Keywords Free Convection, Boundary Layer, Circular Cylinder, Viscoelastic Fluid, Heat Generation I. INTRODUCTION. with previous publication.

Transcription

1 World Academ o Science, Engineering and Technolog 5 Free Convection Boundar Laer Flow o a Viscoelastic Fluid in the Presence o Heat Generation Abdul Rahman Mohd Kasim, Mohd Ari Admon, and Sharidan Shaie Abstract The present paper considers the stead ree convection boundar laer low o a viscoelastics luid with constant temperature in the presence o heat generation. The boundar laer equations are an order higher than those or the Newtonian (viscous) luid and the adherence boundar conditions are insuicient to determine the solution o these equations completel. The governing boundar laer equations are irst transormed into non-dimensional orm b using special dimensionless group. Computations are perormed numericall b using Keller-bo method b augmenting an etra boundar condition at ininit and the results are displaed graphicall to illustrate the inluence o viscoelastic K, heat generation γ, and Prandtl Number, Pr parameters on the velocit and temperature proiles. The results o the surace shear stress in terms o the local skin riction and the surace rate o heat transer in terms o the local Nusselt number or a selection o the heat generation parameter γ (=.,.,.5,.8,.) are obtained and presented in both tabular and graphical ormats. Without eect o the internal heat generation inside the luid domain or which we take γ =., the present numerical results show an ecellent agreement with previous publication. Kewords Free Convection, Boundar Laer, Circular Clinder, Viscoelastic Fluid, Heat Generation I. INTRODUCTION ATURAL convection has been the subject o research or Nman ears due to its importance in the understanding o phenomena appearing in nature and their etensive engineering applications. Studies on the natural convection boundar laer low past a horizontal clinder have been conducted b several researchers. For eample, Saville and Churchill [] investigated the laminar natural convection boundar laer low near horizontal clinders and vertical aismmetric bodies. Merkin [] studied the natural convection boundar laer low over a clinder o elliptic cross section. Besides that, a stud related to this topic was also carried out b Lien et al. [] b eamining the ree convection heat transer o micropolar luid near a horizontal permeable clinder at a non-uniorm thermal condition while Bhattachara and Pop [4] studied the ree convection heat transer rom an elliptical clinder in micropolar luids. Mansour et al. [5] studied the coupled heat and mass transer in the magnetohdrodnamic low o micropolar luid on circular clinders with uniorm heat and mass lu, while Abdul Rahman Mohd Kasim, Mohd Ari Admon, and Sharidan Shaie are with the Department o Mathematics, Facult o Science, Universiti Teknologi Malasia, 8 UTM Johor Bahru, Johor, MALAYSIA ( abdulrahmanmohdkasim@ahoo.com, ariadmon@ahoo.com, ridaie@ahoo.com). Cheng [6] studied the natural convection heat and mass transer rom a horizontal clinder o elliptic cross section in micropolar luid. In addition, Hossain et al. [7] eamined the eect o thermal radiation on natural convection over clinders o elliptic cross section. In recent ears, the natural convection in viscoelastic luids was investigated due to the applications these materials have in industr and geophsics. In the linear stabilit problem o viscoelastic luids, onl overstable convection was investigated as their stationar motion is identical to that o Newtonian luids. Jitchote and Robertson [8] used a perturbation method to analze the viscoelastic second order luid low in curved pipes o circular cross section or the case where the second normal stress dierence is non-zero, as a model o polmeric liquid. Ariel and Teipel [9] investigated the laminar two-dimensional viscoelastic low near a stagnation point using the orthogonal collocation point method with Laguerre polnomials. Meanwhile, the natural convection o a viscoelastic luid with deormable ree surace was studied b L.A. DaÂvalos-Orozco and E. V. Luis []. It was ound that or dierent values o the Galileo number and relaation times that are large enough, the curves o the critical Raleigh numbers are lower than those o stationar convection and those o overstabilit o the Newtonian luid with deormable ree surace. When the lower surace is rigid, maima in the curves o criticalit against the relaation time are ound. Rasmussen and Hassager [] used the Lagrangian Integral Method to model the classical problem o unstead viscoelastic low rom a sphere in a clinder. Conversel, Wood [] investigated the unstead start-up helical lows or Oldrod-B and upper-convected Mawell luids in straight pipes o circular and annular cross-section.in chemical engineering sstems, viscoelastic lows arise in numerous processes in chemical engineering sstems. Such lows possess both viscous and elastic properties and can ehibit normal stresses and relaation eects. Recentl, the numerical stud o transient ree convective mass transer in a Walters-B viscoelastic low with wall suction was investigated b T. B Chang et al. []. Velocit was ound to increase with a rise in viscoelasticit parameter with both time and distances close to the plate surace. An increase in Schmidt number and a separation rom the plate was also observed to signiicantl decrease both velocit and concentration in time. Increasing species o Grasho number boosted the low velocit at all times and caused a signiicant rise primaril near the plate surace.a large number o phsical phenomena also involve 49

