Similarity solution

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

Similarity Solution for Flow of a Micro-Polar Fluid Through a Porous Medium

Abstract: The equations of two dimensional incompressible steady micro-polar fluid flows through a porous medium are studied. Lie group analysis is employed and the solutions corresponding to the translational symmetry are developed. A boundary value problem is investigated and the results are sketched graphically. The effect on the flow of the porosity coefficient of the porous medium and the micro-polar parameters are observed.

Midplane-symmetry breaking in the flow between two counter-rotating disks

Abstract: This paper considers the axisymmetric steady flow driven by exact counter rotation of two co-axial disks of finite radius. At the edges of the rotating disks one of three conditions is (typically) imposed: (i) zero velocity, corresponding to a stationary, impermeable, cylindrical shroud (ii) zero normal velocity and zero tangential fluid traction, corresponding to a (confined) free surface and (iii) an edge constraint that is consistent with a similarity solution of von Karman form. The similarity solution is valid in an infinite geometry and possesses a pitchfork bifurcation that breaks the midplane symmetry at a critical Reynolds number. In this paper, similar bifurcations of the global (finite-domain) flow are sought and comparisons are made between the resulting bifurcation structure and that found for the similarity solution. The aim is to assess the validity of the nonlinear similarity solutions in finite domains and to explore the sensitivity of the solution structure to edge conditions that are implicitly neglected when assuming a self-similar flow. It is found that, whilst the symmetric similarity solution can be quantitatively useful for a range of boundary conditions, the bifurcated structure of the finite-domain flow is rather different for each boundary condition and bears little resemblance to the self-similar flow.

Fluid film sprayed on a stretching surface

Abstract: A liquid film, due to spraying, is deposited on a stretching surface. The physical problem is governed by a nondimensional parameter α, representing the relative importance of spraying rate to stretching rate. After a similarity transform, the Navier-Stokes equations reduce to a nonlinear ordinary differential equation which is solved by perturbation and numerical integration. The results show nonuniqueness and nonexistence of a steady state solution for certain values of α. Velocity profiles and heat transfer characteristics are determined.

On self-similar thermal rupture of thin liquid sheets

Abstract: We consider the dynamics of a symmetrically heated thin incompressible viscous fluid sheet. We take surface tension to be temperature dependent and consequently the streamwise momentum equation includes the effects of thermocapillarity, inertia, viscous stresses, and capillarity. Energy transport to the surrounding environment is also included. We use a long-wave analysis to derive a single nondimensional system which, with appropriate choices of Reynolds number, recovers two previously studied cases. In both cases, we find conditions under which sufficiently large-amplitude initial temperature profiles induce film rupture in finite time, notably without the inclusion of disjoining pressures from van der Waals effects. When the Reynolds number is large, the similarity solution is governed by a balance of inertia and capillarity near the rupture location, analogous to the isothermal case. When the Reynolds number is small, the thermocapillary transients induce the same similarity solution over intermediate times that is found for the drainage of lamellae in foams. For O(1) Reynolds numbers, the dynamics are governed initially by the large Reynolds number evolution, and then a transition over several orders of magnitude in the sheet thickness needs to take place before the small Reynolds number similarity solution is observed.

Similarity solutions and the computation formulas of a nonlinear fractional-order generalized heat equation

Abstract: A generalized nonlinear heat equation with the fractional derivative is proposed, whose similarity solutions are derived from a type of special scalar transformation with two parameters. With the h...