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Modal and non-modal stability analyses are performed for Poiseuille flow of a Bingham fluid overlying an anisotropic and inhomogeneous porous layer saturated with the same fluid. In the case of modal analysis, the resultant Orr–Sommerfeld type eigenvalue problem is formulated and solved via the Chebyshev collocation method, using QZ decomposition. It is found that no unstable eigenvalues are present for the problem, indicating that the flow is linearly stable. Therefore, non-modal analysis is attempted in order to observe the short-time response. For non-modal analysis, the initial value problem is solved, and the response of the system to initial conditions is assessed. The aim is to evaluate the effects on the flow stability of porous layer parameters in terms of depth ratio (ratio of the fluid layer thickness $d$ to the porous layer thickness $d_{m}$), Bingham number, Darcy number and slip coefficient. The effects of anisotropy and inhomogeneity of the porous layer on flow transition are also investigated. In addition, the shapes of the optimal perturbations are constructed. The mechanism of transient growth is explored to comprehend the complex interplay of various factors that lead to intermediate amplifications. The present analysis is perhaps the first attempt at analysing flow stability of viscoplastic fluids over a porous medium, and would possibly lead to better and efficient designing of flow environments involving such flow.

The stability of neutrally and non-neutrally buoyant particles immersed in a plane Poiseuille flow of a yield-stress fluid (Bingham fluid) is addressed numerically. Particles being carried by the yield-stress fluid can behave in different ways: they might (i) migrate inside the yielded regions or (ii) be transported without any relative motion inside the unyielded region if the yield stress is large enough compared to the buoyancy stress and the other stresses acting on the particles. Knowing the static stability of particles inside a bath of quiescent yield-stress fluid (Chaparian & Frigaard, J. Fluid Mech. , vol. 819, 2017, pp. 311–351), we analyse the latter behaviour when the yield-stress fluid Poiseuille flow is host to two-dimensional particles. Numerical experiments reveal that particles lose their stability (i.e. break the unyielded plug and sediment/migrate) with smaller buoyancy compared to the sedimentation inside a bath of quiescent yield-stress fluid, because of the inherent shear stress in the Poiseuille flow. The key parameter in interpreting the present results is the position of the particle relative to the position of the yield surface in the undisturbed flow (in the absence of any particle): the larger the portion of a particle located inside the undisturbed sheared regions, the more likely is the particle to be unstable. Yet, we find that the core unyielded plug can grow locally to some extent to contain the particles. This picture holds even for neutrally buoyant particles, although they are strictly stable when they are located wholly inside the undisturbed plug. We propose scalings for all cases.

The theory of slow viscous flow around a slender body is generalized to the situation where the ambient fluid has a yield stress. The local flow around a cylinder that is moving along or perpendicular to its axis, and rotating, provides a first step in this theory. Unlike for a Newtonian fluid, the nonlinearity associated with the viscoplastic constitutive law precludes one from linearly superposing solutions corresponding to each independent component of motion, and instead demands a full numerical approach to the problem. This is accomplished for the case of a Bingham fluid, along with a consideration of some asymptotic limits in which analytical progress is possible. Since the yield stress of the fluid strongly localizes the flow around the body, the leading-order slender-body approximation is rendered significantly more accurate than the equivalent Newtonian problem. The theory is applied to the sedimentation of inclined cylinders, bent rods and helices, and compared with some experimental data. Finally, the theory is applied to the locomotion of a cylindrical filament driven by helical waves through a viscoplastic fluid.

We extend the classical energy criterion for stability, the Lagrange–Dirichlet theorem, to rotating non-smooth complex fluids. The stability test so developed is very general and may be applied to most rotating non-smooth systems where the spectral method is inapplicable. In the process, we rigourously define an appropriate coordinate system in which to investigate stability – this happens to be the well-known Tisserand mean axis of the body – as well as systematically distinguish perturbations that introduce angular momentum and/or jumps in the stress state from those that do not. With a view to future application to planetary objects, we specialize the stability test to freely rotating self-gravitating ellipsoids. This is then employed to investigate the stability to homogeneous perturbations of rotating inviscid fluid ellipsoids. We recover results consistent with earlier predictions, and, in the process, also reconcile some contradictory conclusions about the stability of Maclaurin spheroids. Finally, we consider the equilibrium and stability of freely rotating self-gravitating Bingham fluid ellipsoids. We find that the equilibrium shapes of most such ellipsoids are secularly stable to homogeneous perturbations that preserve angular momentum, but not otherwise. We also touch upon the effect of shear thinning on stability.

