International Journal of Heat and Mass Transfer
Heat transfer in a viscoelastic boundary layer flow over a stretching sheet
Introduction
Study of laminar boundary layer flow caused by a moving rigid surface was initiated by Sakiadis [1] and later the work was extended to the flow due to stretching of a sheet by Crane [2]. The flow of an incompressible fluid past a moving surface has several engineering applications. The aerodynamic extrusion of plastic sheets, the cooling of a large metallic plate in a cooling bath, the boundary layer along a liquid film in condensation process and a polymer sheet or filament extruded continuously from a die, or a long thread traveling between a feed roll and a wind-up roll are the examples of practical applications of a continuous flat surface. In certain dilute polymer solution (such as 5.4% of polyisobutylene in cetane and 0.83% solution of ammonium alginate in water [3], [4]), the viscoelastic fluid flow occurs over a stretching sheet. Any fluid that does not behave in accordance with the Newtonian constitutive relation is called non-Newtonian [5], [6], [7], [8], [9], [10], [11], [12]. Non-Newtonian fluids have gained considerable importance because the power required in stretching a sheet in a viscoelastic fluid is less than when it is placed in a Newtonian fluid; and the heat transfer rate for a viscoelastic fluid is found to be less than that of Newtonian fluid.
The central problem in non-Newtonian fluid dynamics is the establishment of expressions for the stress tensor T to replace the Newtonian expression. The relation between the stress tensor and various kinematic tensors is called the constitutive equation or the rheological equation of state. Rivlin and Ericksen and Coleman and Noll have presented constitutive relations for the stress tensor as a function of the symmetric part of the velocity gradient and its higher (total) derivatives. Amongst the many models, the Coleman–Noll constitutes equation [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30]: T , which is based on the postulate of gradually fading memory for an incompressible second-order fluid has received special attention. Here T is the stress tensor, p is the pressure, I is the unit matrix, are material constants with and and are the Rivlin–Ericksen tensors defined byand denotes the velocity. The constitutive relations for the second-grade fluids are derived based on first principles and unlike many other ‘phenomenological’ models, there are no curve-fittings or parameters to adjust. However, the material constants in the model are to be evaluated experimentally. Coleman and Markovitz [31], [32] have presented relations for and suggested several practical methods for measuring the material constants . This class of second-grade fluids is also called as the differential fluids of complexity 2. Fluids of grade 1 are in fact the classical Newtonian ones and the expression for the stress tensor, contains only the first two-terms of the above second-grade fluid model. Another class of models is the rate-type fluid models, such as Oldroyd model, which has been modified by Walters. This modified model is referred to as the Walters’ liquid B. The steady two-dimensional boundary layer equations for Walters’ liquid B were derived by Beard and Walters [10] to first-order in elasticity (i.e., for short memory fluids with short relaxation times). Walters’ liquid B considered by Sidappa and Abel [13] exhibit normal stress-differences in simple shear flows. Rajagopal et al. [19] analyzed the effects of viscoelasticity on the flow of a second-order fluid with gradually fading memory and arrived to the boundary layer equations as that in Ref. [13].
Abel et al. [33] have carried out the analysis for viscoelastic fluid flow and heat transfer over a stretching sheet with viscous dissipation and non-uniform heat source. They provided an exact analytical solution to the Walters’ liquid momentum equation, a series solution to the energy equation in terms of Kummer’s function and presented results for the Prandtl numbers of 3 and 4. Dilute polymer solution like 0.83% ammonium alginate in water and 5.4% polyisobutylene in cetane have approximate Prandtl numbers of 440 and 3 respectively. However, the energy equation in their formulation does not contain terms of the work due to deformation. The exclusion of these terms from the energy equation is not in conformity with the inclusion of viscous dissipation [34], [35], [36], [37], [38]. This paper examines the solution of the problem while including the work due to deformation in the energy equation.
Section snippets
Formulation
Consider the flow of an incompressible viscoelastic (Walters’ liquid B model) fluid over a wall coinciding with the plane y = 0, the flow being confined to y > 0. Keeping the origin fixed, two equal and opposite forces are applied along the x-axis, which result in stretching of the sheet and hence, flow is generated. The basic boundary layer equations for the steady flow are:
Solution of the momentum equation
Rajagopal et al. [19] have studied the flow of a viscoelastic fluid over a stretching sheet. McLeod and Rajagopal [20] and Troy et al. [21] have examined the uniqueness of the solution of the problem. Chang [22] has claimed that the solution of the problem is not necessarily unique. For the specified three boundary conditions in Eqs. (12), (13), it is not possible to solve directly the fourth-order non-linear differential equation (10) of the momentum equation. The closed form solutions of the
Solution of the energy equation
The solution of the energy equation (11) with the boundary conditions (14), (16) for the case of prescribed surface temperature (PST-case) isHere It is to be noted that for PST-case, The Kummer’s function is defined as [39]This satisfies the differential equation:The Pochhammer symbols and
Results and discussion
Viscoelastic boundary layer flow over a stretching sheet is examined by solving equations of momentum and heat transfer. Analytical expressions are obtained to generate temperature profiles for two cases viz., the prescribed surface temperature (PST–case) and the prescribed surface heat flux (PHF-case). The physical parameters in the governing boundary layer equations are: viscoelastic parameter , Prandtl number (Pr), Eckert number (E), and heat source/sink parameters ( and ).
Fig. 1,
Concluding remarks
Analytical solutions are obtained to carryout heat transfer analysis for the steady laminar flow of an incompressible viscoelastic fluid past a stretching sheet with power-law surface temperature or power-law surface heat flux, including the effects of viscous dissipation and internal heat generation or absorption. The magnitude of the non-dimensional surface velocity gradient is found to increase with increasing the viscoelastic parameter . The magnitude of the non-dimensional surface
Acknowledgements
The authors wish to thank the reviewers for their valuable suggestions to improve the clarity of presentation.
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