2 World Academ o Science, Engineering and Technolog 5 natural convection that is driven b heat generation. Vajravelu and Hadjinicolaou [4] investigated the heat transer characteristics in a laminar boundar laer low o a viscous luid over a linearl stretching continuous surace with viscous dissipation/rictional heating and internal heat generation, in which the considered the volumetric rate o heat generation, q'''[ w/ m ] as; ( ) Q T T or T T q''' = or T T < where Q is the heat generation constant. The above relation is valid or the state o some eothermic processes having T as the onset temperature. Furthermore, Chamkha and Issa [5] studied the eect o the heat generation or absorption and thermophoresis on a hdromagnetic low with heat and mass transer over a lat plate, while Mendez and Trevino [6] studied the eects o the conjugate conduction-natural convection heat transer along a thin vertical plate with nonuniorm heat generation. Motivated b the work above, this paper investigates the problem o ree convection boundar laer low o viscoelastics luid past a horizontal circular clinder with the constant temperature in the presence o heat generation. The results obtained herein are compared with the solutions b Merkin [7] and Molla [8] with the publications titled ree convection boundar laer on an isothermal horizontal circular clinders and natural convection low rom an isothermal horizontal circular clinder in presence o heat generation respectivel, in order to veri the accurac o the present results. II. PROBLEM FORMULATION The problem studied is the stead ree convection boundar laer low or an isothermal horizontal circular clinder placed in a viscoelastic luid. Figure illustrates the geometr o the problem and the corresponding coordinate sstem. It was assumed that the constant temperature o the surace o the clinder is T w, and that o the ambient luid is T, where Tw > T corresponds to a heated clinder (assisting low) and T w <T corresponds to a cooled clinder (opposing low), respectivel. The phsical coniguration considered is as shown in Fig.. Under the usual Bousinesq and boundar laer approimations, the equations or mass continuit, momentum and energ took the ollowing orm: Continuit equation: u v + = Momentum equation: u u u e u + υ = u u e + ν u u u u k u + v + gβ( T T )sin( / a) Energ equation: T T T Q u + v = α + ( T T ) ρcp Subject to the boundar conditions; u= v=, T= T on =, w u= ue(), u/ =, T= T as Where ( u, v ) are the velocit components along the aes and the velocit outside the boundar laer, while T, ρ, g, β, μ, k, and α are the luid temperature, densit, gravitational acceleration, coeicient o thermal epansion, dnamic viscosit, vorte viscosit and thermal diusivit o the luid respectivel. Below are the ollowing non-dimensional variables: /4 a / a /4 = a, = Gr ( a), u = Gr u, υ= Gr υ ν ν a / ue = Gr ue, θ = ( T T ) ( TW T ) ν Substitution (5) into () to () led to the ollowing nondimensional equations: () () () (4) (5) g T w Continuit equation: u v + = (6) T a O / a Momentum equation:, U T Fig. Phsical model and coordinate sstem 49

