The shock wave equations for a perfect gas often provide more than one solution to a problem. In an attempt to find out which solution appears in a given physical situation, we present a linearized analysis of the equations of motion of a flow field with a shock boundary. It is found that a solution will be stable when there is supersonic flow downstream of the shock, and asymptotically unstable when there is subsonic flow downstream of it. It is interesting that both flows are found to be stable against disturbances of the d'Alembert type which grow from point sources; it is only when larger-scale line sources are considered that one can discriminate between the stabilities of the two types of flow. The results are applicable to supersonic flow over flat plates at incidence, to wedges, and to some cases of regular reflexion, diffraction and refraction of shocks.

The discrete-cilia approach, or cilia sublayer model, is considered for propulsion of ciliated micro-organisms. Natural periodicity assumptions enable us to obtain a readily usable expression for the velocity where dependences on the direction of propulsion of the metachronal wave and on the time are not averaged out. Thus instantaneous interaction of the cilia with the fluid is calculable, and time-dependent velocity profiles are obtained. Calculations show that the velocity above the cilia sublayer is time independent and uniform. Closer to the bases fluid velocity fluctuations are large.

The stability of the conduction regime of natural convection of a fluid contained in a narrow rotating annulus with a heated inner wall has been investigated according to the linear theory. The results include computations for Prandtl numbers P = 0, 0·72 and 6·7 over a large range of the rotational parameters. For low rotation rates the instability sets in as multicellular convection with the cell axes horizontal in the absence of rotation but tilting monotonically towards the vertical as the rotation rate is in creased. Two other types of instability were found at high rotation rates. For large Froude numbers the unstable thermal stratification leads to a Bénard type of convection with vertically oriented rolls. For large Taylor numbers, through a mainly hydrodynamic mechanism, a nearly vertically oriented cellular structure develops in the flow which is destabilized as a result of rotation.

As a model of an almost fully extended macromolecule, a small, flexible, inextensible, nearly straight thread in a shearing flow with weak Brownian motions is considered. The hydrodynamic resistance to motion is included using the slenderbody theory for Stokes flow. The variation of the small transverse displacements along the thread is expressed as a truncated sum of Fourier components, with appropriately chosen modal functions. A diffusion equation is derived in the Fourier space and solved. The expected deformation of the thread is then given for axisymmetric and two-dimensional straining flows. The transverse displacement of the ends and the small shortening of the projected length of the thread are both found to be sensitive to the truncation of the Fourier representation, although it becomes clear on physical grounds that the ratio of the shortening to the typical transverse distortion should increase with the number of degrees of freedom. In simple shear flow the deformation increases as the thread aligns with the flow, until the analysis breaks down when the entire thread is no longer in the extensional quadrants. The influence of the 2:1 ratio of the resistance coefficients from the slender-body theory is found to be a small numerical factor.

The modified Oseen linearization of the swirling-flow boundary layer of a mature hurricane is discussed. Upper and lower solutions for the radial and azimuthal velocity components are constructed in a subdomain by using a comparison theorem. It is shown that the error incurred in these velocity components by using the linear solution is no more than 30% in the subdomain. It is then inferred that the vertical velocity is also approximated to that order.

A body whose boundary C is vertical at the free surface F oscillates at high frequency in some prescribed manner. Asymptotic forms in terms of limit potentials are obtained for the upward force and the moment about a line in F of the pressure forces on the body. The cylindrical waves radiated to infinity are found to be asymptotically equivalent to those obtained in a two-dimensional solution of the Helmholtz equation. Some illustrative examples are given, principally the horizontal ellipsoid, in which case comparison with strip theory is possible.