EFFECTS OF THERMAL RADIATION AND HEAT TRANSFER OVER AN UNSTEADY STRETCHING SURFACE EMBEDDED IN A POROUS MEDIUM IN THE PRESENCE OF HEAT SOURCE OR SINK

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1 THERMAL SCIENCE, Year 011, Vol. 15, No., pp EFFECTS OF THERMAL RADIATION AND HEAT TRANSFER OVER AN UNSTEADY STRETCHING SURFACE EMBEDDED IN A POROUS MEDIUM IN THE PRESENCE OF HEAT SOURCE OR SINK b Elsaed M. A. ELBASHBESHY a and Tarek G. EMAM b* a Department of Mathematics, Facult of Science, Ain Shams Universit, Abbassia, Cairo, Egpt b Department of Mathematics, The German Universit in Cairo, Ne Cairo, Egpt Introduction Original scientific paper UDC: /.97:53.546:536.4 DOI: 10.98/TSCI110477E The effects of thermal radiation and heat transfer over an unstead stretching surface embedded in a porous medium in the presence of heat source or sink are studied. The governing time dependent boundar laer equations are transformed to ordinar differential equations containing radiation parameter, permeabilit parameter, heat source or sink parameter, Prandtl number, and unsteadiness parameter. These equations are solved numericall b appling Nachtsheim-Singer shooting iteration technique together ith Runge-Kutta fourth order integration scheme. The velocit profiles, temperature profiles, the skin friction coefficient, and the rate of heat transfer are computed and discussed in details for various values of the different parameters. Comparison of the obtained numerical results is made ith previousl published results. Ke ords: boundar laer, stretching surface, porous medium, thermal radiation, heat source or sink Boundar laer flo on a continuous moving surface has man practical applications in industrial manufacturing processes. Examples for such applications are: aerodnamic extrusion of plastic sheets, cooling of infinite metallic plate in a cooling bath, the boundar laer along a liquid film in condensation processes, and a polmer sheet or filament extruded continuousl from a de. Applications also include paper production and glass bloing. It is ver important to control the drag and the heat flux for better product qualit. The continuous moving surface heat transfer problem has man practical applications in industrial manufacturing processes. Since the pioneering ork of Sakiadis [1, ], various aspects of the problem have been investigated b man authors. Most studies have been concerned ith constant surface velocit and temperature (see Tsou et al. [3]), but for man practical applications the surface undergoes stretching and cooling or heating that cause surface velocit and temperature variations. *ncorresponding author; tarek.emam@guc.edu.eg

2 478 THERMAL SCIENCE, Year 011, Vol. 15, No., pp Crane [4], Velggaar [5], and Gupta [6] have analzed the stretching problem ith a constant surface temperature, hile Soundalgekar et al. [7] have investigated the constant surface velocit case ith a poer-la temperature variation. Grubka et al. [8] have analzed the stretching problem for a surface moving ith a linear velocit and ith a variable surface temperature. Ali [9] has reported flo and heat characteristics on a stretched surface subject to poer-la velocit and temperature distributions. The flo field of a stretching surface ith poer-la velocit variations as discussed b Banks [10]. Ali [11] and Elbashbesh [1] extended Banks's ork for a porous stretched surface for different values of the injection. Elbashbesh [13] have analzed the stretching problem hich as discussed b Elbashbesh [1] to include a uniform porous medium. The unstead heat transfer problems over a stretching surface, hich is stretched ith a velocit that depends on time are considered b Andersson et al. [14], a ne similarit solution for the temperature field is devised, hich transforms the time dependent thermal energ equation to an ordinar differential equation. Elbashbesh et al. [15] studied the heat transfer over an unstead stretching surface. Recentl, Ishak et al. [16] have studied the heat transfer over an unstead stretching vertical surface. Ishak et al. [17] have also investigated the unstead laminar boundar laer over a continuousl stretching permeable surface, hile Ali et al. [18] have presented a stud of homotop analsis of unstead boundar laer flo adjacent to permeable stretching surface in a porous medium. On the other hand, the effect of thermal radiation on boundar laer flo and heat transform problems can be quite significant at high operating temperature. In vie of this Elbashbesh et al. [19] and Hossain et al. [0, 1] have studied the thermal radiation of a gra fluid hich is emitting and absorbing radiation in non-scattering medium. Later, Chen [] analzed mixed convection of a poer-la fluid past a stretching surface in the presence of thermal radiation and magnetic field. Battaler [3] has recentl studied the effect of thermal radiation on the laminar boundar laer about a flat-plate. Fang et al. [4] have recentl considered the thermal boundar laer over a shrinking sheet. The have obtained an analtic solution for the boundar laer energ equation for to cases including a prescribed poerla all temperature case and a prescribed poer-la all heat flux case. The unstead viscous flo over a continuousl shrinking surface ith mass transfer has been studied b Fang et al. [5]. Ali et al. [6] have studied the problem of unstead fluid and heat flo induced b a submerged stretching surface hile its stead motion is sloed don graduall. Sharma et al. [7] have studied the effects of Ohmic heating and viscous dissipation on stead magnetohdrodnamics flo near stagnation point on an isothermal stretching sheet. The present ork is to stud heat transfer over an unstead stretching surface embedded in a porous medium in the presence of thermal radiation and heat source or sink. It ma be remarked that the present analsis is an extension of and a complement to the earlier papers [15] and [17]. Formulation of the problem Consider an unstead, to dimensional flo on a continuous stretching surface embedded in a porous medium, ith surface temperature T and velocit U = bx/(1 gt) [14]). The x-axis is taken along the continuous surface in the direction of the motion ith the slot as the origin, and -axis is perpendicular to it (see fig. 1). The conservation equations of the laminar boundar laer are:

