This paper is concerned with an analysis of the near-tip region of a fluid-driven fracture propagating in a permeable saturated rock. It focuses on the calculation of the pore fluid pressure in the tip cavity, the region corresponding to the lag between the front of the fracturing fluid and the fracture tip. In contrast to impermeable rocks where the tip cavity can be considered to be at zero pressure, the fluid pressure in the tip cavity is here unknown and not uniform as exchange of pore fluid between the cavity and the porous medium and flow of pore fluid within the cavity is taking place. Solution of the fluid pressure in the tip region requires therefore simultaneous consideration of fracture mechanics (for the aperture of the tip cavity), diffusion theory for the movement of fluid within the porous medium, and viscous flow along the crack. Construction of such a solution within the framework of some simplifying assumptions is the main objective of this paper. It is shown that the problem depends, in general, upon two numbers with the meaning of a permeability and a propagation velocity. For the asymptotic case of large propagation speed, these two numbers merge into a single parameter, while the solution becomes independent of the propagation velocity in the limit of small velocity. The particular case of large velocity is solved analytically, while both the general and the small velocity cases are computed numerically but with different techniques. The paper concludes with a comprehensive analysis of numerical results.

In this paper the experimental study presented in Part 1 is extended to the three-dimensional case. The in-line force coefficients (added mass and damping) of a sphere oscillating horizontally in a uniformly stratified fluid of limited depth and in a smooth pycnocline are evaluated from Fourier-transforms of the experimental records of impulse response functions. The hydrodynamic loads in the three- and two-dimensional cases are shown to be essentially different, notably in the low-frequency limit, where the damping in the three-dimensional case is zero, while in the two-dimensional case it is maximized due to phenomena akin to blocking. The generalization of the experimental results for affinely similar geometries is discussed. It is found that, as the characteristic vertical extent of stratification decreases, the mean power of internal waves radiated by the oscillating sphere reduces and the maximum of the frequency spectrum of wave power shifts toward lower frequency, which is qualitatively similar to the effects observed in the two-dimensional case. Physically, horizontal stratified waveguides act as low-pass filters since internal waves with nearly vertical group-velocity vector cannot be effectively radiated from oscillating bodies.

In its original form the mild-slope equation, which approximates the motion of linear water waves over undulating topography, is a simplified version of the more recently derived modified mild-slope equation. However, the reduced equation does not deal adequately with rapidly varying small-amplitude perturbations about an otherwise slowly varying bedform and it does not produce free-surface profiles that inherit slope discontinuities from the topography, an intrinsic feature of the approximation on which both equations are based. The inconsistency between the two equations is rectified by the derivation of an alternative form of the mild-slope equation, having the simplicity of the standard form and yet containing all of the essential features of the full equation. In the process, a more transparent version of the modified mild-slope equation is identified. The standard and revised mild-slope equations are compared analytically in the context of two-dimensional plane wave scattering and it is found that they lead to values of the reflected wave amplitude that differ at lowest order in the mild-slope parameter, for a general topography. It is also confirmed that the revised mild-slope equation gives the dominant contribution in the solution of the new form of the modified mild-slope equation. Indeed, the two equations differ only by a term that is virtually negligible.

The properties of waves generated by a vertically oscillating sphere in a uniformly stratified fluid are examined both theoretically and experimentally. Existing predictions for the wave amplitude and phase structure are modified to account for the effects of viscous attenuation. As with waves generated by an oscillating cylinder, the main effect of attenuation is to broaden the two peaks of the amplitude envelope on either flank of the wave beam so that far from the sphere the wave beam exhibits a single peak with a maximum along the centreline. The transition distance from bimodal to unimodal wave beam structure is shown to occur closer to the source than the corresponding distance calculated for the oscillating circular cylinder. For laboratory experiments, a recently developed ‘synthetic schlieren’ method is adapted so that quantitative measurements may be made of an axisymmetric wave field. This non-intrusive technique allows us to evaluate the amplitude of the waves everywhere in space and time. Experiments are performed to examine the amplitude of waves generated by small and large spheres oscillating with a range of amplitudes and frequencies. The wave amplitude is found to scale linearly with the oscillation amplitude $A$ for $A/a$ as large as 0.27, where $a$ is the radius of the sphere. Generally good agreement between theory and experiment is found for the small sphere experiments. However, the theory overpredicts both the amplitude and the bimodal-to-unimodal transition distance for waves generated by the large sphere.

