Viscous Dissipation and Heat Absorption effect on Natural Convection Flow with Uniform Surface Temperature along a Vertical Wavy Surface

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1 Aailable at htt://am.ed/aam Al. Al. Math. ISSN: Alications and Alied Mathematics: An International Jornal (AAM) Secial Isse No. (Ma 6),. 8 8th International Mathematics Conference, March,, IUB Cams, Bashndhara Dhaka, Bangladesh Viscos Dissiation and Heat Absortion effect on Natral Conection Flow with Uniform Srface emeratre along a Vertical Wa Srface Sjon Nath* and Nazma Pareen Deartment of Mathematics Bangladesh Uniersit of Engineering and echnolog Dhaka-, Bangladesh * sjon.bet@ahoo.com ABSRAC he effect of iscos dissiation and heat absortion on a stead two-dimensional natral conection laminar flow of iscos incomressible flid with niform srface temeratre along a ertical wa srface has been inestigated. Using the aroriate ariables; the basic eqations are transformed to conenient form and then soled nmericall emloing er efficient method, namel Keller-Bo method (KBM). Nmerical reslts are resented b the shearing stress in terms of the local skin-friction coefficient; the rate of heat transfer in terms of local Nsselt nmber, streamline and isotherms, resectiel for a wide range of the iscos dissiation arameter N and heat absortion arameter Q. Increasing Q and lessening N case the enhancement of heat transfer rate. KEYWORDS: Viscos dissiation; Heat absortion; Natral conection; Keller-Bo method; Wa srface. MSC () No.: 76D, 76R. INRODUCION Man scientists inestigated iscos dissiation for arios roses in science searatel or in resence of heat generation or heat absortion. Gebhart (96) stdied the effects of iscos dissiation in natral conection and then in eternal natral conection flow

2 Sjon Nath and Nazma Pareen together with Mollendorf (969). Natral conection was introdced along a ertical wa srface b Yao (983). Vajrael and Hadjinolao (993) stdied the heat transfer characteristics in the laminar bondar laer of a iscos flid oer a stretching sheet with iscos dissiation or frictional heating and internal heat generation. In this std the 3 considered that the olmetric rate of heat generation, q [ W / m ], shold be q m = Q (- ), for and eqal to zero for, where Q is the heat generation/absortion constant. Hossain and Rees (999) first inestigatedcombined heat and mass transfer in natral conection flow from a ertical wa srface and then Jang et al. (3) inestigated natral conection heat and mass transfer along a ertical wa srface. Molla et al. () stdied natral conection flow along a ertical wa srface with niform srface temeratre in resence of heat generation/absortion. Alam et al. (7) inestigated iscos dissiation effects on MHD natral conection flow oer a shere in the resence of heat generation. Mamn et al. (8) analzed MHD conjgate heat transfer analsis for a ertical flat late in resence of iscos dissiation and heat generation. Dei and Kaalizhi () considered iscos dissiation and radiation effects on the thermal bondar laer flow with heat and mass transfer oer a non-isothermal stretching sheet with internal heat generation embedded in a oros medim. Jha and Ajibade () stdied effect of iscos dissiation on natral conection flow between ertical arallel lates with time-eriodic bondar conditions. Jole heating and iscos dissiation effects were inestigated on MHD flow ast a semi-infinite inclined late with ariable srface temeratre b Palani and Kim (). Recentl, Pareen and Alim () hae obsered the effects of MHD natral conection flow along a ertical wa srface in resence of heat generation/absortion with iscosit deendent on temeratre. From the aboe one thing is clear that et it has not been jstified the effects of iscos dissiation and heat absortion together on natral conection flow along a ertical wa srface. he object of the std is to show the effect of or resent work.. FORMUAION OF HE PROBEM he bondar laer analsis otlined below allows (X ) being arbitrar, bt or detailed nmerical work assmed that the srface ehibits sinsoidal deformations. he wa srface ma be described b Y w nx ( X ) sin, () where is the wae length associated with the wa srface. he geometr of the wa srface and the two-dimensional Cartesian coordinate sstem are shown in Figre.

