The thermodynamics of irreversible processes is normally limited to processes that can be adequately described by linear constitutive relations, like those of Fourier and Newton in a simple gas. In this paper we use thermodynamic arguments to derive the (nonlinear) Burnett equations for a monatomic gas, thus avoiding the complicated kinetic theory by which the equations were discovered and which somewhat obscures the origin of the various terms in the equations. Expressions are given for the entropy, its flux and its production rate correct to second-order in Knudsen number. The theory involves five phenomenological parameters, and as there are eleven coefficients in the second-order terms of Burnett's equations, we are able to deduce several necessary constraints between these coefficients. Compact forms for the equations are found that clarify their physical significance. The general method we have developed is applicable to media other than simple gases.

In a final section we use our theory of Burnett's equations to draw some general conclusions concerning the second law of thermodynamics. It is shown that the Clausius-Duhem inequality holds only for the linear theory of constitutive relations; and that axiomatic generalizations of the inequality to nonlinear processes – common in continuum mechanics–fail because the vital distinction between reversible and irreversible processes is not made.

The instantaneous velocity vector fields which surround the coherent structures of Perry & Lim (1978) in coflowing jets and wakes have been successfully measured and related to the smoke patterns for Reynolds numbers of order 1000. By the use of critical point theory, a qualitative description of the three-dimensional flow field can be made and is applied to the simplest structures which were classified by Perry & Lim. From these results, the convection of smoke and vorticity from the source and the entrainment properties of the structures are discussed.

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.

The flow of a rotating, incompressible fluid over isolated topography whose non-dimensional height (i.e. topographic height divided by the mean fluid depth) is large compared with the Rossby number is studied. Attention is restricted to flow which is sufficiently shallow that the free-surface equations provide an adequate description. The flow is forced laterally by a specified upstream inflow (obtained from solutions of the zonally symmetric model equations) and by a prescribed surface stress. Dissipation is incorporated using a Rayleigh friction acting anti-parallel to the flow.

Steady-state solutions for uniform inflow on an f-plane are found for (a) linear viscous flow, (b) quasi-geostrophic flow with and without friction and (c) inviscid flow with and without a rigid lid. The presence of friction produces an upstream–downstream flow asymmetry over the obstacle and an associated topographic drag while inertial terms produce left-right (relative to an observer looking downstream) asymmetry. The blocking efficiency B (the percentage of the incident mass flux going around the obstacle rather than over it) of a Gaussian obstacle is largest (∼ 100%) for case (a) when viscous effects are small. In contrast quasi-geostrophic theory calculates no flow blocking (B ≡ O). For inviscid inertial theory, B ∼ 10% and is independent of the Rossby number. The presence of a free surface decreases the blocking for small-Rossby-number flow.

Numerical solutions of the appropriate initial, boundary-value problem for the complete model equations confirm these results and extend them to include the effects of (i) horizontal shear in the upstream inflow, (ii) the magnitude and shape of the topography, and (iii) variations in the Coriolis parameter (β-effect).

Measurements of the shock stand-off distance on cones of various base diameters in carbon dioxide and nitrogen flows with dissociative relaxation show that the detachment process occurs more gradually with relaxation than in a perfect gas, as the cone semi-angle is increased. This is in agreement with a prediction which is made on the basis of the behaviour of the sonic surface in the flow field. The phenomenon is similar to that observed previously with wedge flow but shows interesting effects peculiar to the cone flow. The cone experiments also eliminate the end effect and permit easy variation of the relaxation rate parameter without changing the gas. Perfect-gas argon experiments provide a convenient check.

The propagation of finite-amplitude internal waves in a shear flow is considered for wavelengths that are long compared to the shear-layer thickness. Both singular and regular modes are investigated, and the equation governing the amplitude evolution is derived. The theory is generalized to allow for a radiation condition when the region outside the stratified shear layer is unbounded and weakly stratified. In this case, the evolution equation contains a damping term describing energy loss by radiation which can be used to estimate the persistence of solitary waves or nonlinear wave packets in realistic environments. A continuous three-layer model is studied in detail and closed-form expressions are obtained for the phase speed and the coefficients of the nonlinear and dispersive terms in the amplitude equation as a function of Richardson number.

To resolve possible fluctuations about the mean motion of the Desnyansky-Novikov model for Kolmogorov's inertial flow, we have investigated two dynamical systems of the cascade process which are formally derivable from Burgers’ equation. The first cascade model produced no fluctuations, for its trajectory was identical with the Desnyansky–Novikov model's. Disappointingly, the second cascade system, which is similar to the Kerr–Siggia model, has also proved unable to engender fluctuations. This is because the second model when truncated consistently maps an arbitrary initial point into the attainable phase space of the first cascade model. However, when truncated inconsistently the trajectory of second model can exhibit a quite erratic and somewhat sporadic motion, thereby reflecting the apparently random motion of inviscid equilibrium solutions. Therefore, the observation of temporally intermittent fluctuations by a stationary Kerr–Siggia model is due to the inconsistent truncation produced by restricting energy dissipation for all but the upper truncation mode in their model.

The boundary-layer equations of mixed convection are examined in the vicinity of separation. The correlation between the uniform wall temperature case and that of compressible boundary layer flow is outlined. Goldstein–Stewartson–Buckmaster theory is thus appropriate and associated indeterminacies in the theory are evaluated from a numerical integration. The case of uniform heat flux at the wall is then examined theoretically. Significantly it is concluded that the original Goldstein–Stewartson theory is sufficient to describe the structure of the singularity at separation in this case. Indeterminacies associated with the theory are determined via a reconciliation between analytical and numerical representation of skin friction and heat transfer coefficients near separation.

Previous work on irrotational incompressible inviscid flow (Guiraud & Zeytounian 1977) is extended to rotational and compressible flow. A formal proof is given that, within the core, one may avoid computing with the sheet by defining an equivalent continuous flow. One shows how the vorticity and the entropy gradient between the turns of the sheet are transported along trajectories of the equivalent continuous flow.

Using the method of averaged equations, we examine the difference in temperature between the bulk and fixed heated spherical particles under conditions in which ϕ the volume fraction of the particles and ε the Peclet number of the flow past the particles are both small. If ϕ [Lt ] ε2 the particles are effectively isolated, and so their excess temperature has an O(ε) correction to the pure conduction estimate. On the other hand if ϕ [Gt ] ε2, the bulk heating is of sufficient magnitude to produce a significant temperature gradient throughout the fixed bed. This temperature gradient leads to an O(ϕ½) correction to the pure conduction estimate of the excess temperature of the particles, and the correction depends on the details of the flow even though its magnitude is independent of ε. A study of the leading-order terms when ϕ and ε2 are of the same magnitude finds that the two small effects are not simply additive.

The nonlinear stability of gas bubbles in acoustic fields is studied using a multiple-scale type of expansion. In particular the development of a subharmonic or a synchronous perturbation to the flow is investigated. It is shown when an equilibrium non-spherical shape oscillation of a bubble is stable. If the amplitude of the sound field is ε then it is shown that subharmonic perturbations of order ε½ can exist and be stable. Furthermore synchronous perturbations of order ε can exist and be stable. It is shown that synchronous perturbations, unlike the subharmonic case where the bifurcation is symmetric, bifurcate transcritically when the driving frequency is varied and also undergo secondary bifurcations. It is further shown that, in certain cases, the latter properties of the synchronous modes cause the flow to exhibit a hysteresis phenomenon when the driving frequency is varied.