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A hyperbolic two-fluid model for gas–particle flow derived using the Boltzmann–Enskog kinetic theory is generalized to include added mass. In place of the virtual-mass force, to guarantee indifference to an accelerating frame of reference, the added mass is included in the mass, momentum and energy balances for the particle phase, augmented to include the portion of the particle wake moving with the particle velocity. The resulting compressible two-fluid model contains seven balance equations (mass, momentum and energy for each phase, plus added mass) and employs a stiffened-gas model for the equation of state for the fluid. Using Sturm's theorem, the model is shown to be globally hyperbolic for arbitrary ratios of the material densities $Z = \rho _f / \rho _p$ (where $\rho _f$ and $\rho _p$ are the fluid and particle material densities, respectively). An eight-equation extension to include the pseudo-turbulent kinetic energy (PTKE) in the fluid phase is also proposed; however, PTKE has no effect on hyperbolicity. In addition to the added mass, the key physics needed to ensure hyperbolicity for arbitrary $Z$ is a fluid-mediated contribution to the particle-phase pressure tensor that is taken to be proportional to the volume fraction of the added mass. A numerical solver for hyperbolic equations is developed for the one-dimensional model, and numerical examples are employed to illustrate the behaviour of solutions to Riemann problems for different material-density ratios. The relation between the proposed two-fluid model and prior work on effective-field models is discussed, as well as possible extensions to include viscous stresses and the formulation of the model in the limit of an incompressible continuous phase.
We consider the dynamics of a vertically stratified, horizontally forced Kolmogorov flow. Motivated by astrophysical systems where the Prandtl number is often asymptotically small, our focus is the little-studied limit of high Reynolds number but low Péclet number (which is defined to be the product of the Reynolds number and the Prandtl number). Through a linear stability analysis, we demonstrate that the stability of two-dimensional modes to infinitesimal perturbations is independent of the stratification, whilst three-dimensional modes are always unstable in the limit of strong stratification and strong thermal diffusion. The subsequent nonlinear evolution and transition to turbulence are studied numerically using direct numerical simulations. For sufficiently large Reynolds numbers, four distinct dynamical regimes naturally emerge, depending upon the strength of the background stratification. By considering dominant balances in the governing equations, we derive scaling laws for each regime which explain the numerical data.
Previous experimental (Kühnen et al., Flow Turb. Combust., vol. 100, 2018, pp. 919–943) and numerical (Marensi et al., J. Fluid Mech., vol. 863, 2019, pp. 850–875) studies have demonstrated that a streamwise-localised baffle can fully relaminarise pipe flow turbulence at Reynolds numbers of $O(10^4)$ . Optimising the design of the baffle involves tackling a complicated variational problem built around time stepping turbulent solutions of the Navier–Stokes equations which is difficult to solve. Here instead, we investigate a much simpler ‘spectral’ approach based upon maximising the energy stability of the baffle-modified laminar flow. The ensuing optimal problem has much in common with the variational procedure to derive an upper bound on the energy dissipation rate in turbulent flows (e.g. Plasting & Kerswell, J. Fluid Mech., vol. 477, 2003, pp. 363–379) so well-honed techniques developed there can be used to solve the problem here. The baffle is modelled by a linear drag force $-F(\boldsymbol {x}) \boldsymbol {u}$ (with $F(\boldsymbol {x}) \ge 0 \ \forall \boldsymbol {x}$ ) where the extent of the baffle is constrained by an $L_{\alpha }$ norm with various choices explored in the range $1 \leq \alpha \leq 2$ . An asymptotic analysis demonstrates that the optimal baffle is always axisymmetric and streamwise independent, retaining just radial dependence. The optimal baffle which emerges in all cases has a similar structure to that found to work in experiments: the baffle retards the flow in the pipe centre causing the flow to become faster near the wall thereby reducing the turbulent shear there. Numerical simulations demonstrate that the designed baffle can relaminarise turbulence efficiently at moderate Reynolds numbers ( $Re \le 3500$ ), and an energy saving regime has been identified. Direct numerical simulation at $Re=2400$ also demonstrates that the drag reduction can be realised by truncating the energy-stability-designed baffle to finite length.
