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The bizarre behaviour of a cornstarch suspension (sometimes called oobleck) is well known to all of us who have led public engagement events. At the right solids fraction, it flows smoothly at slow speeds, but can be shattered with a quick spoon movement; if you prepare a large enough sample, you can run across the surface (but if you stand still, you will sink). In rheology circles this phenomenon is known as shear thickening, though the flows described above are not necessarily shear-dominated. In recent years there has been a proliferation of research on the mechanism behind true shear thickening, using both experiments and numerical simulations of shear flows. The understanding of the underlying mechanism is improving markedly. But the paper ‘Microstructure and thickening of dense suspensions under extensional and shear flows’ (Seto, Giusteri & Martinello, J. Fluid Mech., vol. 825, 2017, R3) is the first to consider more general flows. We have, for the first time, simulations of thickening in extensional flows, which are a far better description of oobleck with a runner on top – and can begin to quantify the difference between the idealised shear thickening and the extension thickening that happens in practice.
This work aims to provide a systematic experimental study on the wake of two tandem cylinders of unequal diameters. The fluid dynamics around a circular cylinder of diameter $D$ placed in the wake of another circular cylinder with a smaller diameter of $d$ is investigated, including the time-mean drag coefficient ( $C_{D}$ ), the fluctuating drag and lift coefficients ( $C_{D}^{\prime }$ and $C_{L}^{\prime }$ ), the Strouhal number ( $St$ ) and the flow structures. The Reynolds number based on $D$ is kept constant at $4.27\times 10^{4}$ . The ratios $d/D$ and $L/d$ vary from 0.2 to 1.0 and 1.0 to 8.0 respectively, where $L$ is the distance from the upstream cylinder centre to the forward stagnation point of the downstream cylinder. The ratios $d/D$ and $L/d$ are found, based on extensive hotwire, particle imaging velocimetry, pressure and flow visualization measurements, to have a marked influence on the wake dynamics behind the cylinders. As such, the flow is classified into the reattachment and co-shedding flow regimes, the latter being further subdivided into the lock-in, subharmonic lock-in and no lock-in regions. It is found that the critical spacing that divides the two regimes is dictated by the upstream-cylinder vortex formation length and becomes larger for smaller $d/D$ . The characteristic flow properties are documented in each regime and subdivided region, including the flow structure, $St$ , wake width, vortex formation length and the lateral width between the two gap shear layers. The variations in $C_{D}$ , $C_{D}^{\prime }$ , $C_{L}^{\prime }$ and the pressure distribution around the downstream cylinder are connected to the flow physics.
This paper proposes a resolution to the conundrum of the roles of convective and absolute instability in transition of the rotating-disk boundary layer. It also draws some comparison with swept-wing flows. Direct numerical simulations based on the incompressible Navier–Stokes equations of the flow over the surface of a rotating disk with modelled roughness elements are presented. The rotating-disk flow has been of particular interest for stability and transition research since the work by Lingwood (J. Fluid Mech., vol. 299, 1995, pp. 17–33) where an absolute instability was found. Here stationary disturbances develop from roughness elements on the disk and are followed from the linear stage, growing to saturation and finally transitioning to turbulence. Several simulations are presented with varying disturbance amplitudes. The lowest amplitude corresponds approximately to the experiment by Imayama et al. (J. Fluid Mech., vol. 745, 2014a, pp. 132–163). For all cases, the primary instability was found to be convectively unstable, and secondary modes were found to be triggered spontaneously while the flow was developing. The secondary modes further stayed within the domain, and an explanation for this is a proposed globally unstable secondary instability. For the low-amplitude roughness cases, the disturbances propagate beyond the threshold for secondary global instability before becoming turbulent, and for the high-amplitude roughness cases the transition scenario gives a turbulent flow directly at the critical Reynolds number for the secondary global instability. These results correspond to the theory of Pier (J. Engng Maths, vol. 57, 2007, pp. 237–251) predicting a secondary absolute instability. In our simulations, high temporal frequencies were found to grow with a large amplification rate where the secondary global instability occurred. For smaller radial positions, low-frequency secondary instabilities were observed, tripped by the global instability.
The supply of energy to the internal wave field in the ocean is, in total, sufficient to support the mixing required to maintain the stratification of the ocean, but can the required rates of turbulent dissipation in mid-water be sustained by breaking internal waves? It is assumed that turbulence occurs in regions where the field of motion can be represented by an exact solution of the equations that describe waves propagating through a uniformly stratified fluid and becoming unstable. Two instabilities leading to wave breaking are examined, convective instability and shear-induced Kelvin–Helmholtz instability. Models are constrained by data representative of the mid-water ocean. Calculations of turbulent dissipation are first made on the assumption that all the waves representing local breaking have the same steepness, $s$ , and frequency, $\unicode[STIX]{x1D70E}$ . For some ranges of $s$ and $\unicode[STIX]{x1D70E}$ , breaking can support the required transfer of energy to turbulence. For convective instability this proves possible for sufficiently large $s$ , typically exceeding 2.0, over a range of $\unicode[STIX]{x1D70E}$ , while for shear-induced instability near-inertial waves are required. Relaxation of the constraint that the model waves all have the same $s$ and $\unicode[STIX]{x1D70E}$ requires new assumptions about the nature and consequences of wave breaking. Examples predict an overall dissipation consistent with the observed rates. Further observations are, however, required to test the validity of the assumptions made in the models and, in particular, to determine the nature and frequency of internal wave breaking in the mid-water ocean.
