EFFECTS OF VISCOUS DISSIPATION ON FREE CONVECTION BOUNDARY LAYER FLOW TOWARDS A HORIZONTAL CIRCULAR CYLINDER

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1 EFFECTS OF VISCOUS DISSIPATION ON FREE CONVECTION BOUNDARY LAYER FLOW TOWARDS A HORIZONTAL CIRCULAR CYLINDER Muhammad Khairul Anuar Mohamed 1, Norhaizah Md Sari 1, Abdul Rahman Mohd Kasim 1, Nor Aida Zuraimi Md Noar 1, Mohd Zuki Salleh 1 and Anuar Ishak 1 Applied & Industrial Mathematics Research Group, Faculty o Industrial Science & Technology, University Malaysia Pahang, UMP Kuantan, Pahang, Malaysia School o Mathematical Sciences, Faculty o Science & Technology, University Kebangsaan Malaysia, UKM Bangi, Selangor, Malaysia baa_khy@yahoo.com ABSTRACT In this study, the numerical investigation o the viscous dissipation on ree convective boundary layer lo toards a horizontal circular cylinder ith constant all temperature is considered. The transormed partial dierential equations are solved numerically by using an implicit inite-dierence scheme knon as the Keller-box method. Numerical solutions are obtained or the reduced Nusselt number and the reduced skin riction coeicient as ell as the velocity and temperature proiles. The eatures o the lo and heat transer characteristics or various values o the Prandtl number and Eckert number are analyzed and discussed. The results in this paper is original and important or the researchers orking in the area o boundary layer lo and this can be used as reerence and also as complement comparison purpose in uture. Keyords: constant all temperature, ree convection, horizontal circular cylinder, viscous dissipation. 1. INTRODUCTION Convection boundary layer lo is an important topic to be considered in industrial and engineering activities noadays. These conigurations are applied especially in thermal eects managements, hich involves luid as cooling medium in many industrial outputs or example in electronic devices, computer poer supply and also in engine cooling system such as heatsink in car radiator. Air, electrolyte, ater, polymer, and nanoluid are the example o the luid that may be involve in convection process. Because o the large contributions and issues, this topic has attracted many researchers to study and expand the knoledge so that it could be applied in order to handle the thermal problems produced by these industrial outputs [1, ]. Free convection reers to the convection process occur naturally. Physically, ree convection occurs hen the luid motion is generated by gravitational ield and changes in the luid density. It is contrary dierent ith orced convection here the orced convection generated mechanically by the external agents like bloer, an or nozzle. In considering the convection on a horizontal circular cylinder, Blasius [3] is the irst one ho solved the momentum equation o orced convection boundary layer lo. [4] then solved the energy equation or this problem by considering the constant all temperature (CWT). Since then, this topic has attracted many researchers to study the constant all temperature and constant heat lux. Merkin [5] considered the ree convection boundary layer on an isothermal horizontal cylinder and became the irst one ho got the exact solution or this problem. Merkin and Pop [6] studied ree convection boundary layer on a horizontal circular cylinder ith constant heat lux. Next, Nazar et al. [7] extended [5] and [6] in micropolar luid hile Molla et al. [8] investigated the heat generation eects on ree convection lo on an isothermal horizontal circular cylinder. Recently, Salleh and Nazar [9] and Sari et al. [10] updated [7] ith Netonian heating and convective boundary conditions, respectively. Both problems are solved numerically by using the Keller-box method. From literature study, it is ound that Gebhart [11] is the irst person ho studied viscous dissipation in ree convection lo. The viscous dissipation eects on unsteady ree convective lo over a vertical porous plate as then investigated by Soundalgekar [1]. Vajravelu and Hadjinicolaou [13] then studied the viscous dissipation eects on the lo and heat transer over a stretching sheet. Chen [14] and Partha et al. [15] observed the mixed and MHD ree convection heat transer rom a vertical surace and exponentially stretching surace ith Ohmic heating and viscous dissipation, respectively. Recently, Yirga and Shankar [16] considered this topic ith thermal radiation and magnetohydrodynamic eects on the stagnation point lo toards a stretching sheet. It is orth to mention that the viscous dissipations eect is important to study in order to understand the behavior o temperature distributions hen the internal riction is not negligible. Thereore, the purpose o the present study is to investigate the eects o viscous dissipation on ree convection boundary layer lo toards a horizontal circular cylinder. The governing partial dierential equations are solved numerically and the variation o pertinent physical parameters are analyzed and discussed in detail ith the aid o tables and igures.. MATHEMATICAL FORMULATION The horizontal circular cylinder o radius, hich is heated to a constant temperature, embedded in a 758