Interfacial instabilities of multi-layer shear flows may be eliminated by astute positioning of yield stress fluid layers that remain unyielded at the interface(s). We study the initiation, development lengths and temporal stability of such flows in the setting of a Newtonian core fluid surrounded by a Bingham lubricated fluid, within a pipe. Flow initiation is effected by starting the flow with a pipe full of stationary Bingham fluid and injecting both inner and outer fluids simultaneously. Initial instability and dispersive mixing at the front remains localised and is advected from the pipe leaving behind a stable multi-layer configuration, found for moderate Reynolds numbers (Re), for a broad range of interface radii (ri) and for different inlet diameters (Ri), whenever the base flow parameters admit a multi-layer flow with unyielded interface. The established flows have three distinct entry lengths. These relate to: (i) establishment of the first unyielded plug close to the interface (shortest); (ii) establishment of the interface radius; (iii) establishment of the velocity profile (longest). The three entry lengths increase with Re and decrease with both the Bingham number (B) and the viscosity ratio (m). Nonlinear temporal stability to axisymmetric perturbations is studied numerically, considering initial perturbations that are either localised in yielded parts of the flow or that initially break the unyielded plug regions. The aim is to understand structural aspects of the flow stability, not easily extracted from the energy stability results of Moyers-Gonzalez, Frigaard & Nouar (J. Fluid Mech., vol. 506, 2004, p. 117). The initial stages of a stable perturbed flow are characterised by a very rapid decay of the perturbation kinetic energy during which time the unyielded plug reforms (or breaks and reforms). This is followed by slower exponential decay on a viscous timescale (t ~ Re). For smaller Re and moderate initial amplitudes A, the perturbations decay to the numerical tolerance. As either Re or A is increased sufficiently, a number of interesting phenomena arise. The amount of dispersion increases, making the interfacial region increasingly diffuse and limiting the final decay. At larger Re or A, we find secondary flow structures that persist. A first example of these is when the shear stress decays below the yield stress before the velocity perturbation has decayed, leading to freezing in of the interface shape. This can lead to flows with a rigid wavy interface. Secondly, depending on the core fluid radius and thickness of the surrounding plug region, we may observe a range of dispersive structures akin to the pearls and mushrooms of d'Olce et al. (Phys. Fluids, vol. 20, 2008, art. 024104).

In this paper, a weakly nonlinear stability of viscoplastic fluid flow is performed. The system consists of a plane Rayleigh–Bénard–Poiseuille (RBP) flow of a Bingham fluid. The basic flow is characterized by a central plug zone, of 2yb width, in which the stresses are smaller than the Bingham number B, the dimensionless yield stress. The Bingham model assumes that inside this zone the material moves as a rigid solid, and that outside this zone it behaves as a viscous fluid. The aim of this study is to investigate the influence of the yield stress on the instability conditions. The linear stability analysis is performed using a modal method and provides critical values of Rayleigh and wavenumbers, from which the system becomes unstable. The critical mode, i.e. the least stable mode, is also determined. This mode, also called the fundamental mode, creates perturbation harmonics which cannot be neglected above criticality. The weakly nonlinear analysis is performed for small-amplitude perturbations. In this study, the quadratic modes of the perturbation are determined. Results indicate that the nonlinear modes perturbation can attain high maximal values, which is the consequence of the high variations of viscosity in the flow. The characterization of the complex Landau equation sheds light on a transition in terms of the bifurcation nature above a critical Péclet number Pec = O(1). Below Pec, it is found that a supercritical equilibrium state could exist, such as in the Newtonian case, while above Pec, the bifurcation becomes subcritical. One observes a sharp transition from supercritical to subcritical bifurcation as the Péclet value is increased. A dependence of Pec on the yield stress is highlighted since the subcritical bifurcation is first observed for weak values of yb (yb < O(10−1)). For this range of values, the transition is mainly due to the presence of the unyielded region via non-homogeneous boundary conditions at the yield surfaces. Then for yb > O(10−1), the change of the bifurcation nature is due to the variations of the effective viscosity in the unyielded regions.

