A. I. van de Vooren

The Navier-Stokes solution for laminar flow past a semi-infinite flat plate

SummaryA numerical solution joining Carrier and Lins solution near the leading edge to the boundary layer solution at large distance of the leading edge is presented. The solution is valid for any Reynolds number. Results are given for the skin friction, the integrated skin friction, the displacement thickness, the pressure along the plate and the velocity ahead of the plate. The asymptotic value of the integrated skin friction agrees very well with the exact value. The displacement thickness is already different from zero for small distances ahead of the plate.

A finite element stability analysis for the Marangoni problem in a rectangular container with rigid sidewalls

Abstract A two-dimensional rectangular box is partially filled with a fluid containing a solute which evaporates at the upper surface. The system is considered under zero-gravity. For sufficiently large Marangoni number the quiesent state becomes unstable due to surface tension effects. By aid of a Galerkin method using splines the eigenvalues and eigenvectors of the linearized system are determined. Each eigenvalue corresponds to a critical Marangoni number for a certain mode (eigenvector). The eigenvalues have been investigated as functions of the aspect ratio of the box. Two different symmetries are possible for the modes and it is shown that only eigenvalues pertaining to modes of different symmetry can coincide.

INITIAL FLOW DEVELOPMENT DUE TO MARANGONI CONVECTION IN A MASS-TRANSFER SYSTEM

Abstract A linear stability analysis is used to investigate the onset of convection in a mass transfer system due to surface tension differences. Instead of a steady linear concentration distribution as basic state we make a detailed analysis of the mass transfer due to diffusion in the initially quiescent medium and obtain a time-dependent basic concentration distribution. The critical wavenumber, at neutral stability, and the preferred wavenumber, with largest growth factor, are time dependent and both decrease in time.

Incompressible viscous flow near the leading edge of a flat plate admitting slip

SummaryThe shear stress at the leading edge, calculated on basis of the Navier-Stokes equations and the no-slip boundary condition, approaches infinity. However, taking into account the mean free path of the molecules, which implies admitting a certain slip, the shear stress becomes inversely proportional to the square root of the Knudsen numberk ifk→0.k is defined as the ratio between the mean free path and the viscous length. The new boundary condition modifies the shear stress only within the Knudsen region of which the size is of the order of 3 to 4 times the mean free path.

The connection between Ekman and Stewartson layers for a rotating disk

When a disk of finite radius and the surrounding medium rotate coaxially with slightly different angular velocities, a so-called Stewartson layer exists at the edge of the disk. The properties of this layer outside the boundary layer of the disk have been given in a previous publication. In the present paper it is shown how the radial flow of the Ekman boundary layer turns into the axial flow of the Stewartson layer. This happens in a region of which both the radial and axial dimensions are O(E1/2), where E is the Ekman number.

Fluid flow induced by a rotating disk of finite radius

The boundary-layer equations outside a rotating disk of radius a have been solved. It is shown that it is unnecessary to take special precautions for the sudden change in boundary conditions at the edge of the disk except if one is interested in the flow at distances which are smaller than about 10−3a from the edge. The behaviour of the flow at large distances from the disk is investigated analytically with results which are confirmed by the numerical computations.

The flow near the edge of a disc at rest in a rotating fluid

SummaryThe Reynolds number Re being based on the angular velocity of the fluid and the radius of the disc, it is shown that within a distance O(Re-2/3) from the edge of the disc, the flow is determined by the Navier-Stokes equations. The boundary-value problem describing this flow is formulated. The asymptotic behaviour of its solution is investigated analytically and its complete numerical solution is evaluated. Results for various physical quantities, among them the additional torque due to the Navier-Stokes flow, are presented.