3 THERMAL SCIENCE, Year 011, Vol. 15, No., pp u x v 0 (1) u u u u t x K u v u T T T T u v t x c 1 qr Q ( T T ) c c p p p () (3) ith the associated boundar conditions: Figure 1. Phusical model and co-ordinate sstem = 0: u = U (x,t), n = 0, T = T (x,t) : u = 0, T = T here u and v are the velocit components in the x- and -directions, respectivel, t is the time, n the kinematic viscosit, K the permeabilit, T the temperature inside the boundar laer, k the thermal conductivit, r the fluid densit, c p the specific heat at constant pressure, Q the heat source hen Q > 0 or heat sink hen Q < 0, T the surface temperature, and T the free stream temperature. It is assumed that the viscous dissipation is neglected; the phsical properties of the fluid are constants. Using the Rosseland approximation for radiation [], radiative heat flux is simplified as: 4 T 4 qr (5) 3 * here s and a* are the Stefan-Boltzman constant and the mean absorption coefficient, respectivel. We assume that the temperature differences ithin the flo are such that the term T 4 ma be expressed as a linear function of temperature. Hence, expanding T 4 in a Talor series about T and neglecting higher-order terms e get: Using eqs. (5) and (6), the energ eq. (3) becomes: (4) T 4 4T 3T 3T 4 (6) 16 T u v ( T T ) t x c 3 c c T T T T 3 T Q * p p p The equation of continuit is satisfied if e choose a stream function (x,) such that u = /, and v = / x. The mathematical analsis of the problem is simplified b introducing the folloing dimensionless co-ordinates: (7) b (1 t ) (8)

4 480 THERMAL SCIENCE, Year 011, Vol. 15, No., pp b ( x, ) x f ( ) 1 t (9) 1 ( ) T T, T T b T T x (1 t ) 3 Substituting eqs. (8)-(10) into eqs. () and (3), e obtain: 1 (10) f ff f A f f f 0 (11) 4 A 1 Pr f f (3 ) 0 3R (1) ith the boundar conditions: 0 : f 0 f 1, 1 : f 0, 0 here the prime denotes differentiation ith respect to h. A = g/b is the parameter that measures the unsteadiness, l = (n /K)Re x the permeabilit parameter, Re x = U x/n the local Renolds number, R = ka/4st 3 the thermal radiation parameter, Pr = mc p /k the Prandtl number, d = (Qk/mc p )(Re x / Re k ) the dimensionless heat source or sink, m the dnamic viscosit, and Re k =U k k 1/ /n. Numerical solutions and discussions Equations (11) and (1) ith the boundar conditions (13) are solved using Runge- Kutta fourth order technique along ith shooting technique. We first convert the to eqs. (11) and (1) into the folloing simultaneous linear equations of first order: here (13) 1 (14) 3 (15) 3 A( ) 1 3 (16) 4 5 (17) 3RPr 5 [0.5 A (3 4 5 ) ] 4 3R 1 f f, 3 f, 4,, Then the shooting technique is applied to transform the problem into initial value one. Where the initial conditions are: 5 (18) (0) 0, (0) 1, 4 (0) 1, 3 (0) m, 5 (0) n (19)