High-accuracy three-dimensional numerical simulations of miscible displacements with gravity override in homogeneous porous media are carried out for the quarter five-spot configuration. Special emphasis is placed on describing the influence of viscous and gravitational effects on the overall displacement dynamics in terms of the vorticity variable. Even for neutrally buoyant displacements, three-dimensional effects are seen to change the character of the flow significantly, in contrast to earlier findings for rectilinear displacements. At least in part this can be attributed to the time dependence of the most dangerous vertical instability mode. Density differences influence the flow primarily by establishing a narrow gravity layer, in which the effective Péclet number is enhanced owing to the higher flow rate. However, buoyancy forces of a certain magnitude can lead to a pinch-off of the gravity layer, thereby slowing it down. Overall, an increase of the gravitational parameter is found to enhance mostly the vertical perturbations, while larger ${\hbox{\it Pe}}$ values act towards amplifying horizontal disturbances. The asymptotic rate of growth of the mixing length varies only with Péclet number. For large Péclet numbers, an asymptotic value of 0.7 is observed. A scaling law for the thickness of the gravity layer is obtained as well. In contrast to immiscible flow displacements, it is found to increase with the gravity parameter.

Flow structures of an elliptic jet in cross-flow were studied experimentally in a water tunnel using the laser-induced fluorescence technique (LIF), for a range of jet aspect ratio (AR) from 0.3 to 3.0, jet-to-cross-flow velocity ratio (VR) from 1 to 5, and jet Reynolds number from 900 to 5100. The results show that the effects of aspect ratio (or jet exit orientation) are significant only in the near field, and diminish in the far field which depends only on gross jet geometry. For low-aspect-ratio jets, two adjacent counter-rotating vortex pairs (CVP) are initially formed at the sides of the jet column, with the weaker pair subsequently entrained by the stronger pair further downstream. For high-aspect-ratio jets, only one CVP is formed throughout the jet column, but the shear layer develops additional folds along the windward side of the jet. These folds subsequently evolve into smaller scale counter-rotating vortex pairs, which we refer to as windward vortex pairs (WVP). Depending on its sense of rotation, the WVP can evolve into what Haven & Kurosaka (1997) referred to as unsteady kidney vortices or anti-kidney vortices, or, under some circumstances, interconnecting kidney vortices, which have not been reported previously. While Haven & Kurosaka (1997)'s interpretation of the formation of kidney and anti-kidney vortices is topologically feasible, our observation reveals a slightly different formation process. Despite the differences in the near-field flow structures for different jet aspect ratios, the process leading to the formation of the large-scale jet structures (i.e. leading-edge vortices and lee-side vortices) for all cases is similar to that reported by Lim, New & Luo (2001) for a circular jet in cross-flow.

A numerical and analytical investigation is made into the response of a fluid when a two-dimensional structure is forced to move in a prescribed fashion. The structure is constructed in such a way that it supports a trapped mode at one particular frequency. The fluid motion is assumed to be small and the time-domain equations for linear water-wave theory are solved numerically. In addition, the asymptotic behaviour of the resulting velocity potential is determined analytically from the relationship between the time- and frequency-domain solutions. The trapping structure has two distinct surface-piercing elements and the trapped mode exhibits a vertical ‘pumping’ motion of the fluid between the elements. When the structure is forced to oscillate at the trapped-mode frequency an oscillation which grows in time but decays in space is observed. An oscillatory forcing at a frequency different from that of the trapped mode produces bounded oscillations at both the forcing and the trapped-mode frequency. A transient forcing also gives rise to a localized oscillation at the trapped-mode frequency which does not decay with time. Where possible, comparisons are made between the numerical and asymptotic solutions and good agreement is observed. The calculations described above are contrasted with the results from a similar forcing of a pair of semicircular cylinders which intersect the free surface at the same points as the trapping structure. For this second geometry no localized or unbounded oscillations are observed. The trapping structure is then given a sequence of perturbations which transform it into the two semicircular cylinders and the time-domain equations solved for a transient forcing of each structural geometry in the sequence. For small perturbations of the trapping structure, localized oscillations are produced which have a frequency close to that of the trapped mode but with amplitude that decays slowly with time. Estimates of the frequency and the rate of decay of the oscillation are made from the time-domain calculations. These values correspond to the real and imaginary parts of a pole in the complex force coefficient associated with a frequency-domain potential. An estimate of the position of this pole is obtained from calculations of the added mass and damping for the structure and shows good agreement with the time-domain results. Further time-domain calculations for a different trapping structure with more widely spaced elements show a number of interesting features. In particular, a transient forcing leads to persistent oscillations at two distinct frequencies, suggesting that there is either a second trapped mode, or a very lightly damped near-trapped mode. In addition a highly damped pumping mode is identified.

