Available online at ScienceDirect. Procedia Engineering 90 (2014 )

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1 Available online at.sciencedirect.com ScienceDirect Procedia Engineering 9 ( th International Conference on Mechanical Engineering, ICME 13 Effects of volumetric heat source and temperature dependent viscosit on natural convection flo along a av surface M. A. Alim a *, Shahidul Alam b M. Miraj c a Department of Mathematics, Bangladesh Universit of Engineering and Technolog, Dhaka 1, Bangladesh b Department of Mathematics, Northern College Bangladesh, Dhaka, Bangladesh c Department of Mathematics, Dhaka Commerce College, Dhaka-116, Bangladesh Abstract The conjugate effects of volumetric heat source and temperature dependent viscosit on natural convection flo along a av surface have been investigated. The governing boundar laer equations of the present phsical problem are first transformed into non-dimensional form using suitable set of dimensionless variables. The resulting nonlinear sstem of partial differential equations are mapped into the domain of a vertical flat plate and then solved numericall emploing the implicit finite difference method, knon as Keller-bo scheme. The numerical results of the surface shear stress in terms of skin friction coefficient as ell as the rate of heat transfer in terms of local Nusselt number are shon in tabular form and the stream lines as ell as the isotherms are shon graphicall for a selection of parameters set consisting of viscosit variation parameter, internal heat generation parameter Q of volumetric heat source and Prandtl number Pr. 14 The Authors. Published b Elsevier b Elsevier Ltd. This Ltd. is an open access article under the CC BY-NC-ND license Selection ( and peer-revie under responsibilit of the Department of Mechanical Engineering, Bangladesh Universit of Engineering Selection and peer-revie and Technolog under responsibilit (BUET. of the Department of Mechanical Engineering, Bangladesh Universit of Engineering and Technolog (BUET Keords: Natural convection; volumetric heat source; internal heat generation; av surface; variable viscosit; finite difference method. * Corresponding author. Tel.: ; fa: address: maalim@math.buet.ac.bd The Authors. Published b Elsevier Ltd. This is an open access article under the CC BY-NC-ND license ( Selection and peer-revie under responsibilit of the Department of Mechanical Engineering, Bangladesh Universit of Engineering and Technolog (BUET doi:1.116/j.proeng

2 384 M.A. Alim et al. / Procedia Engineering 9 ( Introduction In this stud, the effects of temperature dependent viscosit on natural convection flo along a av surface ith internal heat generation due to volumetric heat source have been focused. The sinusoidal av surface can be vieed as an approimation to match practical geometries like cooling fin or roughened surface in heat transfer. Roughened surfaces are better heat transfer devices than a plain surface and are encountered in several heat transfer devices such as flat plate solar collectors, condensers in refrigerators etc. The effects of such non-uniformities on the vertical convective boundar laer flo of a Netonian fluid are first studied b Yao [1] and using an etended Prantdl s transposition theorem and a finite-difference scheme. Alam et al. [] have studied the problem of free convection from a av vertical surface in presence of a transverse magnetic field. Combined effects of thermal and mass diffusion on the natural convection flo of a viscous incompressible fluid along a vertical av surface have been investigated b Hossain and Rees [3]. Hossain et al. [4] have studied the problem of natural convection of fluid ith temperature dependent viscosit along a heated vertical av surface. Natural convection heat and mass transfer along a vertical av surface have been investigated b Jang et al. []. Molla et al. [6] have studied natural convection flo along a vertical av surface ith uniform surface temperature in presence of heat generation/absorption. Tashtoush and Al-Odat [7] investigated magnetic field effect on heat and fluid flo over a av surface ith a variable heat flu. Yao [8] studied natural convection along a vertical comple av surface. Molla and Gorla [9] studied natural convection laminar flo ith temperature dependent viscosit and thermal conductivit along a vertical av surface. Parveen and Alim [1] investigated Joule heating effect on Magnetohdrodnamic natural convection flo along a vertical av surface ith viscosit dependent on temperature. Hossain et al. [11] investigated the natural convection flo past a permeable edge for the fluid having temperature dependent viscosit and thermal conductivit. The present stud is to incorporate the idea of the effects of volumetric heat source and temperature dependent viscosit on natural convection flo along a uniforml heated vertical av surface.. Formulation of the problem Stead, to dimensional natural convection flo of a viscous and incompressible fluid ith variable viscosit along a vertical av surface is considered. The surface temperature of the vertical av surface T is uniform, here T >T. The boundar laer analsis outlined belo allos ( being arbitrar, but detailed numerical ork assumed that the surface ehibits sinusoidal deformations. The av surface ma be described b ( sinn L (1 g L T T u v Fig. 1. Phsical model and coordinate sstem The geometr of the av surface and the to-dimensional Cartesian coordinate sstem are shon in Fig. 1. The conservation equations on the flo field, the continuit, momentum and energ equations can be ritten as: u v (