We investigate the spatial organization and temporal dynamics of large-scale, coherent structures in turbulent Rayleigh–Bénard convection via direct numerical simulation of a 6.3 aspect-ratio cylinder with Rayleigh and Prandtl numbers of $9.6\times 10^7$ and $6.7$ , respectively. Fourier modal decomposition is performed to investigate the structural organization of the coherent turbulent motions by analysing the length scales, time scales and the underlying dynamical processes that are ultimately responsible for the large-scale structure formation and evolution. We observe a high level of rotational symmetry in the large-scale structure in this study and that the structure is well described by the first four azimuthal Fourier modes. Two different large-scale organizations are observed during the duration of the simulation and these patterns are dominated spatially and energetically by azimuthal Fourier modes with frequencies of 2 and 3. Studies of the transition between these two large-scale patterns, radial and vertical variations in the azimuthal energy spectra, as well as the spatial and modal variations in the system's correlation time are conducted. Rotational dynamics are observed for individual Fourier modes and the global structure with strong similarities to the dynamics that have been reported for unit aspect-ratio domains in prior works. It is shown that the large-scale structures have very long correlation time scales, on the order of hundreds to thousands of free-fall time units, and that they are the primary source for a horizontal inhomogeneity within the system that can be observed during a finite, but a very long-time simulation or experiment.
A numerical model that allows one to study numerically the evolution of waves along the test section of a wind-wave tank is offered. The simulations are directly related to wind-wave tank experiments carried out for a range of steady wind velocities. At each wind forcing condition, the evolving wind-wave field is strongly non-homogeneous, with wave energy growth along the test section accompanied by frequency downshifting. The wave parameters measured at a short fetch serve as a basis for generating numerous realizations of the initial conditions in the Monte Carlo numerical simulations. The computations are based on a modified unidirectional spatial version of the Zakharov equation that accounts for wind input and dissipation and is applicable for the whole range of wind velocities employed. The model contains two empirical parameters that are selected by comparison of the experimental and numerical results; the same values of those parameters are applied for all wind forcing conditions. The availability of an experimentally verified numerical model allows one to study the contributions of nonlinear wave–wave interactions, dissipation and wind input separately. Special attention is given to accounting for the three-dimensional and random nature of wind waves as observed in experiments. The suggested model combines approaches adopted in the wind-wave growth theories by Miles and Phillips.
This paper provides an experimental investigation on the internal shear layers and the edges of the uniform momentum zones (UMZs) in a turbulent pipe flow. The time-resolved stereoscopic particle image velocimetry data are acquired in the cross-section of the pipe, and span the range of Reynolds number $\textit {Re}_\tau =340\text {--}1259$ . In the first part of the study, internal shear layers are detected using a three-dimensional detection method, and both their geometry as well as their fingerprint in the flow statistics are examined. Three-dimensional conditional mean flow analysis revealed a strong low-speed region beneath the average shear layers. This low-speed region is associated with positive wall-normal fluctuations, and it is accompanied by two swirling motions having opposite signs on either side in the azimuthal direction. Moreover, the shear layers are stretched by the two opposite azimuthal motions. In the second part of the study, the shear layers are treated as the continuous edges of the UMZs, which are detected using the histogram method following Adrian et al. (J. Fluid Mech., vol. 422, 2000, pp. 1–54) and de Silva et al. (J. Fluid Mech., vol. 786, 2016, pp. 309–331). For this part, two different orientation of the planes are used, i.e. the wall-normal–streamwise plane and the wall-normal–spanwise plane (cross-section of the pipe). Comparison of the detected structures shows that the shear layers mostly overlap with a UMZ edge (in either plane).
