View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by ePrints@Bangalore University Journal of Applied Fluid Mechanics, Vol. 7, No. 2, pp. 367-374, 2014. Available online at www.jafmonline.net, ISSN 1735-3572, EISSN 1735-3645. Flow and Heat Transfer in a Newtonian Liquid with Temperature Dependent Properties over an Exponential Stretching Sheet P. G. Siddheshwar1, G. N. Sekhar2 and A. S. Chethan3† 1Department of Mathematics, Bangalore University, Central College Campus,560 001, Bangalore, India 2Department of Mathematics, BMS College of Engineering, Bangalore, 560 019, Karnataka, India 3Department of Mathematics, BMS Institute of Technology, Bangalore, 560 064 Karnataka, India †Corresponding Author Email: [email protected] (Received March 28, 2013; accepted May 28, 2013) ABSTRACT The paper presents a study of a forced flow and heat transfer of an electrically conducting Newtonian fluid due to an exponentially stretching sheet. The governing coupled, non-linear, partial differential equations are converted into coupled, non-linear, ordinary differential equations by a similarity transformation and are solved numerically using shooting method. The influence of various parameters such as the Prandtl number, Chandrasekhar number, variable viscosity parameter, heat source (sink) parameter and suction/injection on velocity and temperature profiles are presented and discussed. Keywords: Stretching sheet, Variable viscosity, Prandtl number, Chandrasekhar number, Shooting Method, Heat transport, Heat source. NOMENCLATURE A, D prescribed constants x flow directional co-ordinate along the Cp specific heat at constant pressure stretching sheet H0 applied uniform vertical magnetic field y distance normal to the stretching sheet H heat source (sink) parameter X, Y dimensionless co-ordinates Hsx local heat source (sink) parameter kinematic viscosity k thermal conductivity dynamic viscosity Pr Prandtl number m magnetic permeability Q Chandrasekhar number dimensionless temperature in PEST case Qs heat source parameter dimensionless temperature in PEHF case Q local Chandrasekhar number x density of the fluid t fluid temperature of the moving sheet electrical conductivity t wall temperature w dimensionless stream function t temperature far away from the sheet Subscripts u, v velocity components along x and y m magnetic quantity direction w wall temperature U, V non-dimensional velocity components along x ∞ ambient temperature condition and y direction non-dimensional temperature in PEST V variable viscosity parameter non-dimensional temperature in PEHF P. G. Siddheshwar et al. / JAFM, Vol. 7, No. 2, pp. 367-374, 2014. boundary layer flow due to an exponentially 1. INTRODUCTION stretching sheet. Sekhar and Chethan (2012) analyzed the flow and heat transfer due to an Flows due to a continuously moving surface are exponentially stretching continuous surface in the encountered in several important engineering presence of Boussinesq-Stokes suspension. applications .viz, in the polymer processing unit of a Siddheshwar et al. (2014) extended this problem by chemical engineering plant, annealing of copper including the effect of the transverse magnetic field. wires, glass fiber and drawing of plastic films. In the present work, we study the boundary layer Sakiadis (1961 a, b, c) initiated the theoretical flow behavior and heat transfer of a Newtonian study of these applications by considering the fluid past an exponentially stretching sheet, when boundary layer flow over a continuous solid surface viscosity is a function of temperature and in the moving with constant speed. This problem was presence of external magnetic field. extended by Erickson et al. (1969) to the case where the transverse velocity at the moving surface is non- zero with heat and mass transfer in the boundary 2. MATHEMATICAL FORMULATION layer accounted for. We consider a steady, two-dimensional boundary Crane (1970) studied the steady two-dimensional layer flow of an incompressible, weakly electrically boundary layer flow caused by the stretching sheet, conducting Newtonian fluid due to a stretching which moves in its own plane with a velocity which sheet. The liquid is at rest and the motion is affected varies linearly with the axial distance. There after by pulling the sheet at both ends with equal force various aspects of the above boundary layer parallel to the sheet and with speed u, which varies problem on continuous moving surface were exponentially with the distance x from the origin. considered by many researchers (Vleggar (1977), The boundary layer equations governing the flow Gupta and Gupta (1977), Grubka and Bobba (1985), and heat transfer in a Newtonian fluid over a Chen and Char (1988) Kumaran and Ramanaiah stretching sheet, assuming that the viscous (1996), Siddheshwar et al. (2005) and Sekhar and dissipation is negligible, are Chethan (2010)). uv 0, (2.1) Many metallurgical processes involve the cooling of xy continuous strips or filaments by drawing them through a quiescent fluid. During this process of u u t u 22H drawing the strips are sometimes stretched. The u v m 0 u, (2.2) properties of final product depend on the rate of x y y y cooling. Pavlov (1974) examined the flow of an electrically conducting fluid caused solely by the 2 t t k t (2.3) stretching of an elastic sheet in the presence of a u v Qs t t . x y C y2 uniform magnetic field. Chakrabarthi and Gupta p (1979) considered the flow and heat transfer of an Here u and v are the components of the liquid electrically conducting fluid past a porous stretching velocity in the x and y directions, respectively, t is sheet. Anderson (1992) presented an analytical the temperature of the sheet, t is the temperature of solution of the magnetohydrodynamic flow using a the fluid far away from the sheet, is the dynamic similarity transformation for the velocity and temperature fields. In all the above mentioned viscosity, m is the magnetic permeability, H0 is the studies the physical properties of the ambient fluid applied magnetic field, is the density, is the were assumed to be constants. However, it is well electric conductivity of the fluid, k is the thermal known that these physical properties of the ambient conductivity, Cp is the specific heat at constant fluid may change with temperature (Herwig and pressure and Qs is the heat source coefficient. Wickern (1986), Takhar et al. (1991), Pop et al. The coefficient of viscosity is assumed to be a (1992), Subhash Abel et al. (2002), Pantokratoras reciprocal function of temperature and it is of the (2004), Ali (2006), Andersson and Aaresth (2007), form Prasad et al. (2009), Sekhar and Chethan (2010)). t . Magyari and Keller (2000) studied the heat and 1 tt mass transfer on the boundary layer flow due to an exponentially stretching surface. Elbashbeshy 1 (2001) added new dimension to the study on If is expanded in Taylor’s series about tt exponentially stretching surface. Partha et al. (2004) have examined the mixed convection flow and heat then the scalar appearing in the above expression transfer from an exponentially stretching vertical 1 surface in quiescent liquid using a similarity can be written as . t solution. Heat and mass transfer in a viscoelastic tt boundary layer flow over an exponentially stretching sheet were investigated by Khan and Here ∞ is the coefficient of viscosity far away from Sanjayanand (2005; 2006). Sajid and Hayat (2008) the sheet considered the influence of thermal radiation on the The following boundary conditions are used. 368 P. G. Siddheshwar et al. / JAFM, Vol. 7, No. 2, pp. 367-374, 2014. x Using Eq. (2.10), the boundary layer equations Eqs. L 2x t tw t Ae in PEST case (2.7) and (2.8) can be written as L uUxUe,vv, 3x aty, 0 wc0 t k De2L in PEHF case 3 2 2 y T 2 w 1+VVV T + 1+ T 3YX 2 2 u0 ,T T as y . YYY 2 (2.4) 22 1+V T 1+ V T Q 0, where YXYY x (2.11) AeL in PEST case ttw x 2 DL TTT 1 (2.12) 2L -U.THT e in PEHF case 2 s k Re Y X X Y Pr Y The corresponding boundary conditions in terms of where tw is the temperature of the sheet, U0 is the reference velocity and L is the reference length. the stream function can be written as We now make the equations and boundary T 1 in PEST X conditions dimensionless using the following e , V , at Y 0 , YXc T X definition: e in PEHF Y (2.13) x, y u,v ,v Re Rec Re tt X,Y ,, U,V,Vc T L U0 t 00,T as Y . (2.5) Y UL where Re 0 is the Reynolds number and The following similarity transformation will now be used on Eqs. (2.11) and (2.12). t tw t is the sheet-liquid temperature difference. in PEST case X x, f e , T X ,Y , The boundary layer Eqs. (2.1) to (2.3) on using Eq. in PEHF case (2.5) take the following form. Y eX . UV (2.14) 0, (2.6) XY Using the transformations given by Eq. (2.14) in Eqs. (2.11) and (2.12), we get the following 2 UVTUU V1 U V QU , boundary value problems. 22 XYYYT 1V T 1V Y (i) PST: (2.7) 2 2 2 1+ V f V f 1 + V f f 2 f Q f 0 (2.15) TTT 1 x UVUH (2.8) TT2 s X Y Pr Y (2.16) where Pr f Pr f Pr Hsx 0, V t is the variable viscosity parameter, f (00 ) V , f ( ) 1 , 0 1, cx (2.17) 2 H 2 m 0 is the Chandrasekhar number, f ( ) 00 , . Q Cp (ii) PHF: Pr is the Prandtl number and k 2 1+V f V f1+ V ff 2 fQf 2 0 (2.18) Q x H s is the heat source (sink) parameter.