A rivulet is a narrow stream of liquid located on a solid surface and sharing a curved interface with the surrounding gas. Capillary instabilities are investigated by a linearized stability theory. The formulation is for small, static rivulets whose contact (common or three-phase) lines (i) are fixed, (ii) move but have fixed contact angles or (iii) move but have contact angles smooth functions of contact-line speeds. The linearized stability equations are converted to a disturbance kinetic-energy balance showing that the disturbance response exactly satisfies a damped linear harmonic-oscillator equation. The ‘damping coefficient’ contains the bulk viscous dissipation, the effect of slip along the solid and all dynamic effects that arise in contact-line condition (iii). The ‘spring constant’, whose sign determines stability or instability in the system, incorporates the interfacial area changes and is identical in cases (ii) and (iii). Thus, for small disturbances changes in contact angle with contact-line speed constitute a purely dissipative process. All the above results are independent of slip model at the liquid–solid interface as long as a certain integral inequality holds. Finally, sufficient conditions for stability are obtained in all cases (i), (ii) and (iii).

It is shown that all parallel inviscid shear flows of constant density are unstable to a wide class of initial infinitesimal three-dimensional disturbances in the sense that, according to linear theory, the kinetic energy of the disturbance will grow at least as fast as linearly in time. This can occur even when the disturbance velocities are bounded, because the streamwise length of the disturbed region grows linearly with time. This finding may have implications for the observed tendency of turbulent shear flows to develop a longitudinal streaky structure.

In many polymer processing applications, the generation of heat by viscous losses in the flowing molten polymer is highly significant. The heating reduces the viscosity of the melt sharply and the flow patterns are different from the isothermal case. In this paper, a finite element scheme based on the Galerkin method is developed and is used to explore the effects of thermally induced property changes in extrusion.

In the program we solve simultaneously for the flow and temperature fields at each iteration. To check the program for accuracy and correctness, some simple problems were first attempted. A solution for viscous heating in Poiseuille flow was used to check the variable-viscosity part of the program. The crucial convection (‘radiation’) boundary condition was checked using the solution for cooling of a moving rod. Finally, the swelling of extruded jets with self-heating was investigated. A new phenomenon, thermal extrudate swell, was thereby discovered. We have found extrudate expansions up to 70% of the die diameter in a Newtonian fluid with thermal properties similar to those of low density polyethylene. It is clear that this phenomenon will affect many experimental interpretations of extrudate swelling.

Correction of an error in the matrix elements used by Salwen & Grosch (1972) has brought the results of the matrix-eigenvalue calculation of the linear stability of Hagen–Poiseuille flow into complete agreement with the numerical integration results of Lessen, Sadler & Liu (1968) for azimuthal index n = 1. The n = 0 results were unaffected by the error and the effect of the error for n > 1 is smaller than for n = 1. The new calculations confirm the conclusion that the flow is stable to infinitesimal disturbances.

Further calculations have led to the discovery of a degeneracy at Reynolds number R = 61·452 ± 0·003 and wavenumber α = 0·9874 ± 0·0001, where the second and third eigenmodes have equal complex wave speeds. The variation of wave speed for these two modes has been studied in the vicinity of the degeneracy and shows similarities to the behaviour near the degeneracies found by Cotton and Salwen (see Cotton 1977) for rotating Hagen-Poiseuille flow. Finally, new results are given for n = 10 and 30; the n = 1 results are extended to R = 106; and new results are presented for the variation of the wave speed with αR at high Reynolds number. The high-R results confirm both Burridge & Drazin's (1969) slow-mode approximation and more recent fast-mode results of Burridge.

Miles’ (1977b) model of the Mach reflexion of a solitary wave by a vertical wall is tested by laboratory experiments. The model over-predicts the measured run-up at the wall, and no evidence of the predicted maximum was found. The measurements provide support for the predicted critical angle of incidence at which Mach reflexion is replaced by regular reflexion. It is shown that mass and energy conservation determine the length of the reflected wave in Miles’ model and that this is not consistent with momentum conservation in the neighbourhood of the end point of the reflected wave. It is suggested that the discrepancy between the measurements and the model may result from this failure of the model.

It is shown that for a small sphere freely suspended in a linear shear flow at small Reynolds numbers, the Nusselt number N is given by $N = \{1 - \alpha P^{\frac{1}{2}} + o(P^{\frac{3}{2}})\}^{-1}$, where P is the Péclet number. For any given type of shear flow, the numerical value of the constant α can be obtained from a general expression derived by Batchelor (1979). The corresponding result for a particle of arbitrary shape is N/N0 = {1 − αN0P½ + O(P3/2)}−1, where N0 is the Nusselt number for pure conduction.

We study theoretically the slow viscous motion of a long slender drop placed in a simple shear flow, the drop having a low viscosity compared with that of the suspending fluid. As a simplifying approximation, the cross-section of the drop is taken to be circular. An equilibrium shape with the drop nearly aligned with the flow is found for all shear rates, although the equilibrium is only stable to small disturbances for shear rates below some critical value. The stable equilibria just below the critical shear rate are found to be accessible only if the shear rate is increased slowly.

