Andrew P. Bassom

The Blasius boundary layer flow of a micropolar fluid

We consider the Blasius boundary-layer flow of a micropolar fluid over a flat plate. Due to inadequacies in previous studies a full derivation of the boundary-layer equations is given. The resulting nonsimilar equations are solved using the Keller-box method and solutions for a range of parameters are presented. It is found that a two-layer structure develops as the distance downstream increases. An asymptotic analysis of this structure is presented, and the agreement between the analysis and the numerical solution is found to be excellent.

Accelerated diffusion in the centre of a vortex

The spiral wind-up and diffusive decay of a passive scalar in circular streamlines is considered. An accelerated diffusion mechanism operates to destroy scalar fluctuations on a time scale of order P1/3 times the turn-over time, where P is a Peclet number. The mechanism relies on differential rotation, that is, a non-zero gradient of angular velocity. However if the flow is smooth, the gradient of angular velocity necessarily vanishes at the centre of the streamlines, and the time scale becomes greater. The behaviour at the centre is analysed and it is found that scalar there is only destroyed on a time scale of order P1/2. Related results are obtained for magnetic field and for weak vorticity, a scalar coupled to the stream function of the flow. Some exact solutions are presented.

The linear stability of flat Stokes layers

The linear stability of the Stokes layer generated by an oscillating flat plate is investigated using Floquet theory. The results obtained include the behaviour of the growth rate of the disturbances, part of the corresponding neutral curve and the structure of neutrally stable disturbances. Previously unknown properties of the growth rate cause the neutral curve to have a complicated geometry: the majority of the marginal curve is defined by waves propagating relative to the basic flow and the curve is smooth in character, but for certain very narrow bands of wavenumbers it was found that stationary modes are the first to become unstable. This phenomenon has the consequence that the underlying smooth neutral curve is punctuated by thin finger-like features. The structure of the eigenfunctions showed that the neutrally stable disturbances tend to grow most rapidly just after the wall velocity passes through zero.

The spiral wind-up of vorticity in an inviscid planar vortex

The relaxation of a smooth two-dimensional vortex to axisymmetry, also known as ‘axisymmetrization’, is studied asymptotically and numerically. The vortex is perturbed at t = 0 and dierential rotation leads to the wind-up of vorticity fluctuations to form a spiral. It is shown that for innite Reynolds number and in the linear approximation, the vorticity distribution tends to axisymmetry in a weak or coarse-grained sense: when the vorticity eld is integrated against a smooth test function the result decays asymptotically as t with = 1 +( n 2 +8 ) 1= 2 , where n is the azimuthal wavenumber of the perturbation and n > 1. The far-eld stream function of the perturbation decays with the same exponent. To obtain these results the paper develops a complete asymptotic picture of the linear evolution of vorticity fluctuations for large times t, which is based on that of Lundgren (1982).

The onset of Darcy-Bénard convection in an inclined layer heated from below

SummaryWe present an, account of the linear instability of Darcy-Boussinesq convection in a uniform, unstably stratified porous layer at arbitrary inclinations α from the horizontal. A full numerical solution of the linearized disturbance equations is given and the detailed graphical results used to motivate various asymptotic analyses. A careful study shows that at large Rayleigh numbers two-dimensional instability can only arise when α≤31.30°. However it is also demonstrated that the maximum inclination below which this instability may be possible is the slightly greater value of 31.49° which corresponds to a critical Rayleigh number of 104.30.

Local thermal non-equilibrium effects arising from the injection of a hot fluid into a porous medium

We examine the effect of local thermal non-equilibrium on the infiltration of a hot fluid into a cold porous medium. The temperature fields of the solid porous matrix and the saturating fluid are governed by separate, but coupled, parabolic equations, forming a system governed by three dimensionless parameters. A scale analysis and numerical simulations are performed to determine the different manners in which the temperature fields evolve in time. These are supplemented by a large-time analysis showing that local thermal equilibrium between the phases is eventually attained. It is found that the thickness of the advancing thermal front is a function of the governing parameters rather than being independent of them. This has the implication that local thermal equilibrium is not equivalent to a single equation formulation of the energy equation as might have been expected. When the velocity of the infiltrating fluid is sufficiently large, the equations reduce to a hyperbolic system and a thermal shock wave is formed within the fluid phase. The strength of the shock decays exponentially with time, but the approach to local thermal equilibrium is slower and is achieved algebraically in time.

