JOURNAL OF INTERNATIONAL ACADEMIC RESEARCH FOR MULTIDISCIPLINARY Impact Factor 1.393, ISSN: , Volume 2, Issue 7, August 2014

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1 HOMOTOPY ANALYSIS TO THERMAL RADIATION EFFECTS ON HEAT TRANSFER OF WALTERS LIQUID-B FLOW OVER A STRETCHING SHEET FOR LARGE PRANDTL NUMBERS HYMAVATHI TALLA* P.VIJAY KUMAR** V.MALLIPRIYA*** *Dept. of Mathematics, Adikavi Nannaya University, Rajahmundry, A.P., India **Dept. of Mathematics, Adikavi Nannaya University, Rajahmundry, A.P., India ***Dept. of Mathematics, Adikavi Nannaya University, Rajahmundry, A.P., India ABSTRACT This paper presents the thermal radiation effects on heat transfer of second order fluid flow over a stretching sheet for large Prandtl numbers. Using similarity transformation the governing boundary layer non-linear partial differential equations are converted as non-linear ordinary differential equations. Homotopy Analysis Method is applied to get series solution. The convergence of the obtained series solution is discussed explicitly. The results are displayed graphically and discussed in detailed. The obtained solution by HAM is valid for all parameters involved in the problem. KEYWORDS: Heat Generation/Absorption, MHD, Radiation, Stretching Sheet, Viscoelastic Fluid, Homotopy Analysis Method. 1. INTRODUCTION Industrial importance of non-newtonian fluids specifically viscoelastic fluids demands for the detail study of such fluids. In the applications like extrusion of wires, production of plastic, fibers, papers and electronic chips etc. The characteristics of the final products depend on the process of cooling and stretching. Since past several decades the study of viscoelastic fluids over a stretching sheet gained the attention of many researchers [1-11]. As mentioned by Nataraj et al [12] dilute polymer solutions like.831 ammonium alginate in water has Prandtl numbers of order of 44 emphasise the importance of heat transfer study of fluids with large Prandtl numbers. In heat transfer process thermal radiation also place a vital role and such effects were analysed by Rapits [13-14], Abel [15]. So, we extended the work of Nataraj et al [12] by considering the thermal radiation effects on heat transfer of second order fluid flow over a stretching sheet for large Prandtl numbers. Also, analytical solution is provided using HAM 458

2 [16], [17], [18]. The result obtained i.e., the effect of radiation and of different flow parameters on heat transfer displayed through graphs and are discussed in detailed. 2. Mathematical formulation The Coleman-Noll constitutive equation based on the postulate of gradually fading memory for an incompressible second-order fluid is where T is the stress tensor, p is the pressure, and are defined as are material constants with where denotes velocity. Assume that the fluid is at rest and the flow is generated by stretching the sheet with velocity. Let be the stretching rate and and be the velocity components along and directions respectively. and are the distances along and normal to the surface. The boundary layer equations for momentum and heat transfer for the study laminar flow of an incompressible second order fluid obeying (1) past a semi-infinite stretching sheet are The boundary conditions are at as The fifth term on the right hand side of (6) represents the radiation effect which has been excluded in [12]. 459

3 Here, is the specific heat generation rate, is the temperature, is the surface temperature and is that of ambient fluid. is a positive parameter associated with the viscoelastic fluid called viscoelastic parameter. is the thermal conductivity and is the density. is the kinematic viscosity. is the specific heat at constant pressure. 2.1Conversion of partial differential equations to ordinary differential equations Introducing the stream function as and defining Using Rosseland approximation for radiation Here, is the Stefan-Boltzmann constant and is the absorption constant. Further, we assume that the temperature difference within the flow is such that is expressed in Taylor series form. Hence, expanding about and neglecting higher order terms, we obtain Also, consider a non dimensional temperature variable as Using the equations, equations reduces to where 46

