World Journal of Mechanics, 2015, 5, 165177 Published Online September 2015 in SciRes. http://www.scirp.org/journal/wjm http://dx.doi.org/10.4236/wjm.2015.59017 How to cite this paper: Sharada, K. and Shankar, B. (2015) MHD Mixed Convection Flow of a Casson Fluid over an Exponen tially Stretching Surface with the Effects of Soret, Dufour, Thermal Radiation and Chemical Reaction. World Journal of Me chanics, 5, 165177. http://dx.doi.org/10.4236/wjm.2015.59017 MHD Mixed Convection Flow of a Casson Fluid over an Exponentially Stretching Surface with the Effects of Soret, Dufour, Thermal Radiation and Chemical Reaction Kankanala Sharada, Bandari Shankar Department of Mathematics, Osmania University, Hyd erabad , India Email: kankan.sharada@gmail.com Received 29 July 2015; accepted 26 September 2015; published 29 September 2015 Copyright © 2015 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract The present study deals with MHD (magneto hydrodynamics) mixed convection flow of a Casson fluid over an exponentially stretching sheet with the effects of Soret and Dufour, thermal radiation, chemical reaction. The governing partial differential equations are converted into ordinary dif ferential equations by using similarity transformations. These equations are then solved numeri cally by applying finite difference scheme known as the Keller Box method. The effects of various parameters on velocity, temperature and concentration profiles are presented graphically to in terpret and the results are discussed. Keywords Casson Fluid, Magneto Hydrodynamics, Mixed Convection, Exponentially Stretching Surface, Soret and Dufour Effects, Thermal Radiation, Chemical Reaction 1. Introduction The study of magnet o hy drodyna mics, boundary layer flow with heat and mass transfer from a stretching surface has many applications in industrial and engineering fields. Magnetic fields can be used to manage thermal en ergy in flowing electr ically conducting polymers (Garnier [1]). After the pion eering studies of Sakiadas [2] and Crane [3], several researchers further investigate the problem of stretching sheet boundary layer flow to obtain thermal and kinematic behaviour by considering the various types of stretching velocity. Magyari and Keller [4] examine the similarity solutions of thermal boundary layer and flow over an exponentially stretching surface
K. Sharada, B. Shankar with an exponential temperature distribution. Numerical solution of boundary layer flow over an exponentially stretching sheet includ ing thermal radiatio n is inv estigated by Bidin and Nazar [5], and analytical stud y is inves tigated by Sajid and Hayat [6]. Casson fluid is one type of fluid model for nonNewtonian flu id, which has significant applications in poly mer processing industries and biomechanics. Casson fluid can be defined as a shear thinning liquid which is presumed to have an infinite viscosity at zero rate of shear and a yield stress under which no flow occurs and zero viscosity at an infinite rate of shear. Casson fluid model is used in many foodstuffs and biological materials, especially blood. It describes the steady shear stress, shear rate behaviour of blood. Merill et al. [7] and Mc Donald [8] investigate the b ehaviour of blood. Nadeem et al. [9] investigate the magneto hydrodynamic bound ary layer flow of Casson fluid over exponentially permeable shrinking sheet. Casson fluid over an unsteady stretching sur face is analyzed by Mukhopadhyay [10]. The application of the minimal energy phenomenon to a Casson fluid is studied by John L. McGregor [11]. A similarity solution technique for mixed convection heat transfer from an exponen tially stretching surface with the eff ect of viscous dissipation is investigated by Par tha et al. [12]. Recently Dual pal [13] studies mixed convection heat transfer in the boundary layers over an expo nentially stretching surface including magnetic field. The effects on mixed free forced convective and