We consider the stability of an elastic membrane on the bottom of a uniform horizontal flow of an inviscid and incompressible fluid of finite depth with free surface. The membrane is simply supported at the leading and the trailing edges which attach it to the two parts of the horizontal rigid floor. The membrane has an infinite span in the direction perpendicular to the direction of the flow and a finite length in the direction of the flow. For the membrane of infinite length we derive a full dispersion relation that is valid for arbitrary depth of the fluid layer and find conditions for the flutter of the membrane due to emission of surface gravity waves. We describe this radiation-induced instability by means of the perturbation theory of the roots of the dispersion relation and the concept of negative energy waves and discuss its relation to the anomalous Doppler effect.
We perform large-eddy simulations of turbulent flow in a channel constricted by streamwise periodically distributed hill-shaped protrusions. Two Reynolds number cases, i.e. $Re_h=10\,595$ (Fröhlich et al., J. Fluid Mech., vol. 526, 2005, pp. 19–66) and $Re_h=33\,000$ (Kähler et al., J. Fluid Mech., vol. 796, 2016, pp. 257–284), are repeated and utilized to verify and validate our numerical results, including the pressure and skin friction coefficients on bottom and top walls of the channel, mean velocity profiles and Reynolds stresses. All comparisons show reasonable agreement, providing a measure of validity that enables us to further probe simulation results at higher Reynolds number ( $Re_h=10^5$ ) into aspects of flow physics that are not available from experiments. Effects of variation of Reynolds number are studied, with emphasis on the mean skin friction coefficients, separation bubble size and pressure fluctuations that are related to separation and reattachment. In addition, the main large-scale features of the separation behind the hill, including the scaling of the mean velocity profiles, are discussed. Furthermore, the instantaneous near-wall flow field is analysed in terms of skin friction portraits, and we confirm the existence of the local very small separation bubble on the hill crest as observed in experimental and numerical investigations. The flow field at the top wall, which is generally not given sufficient attention, is evaluated with the empirical friction law and universal logarithmic law as in planar channel flows. It is found that these empirical laws compare well with the large-eddy simulation results, although the hill constrictions behave as a perturbation source and the developed shear layer has some effects on the flow field near the top wall.
We present results on the effect of dispersed droplets in vertical natural convection (VC) using direct numerical simulations based on a two-way fully coupled Euler–Lagrange approach with a liquid phase and a dispersed droplets phase. For increasing thermal driving, characterised by the Rayleigh number, Ra, of the two analysed droplet volume fractions, $\alpha = 5\times 10^{-3}$ and $\alpha = 2\times 10^{-2}$ , we find non-monotonic responses to the overall heat fluxes, characterised by the Nusselt number, Nu. The Nu number is larger when the droplets are thermally coupled to the liquid. However, Nu values remain close to the 1/4-laminar VC scaling, suggesting that the heat transport is still modulated by thermal boundary layers. Local analyses reveal the non-monotonic trends of local heat fluxes and wall-shear stresses: whilst regions of high heat fluxes are correlated to increased wall-shear stresses, the spatio-temporal distribution and magnitude of the increase are non-monotonic, implying that the overall heat transport is obscured by competing mechanisms. Most crucially, we find that the transport mechanisms inherently depend on the dominance of droplet driving to thermal driving that can quantified by (i) the bubblance parameter $b$ , which measures the ratio of energy produced by the dispersed phase and the energy of the background turbulence, and (ii) ${\textit {Ra}}_d/{\textit {Ra}}$ , where ${\textit {Ra}}_d$ is the droplet Rayleigh number, which we introduce in this paper. When $b \lesssim O(10^{-1})$ and ${\textit {Ra}}_d/{\textit {Ra}} \lesssim O(100)$ , the Nu scaling is expected to recover to the VC scaling without droplets, and comparison with $b$ and ${\textit {Ra}}_d/{\textit {Ra}}$ from our data supports this notion.