2 viscous luid ith ambient temperature as shon in Figure-1. The orthogonal coordinates o and are measured along the cylinder surace, starting ith the loer stagnation point and normal to it, respectively. Under the assumptions that the boundary layer approximations is valid, the dimensional governing equations o steady ree convection boundary layer lo are [17, 9]: u v 0, x u u u x u v g( T T )sin, x a T T T u u v x Cp subject to the boundary conditions ux (,0) vx (,0) 0, Tx (,0) T, ux (, ) 0, Tx (, ) T (4) Figure-1. Physical model o the coordinate system. here u and v are the velocity components along the x and y axes, respectively is the dynamic viscosity, is the kinematic viscosity, g is the gravity acceleration, is the thermal diusivity, is the thermal expansion, T it local temperature, is the luid density and C p is the speciic heat capacity at a constant pressure. Next, it is introduced the governing nondimensional variables: x y a 1/ x, ygr, u Gr u, a a a TT v Gr v, ( ). T T Using (5), (1)-(3) becomes u v 0, x, (1) () (3) (5) (6) u u u u v sin x, x 1 u u v Ec, x Pr subject to the boundary conditions ux (,0) 0, vx (,0) 0, ( x,0) 1, ux (, ) 0, (, x) 0 here Pr is the Prandtl number, Gr Ec is an Eckert number and acpt T 3 g ( T T ) a Gr is the Grasho number. In order to solve (6)-(8), the olloing unctions is introduce: x(, x y), (, x y), (10) here is the stream unction deined as u y and v hich identically satisies (6) and is the x rescaled dimensionless temperature o the luid. Substitute (10) into (6)-(8), then the olloing partial dierential equations is obtained: 3 sin x 3 x x, x x 1 Pr x xec x x, ith boundary conditions ( x,0) 0, ( x,0) 0, ( x,0) 1, ( x, ) 0, ( x, ) 0 The physical quantities o interest are the skin riction coeicient C and the local Nusselt number Nu are [8]: C, aq Nu. x u kt ( T ) (7) (8) (9) (11) (1) (13) x (14) 759

3 The surace shear stress and the surace heat lux q are given by u T, q k, y0 y0 ith and k being the dynamic viscosity and the thermal conductivity, respectively. Using (10) and (15), the reduced skin riction C Gr and reduced Nusselt number NuxGr stated as: x (15) CGr x x,0 and Nu Gr x,0. (16) y y Furthermore, the velocity proiles and temperature distributions can be obtained rom the olloing relations: u (, x y), (, x y). (17) 3. RESULTS AND DISCUSSION Eqns. (11) and (1) subject to the boundary conditions (13) ere solved numerically using the Kellerbox method ith to parameters considered, namely the Prandtl number Pr and the Eckert number Ec. The step size 0.0, x and boundary layer thickness y 8 and x are used in obtaining the numerical results. Tables-1 and sho the comparison values o NuxGr and C Gr ith previous results or various values o x, respectively. It has been ound that they are in good agreement. We can conclude that this method orks eiciently or the present problem, and e are also conident that the results presented here are accurate. Table-3 presents the values o NuxGr ith various values o x and Ec. From table, it is suggest that as Ec and Pr are ix, the NuxGr decreases as x increases hich means the reducing in convective heat transer capabilities. As tables goes to the right, it is ound that the increase o Ec also results to the decrease o Nu Gr as o x and Pr are ixed. x Figures- and 3 displayed the temperature and velocity proiles or various values o Pr, respectively. It is ound that the increase o Pr have reduce the thermal boundary layer thickness in Figure-. It is due to decrease in thermal diusivity hich reduced the energy ability and the thermal boundary layer thickness. In Figure-3, the small Pr produced high velocity distribution. It is because o the small Pr usually has lo in viscosity hich high in momentum diusivity. Next, Figures-4 and 5 illustrated the variations o reduced Nusselt number NuxGr and the reduced skin riction coeicient C Gr or various values o Pr against x, respectively. In Figure-4, it is concluded that the NuxGr decreases as x increases. Furthermore, the eects o Pr are more pronounced at a small value o x. In Figure-5, as expected, the small Pr produced large C Gr compared to large Pr. This situation is related ith Figure-3. Physically, the high velocity gradient ill produced high in skin riction coeicient. Lastly, in order to understand the eects o viscous dissipation Ec in the convective boundary layer lo, Figures-6 and 7 are plotted. Figure-6 shos the variations o NuxGr ith various values o Ec against x. It is seen that the viscous dissipation Ec are negligible at the loer stagnation region x 0. The eects o Ec are signiicance as x increase to the middle o cylinder then converged back at the end o the cylinder x. In Figure-7, at the early stage, it is ound that the C Gr is unique or all Ec value. From igure, it is understand that the Ec inluenced a small eects on C Gr Noticed that as Ec increases, the C Gr also increases. Furthermore, rom numerical calculation, it is concluded that Ec does not aect the temperature and velocity proiles. Table-1. Comparison values o NuxGr ith previous published results or various values o x hen Pr 1, Ec