Oscillatory pipe flows of aqueous Carbopol solutions are investigated both experimentally and analytically. Using the PIV technique, the velocity profiles are measured and compared with the numerical solutions based on an elasto-viscoplastic rheological model, in which an elastic spring is serially connected to a regularized Bingham viscoplastic model. The rheological parameters, such as shear modulus of elasticity, yield stress and viscosity, are estimated from steady-shear measurements. Good agreement between the experiments and the model results is observed. It is apparent that the elasticity plays an important role in the unsteady flows of the soft yield-stress fluid studied herein.

It has been recently shown that the flow of a Bingham fluid in a channel is always linearly stable (Nouar et al., J. Fluid Mech., vol. 577, 2007, p. 211). To identify possible paths of transition we revisit the problem for the case in which the idealized base flow is slightly perturbed. No attempt is made to reproduce or model the perturbations arising in experimental environments – which may be due to the improper alignment of the channel walls or to imperfect inflow conditions – rather a general formulation is given which yields the transfer function (the sensitivity) for each eigenmode of the spectrum to arbitrary defects in the base flow. It is first established that such a function, for the case of the most sensitive eigenmode, displays a very weak selectivity to variations in the spanwise wavenumber of the disturbance mode. This justifies a further look into the class of spanwise homogeneous modes. A variational procedure is set up to identify the base flow defect of minimal norm capable of optimally destabilizing an otherwise stable flow; it is found that very weak defects are indeed capable to excite exponentially amplified streamwise travelling waves. The associated variations in viscosity are situated mostly near the critical layer of the inviscid problem. Neutrally stable conditions are found as function of the Reynolds number and the Bingham number, providing scalings of critical values with the amplitude of the defect consistent with previous experimental and numerical studies. Finally, a structured pseudospectrum analysis is performed; it is argued that such a class of pseudospectra provides information well suited to hydrodynamic stability purposes.

In this paper, we investigate the dynamics of muddy-seabed motions induced by a surface solitary wave. The muddy seabed is characterized as Bingham-plastic mud. We focus our attention on the situations where the horizontal scale of the wave-induced mud flow is much larger than the vertical scale. The thickness of the mud layer is also assumed to be much smaller than the water depth above. With these simplifications, the dynamic pressure in the mud column remains a constant and the vertical displacement of the water–mud interface is negligible. The horizontal gradient of the wave-induced dynamic pressure along the water–seabed interface drives the motions in the mud bed. For a Bingham-plastic muddy seafloor, the mud moves either like a solid (plug flow) or like a viscous fluid (shear flow) depending on whether the magnitude of shear stress is in excess of the yield stress. Velocities inside these two different flow regimes and the location(s) of the yield surface(s) vary in time as functions of water–mud interfacial pressure gradient and the properties of the Bingham-plastic mud. A semi-analytical approach is developed in this paper to analyse the motions inside the mud bed under a surface solitary wave loading. Three possible scenarios are discussed to illustrate the complexity of the seafloor responses. The formula for the damping rate caused by the energy dissipation inside the muddy seabed is also derived. Using realistic values of the physical parameters, the present results for damping rate agree qualitatively with the available field observations.

We consider laminar displacement flows in narrow eccentric annuli, oriented horizontally, between two fluids of Herschel–Bulkley type, (i.e. including Newtonian, power-law and Bingham models). This situation is modelled via a Hele-Shaw approach. Whereas slumping and stratification would be expected in the absence of any imposed flow rate, for a displacement flow we show that there are often steady-state travelling wave solutions in this displacement. These may exist even at large eccentricities and for large density differences between the fluids. When heavy fluids displace light fluids, annular eccentricity opposes buoyancy and steady states are more prevalent than when light fluids displace heavy fluids. For large ratios of buoyancy forces to viscous forces we derive a lubrication-style displacement model. This simplification allows us to find necessary and sufficient conditions under which a displacement can be steady, which can be expressed conveniently in terms of a consistency ratio. It is interesting that buoyancy does not appear in the critical conditions for a horizontal well. Instead a competition between fluid rheologies and eccentricity is the determining factor. Buoyancy acts only to determine the axial length of the steady-state profile.