In the preparation of café latte, spectacular layer formation can occur between the espresso shot in a glass of milk and the milk itself. Xue et al. (Nat. Commun., vol. 8, 2017, pp. 1–6) showed that the injection velocity of espresso determines the depth of coffee–milk mixture. After a while, when a stable stratification forms in the mixture, the layering process can be modelled as a double diffusive convection system with a stably stratified coffee–milk mixture cooled from the side. More specifically, we perform (two-dimensional) direct numerical simulations of laterally cooled double diffusive convection for a wide parameter range, where the convective flow is driven by a lateral temperature gradient while stabilized by a vertical concentration gradient. Depending on the strength of stabilization as compared to the thermal driving, the system exhibits different flow regimes. When the thermal driving force dominates over the stabilizing force, the flow behaves like vertical convection in which a large-scale circulation develops. However, with increasing strength of the stabilizing force, a meta-stable layered regime emerges. Initially, several vertically-stacked convection rolls develop, and these well-mixed layers are separated by sharp interfaces with large concentration gradients. The initial thickness of these emerging layers can be estimated by balancing the work exerted by thermal driving and the required potential energy to bring fluid out of its equilibrium position in the stably stratified fluid. In the layered regime, we further observe successive layer merging, and eventually only a single convection roll remains. We elucidate the following merging mechanism: as weakened circulation leads to accumulation of hot fluid adjacent to the hot sidewall, larger buoyancy forces associated with hotter fluid eventually break the layer interface. Then two layers merge into a larger layer, and circulation establishes again within the merged structure.
We use experiments and direct numerical simulations to probe the phase space of low-curvature Taylor–Couette flow in the vicinity of the ultimate regime. The cylinder radius ratio is fixed at $\eta =r_i/r_o=0.91$ , where $r_i \, (r_o)$ is the inner (outer) cylinder radius. Non-dimensional shear drivings (Taylor numbers ${\textit {Ta}}$ ) in the range $10^7\leq {\textit {Ta}}\leq 10^{11}$ are explored for both co- and counter-rotating configurations. In the ${\textit {Ta}}$ range $10^8\leq {\textit {Ta}}\leq 10^{10}$ , we observe two local maxima of the angular momentum transport as a function of the cylinder rotation ratio, which can be described as either ‘co-’ or ‘counter-rotating’ due to their location or as ‘broad’ or ‘narrow’ due to their shape. We confirm that the broad peak is accompanied by the strengthening of the large-scale structures, and that the narrow peak appears once the driving (Ta) is strong enough. As first evidenced in numerical simulations by Brauckmann et al. (J. Fluid Mech., vol. 790, 2016, pp. 419–452), the broad peak is produced by centrifugal instabilities and that the narrow peak is a consequence of shear instabilities. We describe how the peaks change with ${\textit {Ta}}$ as the flow becomes more turbulent. Close to the transition to the ultimate regime when the boundary layers (BLs) become turbulent, the usual structure of counter-rotating Taylor vortex pairs breaks down and stable unpaired rolls appear locally. We attribute this state to changes in the underlying roll characteristics during the transition to the ultimate regime. Further changes in the flow structure around ${\textit {Ta}}\approx 10^{10}$ cause the broad peak to disappear completely and the narrow peak to move. This second transition is caused when the regions inside the BLs which are locally smooth regions disappear and the whole boundary layer becomes active.
We investigate the formation of subaqueous transverse bedforms in turbulent open channel flow by means of direct numerical simulations with fully resolved particles. The main goal of the present analysis is to address the question whether the initial pattern wavelength scales with the particle diameter or with the mean fluid height. A previous study (Kidanemariam & Uhlmann, J. Fluid Mech., vol. 818, 2017, pp. 716–743) has observed a lower bound for the most unstable pattern wavelength in the range 75–100 times the particle diameter, which was equivalent to 3–4 times the mean fluid height. In the current paper, we vary the streamwise box length in terms of the particle diameter and of the mean fluid height independently in order to distinguish between the two possible scaling relations. For the chosen parameter range, the obtained results clearly exhibit a scaling of the initial pattern wavelength with the particle diameter, with a lower bound around a streamwise extent of approximately $80$ particle diameters. In longer domains, on the other hand, patterns are observed at initial wavelengths in the range 150–180 times the particle diameter, which is in good agreement with experimental measurements. Variations of the mean fluid height, on the other hand, seem to have no significant influence on the most unstable initial pattern wavelength. We argue that the observed scaling with the particle diameter is due to the wake effect induced by the seeds which are formed by initially dislodged lumps of particles, in accordance with the ideas of Coleman & Nikora (Water Resour. Res., vol. 45, 2009, W04402). Finally, for the cases with the largest relative submergence, we observe spanwise and streamwise sediment waves of similar amplitude to evolve and superimpose, leading to three-dimensional sediment patterns.
