Similarity solution

About: Similarity solution is a research topic. Over the lifetime, 2074 publications have been published within this topic receiving 59790 citations.

Similarity solutions of the rayleigh problem for ostwald–de wael electrically conducting fluids

Abstract: We investigate in the present paper the magnetic Rayleigh problem, where a semi-infinite flat plate is moving with a power-law velocity, in a non-Newtonian power-law fluid (Ostwald–de Wael model). The non-stationary flow of this electrically conducting fluid in a transverse magnetic field is then analyzed. The solutions of this problem are obtained by means of similarity techniques. The main goal, is to investigate existence, uniqueness and behavior of such solutions, according to the values of the physical parameters.

Interaction of an unconfined vortex with a solid surface

Abstract: The interaction of an open vortex with a solid plane perpendicular to the axis of the vortex is analyzed numerically. We solve the axisymmetric, incompressible Navier-Stokes equations with boundary conditions that far from the axis correspond to the near-inviscid far-field of Long’s similarity solution for an open vortex. Continuation techniques are used to solve the equations of motion for varying Reynolds numbers. When this parameter is large enough, a vortex breakdown phenomenon occurs, producing a small region of reversed flow at the axis. This region increases in size and migrates toward the solid plane for increasing Reynolds numbers. The subsequent intensification of the swirl near both the axis and the surface generates a bifurcation with nonuniqueness of the solution corresponding to a new, more intense, vortex breakdown. At the end, for Reynolds numbers above a critical value, the flow acquires a two-celled structure, with a region of reversed flow all along the axis surrounded by an annular updraft region with intensified swirl. For large Reynolds numbers this flow structure tends, far above from the plane, to Long’s self-similar solution of Type II. Thus, we show that of the two different similarity solutions for high Reynolds numbers found by Long for a given flow force, only that with negative axial velocity at the axis (Type II solution) is compatible with the viscous interaction of the vortex with a solid surface. We also find that the corresponding flow force increases linearly with the Reynolds number, so that the solution tends for large Reynolds numbers to the similarity solution with the most negative axial velocity at the axis. This transition from one-cell to two-cell flow configuration, and its relation to the intensification of the swirl in the flow, is in agreement with observations of intense tornado-like vortices, where the flow at the axis is directed downward, while the rotation of the flow is intensified in an annular updraft.

Similarity solution and Runge Kutta method to a thermal boundary layer model at the entrance region of a circular tube

Abstract: Purpose The purpose of this paper is to re-examine the assumptions implicit in Leveque’s approximation, and the variation of the temperature and the thickness of the boundary layer were illustrated using the developed solution. The analytical solutions are then checked against numerical solution programming by FORTRAN code obtained via using Runge–Kutta fourth-order (RK4) method. Finally, other important thermal results obtained from this analysis, such as approximate Nusselt number in the thermal entrance region, was discussed in detail. After that, the analytical results of the present paper are validated with certain previous investigations which were found in the specialized literature. Design/methodology/approach By defining a similarity variable, the governing equations are reduced to a dimensionless equation with an analytic solution in the entrance region. This paper gives justification for the similarity variable via scaling analysis, details the process of converting to a similarity form and presents a similarity solution. The calculation methodology for numerical resolution is based on the RK4 technique. Findings The profiles of the solutions are provided from which the authors infer that the numerical and exact solutions agreed very well. Another result that the authors obtained from this paper is the number of Nusselt in the thermal entrance region for which a parametric study was carried out and discussed well for the impact of scientific contribution. Originality/value The novelty of this paper is the application of the RK4 with a step size control, as a sequential numerical method of a ODEs system compared with the exact similarity solution of the thermal boundary layer problem.

Universality in the run-up of shock waves to the surface of a star

Abstract: We investigate the run-up of a shock wave from inside to the surface of a perfect fluid star in equilibrium and bounded by vacuum. Near the surface we approximate the fluid motion as plane-symmetric and the gravitational field as constant. We consider the ‘hot’ equation of state P = (? ? 1)?e and its ‘cold’ (fixed entropy, barotropic) form P = K0?? (the latter does not allow for shock heating). We numerically find that the evolution of generic initial data approaches universal similarity solutions sufficiently near the surface, and we explicitly construct these similarity solutions. The two equations of state show very different behaviour because shock heating becomes the dominant effect when it is allowed. In the barotropic case, the fluid velocity behind the shock approaches a constant value, while the density behind the shock approaches a power law in space, as the shock approaches the surface. In the hot case with shock heating, the density jumps by a constant factor through the shock, while the sound speed and fluid velocity behind the shock diverge in a whiplash effect. We tabulate the similarity exponents as a function of the equation of state parameter ? and the stratification index n?

Motion of a Triple Junction

Abstract: The motion of a triple junction is investigated. We consider only the two-dimensional case and assume that initially the fluid is in the form of three inviscid wedges. A similarity solution is determined which accounts for a balance of force at the triple junction. This similarity solution is computed numerically using boundary integral methods. Results are presented for different initial wedge angles and surface tension ratios. In particular the location of the triple junction and the resulting capillary waves along the interfaces are discussed.