3 World Academ o Science, Engineering and Technolog 5 u u ue u u + v = ue + = =, θ = on =, u u u u K( u v ) θ sin + + (7) sin =, =, θ = as (5) Energ equation: aq u θ v θ + = θ + Pr νρc Gr / p Subject to the boundar conditions u= v=, θ = on =, u= u (), u/ =, θ = as e Where Gr = g β ( T W T ) a / v is the Grasho number and was denoted with the viscoelastic parameter, K as 5/ kgr K =, a θ III. SOLUTION PROCEDURES In order to solve (6) to (8) according to the boundar condition (9), the ollowing variables were assumed: ψ = (, ), θ = θ(, ) () Where ψ is the stream unction deined as: ψ ψ u =, v= () B substituting () and () into (6) to (8), the ollowing equations were obtained: u v + = () sin cos + + K Pr θ sin( ) + = / νρcpgr θ θ aq θ θ + + θ = With respect to the ollowing boundar conditions (8) (9) () (4) At the lower stagnation point o the clinder,, () to (4) was reduced to the ollowing ordinar dierential equation: '' + ''' ' + + θ + K '''' ' ''' + ( '') = (6) ( ) { } θ'' + θ' + γθ = (7) Pr with the boundar conditions () = '() =, θ() = (8) '( ) =, ''( ) =, θ( ) = where primes denote the dierentiation with respect to. The phsical quantities o principal interest are shearing stress and the rate o heat transer in terms o skin riction coeicient C and the Nusselt number Nu respectivel, which can be written as: C = τ ρu, w aqw Nu = kt ( ) w T where u u u u u τw = μ ( ) + k u + v + =, = and T qw = k( ) = Using the variables (5) and the boundar condition into (8) resulted in the ollowing: /4 CGr NuGr /4 = ''(,) (9) = θ(,) () The results o the velocit and temperature distributions were then calculated respectivel rom the ollowing relations u=, θ = θ(, ) IV. RESULT AND DISCUSSION The sstems o Equations (6) and (7) were solved numericall or some values o the heat generation γ = (.,.,.5,.8,.) and viscoelastic parameter K, using the implicit inite-dierence method known as Keller-bo method that was ver well described in the book b Cebeci and Bradshaw [9]. In this paper, the case in question is when 494

4 World Academ o Science, Engineering and Technolog 5 the Prandtl number Pr is.7 and. The present results or /4 the skin riction coeicient CGr and the local Nusselt /4 number, NuGr were also compared with those o Merkin [7] and Molla [8] in order to validate the numerical results obtained. The comparison showed that the numerical solutions (see Table I and II) obtained b the present authors concurs ver well with those o previous authors. TABLE I COMPARISON OF THE PRESENT NUMERICAL RESULTS FOR LOCAL SKIN FRICTION WITH THOSE MERKIN [7] AND MOLLA [8] AT γ = FOR PR =. TABLE II COMPARISON OF THE PRESENT NUMERICAL RESULTS FOR LOCAL NUSSELT NUMBER WITH THOSE MERKIN [7] AND MOLLA [8] AT γ = FOR PR =. Besides that, the comparison o the present results with those o previous works on skin riction and heat transer variation was illustrated as shown in Fig., while the comparison on velocit and temperature proile can be seen in Figure ; both illustrations show a strong assent between the comparisons. The present authors are thereore conident that the present results are ver accurate. The variation o local skin riction and local Nusselt number with various values o heat generation γ = (.,.,.5,.8,.) with the ied values o K = and Pr = are illustrated in Fig. 4(a) and 4(b). From these igures, it can be seen that an increase in heat generation parameter, γ, leads to an increase in the local skin-riction coeicient and decrease on the local Nusselt number. This is epected as the heat generation mechanism creates a laer o hot luid near the CGr X Merkin Molla Series solutions Present [7] [ 8] b Molla [8] result..... π / π / π / π / π / / 4 NuGr /4 X Merkin Molla Series solutions Present [7] [8] b Molla [8] result π / π / π / π / π / surace, which subsequentl causes the rate o heat transer rom the surace to decrease. Owing to enhanced temperature, both the viscosit o the luid and the corresponding local skin-riction coeicient increased. A similar trend was also observed b Molla [] on the stud o natural convection low rom a horizontal circular clinder with uniorm heat lu in the presence o heat generation. Fig. 6 illustrates the variation o local skin riction and heat transer with various values o viscoelastic parameter, which indicated that both local skin riction and heat transer decreased with an increase o K. The eect o Pr on the local skin riction and heat transer is illustrated b Fig. 8(a) and 8(b), which shows that an increase in the value o Prandtl number leads to an increase in both the value o the rate o heat transer and the local skin-riction. The absolute maima o the local skin-riction ma also be observed to shit toward the middle o the surace. Fig. 9(a) and 9(b) shows the distribution o velocit and temperature proile respectivel, whereb an increase o Prandtl number is shown to decrease the velocit distribution, while the opposite trend was observed or the temperature proile. Since there are numerous detailed proiles o the transient velocit and temperature ields, the velocit and temperature distributions at the lower stagnation point were given at Pr = with various values o heat generation coeicient and viscoelastics parameter, K. These are illustrated in Fig. 4 and 9 respectivel. Based on Fig. 5(a) and 5(b), it was noticed that the luid velocit increases with γ as it contributed to the acceleration o the low and enhanced the level o local skinriction coeicient at the same time. On the other hand, the temperature proile increased graduall and at γ.5 it eceeded the level o surace temperature (θ = ), along with some critical levels o temperature appearing close to the surace o the clinder. For γ =. the luid temperature nearl doubled the surace temperature. Based on Fig. 7(a), it was noticed that, or γ = the velocit distributions decreased when the value o viscoelastic parameter, K was increased. The values o these proiles are lower or a viscoelastic luid than or a Newtonian luid ( K = ). Thereore, the thickness o the velocit boundar laer or a viscoelastic luid is higher than or a Newtonian luid. Meanwhile, the opposite behaviour was observed or the temperature proiles as shown in Fig. 7(b). This igure shows that the decrease o the velocit and increase o the viscoelastics parameter have a similar trend with the vertical ree lows (see Katagiri []). This behavior relects the coupling o the energ equation to the momentum equation through the temperature dependent bod orces. 495