We consider the periodic, two-dimensional motion of a viscous, incompressible liquid which fills a rectangular container. The motion is due to the periodic motion of the lid which moves in its own plane. If the velocities are sufficiently small the motion will be governed by the linearized Navier–Stokes equations and consequently the dimensionless stream function $\Psi(x,z,t)\,{=}\,\psi(x,z){\rm e}^{{\rm i}t}$ will satisfy the equation $ \nabla^4 \psi - {\rm i}\hbox{\it Re} \nabla^2 \psi\,{=}\,0$, where ${\it Re}$ is the Reynolds number. If we then seek separable solutions for $\psi (x,z)$ that satisfy the no-slip conditions on the sidewalls, it is easy to show that the problem reduces to the eigenvalue problem $$\lambda \tan{\textstyle\frac{1}{2}}\lambda\,{=}\,\sqrt{\lambda^2 - {\rm i} \hbox{\it Re}} \tan {\textstyle{1 \over 2}} \sqrt{\lambda^2 - {\rm i} \hbox{\it Re}}$$ where $\lambda$ is the eigenvalue. A detailed analysis is made of this eigenvalue problem. All the eigenvalues are complex; all eigenvalues with positive real part either belong to a set $\{\lambda_n^u \}$ in the upper half-plane or to another $\{\lambda_n^l\}$ in the lower half-plane. They satisfy the important relationship $\lambda_n^l\,{=}\,\sqrt{\overline{\lambda_n^u}^2 + {\rm i} \hbox{\it Re}}.$ We show by an asymptotic analysis that while the $\lambda_n^l$ move to the neighbourhood of the real axis as $\hbox{\it Re}\rightarrow \infty$, the $\lambda_n^u$ move away from the origin and approach the line $\lambda_i\,{=}\,\lambda_r$ in the complex-$\lambda$-plane. This fact has an important bearing on the damping of gravity waves at high Reynolds numbers. The eigenfunctions derived above are used to write down a formal expansion for the stream function and the coefficients are determined from the boundary conditions using a least-squares procedure. An examination of the resulting streamline patterns reveals interesting inertial effects even at low Reynolds numbers. In particular we examine the mechanism by which the flow field reverses its direction when the lid stops and reverses its direction of motion. If inertial effects are completely negelected, as has been done till now, one would infer an immediate response of the fluid to the changes in the lid motion; for example, one would conclude, wrongly, that when the lid is at rest so is the fluid. Our analysis shows, in fact, a very intricate and beautiful mechanism, involving among other things an apparent engulfing of the corner eddy by the new primary eddy, by which the direction of the circulation is reversed in the fluid. These results should be of importance in the analysis of mixing, where such effects appear to have been ignored till now.

This communication presents the results and conclusions of an experimental study of the near-separation region of a single-stream shear layer. The momentum thickness at separation ($x\,{=}\,0$) was $\theta_0\,{=}\,9.6$ mm, with Reynolds number $\hbox{\itshape Re}_\theta\,{=}\,4650$. Boundary layer separation was caused by a sharp $90^{\circ}$ edge. Detailed single- and multi-point measurements of the velocity field were acquired at the streamwise locations $0\,{<}\,x/\theta_0\,{<}\,100$. This represents the transition region between two of the canonical turbulent shear flows: the zero-pressure-gradient turbulent boundary layer and the single-stream shear layer. From the viewpoint of a separating boundary layer, the results describe how the turbulent flow reacts to a sudden change in wall boundary conditions. From the viewpoint of the developed shear layer, the results describe the transition to the self-similar region. The data acquired suggest that the initial shear layer instability occurs in the region very near separation ($x\,{\approx}\,\theta_0$), and that it involves only the vorticity filaments which originate in the near-wall region of the upstream boundary layer. This ‘near-wall region’ roughly defines the origin of a narrow wedge-shaped domain that was identified from the velocity statistics. This domain is termed the ‘sub-shear layer’. The statistics of the velocity field in the region bounded by the sub-shear layer and the free-stream flow were found to represent the normative continuation of the upstream boundary layer. The sub-shear layer has been found to exhibit many of the standard features observed in fully developed shear layers. For example, velocity measurements on the entrainment side of the shear layer indicate that large-scale motions with spanwise coherence were observed. The streamwise dependence of the dominant frequency, convection velocity, and spanwise velocity correlation have been documented in order to characterize the sub-shear layer phenomenon.