5 THERMAL SCIENCE, Year 011, Vol. 15, No., pp Hence m and n are a priori unknon and have be determined as a part of the numerical solution. Once the problem is reduced to initial value problem, then it is solved using Runge-Kutta fourth order technique. The computations have been carried out for various values of thermal radiation R, heat source or sink parameter d, permeabilit parameter l, Prandtl number Pr, and unsteadiness parameter A. The accurac of the numerical method is checked b performing various comparisons at different conditions ith previousl published orks. The parameters of phsical interest for the present problem are the local skin friction coefficient C f and the local Nusselt number Nu x, hich are defined as: or C f f u C f T 0 0, Nu x U T T Nu x Re x (0), (0). Rex To validate the numerical method used in this stud, the stead-state flo case A = 0, l = 1/d = 1/R = 0, and Pr = 1 as considered and the results for the heat transfer rate at the surface q (0) are compared ith those reported in references [8, 9, 15, and 17]. The quantitative comparison is shon in tab. 1 and found to be in a ver good agreement. Table 1. Comparison of q (0) for A = l = d = 1/R = 0 and Pr = 1 Present results Grubka et al. [8] Ali [9] Elbashbesh [15] Ishak et al. [17] From tab., e note that, the effect of permeabilit parameter,, is to decrease the skin friction due to internal heat absorption parameter 0. It is interesting to note that the rate of heat transfer decreases ith increasing the permeabilit parameter. Table. The values of f (0) and Nu x /(Re x ) 1/ for various values of l ith Pr = 10, A = 0.8, d = 0.5, and R = 0.3 l f (0) Nu x /(Re x ) 1/ Also, e note from tab. 3 that the surface gradient f (0) decreases ith the increase of the unsteadiness parameter A, that that rate of heat transfer decreases ith unsteadiness parameter. Table 3. The values of f (0) and Nu x /(Re x ) 1/ for various values of A ith Pr = 10, l = 0.1, d = 0.5, and R = 0.3 A f (0) Nu x /(Re x ) 1/ x

6 48 THERMAL SCIENCE, Year 011, Vol. 15, No., pp From tables 4, 5, and 6, e note that the effect of the radiation parameter R, is to decrease the rate of heat transfer hile the rate of heat transfer increases ith the heat source parameter and decreases ith heat sink parameter. Further it is noted that the rate of heat transfer at the surface decreases ith Prandtl number. It is also evident that the temperature gradient q (0) is negative for all parameters values considered in this stud hich means that there is a heat flo from the all. Table 4. The values of f (0) and Nu x /(Re x ) 1/ for various values of R ith Pr = 10, l = 0.1, d = 0., and A = 0.8 R f (0) Nu x /(Re x ) 1/ Table 5. The values of f (0) and Nu x /(Re x ) 1/ for various values of d ith Pr = 10, l = 0.1, R = 0.3, and A = 0.8 d f (0) Nu x /(Re x ) 1/ Table 6. The values of f (0) and Nu x /(Re x ) 1/ for various values of Pr ith d = 0., l = 0.1, R = 0.7, and A = 0.4 Pr f (0) Nu x /(Re x ) 1/ Figure. Velocit profiles f (h) for various values of A ith Pr = 10, R = 0.3, l = 0.1, and d = 0.5 Figure 3. Velocit profiles f (h) for various values of l ith Pr = 10, R = 0.3, A = 0.8, and d = 0.5 Figures and 3 present the velocit profiles, for various values of A and l, respectivel, hile the other parameters are kept constant. From these figures, e note that the velocit decreases ith increasing the value of the unsteadiness parameter A and the permeabilit parameter l. Figures 4-7 present the temperature profiles for various values of A, d, R, and Pr, respectivel, hile the other parameters are kept constant. From these figures, e note that the temperature decreases ith the increase of the value of the unsteadiness parameter A, the radiation parameter R, and the Prandtl number Pr, hile the temperature increases ith increasing the value of the heat source or sink parameter d.

