Magnetohydrodynamic Mixed Convection Flow Past a Wedge Under Variable Temperature and Chemical Reaction

Transcription

1 American Journal of Computational and Applied athematics, (): 74-8 DOI: 9/j.ajcam.. agnetohydrodynamic ixed Convection Flo Past a Wedge Under Variable Temperature and Chemical Reaction Rudra Kanta Deka, Sarat Sharma, athematics Department, auhati University, uahati, 7844, India Department of athematical Sciences, Chaiduar College, ohpur, 78468, India Abstract Steady laminar boundary layer free-forced flo of a Netonian fluid past a edge shaped surface is studied. n The temperature of the edge surface is assumed to vary as x and a first order chemical reaction of the fluid species is considered. The governing boundary layer equations are transformed by the Falknar-Skan transformations and proportionate numerical results are tabulated for the Skin friction, Nusselt number and Sherood number. raphical results for the variation in the velocity, temperature and concentration profiles for various values of the temperature exponent, chemical reaction parameter and Schmidt number are presented. Keyords HD, Wedge, ixed Convection, Chemical Reaction. Introduction Boundary layer flo over a heated vertical surface emerges in a variety of engineering applications and it has been the subject of extensive research. When the fluid is electrically conducting and also exposed to a magnetic field, flo and temperature fields are governed both by the buoyancy and the Lorenz forces. ost of the studies concerning such flos ere taken up as steady flos and are considered either for purely free convection or purely forced convection regime only. In dealing ith forced convection flo, it is customary to ignore the effects of free convection. Similarly e assume the forced convection as negligible hile dealing ith free convection flo. The error involved in ignoring natural convection hile studying forced convection is negligible at high velocities but may be considerable at lo velocities. Situations may arise for hich free and forced convection effects are comparable, in hich case it is inappropriate to neglect either process. Therefore it is desirable to have a criterion to assess the relative magnitude of natural convection in the presence of forced convection flos. Such flos situations here both free and forced convection effects are of comparable order belong to the mixed convection regime. In several practical applications of heat transfer theory to the vertical plate problems there exists significant temperature difference beteen the surface of the hot plate and the free stream. This Corresponding author: sarat_sharma@si fy.com (Sarat Sharma) Published online at Copyright Scientific & Academic Publishing. All Rights Reserved temperature difference cause density gradients in the fluid medium and in the presence of a gravitational body force, free convection effects become important. It has generally been recognized that, r λ = here r is the rashof Re number and Re the Reynolds number is the governing parameter for the laminar boundary layer forced-free mixed convection flo hich represents the ratio of the buoyancy forces to the inertial forces inside the boundary layer. Forced convection exists hen λ hich occurs at the leading edge and the free convection limit can be reached if λ becomes large. We kno from authoritative ork in heat transfer[] that free convection is negligible if λ and forced convection is negligible if λ. Hence combined free and forced (or mixed) convection regime is generally one for hich λ. Although the flos ith only one of the to effects involved ill have a self similar character, such situations ill mathematically lead to ordinary differential equations and are easy to solve. But as soon as the to effects occur together, boundary value problem involving partial differential equations ill arise and are not easy to solve by conventional procedures. The physical explanation of the complexities is that the to effects act differently ith respect to the characteristic length l and any combination means that the characteristic length is introduced into the problem (i.e., the length from the leading edge to the separation position).if both the effects are to be considered, pure forced convection alays dominates for l hile direct natural convection dominates for l. ost of the orks concerning external flos past a vertical

