We consider the deformation of two-dimensional drops when immersed in a slow viscous corner flow. The problem is formulated as one of analytic function theory and simplified by assuming that both the drop and the exterior fluid have the same viscosity. An approximate analysis is carried out, in which the conditions at the interface are satisfied in an average sense, and this reveals the following features of the solution. A drop of given physical properties (volume, surface tension and viscosity), when immersed in a corner flow, has no steady equilibrium shape if the rate of strain of the applied flow is too large. On the other hand, if the rate of strain is small enough for a steady solution to exist, then in general there are two possible solutions. These features are confirmed by formulating the exact problem in terms of a nonlinear integro-differential equation, which is solved numerically.

The effect of the angle of divergence ϕ, the magnitude of the area ratio A21 and the specific-heat ratio of the gas γ on the attenuation of a shock in a two-dimensional area expansion has been experimentally determined. The results are compared with current theories relating shock strength and area ratio. At some distance from the expansion the change in shock strength, for shocks with strengths up to 10, is predicted reasonably accurately by an analysis of Chisnell (1957). Close to the area change surprisingly strong shocks were observed. This is shown to result from the diffracted shock undergoing a Mach reflexion at the end of the expansion.

At most air-liquid interfaces stressed by a static perpendicular electric field, a monolayer of charge is induced that shields the field from the liquid. For relatively inviscid and highly insulating liquids, the electrohydrodynamics can dominate in determining the monolayer and field distributions associated with additional dynamic field components. Electric shear stresses lead to a convection of surface charge that distorts the field, much as though the liquid were conducting. A configuration for studying the monolayer dynamics is developed in which a uniform field is used to induce a uniform monolayer on the interface of a liquid layer having thickness c. Superimposed is a travelling wave of potential \[ {\rm Re}\,\hat{V}_0\exp i(\omega t - kz), \] imposed in a plane parallel to, and a t a distance a above, the interface. Mechanisms for charge redistribution are reflected in the frequency response of the field transmitted through the interface to a second plane bounding the liquid layer from below. A model is developed which accounts for the self-consistent electromechanics in terms of the lumped surface parameters of surface charge σ0 and surface mobility bs, and a bulk conductivity σ. According to this model, interfacial convection dominates a t low frequencies in attenuating the field induced below the interface. For layers which are thick compared with the viscous skin depth, there is a resonance in the response a t the frequency \[ \begin{array}{@{}l@{\qquad\qquad}c@{}} &\omega = \omega_i(2^{\frac{1}{3}}),\\ {\rm where} & \omega_i = \{k\sigma^2_0/\epsilon_0(\rho\eta)^{\frac{1}{2}}[\cot {\rm h}\, ka +(\epsilon/\epsilon_0)\cot {\rm h}\,kc]\}^{\frac{2}{3}} \end{array} \] and ρ, η, ε0 and ε are the liquid mass density, liquid viscosity, and permittivities of air and the liquid, respectively. Surface and bulk conduction, characterized by the relaxation time \[ \tau_b = \epsilon_0(\cot h\,ka + (\epsilon/\epsilon_0)\cot {\rm h}\,kc)/[\sigma_0b_sk + \sigma\cot {\rm h}\,kc], \] result in a broadening of the resonance which is appreciable even for ωτb ∼ 5. At frequencies high compared with both wi and i/Tb) the response is uninfluenced by the charge monolayer. Experiments substantiate the electroviscous resonance. A time-average surface force density, and associated steady convection of the liquid, is also theoretically shown to be a consequence of the phase shifts caused by the electroviscous resonance. This is qualitatively demonstrated by measurement of the steady shear stress induced on the plane bounding the liquid from below.

Weis-Fogh (1973) proposed a new mechanism of lift generation of fundamental interest. Surprisingly, it could work even in inviscid two-dimensional motions starting from rest, when Kelvin's theorem states that the total circulation round a body must vanish, but does not exclude the possibility that if the body breaks into two pieces then there may be equal and opposite circulations round them, each suitable for generating the lift required in the pieces’ subsequent motions! The ‘fling’ of two insect wings of chord c (figure 1) turning with angular velocity Ω generates irrotational motions associated with the sucking of air into the opening gap which are calculated in § 2 as involving circulations −0·69Ωc2 and + 0.69Ωc2 around the wings when their trailing edges, which are stagnation points of those irrotational motions, break apart (position (f)). Viscous modifications to this irrotational flow pattern by shedding of vorticity at the boundary generate (§ 3) a leading-edge separation bubble, and tend to increase slightly the total bound vorticity. Its role in a three-dimensional picture of the Weis-Fogh mechanism of lift generation, involving formation of trailing vortices at the wing tips, and including the case of a hovering insect like Encarsia formosa moving those tips in circular paths, is investigated in § 4. The paper ends with the comment that the far flow field of such very small hovering insects should take the form of the exact solution (Landau 1944; Squire 1951) of the Navier-Stokes equations for the effect of a concentrated force (the weight mg of the animal) acting on a fluid of kinematic viscosity v and density p, whenever the ratio mg/pv2 is small enough for that jet-type induced motion to be stable.

