In the present paper, it is intended to give the elementary solutions of three-dimensional unsteady Oseen flow when unsteady concentrated lift and/or drag is applied in the flow field. It is shown that the pressure fields due to concentrated impulsive lift and/or drag can be represented by an impulsive pressure doublet in the direction of the applied force and the corresponding velocity fields by diffusing free doublets in the direction of the external force that are shed from the location of the force application and convected downstream with otherwise uniform velocity. It is also confirmed that combination of the elementary solutions given in the present paper yields the two-dimensional ones.

The two-dimensional steady flow of a non-Newtonian fluid (a dilute polymer solution) is examined. The flow domain is composed of a parallel-walled inflow region, a contraction region in which the walls are rectangular hyperbolae, and a parallel-walled outflow region. The problem is formulated in terms of the vorticity, stream function and appropriate rheological equation of state, i.e. an Oldroyd-type constitutive equation (with no shear-thinning) for the total shear and normal-stress components. Computational results from the numerical solution of the equations are presented. In particular, the molecular extension and pressure distribution along the centre-line are presented as well as contour plots of the different flow variables. The alignment of the molecules with the principal axes of strain rate is shown by a qualitative comparison of the streamwise normal-stress contours with contours of the eigenvalues of the strain-rate matrix.

The deformation and conditions for breakup of a single slender drop placed symmetrically in a uniaxial extensional flow are examined theoretically. For the case of an inviscid drop in zero-Reynolds-number flow, Buckmaster (1972) showed, using slender-body analysis, that the shape of the drop is given by r≡εR(z)=ε(1−|z|ν)/2ν, where εεγ/Gμl and γ is the interfacial tension, G the strength of the extensional flow, μ the viscosity of the suspending fluid and l the drop half-length; also v = ½P − 1, where P is the unknown constant pressure inside the drop rendered dimensionless with respect to Gμ. By requiring that R be analytic at z = 0, Buckmaster then concluded that v had to be an even integer and thereby obtained a countably infinite set of slender profiles for any (large) value of the flow strength G. In the present work, the expression for R(z) shown above is obtained readily using the method of inner and outer expansions, the method failing when |z|[les ]O(ε) and ν is not an even integer. Thus, in general, a new solution is needed to describe the shape within the ‘singular’ region |z|[les ]O(ε). The requirement that the two solutions match in their domain of overlap then leads to the conclusion that v can be either equal to 2 or greater than or equal to 3. However, a stability analysis reveals that only the solution with v = 2 is stable, and hence a unique shape exists.

Next, drops of low viscosity μi = O(ε2μ) are examined in zero-Reynolds-number flow. Here, again, a unique solution is obtained according to which a steady shape cannot exist if (Gμa/γ) (μi/μ)⅙ > 0·148, where $a \equiv(3V4\pi)^{\frac{1}{3}}$ and V is the volume of the drop. This breakup criterion is identical to that found by Taylor (1964). A similar analysis for the case of an inviscid drop in a flow with non-zero Reynolds number shows that drop breakup will occur if $(G\mu a/\gamma) (\rho a\gamma/\mu^2)^{\frac{1}{5}} > 0.284$, where ρ is the density of the suspending fluid. Finally, when μi = O(ε2μ) and inertial effects are neglected within the drop but retained in the surrounding fluid, the critical value of $(G\mu a/\gamma) (\rho a\gamma/\mu^2)^{\frac{1}{5}}$ required for drop breakup is found as a function of the dimensionless group \[ (\rho a\gamma/\mu^2)^{\frac{1}{5}}(\mu/\mu_i)^{\frac{1}{6}}, \] which depends only on the physical properties of the system and the size of the drop. These last two results are the first which take into account inertial effects in determining the deformation and breakup conditions of a drop placed in a shear field.

The flow of a fluid in a narrow spherical annulus is considered. When the outer sphere remains fixed and the angular velocity of the inner one is increased beyond a critical value an instability resembling Taylor vortices appears. This instability is investigated by expanding the solution in powers of the small parameter ε, the ratio of the gap thickness to the radius, and assuming that two length scales, O(1) and O(ε), are important in the latitudinal direction.

The perturbation then takes the form of cells which are of roughly square cross-section, at least near the equator, but whose amplitude decays rapidly with latitude; it is also subject to a slow spatial modulation. The critical value of the Taylor number at which the instability first appears is shown to be that for infinite concentric cylinders plus a correction O(ε) due to secondary motions and a correction not greater than O(ε) due to the domain being bounded.