5 World Academ o Science, Engineering and Technolog O --- M. Molla et al Present γ =.,.,.5,.8,. CGr. / γ =,.,.5,.8.. θ () O --- M. Molla et al Present Fig. (a) Comparison o skin-riction or dierent values o γ with Pr = Fig. (b) Comparison o temperature distribution or dierent values o γ with Pr = NuGr /4 - C O --- M. Molla et al Present γ =,.,.5, γ =,.8,.5,., Fig. (b) Comparison o heat transer or dierent values o γ with Pr = Fig. 4 (a) Variation o the local skin riction or dierent values o γ at Pr = and K= γ =.,.,.5,.8, '() O --- M. Molla et al Present Q W γ =,.,.5,.8, Fig. (a) Comparison o velocit distribution or dierent values o γ with Pr = Fig. 4. (b) Variation o the local heat transer or dierent values o γ at Pr = and K= 496

6 World Academ o Science, Engineering and Technolog γ =,.,.5,.8, '(η).4 Q W K =,.,.5,, η Fig. 5 (a) Velocit distribution or dierent values o γ at.5 Pr = and K= Fig. 6 (b) Variation o the local heat transer coeicient Q w or various value o K at γ = and Pr =.9.8 γ =,.,.5,.8,..7.6 θ(η).5.5 '(η).4 K =,.,, η Fig. 5 (b) Temperature distribution or dierent values o γ at Pr = and K= η Fig. 7 (a) Velocit distribution or dierent values o K at γ = and Pr =.5.5 K =,.,,.5 C K =,.,.5,,.5 θ(η) Fig. 6 (a) Variation o the local skin riction C or various value o K at γ = and Pr= η Fig. 7 (b) Temperature distribution or dierent values o K at γ = and Pr = 497

7 World Academ o Science, Engineering and Technolog C Gr /4.5 Pr =,,,.7 θ ().5.5 Pr =,,, Fig. 8 (a) Variation o the local skin riction or dierent values o Pr at γ = and K= Fig. 9 (b) Temperature distribution or dierent values o Pr at γ = and K= NuGr -/ Pr =.7,,, Fig. 8 (b) Variation o the local heat transer or dierent values o Pr = at γ = and K= '() Pr =,,, Fig. 9 (a) Velocit distribution or dierent values o Pr at γ = and K= V. CONCLUSION The stead ree convection boundar laer low o an incompressible viscoelastic luid past an isothermal horizontal circular clinder has been investigated numericall in this paper. The governing boundar laer equations were transormed into a non-dimensional orm and the resulting nonlinear sstem o partial dierential equations was solved numericall using the Keller-bo method. This paper has revealed how the parameter K, and the Prandtl number, Pr aect the low and heat transer characteristics. From the present investigation, the ollowing conclusions can be drawn:. An increase in the value o Pr leads to a decrease o both the wall temperature distribution () and the velocit distribution '( ). An increase in the value o Prandtl number leads to an increase in the value o the rate o heat transer, while the opposite eect applies or the local skin-riction.. An increase in heat generation parameter,γ, leads to an increase in the local skin-riction coeicient and decrease on the local Nusselt number. 4. In the presence o heat generation, the velocit distributions decrease when the value o viscoelastic parameter, K increases. The values o these proiles are lower or a viscoelastic luid than or a Newtonian luid ( K = ). 5. Velocit distributions decrease when the value o viscoelastic parameter, K is increased while the opposite behaviour is observed or the temperature proiles. 6. When the viscoelastics parameter, K increases it reduces both the values o skin riction and heat transer (local Nusselt number). 498