fldsjfldsjfdsjfdsfjds

A new class of analytic solutions to the problem of two-dimensional potential fnow is presented here. The method of solution has features of both direct and indirect solutions. The bodies about which flow is computed are semi-infinite and have forward regions that either are flat or consist of a circular arc, which may be convex or concave to the flow. Closed-form solutions are obtained for the surface velocity. Afterbody shapes are defined by implicit equations containing a quadrature. Certain analytic properties of the solutions are investigated. An interesting feature of the bodies is the presence of a ‘pseudo corner’ where the slope angle is continuous but the curvature is infinite. The surface velocity becomes logarithmically infinite at these points in contrast to the power-law behaviour at a true corner. One case of the convex circular arc has finite velocity everywhere, and in some sense represents flow over a circular cylinder with a ‘natural’ separation point. This point occurs at 77·45° from the front stagnation point, which is close to the separation point for incompressible laminar boundarylayer flow.

The stability of laminar axisymmetric jets and wakes, the two prominent examples of free shear layers, is investigated with respect to linear azimuthally periodic disturbances. The complete viscous disturbance equations are integrated numerically and the eigenvalues are obtained by matching the numerically advanced solutions to the asymptotic solutions at a large radius. Both spatial and temporal stability are examined for inviscid and viscid flows. It is found that the critical Reynolds number for the jet and the wake are not much different while the amplification rates for the wake become considerably greater than those for the jet as the Reynolds number increases. The axisymmetric shear-layer flows also seem to be more stable than the corresponding plane flows.

Flow and turbulence in a 50 Hz rotating-field MHD system are investigated using the hot-film constant-temperature anemometer. Factors affecting anemometer disturbances and response time are discussed. From measurements of the magnetic field at points within the liquid the distribution of MHD forces is estimated. The mean rotational velocity of the flow is of the expected order of magnitude but much less dependent on the axial co-ordinate than the corresponding MHD force. With the aid of a thermal transit-time anemometer, a weak secondary flow is detected. A note on scale-model studies of MHD systems envisaged in metallurgical applications of magnetohydrodynamics points out some basic difficulties in modelling large high-powered systems on a small scale.

The stability of a potential vortex with a rotating and a non-rotating jet core is analysed. Eigenvalues are calculated numerically for different values of the ratio of the strength of the vortex to the axial velocity. These results show that the potential vortex becomes unstable in the presence of a jet.

An experimental study of distributed air-injection from a porous section of a flat plate into a uniform incompressible airflow is described. The relative mass flow rates of the injection varied between 0·008 and 0·053 (strong injection) and the blowing was fairly uniformly distributed. In the resulting flow field, which was predominantly laminar except near the dividing streamline, where unsteadiness prevailed, velocity profile and pressure measurements were taken and the position of the dividing streamline thereby estimated. Overall the results agree fairly well with the steady laminar theory for strong normal blowing, outlined in §2, although for the strongest blow some signs of separation some way upstream of the blow are apparent.

The effects of swirl on internal turbulent flows are studied by conducting experiments on turbulent pipe flow with variable initial swirl. This first part of the study is primarily concerned with similarity laws. The mean velocity profiles, both away from and close to the wall, are found to admit similarity representations at sufficiently large Reynolds numbers, provided that flow reversal does not take place near the entrance. While the wall law is not sensibly dependent on swirl, the velocity defect law in its extended form is sensitive to swirl. Further, a logarithmic skin-friction law is obtained in which only the additive coefficient depends on swirl. This coefficient is found to vary linearly with the swirl angle in the range of the present experiments.

Turbulent boundary layers along a convex surface of varying curvature were investigated in a specially designed boundary-layer tunnel. A fairly complete set of turbulence measurements was obtained.

The effect of curvature is striking. For example, along a convex wall the Reynolds stress is decreased near the wall and vanishes about midway between the wall and the edge of a boundary layer where there exists a velocity profile gradient created upstream of the curved wall.

A hypersonic gun tunnel has been used to measure the heat transfer to a sharpedged flat plate inclined at various incidences to generate local Mach numbers from 3 to 9. The measurements have been compared with a number of theoretical estimates by plotting the Stanton number against the energy-thickness Reynolds number. The prediction giving the most reasonable agreement throughout the above Mach number range is that due to Fernholz (1971).

The values of the skin-friction coefficient derived from velocity profiles and Preston tube data are also given.

The decay of perturbations in an infinite, thermally radiating gas of perfect electrical conductivity in the presence of magnetic field is studied. Complete solutions for the decay of initial sinusoidal perturbations in the temperature, gas velocity and pressure are determined. The sinusoidal perturbations are superposed to yield solutions for the decay of initial ‘step’ temperature profiles consisting of a constant initial temperature perturbation inside a finite planar region, with zero temperature perturbation outside. For a broad range of small and intermediate Boltzmann numbers the cooling proceeds in time from being a constant-density cooling process to being a constant-pressure cooling process. The magnetic field causes slower temperature decay with time and makes the temperature perturbations tend to attain constant-pressure cooling values. It quickens the decay of velocity and pressure perturbations and thus the transition from a constant-density to a constant-pressure cooling process is hastened. This transition is produced by the magneto-acoustic waves generated near the profile edges by the radiative cooling.