mass transfer steady, laminar boundary layer flow over a vertical plate with temperatur e dependent viscosity are pre sented by Kafoussias and Williams [14]. The mass flux created by temperature gradient is known as thermal diffusion (Soret) effect. The energy flux caused by concentration differences is known as diffusion thermo (Dufour) effect. The Soret effect has been utilized for medium molecular weight (eg. N2 and air) and isotope separation. These effects have several appli cations in engineering problems. Here a survey conducted by Dursunkaya and Worek [15] on Soret and Dufour effects in transient and steady natural convection from a vertical surface. The combined effects of thermal diffu sion and diffusion thermo over stretching surface with free convective heat and mass transfer are investigated by Afify [16]. T. Hayat [17] describes the effects of Soret and Dufour on magneto hydrodynamic flow of Casson fluid. Alam and Rahman [18] study the Dufour and Soret effects on mixed convect ion fl ow wi th var iable suction. Several researchers have studied the effects of thermal radiation, and these include that Pramanik [19] inves tigates Casson fluid flow and heat transfer over an exponentially stretching surface in the presence of thermal radiation and porous medium. Numerical analysis of magneto hydrodynamic stagnation point flow of a Casson fluid includes thermal radiation discussed by Shateyi and Marewo [20]. Anur Ishak [21] present the magneto hydrodynamic boundary layer flow over an exponentially stretching sheet with the effect of radiation. Swathi Mukhopadhyay [22] investigate Casson fluid flow and heat transfer on an unsteady stretching surface subjected to suction/blowing with the effect of thermal radiation. Recently effects of mass transfer on magneto hydrody namic flow of Casson fluid with che mical reactio n are studied by S.A. Shehzad and T. Hayat [23]. This paper provides the solution to the problem of fluid flow, heat and mass transfer of a Casson fluid over an exponentially stretching surface by considering the effect of Soret and D ufo ur, thermal radiatio n, chemical reac tion by adopting the Keller Box method. 2. Mathematical Formulation Consider twodimensi onal ste ady, lamina r, incompressi ble MHD flow of a C as son flui d over a vert i cal exponential surface stretching with velocity , temperature distribution and concentration distribution moving through a fluid of constant temperature and concentration as shown in Figure 1. The xaxis is oriented about the vertical surface in the upward direction and yaxis is normal to it. A magnetic field is applied in the direction perpendicular to the stretching surface. The magnetic Reynolds number is taken to be small enough such that the induced magnetic field is negligible. The rheological equation of state for an iso tropic and incompressible f low of a Casson fluid can be written as: when (1) 2 2 y ij bij c Pe τµ π = + when (2)
K. Sharada, B. Shankar Figure 1. Physical model and coordinate system. is known as yield stress of the fluid, mathematically expressed as , is known as plastic dynamic viscosity of the non Newtonian fluid, π is the product of the component of deformation rate with itself (i.e. ) and is the (i, j)th component of the stress tensor. Considering u, v as velocity components in the directions of x and y respectively in the flow field. The gov erning equations of continuity, momentum, energy and concentration are given by (3 ) () () ( ) 22 * 2 1 1 TC u uuBu u vgTTCC xy y σ υ ββ βρ ∞∞ ∂∂ ∂ += ++−+−− ∂∂ ∂ (4) 2 22 22 11 1 rT p pps q Dk TT TuC uv xycy CyCC yy ν αρβ ∂ ∂∂ ∂∂∂ +=−+ ++ ∂∂∂ ∂ ∂∂ (5) ( ) 22 22 T m DK CC CT uvDkC C xy T yy ∞ ∂∂ ∂∂ + =+−− ∂∂ ∂∂ . (6) where is the density of the fluid, is the electrical conductivity, is the kinematic viscosity, B is the magnetic induction, α is the thermal diffusivity, D is the solutal diffusivity of the medium, and are the coefficient of thermal and solutal expansions, μ is the