4 Table-. Comparison values o CGr ith previous published results or various values o x hen Pr 1, Ec 0. Table-3. Values o NuxGr ith various values o x and Ec hen Pr 7. Figure-4. Reduced Nusselt number NuxGr or various values o Pr. against x Figure-5. Reduced skin riction coeicient CGr against x or various values o Pr. Figure-. Temperature proiles ( y) against y or various values o Pr. Figure-3. Velocity proiles ( y) against y or various values o Pr. Figure-6. Reduced Nusselt number NuxGr or various values o Ec. against x 761

5 [3] Blasius, H Grenzschichten in Flssigkeiten mit kleiner Reibung. Zeitschrit ur angeandte Mathematik und Physik [4] Fr ossling, N Calculating by series expansion o the heat transer in laminar, constant property boundary layers at non isothermal suraces. Archiv or Fysik [5] Merkin, J. H Free convection boundary layer on an isothermal horizontal cylinder, In: ASME/ AIChe Heat Transer Conerence. St.Louis, USA.pp Figure-7. Reduced skin riction coeicient CGr against x or various values o Ec. 4. CONCLUSIONS In this paper, the viscous dissipation eect on ree convection boundary layer lo toards a horizontal circular cylinder is numerically studied. It is shon ho the Prandtl number Pr and the Eckert number Ec aect the values o the reduced Nusselt and the reduced skin riction coeicient as ell as the velocity and temperature proiles. As a conclusion, the increase o Pr result to the decrease o thermal boundary layer thickness, velocity proiles and the reduced skin riction hile the reduced Nusselt number increases. It is because, an increase o Pr means the increase in viscosity but decrease in thermal diusivity hich reduced the energy ability and the thermal boundary layer thickness. Next, the inluenced o Ec on the reduced Nusselt number are negligible at the loer stagnation region x 0, Ec played the role pronouncedly at the middle o the cylinder. Meanhile, the Ec eects are small on the reduced skin riction coeicient. ACKNOWLEDGEMENT The authors grateully acknoledge the inancial supports received in the orm o research grants rom the Universiti Malaysia Pahang (RDU and RDU150101). REFERENCES [1] Pop, I.,Ingham, D. B Convective Heat Transer: Mathematical and Computational Modelling o Viscous Fluids and Porous Medium. Pergamon. Oxord. [] Salleh, M. Z., Nazar, R. M.,Pop, I Numerical Investigation o Free Convection over a Permeable Vertical Flat Plate Embedded in a Porous Medium ith Radiation Eects and Mixed Thermal Boundary Conditions. AIP Conerence Proceedings (1): [6] Merkin, J. H.,Pop, I A note on the ree convection boundary layer on a horizontal circular cylinder ith constant heat lux. Wärme - und Stoübertragung. (1-): [7] Nazar, R., Amin, N.,Pop, I. 00. Free convection boundary layer on an isothermal horizontal circular cylinder in a micropolar luid. Proceedings o Teith Int Heat Transer Conerence. Paris, Elsevier. : [8] Molla, M. M., Hossain, M. A.,Paul, M. C Natural convection lo rom an isothermal horizontal circular cylinder in presence o heat generation. International Journal o Engineering Science. 44 (13 14): [9] Salleh, M. Z.,Nazar, R Free Convection Boundary Layer Flo over a Horizontal Circular Cylinder ith Netonian Heating. Sains Malaysiana. 39 (4): [10] Sari, N. M., Salleh, M. Z., Tahar, R. M.,Nazar, R Numerical solution o the ree convection boundary layer lo over a horizontal circular cylinder ith convective boundary conditions. AIP Conerence Proceedings. (160) [11] Gebhart, B Eects o viscous dissipation in natural convection. Journal o Fluid Mechanics. 14 (0): 5-3. [1] Soundalgekar, V. M Viscous dissipation eects on unsteady ree convective lo past an ininite, vertical porous plate ith constant suction. International Journal o Heat and Mass Transer. 15 (6): [13] Vajravelu, K.,Hadjinicolaou, A Heat transer in a viscous luid over a stretching sheet ith viscous dissipation and internal heat generation. International Communications in Heat and Mass Transer. 0 (3):

6 [14] Chen, C. H Combined heat and mass transer in MHD ree convection rom a vertical surace ith Ohmic heating and viscous dissipation. International Journal o Engineering Science. 4 (7): [15] Partha, M. K., Murthy, P.,Rajasekhar, G. P Eect o viscous dissipation on the mixed convection heat transer rom an exponentially stretching surace. Heat and Mass transer. 41 (4): [16] Yirga, Y., Shankar, B Eects o Thermal Radiation and Viscous Dissipation on Magnetohydrodynamic Stagnation Point Flo and Heat Transer o Nanoluid toards a Stretching Sheet. Journal o Nanoluids. (4): [17] Khan, W. A.,Pop, I Boundary-layer lo o a nanoluid past a stretching sheet. International Journal o Heat and Mass Transer. 53 (11 1):