2 American Journal of Computational and Applied athematics, (): surface are taken up as steady flos and they are either considered for free or forced convection regimes only. Hoever free-fo rced or popularly knon as mixed convection flos received considerable attention in the late 97s to 98s and numerous literature revies are available. Soundalgekar et al[] studied the flo past a semi-infin ite vertical plate under variable temperature. Wilks[] and Raju et al[4] have shon several formulations under hich a flo can be classified as belonging to the mixed convection regime and values of the mixed convection parameters ere used by them. In nature it is rather impossible to find pure fluid unless special efforts are made to obtain it.the most common fluids like ater, air etc. is contaminated ith impurities like CO, C H, H SO etc. and generally e have to consider presence of such foreign masses hile studying flos past different bodies. In such a case the density difference in the fluid is caused by material constitution in addition to temperature differences. The common example of such a flo is the atmospheric flo hich is driven appreciably by both temperature and concentration differences. When such contaminant is present in the fluid under consideration there does occur some chemical reaction e.g. air and benzene react chemically, so also ater and sulfuric acid. During such chemical reactions, there is alays generation of heat. But hen the foreign mass present in the fluid at very lo level, e can assume a first order chemical reaction and the heat generated due to chemical reaction can be very negligible. Several authors have done significant orks by taking into account a first order chemical reaction on flo past vertical surfaces. Das et al[5] studied the effect of mass transfer and chemical reaction on the flo past a vertical plate. Bala[6] considered a porous, moving vertical plate under variable suction and chemical reaction. ahmoud[7] studied mixed convection flo under variable viscosity and chemical reaction effects. Flo past a edge or edge flo belongs to a special class of boundary layer flo problems here the shape of the edge affects the boundary layer mainly by determining the pressure gradient along the surface of the edge. Flo past a edge of angle βπ yields a different pressure profile for each value of β, thereby offering insight into boundary layer behavior in a number of situations. To of the special cases are planar stagnation flo ( β = ) and parallel flo past a flat plate ( β = ).artin and Boyd[8] studied the flo past a edge ith slip boundary conditions using Falkner-Skan transformations together ith a finite difference method. ukhopadhyay[9] studied the effect of radiation and variable viscosity on the flo past a edge using a special form of scaled Lie group of transformations. Nazar and Pop[] considered the case of a moving edge using the Falkner-Skan transformations. Ece[] used the Thomas algorithm to solve the similar boundary layer equations of flo past a edge under mixed thermal boundary conditions. Hsu et. al.[] utilized the series expansion method to study the flo of a second grade fluid past a edge. assoudi[], ureithi and ason[4], Loganathan et. al.[5] employed the u Non-similarity method to solve edge flo problems. Sattar[6] considered the unsteady flo past a edge by the Local similarity method. Pop and Watanabe[7], Kandasamy et al.[8], Kandasami et al.[9], Kandasami et al.[] studied the flo situations past a edge by introducing a pair of variables ( x, η ) instead of η or a dimensionless distance ξ along the plate of the edge. Williams and Rhyne[] have used a group of transformations to encompass both the situations of small and large time in the unsteady flo past a edge by simple modification of the Falkner and Skan transformations in solving the edge equations employing the Thomas algorithm. Kumari and Nath[] have successfully implemented these transformations in the study of unsteady flo past a vertical plate and shon their efficacy. Despite numerous solution techniques and transformation methods available for the governing boundary layer equations arising in the flo past a edge, the Falkner-Skan transformations remains one of the most chosen and