We consider the onset of Rayleigh–Bénard convection from random fluctuations arising within a fluid. In the specific case in which the fluctuations are thermodynamically determined, we reduce the problem to a random initial value problem for the Fourier modes. For the case of weak nonlinear convection, it is possible to truncate the number of modes and this truncated set is solved both by a Monte Carlo technique and by moment methods for various Rayleigh numbers. We find three stages in the evolution of ordered convection from random fluctuations which correspond to time intervals in which the fluctuations and the nonlinearity have different degrees of importance. It is shown that no simple moment truncation method will succeed and that the time for onset of convection is a mean over a distribution of times for which members of an ensemble exhibit appreciable convective transport.

The problem of acoustic radiation generated by instability waves of a compressible plane turbulent shear layer is solved. The solution provided is valid up to the acoustic far-field region. It represents a significant improvement over the solution obtained by classical hydrodynamic-stability theory which is essentially a local solution with the acoustic radiation suppressed. The basic instability-wave solution which is valid in the shear layer and the near-field region is constructed in terms of an asymptotic expansion using the method of multiple scales. This solution accounts for the effects of the slightly divergent mean flow. It is shown that the multiple-scales asymptotic expansion is not uniformly valid far from the shear layer. Continuation of this solution into the entire upper half-plane is described. The extended solution enables the near-and far-field pressure fluctuations associated with the instability wave to be determined. Numerical results show that the directivity pattern of acoustic radiation into the stationary medium peaks at 20 degrees to the axis of the shear layer in the downstream direction for supersonic flows. This agrees qualitatively with the observed noise-directivity patterns of supersonic jets.

The results of an experimental investigation of the entry, into a curved pipe, of the fully developed oscillatory laminar flow in a straight section are presented. Laser anemometry has been used to measure velocity profiles in the plane of the bend at various stations around a 180°-curved section. The flow development is found to depend upon both the frequency parameter of the flow and the amplitude of oscillation.

Results are presented for two values of the frequency parameter α. The first is for small α where the flow can be considered quasi-steady and the flow development is found to proceed as in previous steady-flow studies. The other case, more extensively studied, is where α has a value such that both viscous and inertial effects play important roles in establishing the basic flow at different parts of the pressure cycle. The flow development process around the curve is found to be complicated, but a general trend is found and the results are explained in terms of those already established for steady-flow development in a curved pipe.

A theoretical and experimental investigation of oscillatory flow in curved pipes is presented. The equations for fully developed laminar flow are found to depend on an amplitude parameter G and a frequency parameter α. Initially the Navier–Stokes equations are expanded in terms of G and the resulting linearized equations are solved numerically using finite Hankel integral transforms. A further expansion is used for the case α → 0 and closed-form solutions are presented.

Laser-Doppler anemometry has been used to obtain velocity information in oscillatory air flow in small-diameter curved glass tubes. Using this technique, low-Reynolds-number experiments were carried out and comparison between theory and experiments is presented.

For α [les ] 1 the velocity distributions found are essentially those for steady flow, while for α [ges ] 11·0 the results are not at variance with earlier work. It is for the transition stage between these two regimes that unexpected behaviour is recorded, but a satisfactory explanation is found in terms of general trends within the flow.

The development of a three-dimensional water boundary layer along a heated longitudinal horizontal cylinder is studied by a finite-difference method. The secondary flow is induced in an otherwise axially symmetric laminar boundary layer by the buoyancy force. The development of the boundary layer is studied under two heating conditions: constant wall heat flux and constant wall temperature. In general, close to the leading edge, the magnitude of the secondary flow is small and the boundary-layer flow is forced-convection dominant. The secondary flow grows downstream, and the interaction of the free and forced convection becomes important. The flow becomes free-convection dominant further downstream. The temperature-dependent viscosity of water has the effect of thinning the heated boundary layer. The buoyancy effect and the variable viscosity effect enhance each other over the lower part of the cylinder and compete with each other over the upper part of the cylinder. The numerical results compare with the forced convection dominant asymptotic solution and indicate that the asymptotic solution is only valid when x < 0·1a/ε½. Since the boundary layer is thin compared with the radius of the cylinder, the transverse curvature effect is small and can be neglected. Therefore, the solution can be applied to the entrance region of heated straight pipes as the zeroth-order boundary-layer flow.

Harris, Graham & Corrsin (1977) have measured the properties of the quasi-homo-geneous turbulence field induced by a large mean shear, and in analysing their measurements they neglect the diagonal components of the mean-field contribution to the pressure–strain correlation. Their measurements are re-analysed using the recommendations of Gibson & Launder (1978) for modelling this correlation, the imbalance between production and dissipation being allowed for by the algebraic modelling technique of Rodi (1976): the experimental data give strong support to Rodi's basic assumption.

The data suggest that the only substantial defect in the Gibson–Launder model is its failure to predict the anisotropy measured in the 23 plane normal to the mean flow. The longitudinal predictions are good, and those for the shear (12) component are much improved when measured values of the anisotropy are substituted into the calculation. This analysis does not suggest any clear need for a nonlinear representation of the pressure–strain correlation. However, the most general linear representation of the mean-field term is even more complex than the analysis of Launder, Reece & Rodi (1975) would suggest: their model is disproved by an example.

Attempts to deduce the dissipation directly from the experiments, rather than by energy balance, are not very successful.