Joule Heating Effects on MHD Natural Convection Flows in Presence of Pressure Stress Work and Viscous Dissipation from a Horizontal Circular Cylinder

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1 Journal of Applied Fluid Mechanics, Vol. 7, No., pp. 7-3, 04. Available online at ISSN , EISSN Joule Heating Effects on MHD Natural Convection Flows in Presence of Pressure Stress Work and Viscous Dissipation from a Horizontal Circular Clinder A. S. Bhuian, N. H. M. A. Azim and M. K. Chowdhur 3 Department of Mathematics, Brock Universit, St. Catharines, Ontario, LS 3A, Canada School of Business Studies, Southeast Universit, Dhaka-3, Bangladesh 3 Department of Mathematics, Bangladesh Universit of Engineering and Technolog, Dhaka, Bangladesh Corresponding Author bd_abdussamad@ahoo.com (Received Januar 6, 0; accepted March 7, 03) ABSTRACT The effects of joule heating on MHD natural convection flow from a horizontal circular clinder along the outer surface from the lower stagnation point to the upper stagnation point in presence of pressure stress work and viscous dissipation is investigated. The results have been obtained b transforming the governing boundar laer equations into a sstem of non-dimensional equations and b appling implicit finite difference method together with Newton s linearization approimation. Numerical results for different values of the magnetic parameter, joule heating parameter and Prandtl number have been obtained. The velocit profiles, temperature distributions, skin friction co-efficient and the rate of heat transfer have been presented graphicall for the effects of the aforementioned parameters. Results are compared with previous investigation. Kewords: Natural convection, Viscous dissipation, Pressure stress work, MHD, Joule heating. NOMENCLATURE Cf local skin friction coefficient, c p specific heat at constant pressure, dimensionless Cartesian coordinates dimensional Cartesian coordinates f dimensionless stream function kinematics viscosit Gr local Grashof number viscosit of the fluid g acceleration due to gravit θ dimensionless temperature function J joule heating parameter λ viscous dissipation parameter M magnetic parameter ε pressure stress work parameter Nu local Nusselt number coefficient β co-efficient of thermal epansion Pr Prandtl number β 0 magnetic field strength temperature at the surface of the clinder stream function T w T temperature of the ambient fluid densit of the fluid T temperature of the fluid in the boundar electric conduction uv, the dimensionless and component of the velocit u,v the dimensional and component of the velocit

2 A. S. Bhuian et al. / JAFM, Vol. 7, No., pp. 7-3, 04.. INTRODUCTION The influence and importance of viscous dissipation effects in free convection flows have been eamined b Gebhar (96). Zakerullah (97) has been investigated the viscous dissipation and pressure work effects in aismmetric natural convection flows. Ackrod (974) studied the stress work effects in laminar flat plate natural convection flow. Takhar and Soundalgekar (980) have studied the effects of viscous and joule heating on the problem posed b Sparrow and Cess (96), using the series epansion method of Gebhart. But the investigated generall not in a particular case of stud. Natural convection flow from a horizontal clinder due to thermal buoanc was analzed b a number of researchers Merkin et al. (988), Kuehn et al. (980) and Wang et al. (990). Joshi and Gebhart (98) have shown the effect of pressure stress work and viscous dissipation in some natural convection flows. Effects of pressure stress work and viscous dissipation in natural convection flow along a vertical flat plate with heat conduction have been investigated b Alam et al. (006). Recentl, He et al. (007) have considered the effects of heat and mass transfer on natural convection flows across an isothermal horizontal circular clinder with chemical reaction. MHD flow and heat transfer process are now an important research area due to its potential application in engineering and industrial fields. A considerable amount of research has been done in this field. Wilks et al. (976) studied MHD free convection about a semiinfinite vertical plate in a strong cross field. Takhar and Soundalgekar (980) investigated dissipation effects on MHD free convection flow past a semi-infinite vertical plate. Hossain (99) studied viscous and Joule heating effects on MHD free convection flow with variable plate temperature. Aldoss et al. (996) analzed MHD mied convection from a horizontal circular clinder. El-Amin (003) found out the combined effect of viscous dissipation and Joule heating on MHD forced convection over a non-isothermal horizontal circular clinder embedded in a fluid saturated porous medium. He observed that both the velocit profiles and temperature profiles shifted down for increasing value of magnetic parameter and that are rise up for increasing value of joule heating parameter. Recentl, Molla et al. (0) studied the effect of temperature dependent viscosit on MHD natural convection flow from an isothermal sphere. However, the joule heating effects on MHD natural convection flow in presence of pressure stress work and viscous dissipation has received little attention. Hence, the present stud is attempted.. MATHEMATICAL ANALYSIS Let us consider a stead natural convection flow of a viscous incompressible fluid from an isothermal horizontal circular clinder of radius a placed in a fluid of uniform temperature. A uniform magnetic field having strength B 0 is acting normal to the clinder surface. The effects of pressure stress work, viscous dissipation and joule heating in the flow region and conduction from surface considered in the present stud. The flow configuration and the coordinates sstem are shown in Fig.. Under the balance laws of mass, momentum and energ and with the help of Boussinesq approimation for the bod force term in the momentum equation, the equations governing this boundar-laer natural convection flow can be written as: Continuit equation u v 0 Momentum equation u u u u v g T T sin a Energ equation B0 u T T K T u u v C P C P T P B0 u u CP Fig.. The geometr of the problem () () (3) The phsical situation of the sstem suggests the following boundar conditions u v 0, T Tw at 0, 0 u 0, T T as (4) The governing equations and the boundar conditions Eqs. ()-(4) can be made non-dimensional, using the Grashof number Gr 3 g a ( Tw T) which is assumed large and the following non-dimensional variables:, a va v Gr Gr a 4 4, u T T, T T w ua Gr, (5) Where is the dimensionless temperature. The non dimensional forms of the Eqs. ()-(3) are as follows: u v 0 (6) 8