We consider the injection of a buoyant low viscosity fluid into an aquifer saturated with a higher viscosity fluid. The nose region of the flow, where the thickness of the injected fluid is less than the thickness of the aquifer, grows in proportion to time and as a result fluid continually migrates further into the nose where it has a progressively smaller vertical extent. We explore how the flow in the nose influences the migration of a pulse of tracer. The growth of the nose stretches a pulse of tracer of initial length, $L_0$ , longitudinally to have a length proportional to $L_0 (T/T_E)^{1/2}$ , where $T_E$ is the nose entry time. Diffusion acts at the same rate and the combination of the two processes results in an tracer spreading longitudinally with a length proportional to $(D T\, \log T)^{1/2}$ at long times after entering the nose, where D is the coefficient of diffusion. The results are generalised to consider the case in which the permeability in the aquifer varies with depth. At early times, the tracer is sheared. As the tracer migrates into continually thinner regions of the growing nose, the permeability contrast sampled by the tracer rapidly decays. The role of the shear becomes dominated by the stretching of the nose and ultimately the late-time behaviour is as in a uniform aquifer. However, the effective pulse length of the tracer upon asymptoting to the stretching regime is now given by $L_0={\rm \Delta} U T_E$ , where ${\rm \Delta} U$ is the magnitude of the shear. The spreading in the stretching regime then has a length scale of ${\rm \Delta} U (T_E T)^{1/2}$ , which may be much faster than in the case of a uniform aquifer. If the diffusion is sufficiently fast, there may be an intermediate regime in which Taylor dispersion is important prior to the stretching dominating.
Many permeable aquifers have vertical variations in the permeability, which are correlated over long lateral distances. When a buoyant high viscosity fluid is injected into a permeable aquifer and displaces a less viscous fluid, the interface that develops has finite extent and travels with constant speed. The effect of the permeability variations leads to fluid in the high permeability regions travelling into the front, where it migrates into the lower permeability part. Subsequently, it lags progressively further behind the advancing front. We explore the influence of this cycling on the dispersion of a tracer or additive in the fluid to determine its distribution within the current as a function of time. At early times tracer is sheared owing to the vertically varying permeability. At later times, cross-aquifer diffusion homogenises the tracer distribution, which spreads longitudinally at a faster rate than by diffusion alone in this shear dispersion regime. Eventually, tracer reaches the front of the current. This acts as a no-flux boundary and the concentration profile transitions to a half-Gaussian, with the maximum concentration at the front. The centre of mass of the tracer spreads backwards relative to the fixed nose at a rate proportional to $(DT)^{1/2}$ , where T is time and D is the diffusion coefficient. The initial release of tracer may not be vertically uniform owing to the heterogeneity and we show that this can lead to the centre of mass of tracer initially advancing faster than the mean flow. Although our model is highly idealised, it illustrates how, owing to a vertical variation of permeability, it is possible for a finite pulse of tracer or chemical added after the start of injection to reach the front with important implications for tracer tests and strategies for enhanced recovery.