8 World Academ o Science, Engineering and Technolog 5 ACKNOWLEDGMENT This work is supported b a research grant (Vot FRGS No. 788 & 7845) rom Universiti Teknologi Malasia, UTM. REFERENCES [] D.A. Saville, S.W. Churchill, Laminar ree convection in boundar laers near horizontal clinders and vertical aismmetric bodies, Journal o Fluid Mechanics 9 (967) [] J.H. Merkin, Free convection boundar laer on clinders o elliptic cross-section, ASME, Journal o Heat Transer, 99 (977) [] F.S. Lien, T.M. Chen, C.K. Chen, Analsis o a ree- convection micropolar boundar laer about a horizontal permeable clinder at a non-uniorm thermal condition, ASME Journal o Heat Transer (99) [4] S. Bhattachara, I. Pop, Free convection rom clinders o elliptic cross section in micropolar luids, International Journal o Engineering Science 4 (996). [5] M.A. Mansour, M.A. El-Hakiem, S.M. El Kabeir, Heat and mass transer in magnetohdrodnamic low o micropolar luid on a circular clinder with uniorm heat and mass lu, Journal o Magnetism and Magnetic Materials () [6] C.Y. Cheng, Free convection heat and mass transer rom a horizontal clinder o elliptic cross section in micropolar luids, International Communications in Heat and Mass Transer (6) 8. [7] M.A.Hossain, M.A. Alim,D.A.S. Rees, Eect o thermal radiation on natural convection over clinders o elliptic cross section, Acta Mechanica 9 (998) [8] W. Jitchote, AM. Robertson. Flow o second order luids in curved pipes. J Non-Newtonian Fluid Mech ;9():9 6. [9] PD. Ariel, I. Teipel On dual solutions o stagnation point low o a viscoelastic luid. Zeitschrit ür Angewandte Mathematik und Mechanik (ZAMM) 994;74(8):4 7. [] L.A. DaÂvalos-Orozco and E. VaÂzquez Luis, Natural convection o a viscoelastic luid with deormable ree surace, J. Non-Newtonian Fluid Mech. 85 (999) [] HK. Rasmussen, O. Hassager Simulation o transient viscoelastic low with second order time integration. J Non-Newtonian Fluid Mech 995;56(): [] WP. Wood Transient viscoelastic helical lows in pipes o circular and annular cross-section. J Non-Newtonian Fluid Mech ;(- ):5 6. [] ] T. B. Chang et al. Numerical stud o transient ree convective mass transer in a Walters-B viscoelastic low with wall suction, Commun Nonlinear Sci Numer Simulat, article in press. [4] K. Vajravelu, A. Hadjinicolaou, Heat transer in a viscous luid over a stretching sheet with viscous dissipation and internal heat generation, International Communication Heat Mass Transer (99) [5] A.J. Chamkha, Camille Issa, Eects o heat generation/absorption and the thermophoresis on hdromagnetic low with heat and mass transer over a lat plate, International Journal o Numerical Methods or Heat & Fluid Flow (4) () [6] F. Mendez, C. Trevino, The conjugate conduction- natural convection heat transer along a thin vertical plate with non-uniorm internal heat generation, International Journal o Heat Mass Transer 4 () [7] J.H. Merkin, Free convection boundar laer on an isothermal horizontal circular clinders, ASME/AIChE, Heat Transer Conerence, St. Louis, Mo., August 9- (976). [8] M.M. Molla, M.A. Hossain, M.C. Paul, Natural convection low rom an isothermal horizontal circular clinder in presence o heat generation. International Journal o Engineering Science 44 (6) [9] T. Cebeci, P. Bradshaw, Phsical and Computational Aspects o Convective heat Transer, Springer, New York 984. [] M. M. Molla, M. A. Hossain, R. S. R. Gorla, Natural convection low rom an isothermal horizontal circular clinder with temperature dependent viscosit, Heat and Mass Transer, 4 (5) [] M. Kumari, H.S. Takhar, G. Nath, Nonsimilar mied convection low [] o a non-newtonian luid past a vertical wedge, Acta Mechanica [] (995) 5 -. [4] M. M. Molla et al. Natural Convection Flow rom a Horizontal [5] Circular Clinder with Uniorm Heat Flu in the Presence o Heat [6] Generation, Applied Mathematics Modelling (9) 6-6 [7] M. Katagiri et al. transient ree convection rom an isothermal horizontal circular clinder, Warme-und Stoubertragung (979)