dynamic coefficient of viscosity, and are the specific heat capacity and concentration susceptibility, is the mean fluid temperature, is the thermal diffusion ratio, γ is the chemical reaction parameter and is the Casson parameter. The boun da ry c ondition s c on s i de r ed are defined as: ( ) ( ) 2 0 e ,0,e xx LL ww uU xavTTTTT ∞∞ === ==+− at (7) as (8) where subscripts w and indicate that conditions at the wall and at the external(outer) edge of the boundary layer respectively and is the stretching velocity, and are the exponential temperature and concentration distr ibution respectively. In above conditions “a” is velocity parameter of the stretching surface, is temperature distribution pa
K. Sharada, B. Shankar rameter, is the parameter of the concentration distribution in the stretching surfac e. We introduce stream function ψ as , and defi ne va riables. () ()( ) 22 e ,e,e 22 x xx L LL aa u afyvff LL υ ηηηη η υ ′′ ===−+ ( ) ( ) ( )() ( ) ( ) 22 00 ,e ,,e xx LL T xyTTTCxyCCC θη φη ∞∞∞ ∞ =+−=+ − By using the Rosseland approximation for radiative heat flux, is define d a s: (9) where is the Stephen Boltzman constant and is the mean absorption coeff icient. Sin ce the te mperature differences within the flow field are assumed to be small, and then we linearize and expand into the Taylor series about , which after neglecting higher order forms take the form. To obtain similarity solutions, we assumed that the magnetic field is of the form where is the constant magnetic field and is the exponential reaction, is constant. With the help of above similarity transformations the governing equations are transformed in to ( )() 3 22 1 12 2e0 X f fffNMf λ θφ β − ′′′ ′′′′ ++−++− = (10) ( )( ) 32 2 41 1 10 3 X f NrPr ffPrEcefPrD θ θθφ β ′′′ ′′′′′ ++−+++ = (11) () 0Sc ffScSrSc φφ φθγφ ′′′ ′′′ +−+− = (12) The correspondin g bo undary condition s are () ()()() 0 1,001, 0101 : ,ff θη φ ′ === = = ( )( )( ) :0,0, 0f ηθφ ′ →∞ ∞=∞=∞= where is the Xlocation, observing Equations (10), (11) in MHD mixed convection due to Casson fluid, the velocity and temperature profiles are not similar because the x coordinate cannot be deleted from these equa tions. It is hard to proceed in this case. To reduce the difficulty of analysis, it was considered to find local similarity solutions for the governing equations. Now one can study the effects of different parameters on various profiles at any given Xlocation. is the magnetic parameter, is the mixed convection parameter, is the Reynolds number, is the Grashoff number, is the Buoyancy ratio, is the Prandtl number, is the Eckert number, is the Thermal radia tion parameter, ( ) ( ) 0 0 T fsp DK CC DCCTT ν ∞ ∞ − =− is the Dufour number, ( ) () 0 0 T m DK TT Sr TC C ν ∞ ∞ − =− is the Soret number, is the Schmidt number, is the chemical reaction parameter.
K. Sharada, B. Shankar The wall shear stress, heat and mass trans fer coefficients with constant density are given by 0 2 y wb y Pu y τµ π = ∂ = + ∂ , and where k is the thermal conductivity. The dimensionless form of skin friction . Where is the characteristic velocity, the local Nusselt number , and local Sherwood number are given by ( )( ) 21 4 1 0,10 32 x xx x Re X X Cff NuNr Re θ β ′′ ′ = +=−+ and 3. Method of Solution The governing equations with boundary equations are solved numerically by using finite difference scheme known as Keller Box method which is described by Cebeci and Bradshaw [24]. This method involves the fol lowing steps: Step 1: Reducing higher order ODEs (systems of ODES) in to systems of first order ODEs; Step 2: Writing the systems of first order ODEs into difference equations using central diffe rence scheme ; Step 3: Linearizing the difference equations using Newton’s method and writing it in vector fo rm; Step 4: Solving the system of equations using block elimination method. 