We examine the buoyancy-driven rise of a bubble in a Newtonian or a viscoplastic fluid assuming axial symmetry and steady flow. Bubble pressure and rise velocity are determined, respectively, by requiring that its volume remains constant and its centre of mass remains fixed at the centre of the coordinate system. The continuous constitutive model suggested by Papanastasiou is used to describe the viscoplastic behaviour of the material. The flow equations are solved numerically using the mixed finite-element/Galerkin method. The nodal points of the computational mesh are determined by solving a set of elliptic differential equations to follow the often large deformations of the bubble surface. The accuracy of solutions is ascertained by mesh refinement and predictions are in very good agreement with previous experimental and theoretical results for Newtonian fluids. We determine the bubble shape and velocity and the shape of the yield surfaces for a wide range of material properties, expressed in terms of the Bingham Bn=$\tau_y^{\ast}/\rho^{\ast}g^{\ast} R_b^{\ast}$ Bond Bo =$\rho^{\ast}g^{\ast} R_b^{\ast 2}/\gamma^{\ast}$ and Archimedes Ar=$\rho^{\ast2}g^{\ast} R_b^{\ast3}/\mu_o^{\ast2}$ numbers, where ρ* is the density, μ*o the viscosity, γ* the surface tension and τ*y the yield stress of the material, g* the gravitational acceleration and R*b the radius of a spherical bubble of the same volume. If the fluid is viscoplastic, the material will not be deforming outside a finite region around the bubble and, under certain conditions, it will not be deforming either behind it or around its equatorial plane in contact with the bubble. As Bn increases, the yield surfaces at the bubble equatorial plane and away from the bubble merge and the bubble becomes entrapped. When Bo is small and the bubble cannot deform from the spherical shape the critical Bn is 0.143, i.e. it is a factor of 3/2 higher than the critical Bn for the entrapment of a solid sphere in a Bingham fluid, in direct correspondence with the 3/2 higher terminal velocity of a bubble over that of a sphere under the same buoyancy force in Stokes flow. As Bo increases allowing the bubble to squeeze through the material more easily, the critical Bingham number increases as well, but eventually it reaches an asymptotic value. Ar affects the critical Bn value much less.

We present new results on the laminar displacement of Herschel-Bulkley fluids along narrow eccentric annuli. We adopt a Hele-Shaw modelling approach and consider the possibility that a long displacement finger should advance along the wide side of the annulus. We deduce conditions under which this cannot happen. We also analyse local instability of the interface on wide and narrow sides of the annulus using the Muskat approach

金沢大学理工研究域環境デザイン学系, We investigate the slow spreading of fluid mud over a gently sloped conical surface which may simulate a shallow basin or a hill. The mud is assumed to behave as a Bingham plastic possessing a finite yield stress and the lubrication approximation is used. Because of the finite yield stress, a variety of non-trivial equilibrium profiles can exist, corresponding to the state of deposit at the end of upward or downward motion. Analytical solutions are derived for axially symmetric deposits. It is shown that the front of the final profile in axisymmetric spreading is of a common form in dimensionless variables, independent of the total mud volume. Transient evolutions are then studied numerically by employing a finite-volume scheme for both axially symmetric and asymmetric spreading from a localized source. The characteristic features of the mud pile at different stages of spreading are examined. The final shape in asymmetric spreading is strongly affected by the total volume released and by the rate of discharge. © 2004 Cambridge University Press.

A common problem in multi-layer shear flows, especially from the perspective of process engineering, is the occurrence of interfacial instabilities. Here we show how multi-layer duct flows can in fact be made nonlinearly stable, by using a suitable lubricating fluid. First we show how interfacial instabilities may be eliminated through the introduction of a yield stress fluid as the lubricant and by preserving an unyielded layer adjacent to the interface. Second we show how to treat the nonlinear stability of a two-layer flow, allowing finite motion of the domains. We focus on the simplest practically interesting case of visco-plastically lubricated viscous shear flow: a core–annular pipe flow consisting of a central core of Newtonian fluid surrounded by a Bingham fluid. We demonstrate that this flow can be nonlinearly stable at significant Reynolds numbers and produce stability bounds. Our analysis can be straightforwardly generalized to other flows in this class.

