We show that there exist bounded self-similar solutions to the steady state problem of the MHD stagnation point flow of a power-law fluid over a shrinking sheet. We then discuss the stability of the unsteady solutions about each steady solution, showing that one steady state solution corresponds to a stable solution whereas the other corresponds to an unstable solution. The stable solution corresponds to the physically relevant solution. Further, we obtain numerical results for each solution, which enable us to discuss the features of the respective solutions. Our method of finding dual solutions and analyzing stability is of practical application to those interested in engineering analysis, as it provides one with a way to determine whether a given steady state solution is physically meaningful. Hence, our study is useful not only as a discussion of the problem of the MHD stagnation point flow of a power-law fluid over a stretching or shrinking sheet but as a demonstration of the treatment of fluid flow problems with multiple solutions.

This communication studies the effect of melting heat transfer on the stagnation-point flow of a Jeffrey fluid over a stretching sheet. Heat transfer analysis is carried out in the presence of viscous dissipation. The arising differential system has been solved by the homotopy analysis method (HAM). The results indicate an increase in the velocity and the boundary layer thickness with an increase in the values of the elastic parameter (Deborah number) for a Jeffrey fluid which are opposite to those accounted for in the literature for the other subclasses of rate type fluids. Furthermore, an increase in the melting process corresponds to an increase in the velocity and a decrease in the temperature. A comparative study between the current computations and the previous studies is also presented in a limiting sense.

The steady two-dimensional magnetohydrodynamic (MHD) stagnation-point flow of an electrically conducting incompressible viscous fluid toward a shrinking sheet is investigated. The sheet is shrunk in its own plane with a velocity proportional to the distance from the stagnation-point and a uniform magnetic field is applied normal to the sheet. Velocity component parallel to the sheet is found to increase with an increase in the magnetic field parameter M . A region of reverse flow occurs near the surface of the shrinking sheet. It is shown that as M increases, the tendency of this flow reversal decreases. It is also observed that the nonalignment of the stagnation-point flow and the shrinking sheet considerably complicates the flow structure. The effect of the magnetic parameter M on the streamlines is shown for both aligned and nonaligned cases. The temperature distribution in the boundary layer is found when the surface is held at constant temperature. The analysis reveals that the temperature at a point increases with increasing M in a certain neighborhood of the surface but beyond this, the temperature decreases with increasing M . For fixed M , the surface heat flux decreases with increase in the shrinking rate.

We present a purely analytic solution to the steady three-dimensional viscous stagnation point flow of second grade fluid over a heated flat plate moving with some constant speed. The analytic solution is obtained by a newly developed analytic technique, namely, homotopy analysis method. By giving a comparison with the existing results, it is shown that the obtained analytic solutions are highly accurate and are in good agreement with the results already present in literature. Also, the present analytic solution is uniformly valid for all values of the dimensionless second grade parameter α . The effects of α and the Prandtl number Pr on velocity and temperature profiles are discussed through graphs.

In this study, a general, analytical solution of a steady creeping or Stokes flow impinging on a stationary spherical cap-shaped bubble on a solid flat surface is provided. The phenomena usually take place in bubble∕pore formation in materials and manufacturing processing and MEMS, boiling heat transfer, and nucleation and growth of gas bubbles in tissues of animals and human, etc. In view of high capillary pressure and small liquid pressure, the shape of the bubble in a microscale can be considered as a spherical cap on a surface. In this model, shear stresses associated with the no-slip condition, interfacial mass transport such as condensation and evaporation are absent on the bubble surface. An analytical solution of the Stokes equations for zero Reynolds number flow in a toroidal coordinate system is found by decomposing the flow into a stagnation flow and a flow disturbed by the bubble and applying the separation-of-variables method. The stream function can be expressed in terms of a difference in Legendre functions of the first kind. The effects of impinging velocity and contact angle of the bubble on the flow pattern and pressure distribution are provided.

The problem of plane stagnation point flow with freestream turbulence is examined from a basic theoretical standpoint. It is argued that the singularity which arises in the standard K –ε model results from the use of an inconsistent freestream boundary condition. The inconsistency lies in the implementation of a production-equals-dissipation equilibrium hypothesis in conjunction with a freestream mean velocity field that corresponds to homogeneous plane strain—a turbulent flow for which the standard K –ε model does not predict such a simple equilibrium. The ad hoc adjustment that has been made in the constants of the ε-transport equation to eliminate this singularity is shown to be inconsistent for homogeneous plane-strain turbulence as well as other benchmark turbulent flows. An alternative means to eliminate this singularity—without compromising model predictions in more basic turbulent flows—is proposed based on the incorporation of nonequilibrium vortex stretching effects in the turbulent dissipation rate equation.

A practical one-parameter polynomial type integral method is developed in this paper for laminar incompressible plane and thin axisymmetric boundary layer flow with transpiration and pressure gradient. The method features the use of approximations for the velocity distribution that are based on second and third order polynomial approximations for the distribution in shear stress. These approximations are used to develop solutions to the integral momentum equation for similar and nonsimilar flows. The accuracy of the method is generally within about 3 percent, except near separation where the error can reach 10 to 15 percent. The range of conditions for which the method applies covers a fairly wide range of blowing and suction rates and pressure gradients which encompasses plane and axisymmetric stagnation flows and extends to separation. Because of its fundamental nature, the approach provides a basis for generalization to heat and mass transfer and turbulent flow, and provides a framework for the development of more accurate multiple parameter integral methods for transpired boundary layer flow.

The equations of motion for the three-dimensional nonsteady flow of incompressible viscous fluid in the vicinity of a forward stagnation point are reduced to three ordinary differential equations for a potential flow field chosen to vary inversely as a linear function of time. The resulting ordinary differential equations contain two parameters C and D, the former characterizes the type of curvature of the surface around the stagnation point and the latter the degree of acceleration or deceleration of the potential flow. The simple stagnation-point problems which have been studied previously are obtainable as special cases of the present analysis by assigning particular values to C and D. Exact solutions have been computed numerically for the velocity field and the pressure distribution in the boundary-layer flow around the stagnation point of a three-dimensional blunt body for the values of the parameter C from 0–1.