The primary goal of the study is to identify the selection mechanism responsible for the appearance of a double-helix structure in the pre-breakdown stage of so-called screened swirling jets for which the circulation vanishes away from the jet. The family of basic flows under consideration combines the azimuthal velocity profiles of Carton & McWilliams (1989) and the axial velocity profiles of Monkewitz (1988). This model satisfactorily represents the nozzle exit velocity distributions measured in the swirling jet experiment of Billant et al. (1998). Temporal and absolute/convective instability properties are directly retrieved from numerical simulations of the linear impulse response for different swirl parameter settings. A large range of negative helical modes, winding with the basic flow, are destabilized as swirl is increased, and their characteristics for large azimuthal wavenumbers are shown to agree with the asymptotic analysis of Leibovich & Stewartson (1983). However, the temporal study fails to yield a clear selection principle. The absolute/convective instability regions are mapped out in the plane of the external axial flow and swirl parameters. The absolutely unstable domain is enhanced by rotation and it remains open for arbitrarily large swirl. The swirling jet with zero external axial flow is found to first become absolutely unstable to a mode of azimuthal wavenumber $m\,{=}\,{-}2$, winding with the jet. It is suggested that this selection mechanism accounts for the experimental observation of a double-helix structure.

The behaviour of an axisymmetric inviscid gravity current, which is released from a lock near the outer wall of a circular container and then propagates towards the centre over a horizontal boundary, is considered. Shallow-water and box-model theoretical analyses and experimental results are presented and compared. The resulting motion predicted by the shallow-water model displays interesting differences with the previously reported outward propagation of an axisymmetric current, as well as with propagation in a two-dimensional rectangular geometry. The current initially develops the usual decelerating motion with a nose-up tail-down shape, but when the nose reaches about half of the outer radius the confining geometry opposes the further decrease of the height and velocity of the nose. The box-model approximation, which omits the inclination of the interface, is unable to reproduce the hindering (and eventual reversal) effect of the geometrical confinement on the decrease of the nose velocity during the inward propagation.

We study the problem of body-force-driven shear flows in a plane channel of width $\ell$ with free-slip boundaries. A mini–max variational problem for upper bounds on the bulk time-averaged energy dissipation rate $\epsilon$ is derived from the incompressible Navier–Stokes equations with no secondary assumptions. This produces rigorous limits on the power consumption that are valid for laminar or turbulent solutions. The mini–max problem is solved exactly at high Reynolds numbers $Re = U\ell/\nu$, where $U$ is the r.m.s. velocity and $\nu$ is the kinematic viscosity, yielding an explicit bound on the dimensionless asymptotic dissipation factor $\beta=\epsilon \ell/U^3$ that depends only on the ‘shape’ of the shearing body force. For a simple half-cosine force profile, for example, the high Reynolds number bound is $\beta \le \pi^2/\sqrt{216} = 0.6715\ldots$. We also report extensive direct numerical simulations for this particular force shape up to $Re \approx 400$; the observed dissipation rates are about a factor 3 below the rigorous high-$Re$ bound. Interestingly, the high-$Re$ optimal solution of the variational problem bears some qualitative resemblance to the observed mean flow profiles in the simulations. These results extend and refine the recent analysis for body-forced turbulence in Doering & Foias (2002).

We investigate experimentally and numerically the filling of a collapsible tube, motivated by venous hemodynamics in the lower limbs. The experiments are performed by filling an initially collapsed flexible tube, applying pressure through a hydraulic circuit. The tube law and the tube tension have been previously measured. The tube shape, the flow rate and the pressure at the two ends of the tube are measured continuously. The filling occurs in three stages: a rapid equilibration of the pressure near the tube entry with atmospheric pressure, a quasi-steady filling of the tube with a linearly rising pressure, and a final stage of tube inflation. Our numerical model is the classical one-dimensional collapsible tube equations. Excellent quantitative agreement is found between computations and experimental data. We show experimentally observed shapes near the tube end that indicate possible three-dimensional effects; however these effects do not impair significantly the ability of the one-dimensional model to describe the experiment. Travelling waves of large amplitude are observed in the simulations and the experiments.