A variety of problems in engineering and geology involve spreading cooling non-Newtonian fluids. If the fluid is relatively shallow and spreads slowly, lubrication-style asymptotic approximations can be used to build reduced models for the spreading dynamics. The centrepiece of such models is a nonlinear diffusion equation for the local fluid thickness, and ideally this should become coupled to a correspondingly simple equation determining the local temperature field. However, when heat diffuses relatively slowly as the fluid flows, we cannot usefully reduce the temperature equation, and the asymptotic reduction couples the local thickness equation to an advection diffusion equation that crucially involves diffusion in the vertical. We present an efficient computational algorithm for numerically solving this more complicated type of lubrication model, and describe a suite of solutions that illustrate the dynamics captured by the model in the case of an expanding Bingham fluid with a temperature-dependent viscosity. Based on these solutions, we evaluate two simpler models that further approximate the temperature equation: a vertically isothermal theory, and a ‘skin theory’. The latter is based on the integral-balance method of heat-transfer theory, and demands that the vertical structure of the temperature field has the form of an advancing boundary layer, or skin. The vertically isothermal model performs well when the thermal conductivity is relatively large. The skin theory reproduces the full dynamics qualitatively, if not quantitatively, for all thermal conductivities. The main errors in both models arise near the fluid edge, where the numerical solutions show that chilled fluid is overridden as the fluid expands, creating an underlying collar of cold material. Encouraged by the success of the skin model, we extend the theory by incorporating extensional stresses in the skin, which emerge when cooling induces an extreme rheological change in the material, such as an exponential rise in the viscosity. The model predicts that when skin stresses are sufficiently strong, the skin is brought to rest, whilst hotter fluid expands underneath.

This is a study on the tangential flow and advective mixing of viscoplastic fluids (Bingham plastics) between two eccentric, alternately rotating cylinders. Two geometrical configurations and various rotation modes are considered for a relatively large range of the yield stress. The hp-type finite element method with the mixed formulation is used to solve for the steady velocity and pressure fields. The bi-viscosity and the Papanastasiou models agree quantitatively with each other in predicting the velocity fields and the practically unyielded zones. However, the Papanastasiou model is more robust and economic than the bi-viscosity model in the computation using Newton iteration. In the steady flows, in addition to the motionless zones, we have discovered some plugs with rigid rotation, including rotating plugs stuck onto the outer cylinder and rotating, even counter-rotating, plugs disconnected from both cylinders. The unsteady, periodic flow is composed of a sequence of the steady flows, which is valid in the creeping flow regime. The characteristics of advective mixing in these flows have been studied by analysing the asymptotic coverages of a passive tracer, the distributions of the lineal stretching in the flow and the variations of the mean stretching of the flow with time. The tracer coverage is intuitive but qualitative and, occasionally, it depends on the initial location of the tracer. On the other hand, the distribution of stretching is quantitative and more reliable in reflecting the mixing characteristics. Interestingly, the zones of the lowest stretching in the distribution graphs are remarkably well matched with the regular zones in the tracer-coverage graphs. Furthermore, the mixing efficiency proposed by Ottino (1989) is used to characterize the advective mixing in the two geometrical configurations with various rotation modes. It is important to realize that, for plastic fluids, a major barrier to effective mixing is the unyielded fluid plugs which are controlled by the yield stress and geometrical configurations. Therefore, when designing an eccentric helical annular mixer it is important to pay attention first to the geometric issues then to the operating issues.

金沢大学大学院自然科学研究科, We present a theory for the three-dimensional flow of a Bingham-plastic fluid in a shallow and wide channel. Focusing attention on slow flows appropriate for gentle slopes, low discharge rates or the final stage of deposition, we ignore inertia and apply the long-wave approximation. For steady flows, the velocity distribution, total discharge, and section-averaged flux are obtained analytically in terms of the fluid property and the geometry of the channel cross-section. Nonlinear stationary waves, which connect a uniform depth upstream to another uniform depth downstream, are then investigated, for both wet and dry beds. A numerical scheme is applied to calculate the transient flow evolution. The final development of the stationary wave due to steady discharge upstream is obtained numerically and the relation between the tongue-like shape of the wave front and the fluid property is discussed. The phase speed of the stationary wave is also derived analytically. Finally, the transient spreading of a finite fluid mass released from a reservoir after a dam break is simulated numerically. The transient development of the front and the final extent of deposition are examined.