3 M.A. Alim et al. / Procedia Engineering 9 ( u u 1 p u v 1. ug ( T T v v p u v v 1 1. T T k Q ( T T u v T C C p p (3 (4 ( here (, are the dimensional coordinates along and normal to the tangent of the surface and ( uv, are the velocit components parallel to (,, g is the acceleration due to gravit, p is the dimensional pressure of the fluid, is the densit, is the coefficient of thermal epansion, is the viscosit in the boundar laer, k is the thermal conductivit and C p is the specific heat due to constant pressure. The boundar conditions relevant to the above problem are: u, v, T T at ( and u, T T, p p as (6 here T is the surface temperature, T is the ambient temperature of the fluid and p is the pressure of fluid outside the boundar laer. The variable viscosit chosen in this investigation that is introduced b Charraudeau [1] and used b Hossain et al. [11] as follos: * [1 ( T T ] (7 * here μ is the viscosit of the ambient fluid and T is a constant evaluated at the film 1 f f temperature of the flotf 1( T T. Using Prandtl s transposition theorem to transform the irregular av surface into a flat surface as etended b Yao [] and boundar-laer approimation, the folloing dimensionless variables ere introduced for non-dimensionalizing the governing equations: L L L, Gr, p Gr p, u Gr u, v Gr ( v u, L L d d g ( T T 3 ( T T, Gr L, d d ( T T here is the non-dimensional temperature function and (u, v are the dimensionless velocit components parallel to (,. Introducing the above dimensionless dependent and independent variables, the transformed momentum and energ equations can be ritten as u v 1 4 u u p p u v Gr (1 (1 u (1 u 1 u u p 4 u u ( u v Gr (1 (1 (1 u 1 (1 u v Q Pr here Pr C k p is the Prandtl number, (8 (9 (1 (11 (1 1 Q Q L CpGr is the heat generation parameter and

4 386 M.A. Alim et al. / Procedia Engineering 9 ( * ( T T is the viscosit variation parameter. It can easil be seen that the convection induced b the av surface is described b Eqs. (9 (1. We further notice that, Eq. (11 indicates that the pressure gradient 1 4 along the -direction is OGr (, hich implies that loest order pressure gradient along -direction can be determined from the inviscid flo solution. For the present problem this pressure gradient ( p is zero. 1 4 / Equation (11 further shos that Gr p is O(1 and is determined b the left-hand side of this equation. Thus the eliminating of p/ from the Eqs. (1 and (11 leads to the folloing equation: u u u u u v (1 (1 u (1 The corresponding boundar conditions for the present problem then turn into uv, 1 at and uθ p as (14 No e introduce the folloing transformations to reduce the governing equations to a convenient form: ( f(,,, (, (1 here f(η is the dimensionless stream function, η is the pseudo similarit variable and ψ is the stream function that satisfies the equation (9 and is defined b u, v. Using the transformation Eq. (1 into Eqs. (13 and (1, the folloing sstem of non linear equations are obtained: f f (1 (1 f ff (1 f ( f ( f f f (1 f Q ( f Pr 4 The boundar conditions (14 no take the folloing form: f (, o f(, o, (, o 1, f(,, (, (18 The phsical quantities of principle interest are the skin-friction coefficients C f and the rate of heat transfer in terms of Nusselt number Nu hich can be obtained from: 1 1/4 1/4 f ( Gr / C (1 1 f (, and ( Gr / Nu 1 (, (19 (16 (17 3. Results and discussion Solutions are obtained in terms of the skin-friction coefficient C f and rate of heat transfer Nu respectivel for different values of the relevant phsical parameters. such as viscosit variation parameter, heat generation parameter Q and Prandtl number Pr and these are shon in tabular form in Table 1. Table 1: Skin friction coefficient C f and the local rate of heat transfer Nu hen = 4. for the variation of Prandtl number Pr, heat generation parameter Q and viscosit variation parameter ith =.3. Q =., =. Q =., = 4. Q =., =. Q =., = 4. Pr C f Nu C f Nu C f Nu C f Nu

