Oblique stagnation-point flow of an incompressible visco-elastic fluid towards a stretching surface

An analysis is made of the steady two-dimensional stagnation-point flow of an incompressible viscoelastic fluid over a flat deformable surface when the surface is stretched in its own plane with a velocity cx, where x is the distance from the stagnation-point and c is a positive constant. It is shown that for a viscoelastic fluid of short memory (obeying Walters’ B model), a boundary layer is formed when the stretching velocity of the surface is less than ax, where ax+2by is the inviscid free-stream velocity and y is the distance normal to the plate, a and b being constants and the velocity at a point increases with increase in the elasticity of the fluid. On the other hand an inverted boundary layer is formed when the surface stretching velocity exceeds ax and the velocity decreases with increase in the elasticity of the fluid. A novel result of the analysis is that the flow near the stretching surface is that corresponding to an inviscid stagnation-point flow when a=c. Temperature distribution in the boundary layer is found in three cases, namely: (i) the sheet with constant surface temperature (CST); (ii) the sheet with variable surface temperature (VST) and (iii) the sheet with prescribed quadratic power law surface heat flux (PHF) for various values of non-dimensional parameters. It is found that in all the three cases when a/c>1, temperature at a point decreases with increase in the elasticity of the fluid and when a/c<1, temperature at a point increases with increase in the elasticity of the fluid. Further temperature at a point decreases with increase in the radiation parameter and wall temperature parameter.     

Heat transfer in stagnation-point flow towards a stretching sheet

 Steady two-dimensional stagnation-point flow of an incompressible viscous fluid over a flat deformable sheet is investigated when the sheet is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. It is shown that for a fluid of small kinematic viscosity, a boundary layer is formed when the stretching velocity is less than the free stream velocity and an inverted boundary layer is formed when the stretching velocity exceeds the free stream velocity. Temperature distribution in the boundary layer is found when the surface is held at constant temperature and surface heat flux is determined. Received on 12 July 2000 / Published online: 29 November 2001     

Heat transfer characteristics of the flow of an incompressible viscous fluid over a stretching sheet

The heat transfer characteristics of the three-dimensional flow of an incompressible viscous fluid caused by stretching of an elastic sheet with uniform tension in two horizontal directions are studied subject to the following conditions; (a) the sheet is held at constant temperature; (b) the sheet is exposed to a uniform heat flux. The similarity solutions for temperature exist for both the cases. It is shown that for fixedPr, the temperature at a point decreases with increase in in Case (a) and the sheet temperature also decreases with increasing for Case (b).

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Die Wärmeübertragungseigenschaften eines inkompressiblen viskosen Fluids bei der Strömung über eine dehnbare Folie

Zusammenfassung Es werden die Wärmeübertragungseigenschaften einer dreidimensionalen Strömung eines inkompressiblen viskosen Fluids, verursacht durch Dehnung einer elastischen Folie, mit gleichmäßiger Spannung in zwei horizontale Richtungen, unter Berücksichtigung folgender Bedingungen betrachtet; (a) die Folie wird bei konstanter Temperatur gehalten; (b) die Folie wird einem gleichmäßigen Wärmestrom ausgesetzt. Die Ähnlichkeitslösungen für die Temperatur existieren in beiden Fällen. Es wird gezeigt, daß im Fall (a) die Temperatur bei konstanter Prandtl-Zahl gleichzeitig mit einem anwachsenden fällt, und daß auch im Fall (b) die Folientemperatur mit wachsendem fällt.

     

Magnetohydrodynamic stagnation-point flow of a power-law fluid towards a stretching surface

Steady two-dimensional stagnation-point flow of an electrically conducting power-law fluid over a stretching surface is investigated when the surface is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. We have discussed the uniqueness of the solution except when the ratio of free stream velocity and stretching velocity is equal to 1. The effect of magnetic field on the flow characteristic is explored numerically and it is concluded that the velocity at a point decreases/increases with increase in the magnetic field when the free stream velocity is less/greater than the stretching velocity. It is further observed that for a given value of magnetic parameter M, the dimensionless shear stress coefficient |F^″(0)| increases with increase in power-law index n when the value of the ratio of free stream velocity and stretching velocity is close to 1 but not equal to 1. But when the value of this ratio further differs from 1, the variation of |F^″(0)| with n is non-monotonic.     

Momentum and heat transfer in the magnetohydrodynamic stagnation-point flow of a viscoelastic fluid toward a stretching surface

An analysis is made of the steady two-dimensional stagnation-point flow of an incompressible viscoelastic fluid over a flat deformable surface when the surface is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. It is shown that for a viscoelastic conducting fluid of short memory (obeying Walters’ Bʹ model), a boundary layer is formed when the stretching velocity of the surface is less than the inviscid free-stream velocity and velocity at a point increases with increase in the Hartmann number. On the other hand an inverted boundary layer is formed when the surface stretching velocity exceeds the velocity of the free stream and the velocity decreases with increase in the Hartmann number. A novel result of the analysis is that the flow near the stretching surface is that corresponding to an inviscid stagnation-point flow when the surface stretching velocity is equal to the velocity of the free stream. Temperature distribution in the boundary layer is found when the surface is held at constant temperature and surface heat flux is determined. It is found that in the absence of viscous and Ohmic dissipation and strain energy in the flow, temperature at a point decreases with increase in the Hartmann number.     

