D.B. Ingham

Laminar boundary layer on an impulsively started rotating sphere

The problem of determining the development with time of the flow of a viscous incompressible fluid outside a rotating sphere is considered. The sphere is started impulsively from rest to rotate with constant angular velocity about a diameter. The motion is governed by a coupled set of three nonlinear time‐dependent partial differential equations which are solved by first employing the semi‐analytical method of series truncation to reduce the number of independent variables by one and then solving numerically a finite set of partial differential equations in one space variable and the time. The calculations have been carried out on the assumption that the Reynolds number is very large. The physical properties of the flow are calculated as functions of the time and compared with existing solutions for large and small times. A radial jet is found to develop with time near the equator of the sphere as a consequence of the collision of the boundary layers.

Diffusion of aerosols from a stream flowing through a cylindrical tube

Abstract The problem of steady state mass diffusion of acrosols without axial diffusion in a long cylindrical channel and for small diffusion parameter, Δ, is presented. Results are obtained when the laminar fluid flow is Poiseuille, plug and a combination of Poiseuille and plug, such that an allowance for slip velocity of the fluid at the wall can be taken into account. The asymptotic solution for large Δ, that has been obtained by several previous authors, is matched to asymptotic solutions that have been obtained in this paper for small Δ. A solution for the complete range of values of the parameter Δ is thus presented.

The steady flow due to a rotating sphere at low and moderate Reynolds numbers

The problem of determining the steady axially symmetrical motion induced by a sphere rotating with constant angular velocity about a diameter in an incompressible viscous fluid which is at rest at large distances from it is considered. The basic independent variables are the polar co-ordinates ( r , θ) in a plane through the axis of rotation and with origin at the centre of the sphere. The equations of motion are reduced to three sets of nonlinear second-order ordinary differential equations in the radial variable by expanding the flow variables as series of orthogonal Gegenbauer functions with argument μ = cosθ. Numerical solutions of the finite set of equations obtained by truncating the series after a given number of terms are obtained. The calculations are carried out for Reynolds numbers in the range R = 1 to R = 100, and the results are compared with various other theoretical results and with experimental observations. The torque exerted by the fluid on the sphere is found to be in good agreement with theory at low Reynolds numbers and appears to tend towards the results of steady boundary-layer theory for increasing Reynolds number. There is excellent agreement with experimental results over the range considered. A region of inflow to the sphere near the poles is balanced by a region of outflow near the equator and as the Reynolds number increases the inflow region increases and the region of outflow becomes narrower. The radial velocity increases with Reynolds number at the equator, indicating the formation of a radial jet over the narrowing region of outflow. There is no evidence of any separation of the flow from the surface of the sphere near the equator over the range of Reynolds numbers considered.

A new model for viscous dissipation in porous media across a range of permeability values

In this paper a unified mathematical theory for the viscous dissipation term in the governing Brinkman equation is derived. This term has, unlike other models, the correct asymptotic behaviour in both the fully Darcy and Newtonian fluid flow limits.

Emerging technologies and techniques in porous media

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The flow in industrial cyclones

A simple mathematical model for the flow in a conical cyclone is developed which allows solutions to be obtained in closed form. The flow in the main body of the cyclone is regarded as inviscid but the nature of the fluid entry to the device and the conical geometry ensure that secondary flows develop which make the flow highly rotational. The results of the theory are compared with data from two quite different experimental investigations, and good agreement is obtained.

Finite-difference methods for calculating steady incompressible flows in three dimensions

Abstract A new stable numerical method is described for solving the Navier-Stokes equations for the steady motion of an incompressible fluid in three dimensions. The basic governing equations are expressed in terms of three equations for the velocity components together with three equations for the vorticity components. This gives six simultaneous coupled second-order partial differential equations to be solved. A finite-difference scheme with second-order accuracy is described in which the associated matrices are diagonally dominant. Numerical results are presented for the flow inside a cubical box due to the motion of one of its sides moving parallel to itself for Reynolds numbers up to 100. Several methods of approximation are considered and the effect of different discretizations of the boundary conditions is also investigated. The main method employed is stable for Reynolds numbers greater than 100 but a finer grid size would be required in order to obtain accurate results.

NUMERICAL MODELLING OF THE FLUID AND PARTICLE PENETRATION THROUGH SMALL SAMPLING CYCLONES

Abstract In this paper, the numerical modelling of the fluid flow and the particle dynamics in small sampling cyclones is addressed. The presence of the particles is assumed not to affect the fluid flow in the cyclone due to the light particle loads which are normally encountered in most small air sampling cyclone applications. The governing fluid flow equations, along with the RNG turbulent model equations, are solved using the control volume method in a body-fitted coordinate system. A particle tracking technique is used to track the motion of the particles in the cyclone. We have found reasonable agreement between the present numerical results and the experimental data of Kim and Lee (1990) for the particle separation efficiency of a small sampling cyclone even though the turbulent effects of the fluid flow have not been included in the particle tracking procedure. Further, the effects of changing various geometrical dimensions and operating conditions on the performance of the cyclone have been discussed. The predictions of the cyclone separation efficiency using the Barth (1956) and the Dietz (1981) empirical models are also presented.

An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation

In this paper, the iterative algorithm proposed by Kozlov et al. [Comput. Maths. Math. Phys. 31 (1991) 45] for obtaining approximate solutions to the ill-posed Cauchy problem for the Helmholtz equation is analysed. The technique is then numerically implemented using the boundary element method (BEM). The numerical results confirm that the iterative BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data. An efficient stopping regularising criterion is also proposed.

Flow past a suddenly heated vertical plate in a porous medium

An analysis is presented for the steady free convection flow about a semi-infinite vertical flat plate that is embedded in a saturated porous medium at high Rayleigh numbers. Similarity solutions are obtained for a class of problems where the wall temperature varies as a power of the distance from the leading edge of the plate. The existence and uniqueness of the solutions are considered. The approach to this steady-state solution is also considered by investigating the temporal development of the flow when the temperature of the plate is impulsively increased from that of the surroundings. A numerical solution is presented that matches the small and large time solutions. For some temperature distributions on the plate it is found that the velocity achieves its maximum value within the boundary layer. For these the disturbance from the leading edge of the plate travels fastest within the boundary layer. An asymptotic solution valid at large times is presented and the approach of the numerical solution to this asymptotic solution is illustrated. For the situation in which the plate is impulsively heated to a constant temperature an analysis is presented for the early stages of the departure from the one-dimensional solution.