3 AAM: Intern. J., Secial Isse, No. (Ma 6) 3 X Figre. he coordinate sstem and the hsical model he conseration eqations for the flow characterized with stead, laminar and twodimensional bondar laer; nder the sal Bossinesq aroimation, dimensionless form of the continit, momentm and energ eqations can be written as:, (), (3), (), Pr N Q () where k C Pr is the Prandtl nmber, C Q Q is the heat absortion arameter and ) ( C N w is the iscos dissiation arameter. he following dimensionless ariables are introdced for non-dimensionalizing the goerning eqations,, Y X, P ) (,, U V U w,, ) (, 3 g d d dx d w Y V g w U

4 Sjon Nath and Nazma Pareen where is the non-dimensional temeratre fnction and (, ) are the dimensionless elocit comonents. It can easil be seen that the conection indced b the wa srface is described b eqations () (). We frther notice that, eqation () indicates that the ressre gradient along the -direction is O ( ), which imlies that lowest order ressre gradient along - direction can be determined from the iniscid flow soltion. For the resent roblem this ressre gradient ( ) is zero. Eqation () frther shows that / O() and is determined b the left-hand side of this eqation. hs, the elimination of / from eqations (3) and () leads to is. (6) he corresonding bondar conditions for the resent roblem are:,, at,,, as. (7) Now we introdce the following transformations to redce the goerning eqations to a conenient form: 3 f (, ),, (, ). (8) Introdcing the transformations gien in Eqation (8) and into Eqation (6) and (), the following sstem of nonlinear eqations are obtained 3 f f, f ff f f f (9) 3 f f Q Nf f. Pr he bondar conditions (7) now take the following form: ()

5 AAM: Intern. J., Secial Isse, No. (Ma 6) f (, o) f (, o), (, o), f (, ), (, ). () he local skin friction coefficient C f and the rate of heat transfer in terms of the local Nsselt nmber N takes the following form: C f ( / ) / f (, o), () ( / ) (, ). N o (3) 3. MEHOD OF SOUION he goerning eqations are soled nmericall with the hel of imlicit finite difference method together with the Keller-Bo scheme (97). he discretization of momentm and energ eqations are carried ot with resect to non-dimensional coordinates and η to cone the eqations in finite difference form b aroimating the fnctions and their deriaties in terms of central differences in both the coordinate directions. hen the reqired eqations are to be linearized b sing the Newton s Qasi-linearization method. he linear algebraic eqations can be written in a block matri which forms a coefficient matri. he whole rocedre namel redction to first order followed b central difference aroimations, Newton s Qasi-linearization method and the block homas algorithm, is well known as Keller-bo method.. RESUS AND DISCUSSION Here we hae shown the combined effects of iscos dissiation and heat absortion on natral conection flow of iscos incomressible flid along a ertical wa srface. he skin friction coefficient C f, the rate of heat transfer in terms of Nsselt nmber N, the streamlines as well as the isotherms are shown grahicall in Figres to for different ales of the heat absortion arameter Q and the iscos dissiation arameter N. From the Figres (a) and (b), it is noted that the higher ale of N accelerates the flid flow and increases the temeratre so with the increasing of iscos dissiation arameter N = (.,.,.,.,.), the skin friction coefficient C f increases along the stream direction of the srface and the heat transfer rates in terms of the local Nsselt nmber N decreases.

6 6 Sjon Nath and Nazma Pareen Cf 8 6 N =. N =. N =. N =. (a) N - (b) N =. N =. N =. N = Figre. Effect of N on (a) skin friction coefficient C f and (b) rate of heat transfer N while =.3and Pr =.73 From Figres 3(a) and (b), it is obsered that the heat absortion mechanism does not create a laer of hot flid near the srface so the temeratre of the flid decreases the srface temeratre and temeratre gradient increases. hs the ales of heat absortion arameter Q = (., -.,-., -.8) leads to decrease the local skin friction coefficient C f and increase the local rate of heat transfer N at different osition of.. Q =. Q = -. (a) Q = -. Q = -.8 (b) Q =. Q = -. Q = -. Q = -.8 Cf N Figre 3. Effect of Q on (a) skin friction coefficient C f and (b) rate of heat transfer N while =.3and Pr =.73 Figres and show the with and withot effect of iscos dissiation arameter N and heat absortion arameter Q together on the formlation of streamlines and isotherms resectiel while Pr =.73 and =.3. It is noted that with the effect of N, the elocit bondar laer becomes thicker and when Q increases the oosite reslts obsered. he thermal bondar laer becomes thicker gradall for the effect of N. he temeratre distribtion decreases in resence of heat absortion arameter Q. he maimm ale of ma is 3.86 for N =..

