Similarity solution

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

Newtonian pizza: spinning a viscous sheet

Abstract: We study the axisymmetric stretching of a thin sheet of viscous fluid driven by a centrifugal body force. Time-dependent simulations show that the sheet radius R(t) tends to infinity in finite time. As time t approaches the critical time t∗, the sheet becomes partitioned into a very thin central region and a relatively thick rim. A net momentum and mass balance in the rim leads to a prediction for the sheet radius near the singularity that agrees with the numerical simulations. By asymptotically matching the dynamics of the sheet with the rim, we find that the thickness h in the central region is described by a similarity solution of the second kind, with h ∝ (t∗ − t)α where the exponent α satisfies a nonlinear eigenvalue problem. Finally, for non-zero surface tension, we find that the exponent increases rapidly to infinity at a critical value of the rotational Bond number B = 1/4. For B > 1/4, surface tension defeats the centrifugal force, causing the sheet to retract rather than to stretch, with the limiting behaviour described by a similarity solution of the first kind.

Swirling Axisymmetrical Laminar Jet

Abstract: A similarity solution for the complete Navier‐Stokes equations describing the flow of an incompressible laminar swirling jet was obtained. The solution clearly shows the effect of swirl on the other velocity components, and it is valid for all degrees of swirl. It thus differs from the previous solutions which were of the perturbation type.

A simple method for solving unidirectional methane gas flow in coal seam based on similarity solution

Abstract: The equation used to model the unidirectional flow of methane gas in coal seams is usually formulated as a nonlinear partial differential equation, which needs to be solved numerically with a computer program. Nevertheless, for people without access to the computer program, the conventional numerical method may be inconvenient. Thus, the objective here is to seek some method simpler than the conventional one for solving the flow problem. A commonly used model of the unidirectional methane gas flow is considered, where the methane adsorption is described by the Langmuir isotherm and the free gas is treated as real gas. By introducing the similarity solution, a simple method for solving the flow model is proposed, which can be done on a hand calculator. It is shown by two examples that the gas pressure profile obtained by the proposed method agrees well with the direct numerical solution of the flow model.