4. Numerical Discussions To solve the above differential equations numerically, we adopt Matlab software. In this process the step size, is used to obtain numerical solution with five decimal place accuracy as the criterion of convergence. Table 1 shows that the comparison between wall temperature gradient calculated by the present method and of Magyari and Keller for λ = X = Df = Nr = Sr = γ = N = M = Sc = Ec = 0 and . The numerical vales of , , and for the various values of λ, X, Df , Nr, Sr, γ, β are shown in Ta ble 2. 5. Results and Discussion In the present study, the following values are adopted for the numerical computations N = 0.5, Df = 0.03, Sr = 0.5, λ = 1.0, Pr = 1.0, M = 1.0, Sc = 0.22 , X = 0.5, γ = 0.1 , β = 0.5. Thes e values are used throug hout the calcu la tions unless otherwise mentioned. Figure 2 shows that the nondimensional velocity profile for different values of the mixed convection pa rameter λ. We have observed the results for both aiding flow and apposing flow. It illustrates that the dimen sionless velocity increases, as the value of λ increases. As λ increases, the buoyancy effects increase and hence the fluid flow accelerates. In the case of mixed convection, temperature is low for aiding flow and more for Table 1. Comparison of results for with previous published work. Pr Magyari and Keller [4] Present 0.5 1.0 3.0 5.0 8.0 10.0 −0.59434 −0.95478 −1.86908 −2.50014 −3.24213 −3.66038 −0.5946 −0.9548 −1.8691 −2.5002 −3.2423 −3.6606
K. Sharada, B. Shankar Table 2. Effects of s k i n f r icti o n co ef f icien t, h eat and mass tr an sfer coefficients fo r various values o f λ, X, Df , Nr, Sr, γ and β. Λ X Nr Sr γ Β (1 + 1/β)"(0) −θ'(0) −Ф'(0) 1.0 2.0 3.0 0.5 0.5 0.5 1.0 1.0 1.0 0.5 0.5 0.5 0.03 0.03 0.03 0.1 0.1 0.1 0.5 0.5 0.5 −1.9167 −1.1259 −0.3918 0.6505 0.7017 0.7383 0.4246 0.4560 0.4797 1.0 1.0 1.0 0.2 1.0 2.0 1.0 1.0 1.0 0.5 0.5 0.5 0.03 0.03 0.03 0.1 0.1 0.1 0.5 0.5 0.5 −1.4607 −2.3684 −2.7091 0.6870 0.5841 0.3124 0.4431 0.4044 0.4054 1.0 1.0 1.0 0.5 0.5 0.5 1.5 3.0 4.6 0.5 0.5 0.5 0.03 0.03 0.03 0.1 0.1 0.1 0.5 0.5 0.5 −1.8828 −1.8174 −1.7766 0.5637 0.4204 0.3429 0.4371 0.4593 0.4722 1.0 1.0 1.0 0.5 0.5 0.5 1.0 1.0 1.0 1.0 3.0 5.0 0.03 0.03 0.03 0.1 0.1 0.1 0.5 0.5 0.5 −1.9071 −1.8687 −1.8309 0.6527 0.6612 0.6692 0.3779 0.1894 −0.0019 1.0 1.0 1.0 0.5 0.5 0.5 1.0 1.0 1.0 0.5 0.5 0.5 1.0 2.0 3.0 0.1 0.1 0.1 0.5 0.5 0.5 −1.8774 −1.8372 −1.7967 0.5596 0.4601 0.3535 0.4387 0.4530 0.4675 1.0 1.0 1.0 0.5 0.5 0.5 1.0 1.0 1.0 0.5 0.5 0.5 0.03 0.03 0.03 0.15 0.25 0.35 0.5 0.5 0.5 −1.9203 −1.9266 −1.9326 0.6496 0.6480 0.6466 0.4394 0.4670 0.4927 1.0 1.0 1.0 0.5 0.5 0.5 1.0 1.0 1.0 0.5 0.5 0.5 0.03 0.03 0.03 0.1 0.1 0.1 0.8 1.0 1.5 −1.6085 −1.4976 −1.3415 0.6370 0.6315 0.6233 0.4171 0.4143 0.4101 Figure 2. Velocity profiles for various values of mixed con vection parameter λ. opposing flow. As λ increases, the convection cooling effect increases and therefore temperature decreases. Figure 3, Figure 4 reveal that the dimensionless temperature and concentration reduces with increase of λ. Figure 5 depicts the effects of Casson parameter β on dimensionless velocity profiles for impermeable stretching surface. The velocity is found to decrease with the increasing Casson parameter β. Figure 6, Figure 7 present the temperature and concentration profiles for Casson parameter β. It shows that the both temperature and concentration profiles decreasing with increasing of β. Figure 8 illustrates the velocity profile for different values of the magnetic parameter M. The presence of magnetic field reduces the momentum boundary layer
K. Sharada, B. Shankar Figure 3. Temperature profiles for various values of mixed convection parameter λ. Figure 4. Concentration profiles for various values of mixed convection parameter λ. Figure 5. Velocity profiles for various values of Casson pa rameter β.
K. Sharada, B. Shankar Figure 6. Temperature profiles for various values of Casson parameter β. Figure 7. Concentration profiles for various values of Casson parameter β. Figure 8. Velocity profile for various values of magnetic pa rameter M.