3 f ' A. S. Bhuian et al. / JAFM, Vol. 7, No., pp. 7-3, 04. u u u u v Mu sin (7) u v Pr ga T ( TW T ) u Ju cp TW T (8) Where / M ( a B0 ) / ( Gr ) parameter, / J ( B0 Gr ) /{ cptw T} is the magnetic is the joule heating parameter and Pr c p / is the Prandtl number. The boundar condition Eq. (4) can be written as in the following dimensionless form: u v 0, at 0 u 0, 0 as, (9) To solve Eqs. (6)-(8), subject to the boundar condition Eq. (9), we assume following transformations f,, (, ) (0) Where is the stream function usuall defined as u /, v / () Substituting Eq. () into the Eqs. (6)-(9), the new forms of the dimensionless Eq. (7) and Eq. (8) are sin f f f ff f Mf f f T f f f Pr TW T f J f f f () (3) In the above equations primes denote differentiation with respect to. The corresponding boundar conditions take the following form: f (,0) f (,0) 0, at 0 f (, ) 0, (, ) 0 as 3. METHOD OF SOLUTION (4) Equations () and (3) are solved numericall based on the boundar conditions as described in Eq. (4) using one of the most efficient and accurate methods known as implicit finite difference method with Keller bo scheme. 4. RESULT AND DISCUSSION Joule heating effects on magneto-hdrodnamic natural convection flow in presence of pressure stress work and viscous dissipation from a horizontal circular clinder has been investigated. The velocit profiles, temperature distributions, local skin-friction and the local rate of heat transfer obtained b the finite difference method for various values of the governing parameters. The aims of the figures are to displa how the profiles var with the scaled stream wise coordinate. From Fig. (a), it is observed that the velocit increases as the values of the joule heating parameter J increase. The velocit increases significantl along and becomes maimum and then decreases slowl and finall approaches to zero, the asmptotic value. The maimum values of the velocit are , , and for J = 0., 0.3, 0.5 and 0.9 respectivel which occur at =.80 for first, second maimum values, at =.45 for third and fourth maimum values. Here it is observed that the velocit increase b 5.3% as J increases from 0. to 0.9. From Fig. (b), it is seen that when the values of joule heating parameter J increase, the temperature also increases J = 0. Fig. (a). Variation of velocit profile against for varing of J with M = 0., 0.5, 0.5 and Pr = J = 0. Fig. (b). Variation of temperature against for varing of J with M = 0., 0.5, 0.5 and Pr=.0. Fig. 3(a) and Fig. 3(b) displa results for the velocit and temperature profiles for different values of magnetic parameter M (M = 0., 0.3, 0.5, 0.9) having Prandtl number Pr =.0, J = , 0.5. It is observed that, as the magnetic parameter M increases, the velocit profile decreases between 0 5and then increases with ver small difference and finall approaches to zero along direction. The temperature 9