5 9. 1. M.A. Alim et al. / Procedia Engineering 9 ( It is observed from Table that as the Prandtl number Pr increases, skin friction coefficient C f decreases and the rate of heat transfer Nu increases but the skin friction (C f rise up and the rate of heat transfer Nu reduces for higher values of viscosit variation parameter. It is also found that the for the internal heat generation due to volumetric heat source all friction becomes higher and the rate of heat transfer falls don significantl. 1 (a (b. Q = Q =.. (c (d Fig. Streamlines for (a =., Q =.; (b = 4., Q =.; (c =., Q =. and (d = 4., Q =. hile Pr =.7 and =.3 Q =. 1 (a (b Q =. 1 (c (d Fig. 3 Isotherms for (a =., Q =.; (b = 4., Q =.; (c =., Q =. and (d = 4., Q =. hile Pr =.7 and =.3

6 388 M.A. Alim et al. / Procedia Engineering 9 ( The combined effects of viscosit variation parameter and heat generation parameter Q on the development of streamlines hich are displaed in Figs. (a-(d ith amplitude of the av surface =.3 and Prandtl number Pr =.7. It is observed from Fig. (a that the maimum value of stream function ψ ma is 1.46 for Q =. and =..Figure (b displas the results that an increasing values of, the momentum boundar laer thickness enhances. In this case the maimum value of stream function is ψ ma = For higher values of heat generation parameter Q the boundar laer becomes thinner and the maimum value of stream function ψ ma is.1 that is shon in Fig. (c. The effects of viscosit variation parameter (=. and 4. and the heat generation parameter Q (=. and. on the isotherms for =.3 and Pr =.7 are shon in Figs. 3(a-3(d. From these figures it is observed that temperature of the fluid rise up significantl due to volumetric heat generation and temperature as ell as the thermal boundar laer thickness increase for the combined effect of the viscosit variation parameter and heat generation parameter Q. 4. Conclusion The effects of temperature dependent viscosit variation parameter, heat generation parameter Q and Prandtl number Pr on momentum and heat transfer have been studied numericall. From the present investigation the folloing conclusions ma be dran: The frictional force at the all enhances for the higher values of heat generation parameter Q, the viscosit variation parameter and the Prandtl number Pr over the hole boundar laer but the rate of heat transfer reduces significantl for all these cases. The skin friction reduces and the rate of heat transfer rise up significantl for higher values of the Prandtl number ithout effect of viscosit variation and heat generation. References [1] L. S.Yao, Natural Convection along a Vertical Wav Surface, ASME J. Heat Transfer 1 ( [] K. C. A. Alam, M. A. Hossain and D. A. S. Rees, Magnetohdrodnamic Free Convection along a Vertical Wav Surface, Int. J. Appl. Mech. Engg. 1 ( [3] M. A. Hossain and D. A. S. Rees, Combined Heat and Mass Transfer in Natural Convection Flo from a Vertical Wav Surface, Acta Mechanica 136 ( [4] M. A. Hossain, S. Kabir and D. A. S. Rees, Natural Convection of Fluid ith Temperature Dependent Viscosit from Heated Vertical Wav Surface, Z. Ange. Math. Phs. 3 ( [] J. H. Jang, W. M. Yan and H. C. Liu, Natural Convection Heat and Mass Transfer along a Vertical Wav Surface, Int. J. Heat Mass Transfer 46 ( [6] M. M. Molla, M. A. Hossain and L. S. Yao, Natural Convection Flo along a Vertical Wav Surface ith Uniform Surface Temperature in Presence of Heat Generation/Absorption, Int. J. Therm. Sci. 43 ( [7] B. Tashtoush and M. Al-Odat, Magnetic Field Effect on Heat and Fluid flo over a Wav Surface ith a Variable Heat Flu, J. Magn. Magn. Mater 68 ( [8] L. S.Yao, Natural Convection along a Vertical Comple Wav Surface, Int. J. Heat Mass Transfer 49 ( [9] M. Molla and R. S. R. Gorla, Natural Convection Laminar Flo ith Temperature Dependent Viscosit and Thermal Conductivit Along a Vertical Wav Surface, Int. J. Fluid Mech. Res. 36 ( [1] N. Parveen and M. A. Alim, Joule Heating Effect on Magnetohdrodnamic Natural Convection flo along a Vertical Wav Surface ith Viscosit Dependent on Temperature, Int. J. Energ & Tech. 3 ( [11] M. A. Hossain, M. S. Munir and D. A. S. Rees, Flo of Viscous Incompressible Fluid ith Temperature Dependent Viscosit and Thermal Conductivit past a Permeable Wedge ith Uniform Surface Heat flu, Int. J. Therm. Sci. 39 ( [1] J. Charraudeau, Influence De Gradients De Properties Phsiques En Convection Force Application Au Cas Du Tube, Int. J. Heat Mass Transfer 18 (