used for their simplicity and computational ease. All the above cited orks concerning flo past a edge ere carried out by assuming a constant temperature of the edge surface. Here e propose to undertake this study by adopting the popular Falkner-Skan transformations to solve our problem by considering a variable temperature along the surface of the edge under a first order chemical reaction. We ill also study the variation in velocity, temperature and concentration profiles for various values of Schmidt numbers.. Formulation of the Problem Let us consider the steady to-dimensional hydro magnetic mixed convection boundary layer flo past the surface of a edge. We formulate the problem in a body fixed rectangular coordinate system in hich x is measured along the surface of the edge, from the apex, and y is measured normal to the edge surface. The flo configuration is shon in fig. The fluid is assumed to be Netonian and its physical property variations due to temperature is limited to density only. A uniform transverse magnetic field of strength B is applied parallel to the y-axis and induced magnetic field is neglected. The concentration of diffusing species is very small in comparison to other chemical species and far fro m the surface of the edge the concentration is assumed infinitesimally small and hence the Dufour and Soret effects are neglected. Also there is first order chemical reaction taking place in the flo. Assuming Boussinesq approximation, the boundary layer governing equations of momentum, energy and diffusion for the flo problem are: u u + = () x y

3 76 Rudra Kanta Deka et al.: agnetohydrodynamic ixed Convection Flo Past a Wedge Under Variable Temperature and Chemical Reaction u u u du u + v = + u x y y dx Ω [ gβ( T T) + gβ ( C C) ]sin σ B ( u u ) () ρ u + v = α x y y T T T C C C u + v = D k C C x y y The boundary conditions are: ( ) uxy (, ) = vxy (, ) =, Txy (, ) = T, Cxy (, ) = C t < For t, u( x, ) = vx (,) =, ux (, ) = u for n T( x, ) = T, T( x,) = T ( x) = bx, C( x, ) = C, () (4) Cx (,) = C, b>, n (5) Here x and y are respectively, the distances along and perpendicular to the surface of the edge, u and v are the velocity components along x and y directions respectively, t is the time, is the kinematic viscosity, g is the acceleration due to gravity, β is the volumetric coefficient of thermal expansion, β is the volumetric coefficient of is the density, α is the thermal diffusivity, D is the effective diffusion coefficient and the subscripts and denote conditions at the surface of the edge and in the free stream respectively. We no introduce the folloing Falkner-Skan transformations for edge flo as given in[] and[] ith concentration, σ is the electrical conductivity, ρ η m + u = y, ( x, y) xu f ( ), x ψ = m η + Txy (, ) T = ( T( x) T) θη ( ), Cxy (, ) C = ( C( x) C) ϕη ( ) (6) here the potential flo velocity can be ritten as u ( x) ax, β m = = (7) + m m Where a is a constant and β is the Hartree pressure Ω π gradient parameter that corresponds to β = for a total angle Ω of the edge. To important cases are the flat plate flo (m=) and the stagnation-point flo (m=). For ( m >, β > ) velocity profiles accelerated flos exhibit no point of inflexion but for retarded flos ( m <, β < ) there exists a point of inflexion. The case m =.9, β =.99 is a limit ing case, no solution to the Falkner-Skan equation for accelerated forard flo exist for β <.99 hich corresponds to the velocity profile ith vanishing all shear stress (separation). Numerical results from Hartree are plotted for various edge angles in[]. The velocity components are given in terms of the stream function ψ by the relations: uxy (, ) ψ = and vxy (, ) y Fi gure. Flo Configuration ψ = (8) x The continuity equation () is readily satisfied by (8) and using (6) and (7), equations ()-(4) and the boundary conditions (5) are reduced to: Ω f + ff + β[ ( f ) ] + [ λθ + λϕ]sin m + ( ) m f = + n (Pr) θ + fθ f θ = m + ( Sc) ϕ + f ϕ + γϕ = m + The boundary conditions for equations (9)-() are: Here number, f f θ ϕ () = () =, () =, () =, (9) () () f ( ) =, θ( ) =, ϕ( ) = () gβ ( T T ) x gβ ( T T ) x ux λ = is Re = is the thermal rashof = is the species rashof number, Re = is the Reynolds number,