K. Sharada, B. Shankar thickness. Thus the velocity decreases as the magnetic parameter increases. Increasing the values of the Ec generates heat in the fluid due to frictional heating. Figure 9 demonstrates that influence of the Ec on the profile of temperature. It is observed that the temperature profile increases as the value of the Ec increases. The effects of Prandtl number on temperature profiles are shown in Figure 10. Prandtl number can be used to increase the rate of cooling conducting fluids. Prandtl number signifies the ratio of mo mentum diffusivity to thermal diffus ivity. Temperature is found to decrease w ith increasing Pr. Pr = 1, indicates that the velocity boundary layer and thermal boundary layer are approximately equal. Figure 11 displays the effect of thermal radiation of the temperature profiles. From this we observe that the temperature increases with increasing value of the parameter Nr. Figure 12 shows that the effect of Soret on concentration profile. An increase in the Soret number increases the concentration and the boundary layer thickness. It is observed that the concentration profile increases as the Sr increases. Figures 1315 explain the effects of the Xlocation on the velocity, temperature and concentration profiles. From Figure 13, it is observed that the velocity profile decreases with an increase in X in the boundary layer. Figu re 14 depicts that as the value of X increases, the temperature increases. The concentration profile increases with as increasing the value of X. The effect of on temperature profile is de picted in Figure 16. The temperature profile increases as increases. Figure 17 displays the nature of concentration profiles for various values of chemical reaction Figure 9. Temperature profile for various values of Eckert number Ec. Figure 10. Temperature profile for various values of number Pr.
K. Sharada, B. Shankar Figure 11. Temperat ure profile for various values of radiation parameter Nr. Figure 12. Concentration profile for various values of Soret number Sr. Figure 13. Velocity profile for various values of Xlocation.
K. Sharada, B. Shankar Figure 14. Temperature profile for various values of Xloca tion. Figure 15. Concentration profile for various values of Xlo cation. Figure 16. Temperature profile for various values of Dufour number Df.
K. Sharada, B. Shankar Figure 17. Concentration profile for various values of chem ical reaction parameter γ. parameter γ. Chemical reaction increases the rate of interfacial mass transfer and reduces the local concentration. Therefore the concentration profile decreases with an increase of the parameter γ. 6. Conclusions The numerical solutions for MHD mixed convection flow of a Casson fluid over an exponentially stre tchin g sur face with Soret and Dufour, thermal radiation , chemical reaction are analyzed. The results are presented graphi cally with various parameters and the values are agreed with previously published work. From the graphical re presentations, we have the foll owing obser vations: • The effect of Casson fluid parameter when treated as fluid with variable plastic dynamic viscosity, the veloc ity profile decreases, temperature and concentration distributions are increase d. • The temperature increases with increasing values of the radiation parameter and Dufour number . • An increase in the mixed convection parameter λ increases the velocity p rofiles, and decreases the tempera ture profiles and concent ra tion profi les. • An increase in the Xlocation reduces the nondimensional velocity increase in the temperature and concen tration distribution. • With increasing values of chemical reaction parameter γ, the velocity profile decreases. An increase in the Soret number increases the concentration profile and the boundary layer thickness. References [1] Garnier, M. (1962) Magneto Hydrodynamics in Materials Processing. Philosophical Transactions of the Royal Society A, 344, 249263. http://dx.doi.org/10.1098/rsta.1993.0090 [2] Sakiadas, B.C. (1961) BoundaryLayer Behavior on Continuous Solid Surfaces. AIChE Journal, 7, 2628. http://dx.doi.org/10.1002/aic.690070108 [3] Crane, L.J. (1970) Flow Past a Stretching Plate. Zeitschrift für Angewandte Mathematik und Physik, 21, 590595. http://dx.doi.org/10.1007/BF01587695 [4] Magyari , E. and Keller, B. (1999) Heat and Mass Transfer in the Boundary Layers on an Exponentially Stretching Continuous Surface. Journal of Physics D, 32, 577585. http://dx.doi.org/10.1088/00223727/32/5/012 [5] Bidin, B. and Nazar, R. (2009) Numerical Solution of the Boundary Layer Flow over an Exponentially Stretching Sheet with Thermal Radiation. European Journal of Scientific Research, 33, 710717. [6] Sajid, M. and Hayat, T. (2008) Influence of Thermal Radiation on the Boundary Layer Flow Due to an Exponentially Stretching Sheet. International Communications in Heat and Mass Transfer, 35, 347356. http://dx.doi.org/10.1016/j.icheatmasstransfer.2007.08.006 [7] Merrill, F.W. (1965) Rheol ogy of Human Blood and Some Speculations on Its Role in Vascular Homeostatics Biome chanical Mechanisms in Vascular Homeostatics and Intravascular Thrombosis. In: Sawyer, P.N., Ed., Appleton Cen
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