4 f ' f ' A. S. Bhuian et al. / JAFM, Vol. 7, No., pp. 7-3, 04. profile increases with increasing magnetic parameter M. The maimum values of the velocit are recorded as , , and for M = 0., 0.3, 0.5 and 0.9 respectivel which occur at =.43 for st, nd, 3rd and 4th maimum values. It is found that the velocit decreases b 6.75% as the magnetic parameter M increases from 0. to M = 0. Fig. 3(a). Variation of velocit profile against for varing of M with J = 0., 0.5, 0.5 and Pr= coefficient Cf decreases. Owing to increasing values of M in the presence of viscous dissipation and pressure stress work, the fluid temperature within the boundar laer increases and the associate thermal boundar laer becomes thicker. For increasing fluid temperature, the temperature difference between fluid and surface decreases and the corresponding rate of heat transfer decreases Pr = 0.7 Pr =.00 Pr =.44 Pr =.74 Fig. 4(a). Variation of velocit profile against for varing of Pr with M= 0., 0.5, 0.5 and J= M = 0. Pr = 0.7 Pr =.00 Pr =.44 Pr = Fig. 3(b). Variation of temperature against for varing of M with J = 0., 0.5, 0.5 and Pr=.0. Figures 4(a) and 4(b) indicate the effects of the Prandtl number Pr with M = 0., 0.5 J = 0. and 0.5 on the velocit profiles and the temperature profiles. From Fig. 4(a) it is observed that the increasing values of Prandtl number Pr leads to the decrease in the velocit profiles. The maimum values of the velocit are , , and for Pr = 0.7,.0,.44 and.74 respectivel which occur at =.43, =.36, =.30 and =.6 for the first, second, third and fourth maimum value. Here it is depicted that the velocit decreases b 3.63% as Pr increases from 0.7 to.74. From Fig. 4(b) it is observed that the temperature profiles decreases with the increasing values of Prandtl number Pr. It can easil be seen that the effect of the magnetic parameter M leads to a decrease in the local skin friction coefficient and the local Nusselt number Cf Nu in Fig. 5(a) and Fig. 5(b). This phenomenon can easil be understood from the fact that the magnetic parameter M opposes the flow, therefore decreases the velocit gradient and hence the local skin friction Fig. 4(b). Variation of temperature profile against for varing of Pr with M =0., 0.5, 0.5 and J=0.. Cf M = Fig. 5(a). Variation of skin friction against for varing of M with J = 0., 0.5, 0.5 and Pr=.0. 0

5 A. S. Bhuian et al. / JAFM, Vol. 7, No., pp. 7-3, 04. Nu 0.0 M = Fig. 5(b). Variation of heat transfer against for varing of M with J = 0., 0.5, 0.5 and Pr =.0. The variation of the reduced local skin friction coefficient and the local rate of heat transfer for different values of the joule heating parameter J (J = 0., 0.3, 0.5, 0.9) are illustrated in Fig. 6(a) and Fig. 6(b) with M = 0., 0. 5 and 0.5 and Prandtl number Pr =.0. From the figures it can be seen that the increase of the joule heating parameter J leads to an increase in the local skin-friction coefficient Cf and a decrease in the local Nusselt number Nu. These are epected, since the joule heating mechanism in presence of viscous dissipation and pressure stress work creates a laer of hot fluid near the surface, and finall the resultant temperature of the fluid eceeds the surface temperature. For this reason the rate of heat transfer from the surface decreases. Owing to the enhanced temperature, the viscosit of the fluid increases and the corresponding local skin-friction coefficient increases. In order to verif the accurac of the present work, the numerical values of the local Nusselt number Nu for M = 0.0, J = 0.0, 0.0, 0.0 and Pr =.00 in different position of are compared with those reported b Merkin (976), Nazar et al. (00) and He et al. (007) as presented in table The results are found to be in ecellent agreement. Cf J = 0. Nu Cf 0.0 J = Fig. 6(b). Variation of heat transfer against for varing of J with M = 0., 0.5, 0.5 and Pr= Pr = 0.7 Pr =.00 Pr =.44 Pr = Fig. 7(a). Variation of skin friction against for varing of Pr with M = 0., 0.5, 0.5 and J=0. Nu 0.0 Pr = 0.7 Pr =.00 Pr =.44 Pr = Fig.7(b). Variation of heat transfer against for varing of Pr with M = 0., 0.5, 0.5, and J= Fig. 6(a). Variation of skin friction against for varing of J with M = 0., 0.5, 0.5 and Pr =.0