7 AAM: Intern. J., Secial Isse, No. (Ma 6) 7. (a) (c) (b) (d) Figre. Streamlines for (a) N =., Q =. (b) N =., Q =. (c) N =., Q = -.8 and (d) N =.,Q = -.8 while Pr =.73 and =.3 (a) (c) (b) Figre. Isotherms for (a) N =., Q =. (b) N =., Q =. (c) N =., Q = -.8 and (d) N =., Q = -.8 while Pr =.73 and =.3 (d) CONCUSION he effects of the heat absortion arameter Q and the iscos dissiation arameter N on natral conection flow of iscos incomressible flid along a ertical wa srface hae been inestigated. From the resent inestigation the following conclsions ma be drawn:

8 8 Sjon Nath and Nazma Pareen he skin friction coefficient C f has increased and the rate of heat transfer in terms of Nsselt nmber N has decreased for increasing ales of N and oosite reslts has haened for heat absortion arameter Q. Streamlines hae changed slightl too er and the same reslts are obsered for thermal bondar laer thickness with the increasing ales of iscos dissiation arameter N and oosite reslts hae obsered for heat absortion arameter Q. REFERENCES Alam, M.M., Alim, M.A. and Chowdhr, M.M.K. (7). Viscos dissiation effects on MHD natral conection flow oer a shere in the resence of heat generation, Nonlinear Analsis: Modelling and Control, Vol. (), 7-9. Dei, S.P.A. and Kaalizhi, M. (). Viscos dissiation and radiation effects on the thermal bondar laer flow with heat and mass transfer oer a non-isothermal stretching sheet with internal heat generation embedded in a oros medim, Int. J. Energ echnolog, Vol. (),. -. Gebhart, B. (96). Effects of iscos dissiation in natral conection, J. Flid Mech., Vol.,. 3. Gebhart, B. and Mollendorf, J. (969). Viscos dissiation in eternal natral conection flows, J. Flid Mech., Vol. 38(), Hossain, M.A. and Rees, D.A.S. (999).Combined heat and mass transfer in natral conection flow from a ertical wa srface, Acta Mechanica, Vol. 36,. 33. Jha, B.K. and Ajibade, A.O. (). Effect of iscos dissiation on natral conection flow between ertical arallel lates with time-eriodic bondar conditions, J. hermohsics and Aeromechanics, Vol. 8(), Jang, J.H., Yan, W.M. and i, H.C. (3). Natral conection heat and mass transfer along a ertical wa srface, Int. J. Heat Mass ransfer, Vol.6, Keller, H.B. (978). Nmerical methods in bondar laer theor, Ann. Re. Flid Mech., Vol., Mamn, A.A., Chowdhr, Z.R., Azim, M.A. and Molla, M.M. (8). MHD conjgate heat transfer analsis for a ertical flat late in resence of iscos dissiation and heat generation, Int. comm. Heat ransfer, Vol. 3, Molla, M.M., Hossain, M.A. and Yao,.S. (). Natral conection flow along a ertical wa srface with niform srface temeratre in resence of heat generation/absortion, Int. J. herm. Sci., Vol.3, Palani, G. and Kim, K.Y. (). Jole heating and iscos dissiation effects on MHD flow ast a semi-infinite inclined late with ariable srface temeratre, J. Engineering hermohsics, Vol. (),. -7. Pareen, N. and Alim, M.A. (). MHD natral conection flow along a ertical wa srface in resence of heat generation/absortion with iscosit deendent on temeratre, J. Mechanical Engineering, Vol. ME (),. 7. Vejrael, K. and Hadjinicolao, A. (993). Heat transfer in a iscos flid oer a stretching sheet with iscos dissiation and internal heat generation, Int. comm. Heat ransfer, Vol., Yao,.S. (983). Natral conection along a ertical wa srface, ASME J. Heat ransfer, Vol.,