7 THERMAL SCIENCE, Year 011, Vol. 15, No., pp Figure 4. Velocit profiles q(h) for various values of A ith Pr = 10, R = 0.3, l = 0.1, and d = 0.5 Figure 5. Velocit profiles q(h) for various values of d ith Pr = 10, R = 0.3, l = 0.1, and A = 0.8 Figure 6. Temperature profiles q(h) for various values of R ith Pr = 10, d = 0., l = 0.1, and A = 0.4 Figure 7. Temperature profiles q(h) for various values of Pr ith R = 0.7, l = 0.1, d = 0., and A = 0.4 Conclusions Numerical solutions have been obtained for the effects of the thermal radiation and heat transfer over an unstead stretching surface embedded in a porous medium in the presence of heat source or sink. An appropriate similarit transformed as used to transform the sstem of time dependent partial differential equations to a set of ordinar differential equations. These equations are solved numericall b appling the Nachtsheim-Singer shooting technique together ith Runge-Kutta fourth order integration scheme. Numerical computations sho that the present values of the rate of heat transfer are in a close agreement ith those obtained b previous investigation in the absence of porous medium, thermal radiation, heat source or sink, and unsteadiness parameter. The folloing results are obtained: the velocit decreases ith an increase in the value of unsteadiness parameter and permeabilit parameter, the temperature decreases ith an increase in the value of unsteadiness parameter, radiation parameter, and Prandtl number hile it increases ith an increase in the value of heat source or sink parameter,

8 484 THERMAL SCIENCE, Year 011, Vol. 15, No., pp the surface gradient f (0) is negative and decreases ith increasing unsteadiness parameter and permeabilit parameter, and the rate of heat transfer q (0) is negative decreases ith increasing the unsteadiness parameter, permeabilit parameter, and radiation parameter and increases ith increasing heat source or sink parameter. Nomenclature A unsteadiness parameter (= g/b), [ ] b positive constant, [s 1 ] C f local skin-friction coefficient, (= t W /r U ) c P specific heat due to constant pressure, [Jkg 1 K 1 ] f stream function, [ ] K Permeabilit, [m ] Nu x Nusselt number, [ ] Pr Prandtl number (= mc p /k), [ ] Q heat source or sink q r radiation heat flux, [= ( 4s/3a*)( T 4 / )], [kgm ] R thermal radiation parameter, (= ka/4st 3 ), [ ] Re x local Renolds number (= U W x/n), [ ] T temperature of the fluid, [K] t time, [s] T surface temperature, [K] T free stream temperature, [K] U surface velocit, [ms 1 ] u fluid velocit in x-direction, [ms 1 ] v fluid velocit in -direction, [ms 1 ] x, Cartesian co-ordinates along the surface and normal to it, respectivel, [m] Greek smbols * mean absorption coefficient, [m 1 ] stretching rate, [s 1 ] heat source or sink, [= (Qn /mc p )(Re x / U m )], [ ] similarit variable, [ ] similarit temperature function, [ ] thermal conductivit, [kgms 3 K 1 ] permeabilit parameter, (= n /KRe x ), [m s ] dnamic viscosit of the fluid, [Nsm ] kinematic viscosit, (= m/r), [m s 1 ] densit of fluid, [kgm 3 ] Stefan-Boltzman constant, [kgm K 4 ] stream function, [m s 1 ] Superscript differentiation ith respect to h Subscripts surface conditions conditions far aa from the surface References [1] Sakiadis, B. C., Boundar Laer Behaviour on Continuous Solid Surface I. Boundar-Laer Equations for To Dimensional and Axismmetric Flo, AIChE J., 7 (1961), 1, pp. 6-8 [] Sakiadis, B. C., Boundar Laer Behaviour on Continuous Solid Surface II, Boundar Laer Behaviuor on Continuous Flat Surface, AIChE J., 7 (1961) 1, pp [3] Tsou, F. K., Sparro, E. M., Goldstien, R. J., Flo and Heat Transfer in the Boundar Laer on a Continuous Moving Surface, Int. J Heat Mass Transfer, 10 (1967),, pp [4] Crane, L. J., Flo Past a Stretching Plate, Z Ange, Math. Phs., 1 (1970), 4, pp [5] Vleggaor, J., Laminar Boundar Laer Behaviour on Continuous Accelerating Surfaces, Chem. Eng. Sci., 3 (1977), 1, pp [6] Gupta, P. S., Gupta, A. S., Heat and Mass Transfer on a Stretching Sheet ith Suction or Bloing, Canadian J. Chem. Eng., 55 (1979), 6, pp [7] Soundalgekar, V. M., Ramana, T. V., Heat Transfer Past a Continuous Moving Plate ith Variable Temperature, Wärme- ünd Stoffübertragung, 14 (1980),, pp [8] Grubka, L. J., Bobba, K. M., Heat Transfer Characteristics of a Continuous Stretching Surface ith Variable Temperature, J. Heat Transfer, 107 (1985), 1, pp [9] Ali, M. E., Heat Transfer Characteristics of a Continuous Stretching Surface, Wärme und Stoffübertragung, 9 (1994), 4, pp. 7-34