6 A. S. Bhuian et al. / JAFM, Vol. 7, No., pp. 7-3, 04. Table Numerical values of Nu for different values of while Pr=.0, M = 0.0, J=0.0, 0.0 and 0.0 Merkin Nazar et al. He et al. Present (976) (00) (007) / / / / / CONCLUSION We have studied the joule heating effects on magnetohdrodnamic (MHD) natural convection flow in presence of viscous dissipation and pressure stress work from a horizontal circular clinder. The transformed non-similar boundar laer governing equations of the flow together with the boundar conditions were solved numericall using implicit finite difference method together with Keller bo scheme. The coupled effect of natural convection that the temperature and the rate of heat transfer is continuous at the surface. From the present investigation, the following conclusions ma be drawn: The local skin friction coefficients and the rate of heat transfer along the surface of the clinder decrease for the increasing value of magnetic parameter M. An increase in values of M leads to decrease the velocit distribution but slightl increase the temperature distribution. For increasing values of joule heating parameter J, the skin-friction coefficient increases but the Nusselt number decreases significantl within the boundar laer. With the effect of joule heating parameter J, both the velocit and temperature distributions increase significantl the thickness of the thermal boundar laer. An increasing value of Prandtl number Pr leads to decrease in the velocit and the temperature distributions as epected. REFERENCES Ackrod, J. A. D. (974). Stress work effets in laminar flat-plate natural convection. J. of Fluid Mech., 6, Alam, M., A. Alim and M.K. Chowdhur (006). Effect of pressure stress work and viscous dissipation flow along a vertical flat plate with heat conduction. Journal of Naval Architecture and Marine Engineering, 3(), Aldoss, T.K., Y.D. Ali and M.A. Al-Nimr (996). MHD mied convection from a horizontal circular clinder. Numerical Heat Transfer, Part A, 30, Cebeci, T. and P. Bradshaw (984). Phsical and computational aspects of convective heat transfer, Springer, New York. El-Amin, M.F. (003). Combined effect of viscous dissipation and Joule heating on MHD forced convection over a non-isothermal horizontal clinder embedded in a fluid saturated porous medium. Journal of Magnetism and Magnetic materials, 63, Gebhart, B. (96). Effect of dissipation in natural convection J of Fluid Mech.,4, 5-3. He, M.A, M.M. Molla and M.H. Khan (007). Conjugate effects of heat and mass transfer on natural convection flow across an isothermal horizontal circular clinder with chemical reaction. Int. J. of Non-linear modeling and control,, 9-0. Hossain, M. A. (99). Viscous and Joule heating effects on MHD-free convection flow with variable plate temperature. Int. J. heat Mass Transfer, 35, Joshi, Y. and B. Gebhart(98). Effect of pressure stress work and viscous dissipation in some natural convection flows. Int. J. of Heat Mass Transfer, 4, Keller, H.B. (978). Numerical methods in the boundar laer theor. Annual Reviews of Fluid Mechanics, 0, Kuehn, T.H. and R.J. Goldstein (980). Numerical solution to the Navier-Stokes equations for laminar natural convection about a horizontal isothermal circular clinder. Int. J. heat Mass Transfer, 3, Merkin, J. H. and I. Pop (988). A note on the free convection boundar laer on a horizontal circular clinder with constant heat flu. Wärme und Stoffübertragung,, Sparrow, E.M. and R.D. Cess (96). Effect of magnetic field on free convection heat transfer. Int. J of Heat and Mass Transfer, 3, Takhar, H.S. and V.M. Soundalgekar (980). Dissipation effects on MHD free convection flow past a semi-infinite vertical plate. Applied Science Research, 36, 63-7.

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