The spreading and sediment deposit of a two-dimensional, unsteady, laminar mud flow from a constant-volume source on a relatively steep slope is studied theoretically and experimentally. The mud under consideration has the rheological properties of a Herschel–Bulkley fluid. The flow is of low-Reynolds-number type and has a well-formed wave front moving a substantial distance downslope. Due to the nonlinear rheological characteristics, a set of nonlinear partial differential equations is needed for this transient problem. Depth-integrated continuity and momentum equations are derived by applying von Karman's momentum integral method. A matched-asymptotic perturbation method is implemented analytically to get asymptotic solutions for both the outer region away from, and the inner region near, the wave front. The outer solution gives accurate results for spreading characteristics, while the inner solution, which is shown to agree well with experimental results of Liu & Mei (1989) for a Bingham fluid, predicts fairly well the free-surface profile near the wave front. A composite solution uniformly valid over the whole spreading length is then achieved through a matching of the inner and outer solutions in an overlapping region. The range of accuracy of the solution and the size of the inner and overlapping regions are quantified by physical scaling analyses. Rheological and dynamic measurements are obtained through laboratory experiments. Theoretical predictions are compared with experimental results, showing reasonable agreement. The impact of shear thinning on the runout characteristics, free-surface profiles and final deposit of the mud flow is examined. A mud flow with shear thinning spreads beyond the runout distance estimated by a Bingham model, and has a long and thin deposit.

The stability to linearized two-dimensional disturbances of plane Poiseuille flow of a Bingham fluid is considered. Bingham fluids exhibit a yield stress in addition to a plastic viscosity and this description is typically applied to drilling muds. A non-zero yield stress results in an additional parameter, a Bingham number, and it is found that the minimum Reynolds number for linear instability increases almost linearly with increasing Bingham number.

We examine the effects of a fluid yield stress on the classical Rayleigh-Bénard instability between heated parallel plates. The focus is on a qualitative characterization of these flows, by theoretical and computational means. In contrast to Newtonian fluids, we show that these flows are linearly stable at all Rayleigh numbers, Ra, although the usual linear modal stability analysis cannot be performed. Below the critical Rayleigh number for energy stability of a Newtonian fluid, RaE, the Bingham fluid is also globally asymptotically stable. Above RaE, we provide stability bounds that are conditional on Ra - RaE, as well as on the Bingham number Pr, the Prandtl number Ra, and the magnitude of the initial perturbation. The stability characteristics therefore differ considerably from those for a Newtonian fluid. A second important way in which the yield stress affects the flow is that when the flow is asymptotically stable, the velocity perturbation decays to zero in a finite time. We are able to provide estimates for the stopping time for the various types of stability. A consequence of the finite time decay is that the temperature perturbation decays on two distinctly different time scales, i.e. before/after natural convection stops. The two decay time scales are clearly observed in our computational results. We are also able to determine approximate marginal stability parameters via computation, when in the conditional stability regime, although computation is not ideal for this purpose. When just above the marginal stability limits, perturbations grow into a self-sustained cellular motion that appears to resemble closely the Newtonian secondary motion, i.e. Rayleigh-Bénard cells. When stable, however, the decaying flow pattern is distinctly different to that of a Newtonian perturbation. As t → ∞, a stable Newtonian perturbation decays exponentially and asymptotically resembles the least stable eigenfunction of the linearized problem. By contrast, as Ra approaches its stopping value, the Bingham fluid is characterized by growth of a slowly rotating (almost) unyielded core within each convection cell, with fully yielded fluid contained in a progressively narrow layer surrounding the core. Finally, preliminary analyses and remarks are made concerning extension of our results to inclined channels, stability of three-dimensional flows and the inclusion of residual stresses in the analysis. © 2006 Cambridge University Press.