The evaporation of multicomponent droplets is relevant to various applications but challenging to study due to the complex physicochemical dynamics. Recently, Li etal. (Phys. Rev. Lett., vol. 120, 2018, 224501) reported evaporation-triggered segregation in 1,2-hexanediol–water binary droplets. In this present work, we added 0.5 wt % silicone oil to the 1,2-hexanediol–water binary solution. This minute silicone oil concentration dramatically modifies the evaporation process, as it triggers an early extraction of the 1,2-hexanediol from the mixture. Surprisingly, we observe that the segregation of 1,2-hexanediol forms plumes, rising up from the rim of the sessile droplet towards the apex during droplet evaporation. By orientating the droplet upside down, i.e. by studying a pendent droplet, the absence of the plumes indicates that the flow structure is induced by buoyancy, which drives a Rayleigh–Taylor instability (i.e. driven by density differences and gravitational acceleration). From micro particle image velocimetry measurement, we further prove that the segregation of the non-volatile component (1,2-hexanediol) hinders the evaporation near the contact line, which leads to a suppression of the Marangoni flow in this region. Hence, on long time scales, gravitational effects, rather than Marangoni flows, play the dominant role in the flow structure. We compare the measurement of the evaporation rate with the diffusion model of Popov (Phys. Rev., vol. 71, 2005, 036313), coupled with Raoult's law and the activity coefficient. This comparison indeed confirms that the silicone-oil-triggered segregation of the non-volatile 1,2-hexanediol significantly delays the evaporation. With an extended diffusion model, in which the influence of the segregation has been implemented, the evaporation can be well described.
Free-stream turbulence (FST) and its effect on boundary-layer transition is an intricate problem. Elongated unsteady streamwise streaks of low and high speed are created inside the boundary layer and their amplitude and spanwise wavelength are believed to be important for the onset of transition. The transitional Reynolds number is often simply correlated with the turbulence intensity ( ${Tu}$ ), and the characteristic length scales of the FST are often considered to have a small to negligible influence on the transition location. Here, we present new results from a large experimental measurement campaign, where both the ${Tu}$ and the integral length scale ( $\Lambda _x$ ) are varied ( $1.8\,\% < {Tu}< 6.2\,\%$ ; $16\ \textrm {mm}< \Lambda _x < 26\ \textrm {mm}$ ). In the current experiments it has been noted that on the one hand, for small $Tu$ , an increase in $\Lambda _x$ advances transition, which is in agreement with established results. On the other hand, for large $Tu$ , an increase in $\Lambda _x$ postpones transition. This trend can be explained by the fact that an optimal ratio between FST length scale and boundary-layer thickness at transition onset exists. Furthermore, our results strengthen the fact that the streaks play a key role in the transition process by showing a clear dependence of the FST characteristics on their spanwise scale. Our measurements show that the aspect ratio of the streaky structures correlates with an FST Reynolds number and that the aspect ratio can change by a factor of two at the location of transition. Finally, we derive a semi-empirical transition prediction model, which is able to predict the influence of $\Lambda _x$ for both small and high values of ${Tu}$ .
We show how settling and phase change can combine to drive an instability, as a simple model for the formation of mammatus clouds. Our idealised system consists of a layer (an ‘anvil’) of air mixed with saturated water vapour and monodisperse water droplets, sitting atop dry air. The water droplets in the anvil settle under gravity due to their finite size, evaporating as they enter dry air and cooling the layer of air just below the anvil. The colder air just below the anvil thus becomes denser than the dry air below it, forming a density ‘overhang’, which is unstable. The strength of the instability depends on the density difference between the density overhang and the dry ambient, and the depth of the overhang. Using linear stability analysis and nonlinear simulations in one, two and three dimensions, we study how the amplitude and depth of the density layer depend on the initial conditions, finding that their variations can be explained in terms only of the size of the droplets making up the liquid content of the anvil and by the total amount of liquid water contained in the anvil. We find that the size of the water droplets is the controlling factor in the structure of the clouds: mammatus-like lobes form for large droplet sizes; and small droplet sizes lead to a ‘leaky’ instability resulting in a stringy cloud structure resembling the newly designated asperitas.