9 THERMAL SCIENCE, Year 011, Vol. 15, No., pp [10] Banks, W. H. H., Similarit Solutions of the Boundar Laer Equation for a Stretching Wall, J. Mac. Theo. Appl., (1983), 3, pp [11] Ali, M. E., On Thermal Boundar Laer on a Poer La Stretched Surface ith Suction or Injection, Int. J. Heat Mass Flo, 16 (1995), 4, pp [1] Elbashbesh, E. M. A., Heat Transfer Over a Stretching Surface ith Variable Heat Fux, J. Phsics D: Appl. Phsics., 31 (1998), 16, pp [13] Elbashbesh, E. M. A., Bazid, M. A. A., Heat Transfer over a Continuousl Moving Plate Embedded in a Non-Darcian Porous Medium, Int. J. Heat and Mass Transfer, 43 (000), 17, pp [14] Andersson, H. T., Aarseth, J. B., Dandapat, B. S., Heat Transfer in a Liquid Film on an Unstead Stretching Surface, Int. J. Heat Transfer, 43 (000), 1, pp [15] Elbashbesh, E. M. A., Bazid, M. A. A., Heat Transfer over an Unstead Stretching Surface, Heat Mass Transfer, 41 (004), 1, pp. 1-4 [16] Ishak, A., Nazar, R., Pop, I., Heat Transfer over an Unstead Stretching Surface ith Prescribed Heat Flux, Can. J. Phs., 86 (008), 6, pp [17] Ishak, A., Nazar, R., Pop, I., Heat Transfer over an Unstead Stretching Permeable Surface ith Prescribed Wall Temperature, Non-linear Analsis: Real World Applications, 10 (009), 5, pp [18] Ali, A., Mehmood, A., Homotop Analsis of Unstead Boundar Laer Flo Adjacent to Permeable Stretching Surface in a Porous Medium, Commun. Nonlinear Sci. Numer. Simul., 13 (008),, pp [19] Elbashbesh, E. M. A., Dimian, M. F., Effects of Radiation on the Flo and Heat Transfer over a Wedge ith Variable Viscosit, Appl. Math. Comp., 13 (00), -3, pp [0] Hossain, M. A., Alim, M. A., Rees, D., The Effect of Radiation on Free Convection from a Porous Vertical Plate, Int. J. Heat Mass Transfer, 4 (1999), 1, pp [1] Hossain, M. A., Khanfer, K., Vafai, K., The Effect of Radiation on Free Convection Flo of Fluid ith Variable Viscosit from a Porous Vertical Plate, Int. J. Therm. Sci., 40 (001),, pp [] Chen, C. H., MHD Mixed Convection of a Poer-La Fluid Past a Stretching Surface in the Presence of Thermal Radiation and Internal Heat Generation /Absorption, Int. J. of Nonlinear Mechanics, 44 (008), 6, pp [3] Bataller, R. C., Radiation Effects in the Blasius Flo, Appl. Math. Comput., 198 (008), 1, pp [4] Fang, T, Zhang, J., Thermal Boundar Laers over a Shrinking Sheet: An Analtical Solution, Acta Mechanica, 09 (010), 3-4, pp [5] Fang, T.-G., Zhang, J., Yao, S.-S., Viscous Flo over an Unstead Shrinking Sheet ith Mass Transfer, Chin. Phs. Lett., 6 (009), 1, pp [6] Ali, M. E., Magari, E., Unstead Fluid and Heat Flo Induced b a Submerged Stretching Surface hile its Stead Motion is Sloed Don Graduall, Int. J. Heat Mass Transf., 50 (007), 1-, pp [7] Shamara, P. R., Effects of Ohmic Heating and Viscous Dissipation on Stead MHD Flo Near a Stagnation Point on an Isothermal Stretching Sheet, Thermal Science, 13 (010), 1, pp. 5-1 Paper submitted: June 6, 009 Paper revised: March 16, 010 Paper accepted: March 5, 010