Analytical solutions are obtained by expanding in powers of δ, the ratio of the difference in radii of the bounding cylinders to their mean. The solution over most of the annulus is similar to that in a slot of uniform width, containing a central plug-like region over which the velocity is independent of the radial variable

A solid sphere falling through a Bingham plastic moves in a small envelope of fluid with shape that depends on the yield stress. A finite-element/Newton method is presented for solving the free-boundary problem composed of the velocity and pressure fields and the yield surfaces for creeping flow. Besides the outer surface, solid occurs as caps at the front and back of the sphere because of the stagnation points in the flow. The accuracy of solutions is ascertained by mesh refinement and by calculation of the integrals corresponding to the maximum and minimum variational principles for the problem. Large differences from the Newtonian values in the flow pattern around the sphere and in the drag coefficient are predicted, depending on the dimensionless value of the critical yield stressYgbelow which the material acts as a solid. The computed flow fields differ appreciably from Stokes’ solution. The sphere will fall only whenYgis below 0.143 For yield stresses near this value, a plastic boundary layer forms next to the sphere. Boundary-layer scalings give the correct forms of the dependence of the drag coefficient and mass-transfer coefficient on yield stress for values near the critical one. The Stokes limit of zero yield stress is singular in the sense that for any small value ofYgthere is a region of the flow away from the sphere where the plastic portion of the viscosity is at least as important as the Newtonian part. Calculations For the approach of the flow field to the Stokes result are in good agreement with the scalings derived from the matched asymptotic expansion valid in this limit.

To study the dynamics of fluid mud with a high concentration of cohesive clay particles, we present a theory for a thin sheet of Bingham-plastic fluid flowing slowly on an inclined plane. The physics is discussed on the approximate basis of the lubrication theory. Because of the yield stress, the free surface need not be horizontal when the Bingham fluid is in static equilibrium, nor parallel to the plane bed when in steady flow. We then show that there is a variety of gravity currents that can advance at a constant speed and with the same profile. Experimental confirmation of one type is presented. By solving a nonlinear partial differential equation, transient flows due either to a steady upstream discharge or to the sudden release of a finite fluid mass on another fluid layer are studied. In the first case there is a mud front which ultimately propagates as a constant speed as a steady gravity current. In the second case, when the ambient layer is sufficiently shallow that there is no initial motion, the flow induced by the new fluid can terminate after the disturbance has travelled a finite distance. The extent of the final spread is examined. Disturbances due to an external pressure travelling parallel to the free surface are also examined. It is found in particular that a travelling localized pulse of pressure gradient not only generates a localized mud disturbance which travels along with the forcing pressure, but further leaves behind a permanent footprint.

The paper provides a model of turbulence which effects closure through approximated transport equations for the Reynolds stress tensor$\overline{u_iu_j}$and for the turbulence energy-dissipation rate ε. In its most general form the model thus entails the solution of seven transport equations for turbulence quantities but contains only six constants to be determined by experiment. It is demonstrated that the proposed approximation to the pressure-rate-of-strain correlations leads to satisfactory predictions of the component stress levels in plane homogeneous turbulence, including the non-equality of the lateral and transverse normal-stress components.For boundary-layer flows a simpler version of the model is derived wherein transport equations are solved only for the shear stress$-\overline{u_1u_2}$the turbulence energy κ and ε. This model has been incorporated in the numerical solution procedure of Patankar & Spalding (1970) and applied to the prediction of a number of boundary-layer flows including examples of flow remote from walls, those developing along one wall and those confined within ducts. Three of the flows are strongly asymmetric with respect to the surface of zero shear stress and here the turbulent shear stress does not vanish where the mean rate of strain goes to zero. In most cases the predicted profiles and other quantities accord with the data within the probable accuracy of the measurements.

Two problems representative of the gravity flow of granular materials are considered in the context of a theory presented by Goodman (1970). The problems consist of steady fully-developed flow of a granular material down an inclined plane and between vertical parallel plates. It is shown that the dynamical behaviour of these materials is quite different from that of a viscous fluid. For the inclined flow problem, the normal stresses are not only unequal but vary non-linearly with depth. Also the maximum value of the mass flux distribution does not necessarily occur at the upper surface. For the vertical channel-flow problem, the material behaves somewhat like a Bingham fluid in that a plug region exists in the central part of the channel. The interesting feature of this problem is that the concentration of material volume in the shearing region outside the plug may either increase or decrease from the plug to the channel wall, depending on the boundary conditions. Experimental evidence for these phenomena in real granular materials is cited.The results of this investigation suggest that the gravity flow of granular materials is essentially governed by two factors-a material characteristic length, which is possibly related to the grain size, and the externally imposed constraints such as the gravity field or the pressure exerted upon the granular material from the confining plates.