4 is the Reynolds number; is the Eckert number; is the Prandtl number; is the heat source/sink parameter, is the viscoelastic parameter ; and are characteristic length and velocity respectively. Prime denotes the differentiation with respect to. indicates that the order of the momentum boundary layer thickness is. 3. Homotopy analysis solution For finding the analytical solution of the equations the initial guess and the auxiliary operator in the following form using HAM, choose with the property where are arbitrary constants. If denotes the embedding parameter, and are the non-zero auxiliary parameters, then the zeroth-order deformation equations are subject to the boundary conditions where we define as 461

5 For, we have When p increases from to 1, varies from to and varies from to. By Taylor s theorem and using we can write where Assume that and are selected in such a way that the two series are convergent at. Therefore through equations we have By differentiating the equations times with respect to,dividing by and then setting we get mth order deformation equations with the following boundary conditions where 462

6 and Now using MATHEMATICA we can solve the equations boundary conditions with the 4. Convergence of HAM solution As mentioned by Liao [16], the convergence and the rate of approximation of the two series strongly depends upon the auxiliary parameters and. To choose and, curves have been plotted for different values of the parameters. It is clear from Figs. 1-4 that the range for the admissible values of and are - (Fig. 1), (Fig. 2), (Fig.3 for large ), (Fig.4 for Large ). Our computations indicate that the two series converge in the whole region of when and are chosen from the range described before. Besides, the employed values of and for each case are mentioned in the figures caption. 463

7 6 4 =.1 =.5 2 f''() Fig. 1. -curve for the 15 th order approximations for different values of 6 4 =.3, =-.2, Ec=.5 =.5, =.2, Ec=1. 2 '() Fig. 2. -curve for the 15 th order approximations for different values of when 4 2 Nr=5. Nr=7. '() Fig. 3. -curve for the 15 th order approximations for different value of when 15 1 Pr=5. Pr=85. '() Fig. 4. -curve for the 15 th order approximations fir different values of when 464

8 5. Results and Discussion The momentum and heat transfer equations are characterised by the viscoelastic parameter, Prandtl number, Eckert number, heat source/sink parameter and radiational parameter. Fig. 5 shows the variation of dimensionless velocity gradient with for several values of the elastic parameter. It can be seen that the velocity decreases with an increase in viscoelastic parameter of the fluid. Fig. 6 is drawn for temperature profiles for various values of visco elastic parameter keeping the other parameters fixed. It is observed from the figure that the temperature at a point increases by increasing the viscoelastic parameter. The effect of Prandtl number on heat transfer may be analysed from Fig.7. This graph reveals that the increase of Prandtl number results in the decrease of temperature distribution. This is because there would be a decrease of thermal boundary layer thickness with the increase in the values of Prandtl number. The increase of Prandtl number means slow rate of thermal diffusion. Fig. 8 is a graphical representation which depicts the effect of Eckert number on dimensionless temperature profile. It is found that the effect of Eckert number is to increase the wall temperature due to the heat addition by means of frictional heating. If the Eckert number is large enough the heat transfer may reverse direction. The effect of heat source/sink parameter is shown in Fig. 9. The fluid temperature is greater when there is heat generation. For positive values of through the field and is lower for negative the temperature distribution is higher Fig. 1 depict that temperature increases with the increase of radiation parameter throughout the boundary layer. The increase of radiation parameter implies the release of heat energy from the flow region by means of radiation; this can also be explained by the fact that the effect of radiation is to increase the rate of energy transport to the fluid and accordingly increase the fluid temperature. 465

9 f'() =. =.2 =.4 = Fig. 5. Horizontal velocity profile for different values of =. =.2 =.4 =.6 () Fig. 6. Temperature profile for various values of with 1.8 Pr=1. Pr=2. Pr=3..6 () Fig. 7. Temperature profile for various values of with 466

10 1.8.6 Ec=. Ec=.5 Ec=1. Ec=1.5 () Fig. 8. Temperature profile for various values of with =-.5 =-.1 =. =.1 () Fig. 9. Temperature profile for various values of with Nr=1. h 2 =-1. Nr=5. h 2 =-.2 Nr=9. h 2 =-.2 () Fig. 1. Temperature profile for various values of with The results presented in Table 1 indicate that the magnitude of the dimensional temperature profile increases with the Prandtl number,. The obtained results are compared with the results of Abel [12]. 467