In two recent papers Itoh has developed a finite amplitude stability theory which indicates that nonlinearity increases the damping rate of a small but finite amplitude disturbance to flow in a circular pipe when the disturbance is concentrated near the axis of the pipe. For such a centre mode, which is the only mode considered by Itoh, Davey & Nguyen found, in an earlier paper, the opposite result that nonlinearity decreases the damping rate. We examine the reasons for this discrepancy and we explain the subtle difference between Itoh's method and the method of Reynolds & Potter, which was used by Davey & Nguyen.

We suggest that for the centre mode of pipe flow neither Itoh's result nor Davey & Nguyen's result is a reliable guide to the true situation. However, for the wall mode of pipe flow, and especially for plane Couette flow, both methods give very similar results and we suggest that this similarity indicates that in these cases the damping rate is decreased by nonlinearity. For a particular problem we believe that it is only when the results of the two methods are very similar that either method is likely to be useful.

The problem of fluid transport by cilia is investigated using the Green's function for a Stokeslet between two parallel plates. The discrete-cilia approach is used in building the model, and a readily usable expression for the velocities is obtained. Dependence on the direction of the metachronal wave and on time is not averaged out. Velocity fields, pressure fields and fluxes due to a single Stokeslet and to an infinite line of Stokeslets are discussed. It is found that the flux associated with Stokeslets in between two parallel plates is always zero, in contrast to a Stokeslet parallel to, and above, one plate. In the model one also has to add a plane Poiseuille flow, which incorporates non-zero flux. The flow due to the Stokeslet solution imposes a positive pressure gradient downstream, and the Poiseuille flow a negative pressure gradient. Calculated velocity profiles, in the pumping range, are seen to be time-independent in the centre of the channel and vary between a negative parabolic profile and a plug flow. The reason for these profiles and some possible biological applications are discussed.

Velocity and pressure fields for Stokes flow due to a force singularity (Stokeslet) of arbitrary orientation and at arbitrary location inside an infinite circular pipe are obtained. Two alternative expressions for the solution, one in terms of a Fourier-Bessel type expansion, and the other as a doubly infinite series, are given. The latter is especially suitable for computational purposes as it is shown to be an exponentially decaying series. From the series it is found that all velocity components decay exponentially to zero up- or downstream away from the Stokeslet. This is also true for pressure fields of Stokeslets perpendicular to the axis of the pipe. A Stokeslet parallel to the axis of the pipe raises the pressure difference between − ∞ to + ∞ by a finite non-zero amount. Some numerical results for a Stokeslet parallel to the axis are given. Comparison of the results with flow in a two-dimensional channel is also discussed.

The concept of diffusion by bulk convection formulated by Bradshaw is applied to the transport equations for the turbulent kinetic energy, turbulent shear stress and an integral length scale. The resulting set of hyperbolic partial differential equations is solved by an explicit finite-difference scheme for the cases of incompressible axisymmetric wakes and jets in a coflowing air stream. It is found that the profiles of mean velocity and shear stress are almost insensitive to the empirical input whereas the profiles of kinetic energy are very sensitive.

A two-dimensional mathematical model is described for the calculation of the depth-averaged velocity and temperature or concentration distribution in open-channel flows, an essential feature of the model being its ability to handle recirculation zones. The model employs the depth-averaged continuity, momentum and temperature/concentration equations, which are solved by an efficient finite-difference procedure. The ‘rigid lid’ approximation is used to treat the free surface. The turbulent stresses and heat or concentration fluxes are determined from a depth-averaged version of the so-called k, ε turbulence model which characterizes the local state of turbulence by the turbulence kinetic energy k and the rate of its dissipation ε. Differential transport equations are solved for k and ε to determine these two quantities. The bottom shear stress and turbulence production are accounted for by source/sink terms in the relevant equations. The model is applied to the problem of a side discharge into open-channel flow, where a recirculation zone develops downstream of the discharge. Predicted size of the recirculation zone, jet trajectories, dilution, and isotherms are compared with experiments for a wide range of discharge to channel velocity ratios; the agreement is generally good. An assessment of the numerical accuracy shows that the predictions are not influenced significantly by numerical diffusion.

Periodic permanent waves in shallow water are stable to periodic disturbances in the same direction, but are unstable to certain oblique periodic disturbances. A computer-assisted stability analysis is made of such waves for oblique disturbances with wavelengths comparable to and long compared with the fundamental wavelength. Regions of instability are calculated, and an explanation is given for the occurrence of instability. It is shown that disturbances in the same direction with a small margin of stability may cause a greater modification to the permanent wave in practice than do oblique unstable disturbances.