4 American Journal of Computational and Applied athematics, (): the buoyancy parameter due to temperature, the buoyancy parameter due to mass, Pr Prandtl number, γ σ B x λ = is Re = is the α Sc = is the Schmidt number, D = ρu is the magnetic parameter and kx u = is the chemical reaction parameter. Here η is the transformed similarity variable, ψ is the dimensional and f(η) is the dimensionless stream function, θ(η) is the dimensionless temperature, φ(η) is the dimensionless concentration and primes denote differentiation ith respect to η. Knoing the velocity field, the physical quantities of τ interest to our study are the Skin friction C f = ρu hich is given by the all shear stress u τ = µ y y= xq Nu = k( T T ) T q = k y y= x Sh = D( C C ) C = D y y=, the Local Nusselt number given by the heat flux and the Local Sherood number given by the mass flu x In vie of (), the Skin Friction coefficient to. m + Cf = ( Re) f ( ) the Nusselt number Nu reduces to Nu m + = ( Re) θ ( ) the Sherood number Sh reduces to C f reduces () (4) Sh m + = ( Re) φ ( ) (5) We have compared our results ith a trivial solution given in[] for the skin-friction to the Falkner-Skan equation for various edge angles by setting =, C =, γ =, n= in our code hich are in excellent agreement (Table). β Ta ble. comparison ith standard ork m (Schlichting) f ( ) Numerical Solution (Present ork) f ( ) We propose to solve equations (9) () under the boundary conditions () using the continuation method ith the help of atlab7 hich implements a finite difference code. Continuation method is an improved shooting method hich reduces the chance of carrying on the computational errors due to poor guesses of the unknon values and ensures greater accuracy. In continuation method firstly e find the unknon boundary conditions for η = for a guess and put those values for η = and carry on the process until the values converge to fixed values. Finally the converged initial values are put as the initial values in solving the boundary value problem for the entire range η [,6].We have tabulated the values in Table-Table5 of f ( ), θ ( ) and φ ( ) hich are proportional to the Skin friction, Nusselt number and Sherood number respectively under magnetic, plate temperature variation, chemical reaction and Schmidt number variation. We have kept other parameters as constant hile analyzing the effects of one parameter hich ill highlight the changes (if any) more clearly. The velocity, temperature and concentration profiles are presented in Figure-Figure5. Ta ble. Flo separation for various values of magnetic, chemical reaction and temperature exponent.7..4 γ n Sc Pr λ λ β

5 78 Rudra Kanta Deka et al.: agnetohydrodynamic ixed Convection Flo Past a Wedge Under Variable Temperature and Chemical Reaction Ta ble 4. values of skin-frict ion, heat flux and mass flux for various Schmidt numbers Sc n γ f ( ) θ ( ) φ ( ) Fi gure. agnetic effect on velocity, temperature and concentrat ion profile Fi gure 4. Effect of chemical reaction on velocity, temperat ure and concentration profile Fi gure. Effect of variable temperat ure on velocity, t emperature and concentration profile Ta ble. values of skin-friction, heat flux and mass flux for magnetic, variat ion in temperat ure and chemical react ion n γ f ( ) θ ( ) φ ( ) Fi gure 5. Effect of Schmidt number on velocity, temperat ure and concentration profile One of the frequently encountered problems in edge flo is the flo separation. It has several undesirable effects in so far as it leads to an increase in the drag on a body

6 American Journal of Computational and Applied athematics, (): immersed in the flo and adversely affects the heat transfer from the surface of the body. We have calculated the value of β for different values of λ, λ,, γ, Sc, n (Table), for hich separation of flo occurs. The result shos that flo separation can effectively be controlled by introduction of magnetic, chemical reaction and plate temperature variation effects into the flo field. 