Similarity solution

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

Waiting Time Solutions of the Shallow Water Equations

Abstract: In this paper we present new solutions of the shallow water equations with waiting time properties. These are shown to represent the local behaviour of the solution of an initial-value problem for certain types of initial data which models the evolution of an initially stationary inviscid fluid layer of nonuniform thickness possessing bounded support. In this context the motion of the interfaces, where the layer thickness is zero, is shown to be governed solely by the local properties of the initial data which may remain stationary for a finite non-zero time before spreading. For other types of initial data we show that the interfaces may move immediately, the local behaviour being governed by another type of similarity solution. This behaviour is remarkably similar to that exhibited by degenerate parabolic equations which have also been used to model viscous thin film flows in the lubrication approximation.

Centrifugal Convection due to Mass Transfer Near a Rotating Disk at High Schmidt Number

Abstract: The centrifugally-driven flow due to a density gradient between the surface of an infinite disk and the ambient fluid in a rotating system with mass transfer is studied for the case of high Schmidt number. Under certain assumptions the velocity and density fields exhibit a similarity like the classical von Karman disk flow, and the governing equations reduce to a nonlinear system of ordinary differential equations. These equations are solved by boundary layer technique or numerically, for high Schmidt number σ = v/D and finite or small density difference eρ = (ρd - ρ∞ )/ρ∞ . In the latter case it is shown that the major scaling parameter is the product σeρ . For σρ ≫ 1 the flow field consists of a constant density (ρ∞ ), linear Ekman layer driven by a buoyancy sublayer of relative thickness (σeρ )-1/4 in which ρ varies from ρd to ρ∞ . The representative Rossby number of the buoyancy driven flow is (σeρ )-1/2 . The general case eρ = O(1), σ ≫ 1 shows similar trends, i.e., a σ-1/4 sublayer. The case of simultaneous driving by density difference and angular velocity difference ev = (Ωd - Ω∞ )/Ω∞ is also discussed.

Development of Boundary Layer of Highly Elastic Flow of the Upper Convected Maxwell Fluid over a Stretching Sheet

Abstract: High Weissenberg boundary layer flow of viscoelastic fluids on a stretching surface has been studied. The flow is considered to be steady and two dimensional. Flows of viscoelastic liquids at high Weissenberg number exhibit stress boundary layers near walls. These boundary layers are caused by the memory of the fluid. Upon proper scaling and by means of an exact similarity transformation, the non-linear momentum and constitutive equations of each layer transform into the respective system of highly nonlinear and coupled ordinary differential equations. Effects of variation in pressure gradient and Weissenberg number on velocity profile and stress components are investigated. It is observed that the value of stress components decrease by Weissenberg number. Moreover, the results show that increasing the pressure gradient results in thicker velocity boundary layer. It is observed that unlike the Newtonian flows, in order to maintain a potential flow, normal stresses must inevitably develop in far fields.