We consider the evaporating meniscus of a perfectly wetting liquid in a channel whose superheated walls are at common temperature. Heat flows by pure conduction from the walls to the phase interface; there, evaporation induces a small-scale liquid flow concentrated near the contact lines. Liquid is continually fed to the channel, so that the interface is stationary, but distorted by the pressure differences caused by the small-scale flow. To determine the heat flow, we make a systematic analysis of this free-boundary problem in the limit of vanishing capillary number based on the velocity of the induced flow. Because surface tension is then large, the induced flow can distort the phase interface only in a small inner region near the contact lines; the effect is to create an apparent contact angle $\mtheta$ depending on capillary number. Though, in general, there can be significant heat flow within that small inner region, the presence of an additional small parameter in the problem implies that, in practice, heat flow is significant only within the large outer region where the interface shape is determined by hydrostatics and $\mtheta$. We derive a formula for the heat flow, and show that the channel geometry affects the heat flow only through the value of the interface curvature at the contact line. Consequently, the heat flow relation for a channel can be applied to other geometries.

Two- and three-dimensional numerical simulations are performed to study interfacial waves in a periodic domain by imposing a source term in the horizontal momentum equation. Removing the source term before breaking generates a stable interfacial wave. Continued forcing results in a two-dimensional shear instability for waves with thinner interfaces, and a convective instability for waves with thick interfaces. The subsequent three-dimensional dynamics and mixing is dominated by secondary cross-stream convective rolls which account for roughly half of the total dissipation of wave energy. Dissipation and mixing are maximized when the interface thickness is roughly the same size as the amplitude of the wave, while the mixing efficiency is a weak function of the interface thickness. The maximum instantaneous mixing efficiency is found to be $0.36\pm 0.02$.

Experimentally observed periodic structures in shallow (i.e. bounded) wake flows are believed to appear as a result of hydrodynamic instability. Previously published studies used linear stability analysis under the rigid-lid assumption to investigate the onset of instability of wakes in shallow water flows. The objectives of this paper are: (i) to provide a preliminary assessment of the accuracy of the rigid-lid assumption; (ii) to investigate the influence of the shape of the base flow profile on the stability characteristics; (iii) to formulate the weakly nonlinear stability problem for shallow wake flows and show that the evolution of the instability is governed by the Ginzburg–Landau equation; and (iv) to establish the connection between weakly nonlinear analysis and the observed flow patterns in shallow wake flows which are reported in the literature. It is found that the relative error in determining the critical value of the shallow wake stability parameter induced by the rigid-lid assumption is below 10% for the practical range of Froude number. In addition, it is shown that the shape of the velocity profile has a large influence on the stability characteristics of shallow wakes. Starting from the rigid-lid shallow-water equations and using the method of multiple scales, an amplitude evolution equation for the most unstable mode is derived. The resulting equation has complex coefficients and is of Ginzburg–Landau type. An example calculation of the complex coefficients of the Ginzburg–Landau equation confirms the existence of a finite equilibrium amplitude, where the unstable mode evolves with time into a limit-cycle oscillation. This is consistent with flow patterns observed by Ingram & Chu (1987), Chen & Jirka (1995), Balachandar et al. (1999), and Balachandar & Tachie (2001). Reasonable agreement is found between the saturation amplitude obtained from the Ginzburg–Landau equation under some simplifying assumptions and the numerical data of Grubĭsić et al. (1995). Such consistency provides further evidence that experimentally observed structures in shallow wake flows may be described by the nonlinear Ginzburg–Landau equation. Previous works have found similar consistency between the Ginzburg–Landau model and experimental data for the case of deep (i.e. unbounded) wake flows. However, it must be emphasized that much more information is required to confirm the appropriateness of the Ginzburg–Landau equation in describing shallow wake flows.