4. Results and Discussion In our effort to analyze the effects of transverse magnetic field on the velocity, temperature and concentration profiles, e have fixed the values of the plate temperature exponent at and the chemical reaction parameter at. The values of f ( ), θ ( ) and φ ( ) are tabulated in Table. The velocity, temperature and concentration profiles for the flo are presented Figure. It is seen from Figure that magnetic field induces considerable influence on the flo field. Increasing magnetic field strength has a enhancing effect on the velocity of the fluid under consideration. agnetic field tends to decrease the temperature and concentration at equal proportion for the flo. In Table, values of skin friction, rate of heat flux and rate of mass flux sho a increasing trend ith the increase of magnetic field strength. Effects of variation in the plate temperature upon the velocity, temperature and concentration of the fluid is computed by keeping the magnetic field strength as and chemical reaction parameter as. The values of f ( ), θ ( ) and φ ( ) are tabulated in Table. The velocity and temperature profiles for the flo are presented for comparison in Figure. For constant plate temperature the velocity of the flo is at a high level. But as e increase the plate temperature exponent the velocity becomes lesser gradually. For temperature, there is a uniform loering effect for the flo ith the increase of plate temperature but for concentration the change though minimally loering, is not significant. From Table, it is seen that increase in plate temperature reduces the skin friction and mass flux of the flo but heat flux increases. To study the chemical reaction effects e have taken the magnetic field strength as and the plate temperature exponent as. The values of f ( ), θ ( ) and φ ( ) are tabulated in Table.The velocity, temperature and concentration profiles for flo are presented in Figure4. Presence of a first order chemical reaction at lo level increases the velocity and concentration of the fluid but temperature comes don. Table shos that in the absence of chemical reaction, the rate of skin friction is comparatively more than the heat and mass flux. For small change in the chemical reaction strength, skin friction and heat flux increases but mass flux decreases. For higher chemical reaction level concentration level increases considerably. Effects of variation in the species of diffusing species in terms of their Schmidt numbers is studied by taking the magnetic field strength as, the plate temperature exponent as and for a lo level of chemical reaction γ =. The values of f ( ), θ ( ) and φ ( ) are tabulated in Table4.The velocity, temperature and concentration profiles for flo are presented in Figure5. Lighter diffusing species has a higher velocity and concentration compared to the heavier species. Temperature is not changed ith a change in the Schmidt number.table4 shos that in the presence of nominal chemical reaction, the skin friction, heat flux and mass flux gradually decreases ith increase in Schmidt numbers. 5. Conclusions (i) agnetic field accelerates the velocity and loers the temperature and concentration. (ii) Skin-friction, heat flux and mass flux increase due to magnetic field. (iii) Increasing plate temperature loers the velocity and temperature considerably but a minor loering effect for concentration is found. (iv) Skin-friction and mass flux decrease but heat flux increases ith plate temperature. (v) At lo level of chemical reaction velocity, temper ature and concentration increase gradually but for higher level the reverse is true. (vi) For lo level of chemical reaction skin-friction, heat flux increase but mass flux decreases. For higher level the reverse is true. (vii) For lo Schmidt number the velocity and concentration is higher but temperature remains same. (viii) In the presence of nominal chemical reaction the skin-friction, heat flux and mass flux increases ith increase in Schmidt number. 6. Nomenclature r rashof number, Re Reynolds number, β coefficient of thermal expansion, β coefficient of thermal expansion ith concentration, stream function, ψ β Hartree parameter, thermal rashof number, mass rashof number, B magnetic field strength, η similarity variable, ( uv, ) dimension less velocity components, u free stream fluid velocity,