An electrical method of detecting and measuring the breakup of liquid jets is applied to the turbulent case. New data produced by this means, together with previous data, support the conjecture that the theory and understanding that were developed in connexion with the breakup of laminar jets can be used as a guide for turbulent jets as well.

The interaction of a vortex ring with a sharp density interface is investigated in the laboratory. Attention is restricted to the case where the Froude number based on the density difference across the interface, the velocity of propagation of the ring normal to the interface and the diameter of the ring is less than unity. It is found that the depth of maximum penetration of the ring, and the diameter of the region of contact between the ring and the interface, are functions of the Froude number. A simple model of the ring-interface interaction which accounts for the observed motion is proposed. This model is then used to calculate the volume rate of entrainment produced by the vortex rings. It is found that this rate of entrainment is proportional to the cube of the Froude number, a result which agrees with measurements of entrainment across density interfaces caused by grid-generated turbulence (Turner 1968) and by a plume incident on the interface (Baines 1973). Thus the vortex ring would appear to be a good approximation to a turbulent eddy in these situations. The main feature of the model is that it identifies the way in which the kinetic energy of the turbulence is converted into potential energy by entraining fluid across the interface. In particular, it indicates that the essential force balance is inertial, and that it is possible to discuss entrainment across a sharp density interface without explicitly invoking either viscosity or molecular diffusion.

An expression is derived for the (low Reynolds number) additional pressure drop created by a relatively small sphere moving near the wall of a circular tube through which there is a Poiseuille flow. Two specific applications are examined: (i) the sedimentation of a homogeneous non-neutrally buoyant sphere in a quiescent fluid; and (ii) the motion of a neutrally buoyant sphere. In the latter case a pronounced increase in the additional pressure drop is predicted when the separation between the sphere and the tube wall is reduced to zero.

This analysis, which includes the behaviour for a sphere in contact with the tube wall, supplements previous ‘method of reflexions’ treatments valid only when the distance from the sphere centre to the wall is large compared with the sphere radius.

Water jets are produced by vertically accelerating a rotating cone partially filled with water. It is shown that the acceleration of the parabolic meniscus results in a motion similar to that observed in a shaped explosive charge (Monroe jet). Acceleration of the cone is effected by means of an inductive electromagnetic accelerating device (conical pinch) whose theory is developed in terms of the WKB approximation. A second-order inviscid theory for the motion of the fluid in the cone in terms of the Penney-Price linearization procedure is presented and it is shown that good agreement for the jet head velocity can be achieved for low velocities. At higher velocities, experimental results appear to lag behind the theoretical ones, probably owing to the dispersal of the jet head through viscous drag with the surrounding atmosphere. The shape of the jet at early times is well represented by first-order theory.

The heat transfer due to forced convection from an isothermal sphere in a steady stream of viscous incompressible fluid is calculated for low values of the Reynolds number and Prandtl numbers of O(1). The mean Nusselt number is compared with the results of experimental measurements. At very low Reynolds numbers, both the local and mean Nusselt numbers are compared with the results obtained from the theory of matched asymptotic expansions.

Additional experimental studies of the structure of Reynolds stress which supplement our previous work (Willmarth & Lu 1971) are reported. The velocity at the edge of the viscous sublayer is again used as a detector signal for bursts and sweeps. The signal uv obtained from an X-wire probe at various locations is conditionally sampled and sorted into four quadrants of the u, v plane. Using this method it is found that, when the velocity uw at the edge of the viscous sublayer becomes low and decreasing, a burst occurs. On the other hand, a sweep occurs when uw becomes large and increasing. The convection speeds of the bursts and the sweeps are found to be equal and are about 0·8 times the local mean velocity and 0·425 times the free-stream velocity at a distance y ≈ 0·15δ* from the wall (δ* is the displacement thickness). Throughout the turbulent boundary layer, the bursts are the largest contributors to $\overline{uv}$ with the sweeps the second largest. On average, the bursts account for 77% of $\overline{uv} $, while the sweeps provide 55%; the excess percentage over 100% is due to the other small negative contributions.

Characteristic mean time intervals are obtained for both bursts and sweeps from certain unique features of the measurements of fractional contributions to $\overline{uv}$ from different events. Both mean time intervals are approximately equal and constant for most of the turbulent boundary layer. The scaling of the mean time interval between bursts with outer flow variables is confirmed. It is suggested that many of the features of the fluctuating flow revealed by the measurements may be explained by convection past the measuring station of an evolving deterministic flow pattern such as the hairpin vorticity model of Willmarth & Tu (1967).