We collected all of the published data we could find on the rise velocity of long gas bubbles in stagnant fluids contained in circular tubes. Data from 255 experiments from the literature and seven new experiments at PDVSA Intevep for fluids with viscosities ranging from 1 mPa s up to 3900 mPa s were assembled on spread sheets and processed in log–log plots of the normalized rise velocity, $\hbox{\it Fr} \,{=}\,U/(gD)^{1/2}$ Froude velocity vs. buoyancy Reynolds number, $R\,{=}\,(D^{3}g (\rho_{l}-\rho_{g}) \rho_{l})^{1/2}/\mu $ for fixed ranges of the Eötvös number, $\hbox{\it Eo}\,{=}\,g\rho_{l}D^{2}/\sigma $ where $D$ is the pipe diameter, $\rho_{l}$, $\rho_{g}$ and $\sigma$ are densities and surface tension. The plots give rise to power laws in $Eo$; the composition of these separate power laws emerge as bi-power laws for two separate flow regions for large and small buoyancy Reynolds. For large $R$ ($>200$) we find \[\hbox{\it Fr} = {0.34}/(1+3805/\hbox{\it Eo}^{3.06})^{0.58}.\] For small $R$ ($<10$) we find \[ \hbox{\it Fr} = \frac{9.494\times 10^{-3}}{({1+{6197}/\hbox{\it Eo}^{2.561}})^{0.5793}}R^{1.026}.\] The flat region for high buoyancy Reynolds number and sloped region for low buoyancy Reynolds number is separated by a transition region ($10\,{<}\,R\,{<}\, 200$) which we describe by fitting the data to a logistic dose curve. Repeated application of logistic dose curves leads to a composition of rational fractions of rational fractions of power laws. This leads to the following universal correlation: \[ \hbox{\it Fr} = L[{R;A,B,C,G}] \equiv \frac{A}{({1+({{R}/{B}})^C})^G} \] where \[ A = L[\hbox{\it Eo};a,b,c,d],\quad B = L[\hbox{\it Eo};e,f,g,h],\quad C = L[\hbox{\it Eo};i,j,k,l],\quad G = m/C \] and the parameters ($a, b,\ldots,l$) are \begin{eqnarray*} &&\hspace*{-5pt}a \hspace*{-0.8pt}\,{=}\,\hspace*{-0.8pt} 0.34;\quad b\hspace*{-0.8pt} \,{=}\,\hspace*{-0.8pt} 14.793;\quad c\hspace*{-0.8pt} \,{=}\,\hspace*{-0.6pt}{-}3.06;\quad d\hspace*{-0.6pt} \,{=}\, \hspace*{-0.6pt}0.58;\quad e\hspace*{-0.6pt} \,{=}\,\hspace*{-0.6pt} 31.08;\quad f\hspace*{-0.6pt} \,{=}\, \hspace*{-0.6pt}29.868;\quad g\hspace*{-0.6pt}\,{ =}\,\hspace*{-0.6pt}{ -}1.96;\\ &&\hspace*{-5pt}h = -0.49;\quad i = -1.45;\quad j = 24.867;\quad k = -9.93;\quad l = -0.094;\quad m = -1.0295.\end{eqnarray*} The literature on this subject is reviewed together with a summary of previous methods of prediction. New data and photographs collected at PDVSA-Intevep on the rise of Taylor bubbles is presented.

The standard textbook model of a helicopter rotor in vertical translation, a disk loaded with a uniform pressure jump in inviscid fluid, is revisited in search of correct descriptions of the far-field velocity and of the vortex sheet, allowing a rigorous control-volume analysis. The translation rate is not required to be large compared with the induced velocity. The classical results for induced power are unchanged, and now have a strong foundation: they are exact within the steady inviscid problem statement, instead of depending on a quasi-one-dimensional approximation as in the literature. Conversely, even with a uniform pressure jump the induced velocity is far from uniform over the disk, again in conflict with common beliefs and with any quasi-one-dimensional argument: the flow is upwards near the rim, both inside and outside it. The cross-section of the vortex sheet probably begins with a 45° spiral, as opposed to the smooth funnel shape that has been sketched, in the literature and below. A viscous numerical solution supports this conjecture. Plausible boundaries between the translation rates that produce the two ‘clean’ streamtube flow types, namely climb/hover and rapid descent, and those in-between that produce the vortex-ring state are also discussed.

Tubes, Sheets and Singularities in Fluid Dynamics. Edited by K. BAJER & H. K. MOFFATT. Kluwer, 2002. 379 pp. ISBN 1402009801. £ 72.00.

Ferrofluids: Magnetically Controlled Fluids and Their Applications. Edited by S. ODENBACH. Springer Lecture Notes in Physics, 2002. 251pp. ISBN 3540439781. £ 47.

Large-Scale Atmosphere-Ocean Dynamics, Volume 1: Analytical methods and numerical models. Edited by J. NORBURY & I. ROULSTONE. Cambridge University Press, 2002. 370pp. ISBN 052180681. £ 50.

Vascular Grafts: Experiment and Modelling. Edited by A TURA. WIT Press, Advances in Fluid Mechanics, 2003. 421pp. ISBN 1853129003. £ 138.00.