Heat transfer in the flow of a viscoelastic fluid over a stretching surface

An analysis is made of heat transfer in the boundary layer of a viscoelastic fluid flowing over a stretching surface. The velocity of the surface varies linearly with the distance x from a fixed point and the surface is held at a uniform temperature T _w higher than the temperature T _∞ of the ambient fluid. An exact analytical solution for the temperature distribution is found by solving the energy equation after taking into account strain energy stored in the fluid (due to its elastic property) and viscous dissipation. It is shown that the temperature profiles are nonsimilar in marked contrast with the case when these profiles are found to be similar in the absence of viscous dissipation and strain energy. It is also found that temperature at a point increases due to the combined influence of these two effects in comparison with its corresponding value in the absence of these two effects. A novel result of this analysis is that for small values of x, heat flows from the surface to the fluid while for moderate and large values of x, heat flows from the fluid to the surface even when T _w >T _∞. Temperature distribution and the surface heat flux are determined for various values of the Prandtl number P, the elastic parameter K ₁ and the viscous dissipation parameter a. Numerical solutions are also obtained through a fourth-order accurate compact finite difference scheme. Received on 14 October 1997     

Stagnation-point flow of a viscoelastic fluid towards a stretching surface

An analysis is made of the steady two-dimensional stagnation-point flow of an incompressible viscoelastic fluid over a flat deformable surface when the surface is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. It is shown that for a viscoelastic fluid of short memory (obeying Walters’ B′ model), a boundary layer is formed when the stretching velocity of the surface is less than the inviscid free-stream velocity and velocity at a point increases with increase in the elasticity of the fluid. On the other hand, an inverted boundary layer is formed when the surface stretching velocity exceeds the velocity of the free stream and the velocity decreases with increase in the elasticity of the fluid. A novel result of the analysis is that the flow near the stretching surface is that corresponding to an inviscid stagnation-point flow when the surface stretching velocity is equal to the velocity of the free stream. Temperature distribution in the boundary layer is found when the surface is held at constant temperature and surface heat flux is determined. It is found that temperature at a point decreases with increase in the elasticity of the fluid.     

MHD stagnation-point flow and heat transfer towards stretching sheet with induced magnetic field

The problem of the steady magnetohydrodynamic (MHD) stagnation-point flow of an incompressible viscous fluid over a stretching sheet is studied. The effect of an induced magnetic field is taken into account. The nonlinear partial differential equations are transformed into ordinary differential equations via the similarity transformation. The transformed boundary layer equations are solved numerically using the shooting method. Numerical results are obtained for various magnetic parameters and Prandtl numbers. The effects of the induced magnetic field on the skin friction coefficient, the local Nusselt number, the velocity, and the temperature profiles are presented graphically and discussed in detail.     

Heat transfer from non-isothermal surfaces in the stagnation-point flow of a micropolar fluid

Summary This paper contains a numerical study of the development of the thermal boundary layer on an impermeable flat surface, following a step change in surface temperature, for stagnation-point flow of a micropolar fluid. A study is made of the relative effects of the individual parameters in the equations, and a comparison of certain integral parameters, representative of the flow, is used to assess the accuracy of the results.

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Zusammenfassung Diese Arbeit enthält eine numerische Untersuchung der Entwicklung der thermischen Grenzschicht auf einer undurchlässigen ebenen Oberfläche nach stufenförmigem Sprung in der Oberflächentemperatur für die Staupunktsströmung einer mikropolaren Flüssigkeit. Die relativen Auswirkungen der einzelnen Parameter in den Gleichungen werden untersucht und ein Vergleich von gewissen, für die Strömung repräsentativen integralen Parametern wird durchgeführt, um die Genauigkeit der Ergebnisse abzuschätzen.

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With 4 figures and 1 table     

Slip effects and heat transfer analysis in a viscous fluid over an oscillatory stretching surface

In this study, we investigate the heat transfer problem in a viscous fluid over an oscillatory infinite sheet with slip condition. The sheet is moved back and forth in its own plane. The derived problem involves a dimensionless parameter indicating the relative magnitude of frequency to sheet stretching rate. A system of non‐linear partial differential equations is solved numerically using the finite‐difference scheme, in which a coordinate transformation is employed to transform the semi‐infinite physical space to a bounded computational domain. The physical features of interesting parameters on the velocity and temperature distributions are shown graphically and discussed. The values of the skin‐friction coefficient and the local Nusselt number are given in tabular form. Copyright © 2008 John Wiley & Sons, Ltd.     