7 8 Rudra Kanta Deka et al.: agnetohydrodynamic ixed Convection Flo Past a Wedge Under Variable Temperature and Chemical Reaction T free stream temperature, T temperature at the plate, magnetic parameter, dimensionless temperature, T ρ density, θ dimensionless temperature, φ dimensionless species concentration, C species concentration in the free stream, C concentration at the all, g acceleration due to gravity, Pr Prandtl number, γ chemical reaction parameter, Sc Schmidt number, kinematic viscosity, k chemical reaction constant, α thermal diffusivity, D mass diffusion coefficient, n temperature exponent REFERENCES [] H Schlichting and K ersten, Boundary Layer Theory,8 th ed., Springer, 999 [] Soundalgekar, V.., Takhar H. S. and Vighnesam,N. V., 988, Combined free and forced convection flo past a semi-infinite vertical plate ith variable surface temperature, Nuclear Engineering and Design,, [] Wilks, raham, 97,Combined forced and free convection flo on vertical surfaces, Int. J. of Heat and ass Transfer, 6, [4] Raju,. S., Liu, X. Q. and La, C. K., 984, A formulation of combined forced and free convection past horizontal and vertical surfaces, Int. J. of Heat and ass Transfer, 7(), 5-4 [5] Das, U. N., Deka, R. K. and Soundalgekar, V.., 998, Effect of mass transfer on flo past an impulsively started infinite vertical plate ith chemical reaction, The Bulletin, U A,5, - [6] Bala A. Jyothi and Varma Vijaya Kumar,, Unsteady HD heat and mass transfer flo past a semi-infinite vertical porous moving plate ith variable suction in the presence of heat generation and homogeneous chemical reaction, Int. J. of Appl. ath. And ech.,7(7), -4 [7] ostafa, A. A. ahmoud, 7, A note on variable viscosity and chemical reaction effects on mixed convection heat and mass transfer along a semi-infinite vertical plate, athematical Problems in Engineering, 7,-7 [8] artin ichael J. and Boyd Iain D.,, Falkner-Skan flo over a edge ith slip boundary conditions, Journal of Thermo physics and Heat Transfer.,4(), 6-7 [9] ukhopadhyay, S., 9, Effect of radiation and variable fluid viscosity on flo and heat transfer along a symmetric edge, Journal of Applied Fluid echanics., (), 9-4 [] Nazar A. I. R. and Pop, I., 7, Falkner-Skan equation for flo past a moving edge ith suction or injection, J of Applied ath. and Computing,5(-), 67-8 [] Ece. C., 5, Free convection flo about a edge under mixed thermal boundary conditions and a magnetic field, Heat and ass Transfer, 4, 9-97 [] Hsu Cheng-Hsing, Chen Chii-Sheng and Teng Jyh-Tong, 997, Temperature and flo fields for the flo of a second grade fluid past a edge, Int. J. Non-Linear echanics, (5), [] assoudi,,, Local non-similarity solutions for the flo of a non-netonian fluid over a edge, Int. J. of Non-Linear echanics, 6(6), [4] ureithi E.W. and ason D. P.,, Local non-similarity solutions for a forced-free boundary layer flo ith viscous dissipation, athematical and Computational Applications, 5(4), [5] Loganathan P., Puvi-arasu P. and Kandasamy R.,, Local non-similarity solution to impact of chemical reaction on HD mixed convection heat and mass transfer flo over porous edge in the presence of suction/injection, Applied athematics and echanics (Eng. ed.),(), [6] Sattar. A.,, A local similarity transformation for the unsteady to dimensional hydrodynamic boundary layer equations of a flo past a edge, Int. J. of Appl. ath and ech., 7(), 5-8 [7] Pop H. and Watanabe T., 99, The effects of suction or injection in boundary layer flo and heat transfer on a continuous moving surface, Technische echanik,, [8] Kandasami R., Hashim I., uhaimin and Ruhaila, 7, Effect of variable viscosity, heat and mass transfer on nonlinear mixed convection flo over a porous edge ith heat radiation in the presence of homogeneous chemical reaction, ARPN J.of Engineering and Applied Sciences, ( 5), 44-5 [9] Kandasami R., Hashim I., uhaimin and Seripah, 7, Nonlinear HD mixed convection flo and Heat and mass transfer of first order chemical reaction over a edge ith variable viscosity in the presence of suction or injection, Theoretical Applied echanics, 4(), -4 [] Kandasamy R., Hashim I., Khamis A. B. and uhaimin I., 7, Combined heat and mass transfer in HD free convection from a edge ith ohmic heating and viscous dissipation in the presence of suction or injection, Iranian Journal of Science and Technology, Trans. A, (A), 5-6 [] Williams III J. C., Rhyne T. B., 98, Boundary layer development on a edge impulsively set into motion, SIA J Appl. ath, 8(), 5-4 [] Kumari., Nath., 997, Development of mixed convection flo over a vertical plate due to an impulsive motion, Heat and ass Transfer, 4(), 8-88 [] W Deen, Analysis of Transport Phenomena, Oxford University Press, Ne York, 998