscous dissipative heat causes a rise in the velocit profiles. The secondar velocit profiles w are shown graphicall in fig. (b) fig.2 (b) and fig.3 (b). The secondar velocit profiles w increase with time t Hall parameter m and the magnetic field parameter M. But decreases owing to increase of transpiration parameter λ it is observed from figures. It also can be observed that the secondar velocit profiles w increases as the Grashof number Gr (for heat transfer) and Gc (for mass transfer) increases but it decreases as Prandtl number Pr and Schmidt number Sc increases. It also seen that effects of viscous dissipation increases the secondar velocit profiles w. From the fig.5 it is evident that the temperature of the fluid increases with the increase of time t. But it decreases with the increase of transpiration parameter λ and Prandtl number Pr. The viscous dissipation effect on temperature profiles shown in fig.3(c). The viscous dissipation heat causes a rise in the temperature which is shown from the figure. Fig.6 is drawn for the concentration profiles. From the figure it can be seen that the concentrations profiles are increases with the increase of time t while it is decreases with the increase of transpiration parameter λ and Schmidt number Sc. The effect of chemical reaction parameter on primar velocit and secondar velocit profiles temperature and concentration profiles are shown in fig.4 (a)-(d) respectivel. From the fig.4 (a) and (b) it can be observed that the effect of leads to decreases the primar velocit profiles as well as secondar velocit profiles. Fig4(c) reviles that an increase of chemical reaction parameter (28) International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 982

7 Chemical Reaction and Viscous Dissipation Effects on an Unstead MHD Flow of Heat and Mass Transfer along a Porous Flat Plate leads to decrease the temperature profiles. Fig.4 (d) shows that there is fall in concentration profiles due to the increasing the value of chemical reaction parameter u S.No t λ m M Pr=.7 Sc=.22 Gr=. Gc = Fig.(a). The Primar velocit profiles S.No t λ m M W Pr=.7 Sc=.22 Gr=. Gc= Fig. (b). The Secondar velocit profiles S.No Gr Gc Pr Sc u. 5 λ =.5 m=.2 M=. t= Fig.2 (a). The Primar velocit profiles International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 983

8 Venkat Redd. V et al S.No Gr Gc Pr Sc W.2. λ =.5 m=.2 M=. t= Fig.2 (b). The Secondar velocit profiles.2.5 t=. λ=.5 m=.2 M= Pr=.7 Sc=.22 Gr=. Gc=. u. Ec= Fig.3 (a). The Primar velocit profiles for different values of Ec.25.2 t=. λ=.5 m=.2 M= Pr=. 7 Sc=.22 Gr=. Gc=. w.5. Ec= Fig.3 (b). The Secondar velocit profiles for different values of Ec International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 984

9 Chemical Reaction and Viscous Dissipation Effects on an Unstead MHD Flow of Heat and Mass Transfer along a Porous Flat Plate.8 θ.6.4 t=. λ=.5 m=.2 M= Ec= Fig.3 (c). Temperature Profiles for different values of Ec.6.45 Sc=.22; Pr=.7; t=.; Ec=.; Gr=2; Gc=2; m=.5; M=.. K = 2 3 u Fig.4 (a). The Primar Velocit Profile for different values of K.2 Sc=.22; Pr=.7; t=.; Ec=.; Gr=2; Gc=2; m=.5; M=...5 K = 2 3 w Fig.4 (b). The Secondar Velocit Profile for different values of K International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 985

10 Venkat Redd. V et al..8 Sc=.22; Pr=.7; t=.; Ec=.; Gr=2; Gc=2; m=.5; M=.. θ.6.4 K = Fig.4(c). The Temperature Profile for different values of K.75.5 K = 2 3 Sc=.22; t=.. c Fig.4 (d). Concentration Profile for different values of K.8.6 S.No λ t Pr θ Fig.5. The Temperature Profiles International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 986

11 Chemical Reaction and Viscous Dissipation Effects on an Unstead MHD Flow of Heat and Mass Transfer along a Porous Flat Plate c S.No λ t Sc Fig.6. The Concentration Profiles REFERENCES [] Das U.N. Deka R.K. and Soundalgekar V.M Effects of mass transfer past an impulsivel started vertical infinite plate with constant heat flux and chemical reaction. Forschung in ingenieurwesen Engineering Research Vol.6 pp [2] Muthukumarswam R. and Ganesan P Flow past an impulsivel started infinite plate with constant heat flux and mass transfer Comp. Method in applied Mechanics and Engg Vol.87 (-2):pp [3] Muthukumarswam R Effect of a chemical reaction on moving isothermal vertical surface with suction Acta Mech; Vol.55pp ). [4] Sato H The Hall Effect in the Viscous Flow of Ionized Gas between Parallel plates under Transverse Magnetic Field. Journal of the Phsical Societ of Japan Vol. 6(7) pp [5] Sherman A G.W. Sutton Engineering Magnetohdrodnamics McGraw-Hill New York 965. [6] Hossain M A Effect on of Hall current on unstead hdromagnetic Free-convection flow near an infinite vertical porous Plate Journal of the Phsical Societ of Japan Vol. 55(7) pp [7] Ram P.C Hall effects on free convention flow and mass transfer through a porous medium Warme Stoffubertrag. Vol.22 pp [8] Pop I The effect of Hall currents on hdromagnetic flow near an accelerated plate J.Math.Phs.Sci.Vol. 5 pp [9] Katagiri M The Effect of Hall Current on the Magneto hdrodnamic Boundar laer Flow Past a Semi-infinite Fast Plate. Journal of the Phsical Societ of Japan Vol. 27(4) pp [] Agarwal R. S Bhargava R. Balaji A.V. S. Finite Element Solution of Flow and Heat Transfer of a Micro Polar Fluid over a Stretching Sheet Int. J. Engg. Sci.Vol. 27() pp [] Sri Ramulu A Kishan N AnandaRao J. Effect of Hall Current on MHD Flow and Heat Transfer along a Porous Flat Plate with Mass Transfer. IE (I) Journal-MC Vol. 87 pp Jan-27. [2] Hossain M A and Rashi Hall Effect on Hdromagnetic Free-convection flows along a porous Flat Plate with Mass Transfer. Journal of the Phsical Societ of Japan Vol. 56() pp International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 987

12 Venkat Redd. V et al. [3] Lodtha A. Tak S. S. Heat Transfer with Viscos Dissipation in Hdromagnetic Flow over a Stretching Sheet Ganita Sandesh Vol 9 no 2 pp [4] GuptaA.S. Hdromagnetic Flow Pas a Porous Plate with Hall Effects Actamechanica Vol. 22(3-4) pp [5] Md A Sattar and K A Maleque. Numerical Solutions of MHD Free-convection and mass Transfer Flow with Hall Current viscous Dissipation and joule heating. Journal of Energ Heat Mass Transfer Vol. 22 pp [6] Bala Siddulu Malga Naikoti Kishan Viscous Dissipation Effects on Unstead free convection and Mass Transfer Flow past an Accelerated Vertical Porous Plate with Suction.Advances in Applied Science Research Vol. 2(6) pp International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 988

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