Cylinder in viscous incompressible fluid flow

We present a technique for calculating the temperature field in the vicinity of a cylinder in a viscous incompressible fluid flow under given conditions for the heat flux or the cylinder surface temperature. The Navier-Stokes equations and the energy equation for the steady heat transfer regime form the basis of the calculations. The numerical calculations are made for three flow regimes about the cylinder, corresponding to Reynolds numbers of 20, 40, and 80. The pressure distribution, voracity, and temperature distributions along the cylinder surface are found.It is known that for a Reynolds number R>1 the calculation of cylinder drag within the framework of the solution of the Oseen and Stokes equations yields a significant deviation from the experimental data. In 1933 Thom first solved this problem [1] on the basis of the Navier-Stokes equations. Subsequently several investigators [2, 3] studied the problem of viscous incompressible fluid flow past a cylinder.It has been established that a stable solution of the Navier-Stokes equations exists for R40 and that in this case the calculation results are in good agreement with the experimental data. According to [2], a stable solution also exists for R=44. The possibility of obtaining a steady solution for R>44 is suggested.Analysis of the results of [2] permits suggesting that the questions of constructing a difference scheme with a given order of approximation of the basic differential relations which will permit obtaining the sought solution over the entire range of variation of the problem parameters of interest are still worthy of attention.Calculation of the velocity field in the vicinity of a cylinder also makes possible the calculation of the cylinder temperature regime for given conditions for the heat flux or the temperature on its surface. However, we are familiar only with experience in the analytic solution of several questions of cylinder heat transfer with the surrounding fluid for large R within the framework of boundary layer theory [4].     

The boundary layers of an unsteady incompressible stagnation-point flow with mass transfer

In the current work, the boundary layers of an unsteady incompressible stagnation-point flow with mass transfer were further investigated. Similarity transformation technique was used and the similarity equation group was solved using numerical methods. Interesting observation is that there are multiple solutions seen for negative unsteadiness parameters, β. The influences of mass transfer, unsteadiness parameter, and Prandtl numbers on velocity and temperature profiles, wall drag, and wall heat fluxes were investigated and analyzed. The asymptotic behaviors for the similarity equations in limiting situations were theoretically analyzed. It is found that solutions exist for all mass transfer parameters for β≥−1. For a certain mass transfer parameter, there are two solutions when β_c<β<0; there is one solution for (β=β_c)∪(β≥0); there is no solution for β<β_c, where β_c is a critical unsteadiness parameter dependent on mass transfer parameter.     

Heat transfer over an unsteady stretching surface

Similarity solution of the laminar boundary layer equations corresponding to an unsteady stretching surface have been studied. The governing time-dependent boundary layer are transformed to ordinary differential equations containg Prandtl number and unsteadiness parameter. The effect of various govern-ing parameters such as Prandtl number and unsteadiness param-eter which determine the velocity and temperature profiles and heat transfer coefficient are studied.     

Transient free convection flow of an incompressible viscous dissipative fluid

A finite-difference solution of transient free convection flow of a viscous dissipative fluid past an infinite vertical plate, on taking into account viscous dissipative heat is presented. Velocity profiles, temperature profiles are shown for different values of Pr, the Prandtl number and E, the Eckert number. The numerical values of the skin-friction and the rate of heat transfer are entered in a Table. It is observed that greater viscous dissipative heat causes a rise in the velocity, temperature and the skin-friction and a fall in the rate of heat transfer. An increase in Pr leads to a fall in the velocity, temperature and the skin-friction but the rate of heat transfer increases with increasing Pr.     

Unsteady flow of an incompressible viscous fluid around a deformable spherical body

An unsteady flow of a viscous incompressible fluid around a deformable spherical body is considered in the approximation of low Reynolds numbers with a predetermined flow velocity. The hydrodynamic impact of the flow incoming onto the body is determined with allowance for small radial displacements of the body surface. The effect of spherical body surface deformation on the magnitude of the incoming flow impact force is taken into account, in particular, the dependence of small radial displacements of the body surface on the time is found, which makes it possible to minimize the physical impact of the incident flow.     

Heat transfer analysis of Williamson fluid over exponentially stretching surface

This study explores the effects of heat transfer on the Williamson fluid over a porous exponentially stretching surface. The boundary layer equations of the Williamson fluid model for two dimensional flow with heat transfer are presented. Two cases of heat transfer are considered, i.e., the prescribed exponential order surface temperature （PEST） case and the prescribed exponential order heat flux （PEHF） case. The highly nonlinear partial differential equations are simplified with suitable similar and non-similar variables, and finally are solved analytically with the help of the optimal homotopy analysis method （OHAM）. The optimal convergence control parameters are obtained, and the physical fea- tures of the flow parameters are analyzed through graphs and tables. The skin friction and wall temperature gradient are calculated.