The linear stability of high-frequency oscillatory flow in a channel

The linear stability of the Stokes layers generated between a pair of synchronously oscillating parallel plates is investigated. The disturbance equations were studied using Floquet theory and pseudospectral numerical methods used to solve the resulting system. Neutral curves for an extensive range of plate separations were obtained and when the plate separation is large compared to the Stokes layer thickness the linear stability properties of the Stokes layer in a semi-infinite fluid were recovered. A detailed analysis of the damping rates of disturbances to the basic flow provides a plausible explanation of why several previous studies of the problem have failed to detect any linear instability of the flow. To compare more faithfully with experimental work the techniques used for the channel problem were modified to allow the determination of neutral curves for axisymmetric disturbances to purely oscillatory flow in a circular pipe. Critical Reynolds numbers for the pipe flow tended to be smaller than their counterparts for the channel case but the smallest critical value was still almost twice the experimentally reported result.

Strongly nonlinear convection cells in a rapidly rotating fluid layer

We investigate the properties of some strongly nonlinear convection cells which may occur in a rapidly rotating fluid layer. Although the stability properties of such layers have been extensively studied, most of the theoretical work concerned with this topic has been based upon either linear or weakly nonlinear analyses. However, it is well known that weakly nonlinear theory has a limited domain of validity for if the amplitude of the convection cells becomes too large then the mean temperature profile within the layer is dramatically perturbed away from its undisturbed state and the assumptions underpinning weakly nonlinear theory break down. It is the case for most fluid stability problems that when the stage is reached that the mean flow is significantly altered by the presence of instability modes, then analytical progress becomes impossible. The problem can then only be resolved by a numerical solution of the full governing equations but we show that for the case of convection rolls within a rapidly rotating layer this sequence of events does not arise. Instead, the properties of large amplitude convection rolls (which are sufficiently strong so as to completely restructure the mean temperature profile) can be determined by analytical methods. In particular, the whole flow structure can be deduced once a single, very simple eigenproblem has been solved. This solution enables us to discuss how large amplitude cells can significantly affect the characteristics of the flow leading to greatly enhanced heat transfer across the layer.

Ermakov systems of arbitrary order and dimension: structure and linearization

Ermakov systems of arbitrary order and dimension are constructed. These inherit an underlying linear structure based on that recently established for the classical Ermakov system. As an application, alignment of a (2 + 1)-dimensional Ermakov and integrable Ernst system is shown to produce a novel integrable hybrid of a (2 + 1)-dimensional sinh - Gordon system and of a conventional Ermakov system.

Non-stationary cross-flow vortices in three-dimensional boundary-layer flows

The upper-branch neutral stability of three-dimensional disturbances imposed on a three-dimensional boundary-layer profile is considered and in particular we investigate non-stationary cross-flow vortices. The wave speed is taken to be small initially and with a disturbance structure analogous to that occurring in two-dimensional boundary-layer stability the linear and nonlinear eigenrelations are derived for profiles with more than one critical layer. For the flow due to a rotating disc we show that linear viscous neutral modes exist for all wave angles between 10.6° and 39.6°. As the extremes of this range are approached the flow structure evolves to another, either the viscous mode of Hall (Proc. R. Soc. Lond. A 406, 93(1986)) or the non-stationary inviscid modes considered by Stuart in Gregory et al. (Phil. Trans. R. Soc. Lond. A 248, 155 (1955)). In the former case, corresponding to wave angles of 39.6°, the waves become almost stationary and in the latter case, with wave angles of 10.6° the waves are travelling much faster with a disturbance structure based on the Rayleigh scalings. The analysis is extended to include O(1) wavespeeds and we show that as the wavespeed of the cross-flow vortex approaches the tree-stream value, the corresponding disturbance amplitude increases, the growth here being slower than that for two-dimensional boundary layers.