We perform interface-resolved simulations to study the modulation of statistically steady-state homogeneous shear turbulence by neutrally buoyant finite-size particles. We consider two shapes, spheres and oblates, and various solid volume fractions, up to 20%. The results show that a statistically steady state is not exclusive to single-phase homogeneous shear turbulence as the production and dissipation rates of the turbulent kinetic energy are also statistically in balance in particle-laden cases. The turbulent kinetic energy shows a non-monotonic behaviour with increasing solid volume fraction: increasing turbulence attenuation up to a certain concentration of solid particles and then enhancement of the turbulent kinetic energy at higher concentrations. This behaviour is observed at lower volume fractions for oblate particles than for spheres. The attenuation of the turbulence activity at lower volume fractions is explained through the enhancement of the dissipation rate close to the surface of particles. At higher volume fractions, however, particle pair interactions induce regions of high Reynolds shear stress, resulting in the enhancement of the turbulence activity. We show that the oblate particles of the considered size have larger rotational rates than spheres with no preferential orientation. This is in contrast to previous studies in wall-bounded flows where preferential orientation close to the wall and reduced rotation rates result in turbulence attenuation and thus drag reduction. Our results shed some light on the effect of rigid particles, smaller than the near-wall turbulent structures but still comparable to the viscous length scale, on the dynamics of the equilibrium logarithmic layer in wall-bounded flows.
For flows in microchannels, a slip on the walls may be efficient in reducing viscous dissipation. A related issue, addressed in this article, is to decrease the effective viscosity of a dilute monodisperse suspension of spheres in Poiseuille flow by using two parallel slip walls. Extending the approach developed for no-slip walls in Feuillebois et al. (J. Fluid Mech., vol. 800, 2016, pp. 111–139), a formal expression is obtained for the suspension intrinsic viscosity $[\mu ]$ solely in terms of a stresslet component and a quadrupole component exerted on a single freely suspended sphere. In the calculation of $[\mu ]$ , the hydrodynamic interactions between a sphere and the slip walls are approximated using either the nearest wall model or the wall-superposition model. Both the stresslet and quadrupole are derived and accurately calculated using bipolar coordinates. Results are presented for $[\mu ]$ in terms of $H/(2a)$ and $\tilde{\lambda}=\lambda /a\leq 1$ , where $H$ is the gap between walls, $a$ is the sphere radius and $\lambda$ is the wall slip length using the Navier slip boundary condition. As compared with the no-slip case, the intrinsic viscosity strongly depends on $\tilde{\lambda}$ for given $H/(2a)$ , especially for small $H/(2a)$ . For example, in the very confined case $H/(2a)=2$ (a lower bound found for practical validity of single-wall models) and for $\tilde{\lambda}=1$ , the intrinsic viscosity is three times smaller than for a suspension bounded by no-slip walls and five times smaller than for an unbounded suspension (Einstein, Ann. Phys., vol. 19, 1906, pp. 289–306). We also provide a handy formula fitting our results for $[\mu ]$ in the entire range $2\leq H/(2a)\leq 100$ and $\tilde{\lambda}\leq 1$ .
When a viscous liquid bridge between two parallel substrates is stretched by accelerating one substrate, its interface on the plates recedes in the radial direction. In some cases the interface becomes unstable. Such instability leads to the emergence of a network of fingers. In this study, the mechanisms of such fingering are studied experimentally and analysed theoretically. The experimental set-up allows a constant acceleration of a movable substrate at up to 180 m s $^{-2}$ . The phenomena are observed using two high-speed video systems. The number of fingers is measured for different liquid viscosities, liquid bridge sizes and wetting conditions. Linear stability analysis of the bridge interface takes into account the inertial, viscous and capillary effects in the liquid flow. The theoretically predicted maximum number of fingers, corresponding to an instability mode with the maximum amplitude, and a threshold for the onset of finger formation are proposed. Both models agree well with the experimental data up to the start of emerging cavitation bubbles.