11 Table 1: Comparison of non-dimensional temperature previously published work and present results with. distribution between Abel [15] results HAM results (26 th order) Conclusion In this article, the flow and heat transfer characteristics are analysed for second order fluid flow over a stretching sheet. The effect of various parameters including viscoelastic parameter, Prandtl number, Eckert number, heat source/sink parameter, and the radiation parameter are shown graphically and discussed briefly. The main results of the present analysis are as follows: 1. The velocity component decreases with the increase of viscoelastic parameter 2. The temperature profile increases with the increase viscoelastic parameter. This is due to the increase of thermal boundary thickness with the increase of viscoelastic parameter. 3. Temperature distribution decreases with the increase of Prandtl number. This is because of decrease of thermal boundary thickness with the increase of. 4. The effect of heat source/sink parameter is to generate temperature for increasing positive values and absorb temperature for decreasing negative values. 5. Temperature increases with increase of 468

12 References 1. Abel, S., Prasad, K. V., and Mahaboob, A. (25). Buoyancy force and thermal radiation effectsin MHD boundary layer visco-elastic fluid flow over continuously moving stretching surface, International Journal of Thermal Sciences, 44, Abo-Eldahab, E. M. (24). The effects of temperature-dependent fluid properties on freeconvective flow along a semi-infinite vertical plate by the presence of radiation, International journal of Heat Mass Transfer, 41, Abramowitz, M. and Stegun, I. A., ed. (1972). Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. Dover, New York. 4. Republication of #55,National Bureau of Standards, Applied Mathematics Series (1964) corrected ed., U.S.Dept. of Commerce, National Bureau of Standards, Washington, D.C. 5. Anderson, H. I. (1992). MHD flow of a viscoelastic fluid past a stretching surface, ActaMechania, 95, Beard, D. W. and Walters, K. (1964). Elasto-viscous boundary layer flow, Proceedings - Cambridge Philosophical Society, 6, Brewster, M. Q. (1992). Thermal Radiative Transfer Properties, John Wiley, New York. 8. Bujurke, N. M., Birader, S. N., and Hiremath, P. S. (1987). Second order fluid flow past astretching sheet with heat transfer, Zeitschriftfürangewandte Mathematik und Physik, 38, Chang, W. D. (1989). The nonuniqueness of the flow of a visco-elastic fluid over Stretchingsheet, Quarterly journal of Applied Mathematics, 47, Chang, W. D., Kazarijnoff, N. D., and Lu, C. (1991). A new family of explicit solutions for thesimilarity equations modeling flow of non-newtonian fluid over a stretching sheet, Archive for Rational Mechanics and Analysis,113, Chen, C. K. and Char, M. I. (1988). Heat transfer of a continuous stretching surface with suction or blowing, Journal of Mathematical Analysis, 135, Coleman, B. D. and Markovitz, H. (1964). Normal stress effects in second order fluids, Journal of Applied Physics, 35, Nataraja, H. R., Sarma, M. S., and Rao, B. N. (1977). On the solution of the energy equation for large Prandtl number in second order fluid flow over a stretching sheet, Zeitschriftfürangewandte Mathematik und Physik,48, Raptis, A. (1998). Technical note: Flow of a micropolar fluid past a continuously moving plateby the presence of radiation, International journal of Heat Mass Transfer, 41, Raptis, A. (1999). Radiation and viscoelastic flow, International Communications in Heat Mass Transfer, 26, Subhas Abel, M., Gousia Begum. (28). Heat transfer in MHD viscoelastic fluid flow on stretching sheet with heat source/sink, viscous dissipation, stress work and radiation for the case of large Prandtl number, Chemical Engineering Communications, 195, Liao, S. J. (23). Beyond Perturbation: Introduction to Homotopy Analysis Method, Chapman & Hall, CRC Press, Boca Raton.Sajid, M., Hayat, T. (29). 18. The application of homotopy analysis method for MHD viscous flow due to a shrinking sheet, Chaos, Solitons and Fractals,39, Nadeem, S., Anwar Hussain. (29). MHD flow of a viscous fluid on a nonlinear porous shrinking sheet with homotopy analysis method, Applied Mathematics and Mechanics, 3,