This paper investigates the effects of finite flat porous extensions to semi-infinite impermeable flat plates in an attempt to control trailing-edge noise through bio-inspired adaptations. Specifically the problem of sound generated by a gust convecting in uniform mean steady flow scattering off the trailing edge and the permeable–impermeable junction is considered. This set-up supposes that any realistic trailing-edge adaptation to a blade would be sufficiently small so that the turbulent boundary layer encapsulates both the porous edge and the permeable–impermeable junction, and therefore the interaction of acoustics generated at these two discontinuous boundaries is important. The acoustic problem is tackled analytically through use of the Wiener–Hopf method. A two-dimensional matrix Wiener–Hopf problem arises due to the two interaction points (the trailing edge and the permeable–impermeable junction). This paper discusses a new iterative method for solving this matrix Wiener–Hopf equation which extends to further two-dimensional problems, in particular those involving analytic terms that exponentially grow in the upper or lower half-planes. This method is an extension of the commonly used ‘pole removal’ technique and avoids the need for full matrix factorisation. Convergence of this iterative method to an exact solution is shown to be particularly fast when terms neglected in the second step are formally smaller than all other terms retained. The new method is validated by comparing the iterative solutions for acoustic scattering by a finite impermeable plate against a known solution (obtained in terms of Mathieu functions). The final acoustic solution highlights the effects of the permeable–impermeable junction on the generated noise, in particular how this junction affects the far-field noise generated by high-frequency gusts by creating an interference to typical trailing-edge scattering. This effect results in partially porous plates predicting a lower noise reduction than fully porous plates when compared to fully impermeable plates.
In a previous paper, an inviscid vortex force map approach was developed for the normal force of a flat plate at arbitrarily high angle of attack and leading/trailing edge force-producing critical regions were identified. In this paper, this vortex force map approach is extended to viscous flows and general airfoils, for both lift and drag forces due to vortices. The vortex force factors for the vortex force map are obtained here by using Howe’s integral force formula. A decomposed form of the force formula, ensuring vortices far away from the body have negligible effect on the force, is also derived. Using Joukowsky and NACA0012 airfoils for illustration, it is found that the vortex force map for general airfoils is similar to that of a flat plate, meaning that force-producing critical regions similar to those of a flat plate also exist for more general airfoils and for viscous flow. The vortex force approach is validated against NACA0012 at several angles of attack and Reynolds numbers, by using computational fluid dynamics.
The onset of unsteadiness in a boundary-layer flow past a cylindrical roughness element is investigated for three flow configurations at subcritical Reynolds numbers, both experimentally and numerically. On the one hand, a quasi-periodic shedding of hairpin vortices is observed for all configurations in the experiment. On the other hand, global stability analyses have revealed the existence of a varicose isolated mode, as well as of a sinuous one, both being linearly stable. Nonetheless, the isolated stable varicose modes are highly sensitive, as ascertained by pseudospectrum analysis. To investigate how these modes might influence the dynamics of the flow, an optimal forcing analysis is performed. The optimal response consists of a varicose perturbation closely related to the least stable varicose isolated eigenmode and induces dynamics similar to that observed experimentally. The quasi-resonance of such a global mode to external forcing might thus be responsible for the onset of unsteadiness at subcritical Reynolds numbers, hence providing a simple explanation for the experimental observations.
The effects of a surface trapped steady background current on internal waves generated by tidal currents oscillating over a small symmetric ridge are investigated using a two-dimensional primitive equation model. A rigid lid is used with a linearly stratified fluid and the effects of rotation are not considered. We consider uni-directional background currents $\bar{U}(z)\geqslant 0$ confined to a surface layer lying well above the ridge. The current introduces asymmetries in the generated wave field. For sufficiently narrow ridges the upstream energy flux is larger than the downstream flux while the opposite is the case for sufficiently wide ridges. The total energy flux radiating away from the ridge is not significantly affected by the current. Mean second-order currents and pressure fields are shown to make important contributions to the total energy flux. A first-order linear theory, valid for a general stratification and surface current, which accurately predicts the wave field is also developed.
A convective velocity must be specified when using Taylor’s frozen eddy hypothesis to relate temporal and spatial fluctuations. Depending on the quantity of interest, using different convective velocities (i.e. time-mean velocity, global convective velocity, etc.) may lead to different conclusions. Often, using Taylor’s hypothesis, the relation between temporal and spatial fluctuations is simplified by assuming a temporally averaged velocity as the convection velocity. In flows where turbulence fluctuations are much smaller than the mean flow velocity, the above treatment does not bring in much error (at least for short periods of time). However, when turbulence fluctuations are comparable to the mean velocity, using a constant convective velocity for fluid motions of all scales can sometimes be problematic. In the context of wall-bounded flows, turbulence fluctuations are comparable to the mean flow in the near-wall region, and as a result, using a constant global convective velocity for converting temporal signals to spatial ones distorts the spatial eddies. Although such distortion will not significantly affect measurements of flow quantities including central moments and power spectra, the significance of amplitude modulation is largely overestimated. Here, we show that if temporal hot-wire data are to be used for studying spatial amplitude modulation, the local fluid velocity must be used as the local convective velocity. The impact of amplitude modulation on power spectra and skewness are reconsidered using the proposed correction.
A study of the propagation of a mode-2 internal solitary wave over a slope-shelf topography is presented. The methodology is based on a variable-coefficient Korteweg–de Vries (vKdV) equation, using both analysis and numerical simulations, and simulations using the MIT general circulation model (MITgcm). Two configurations are considered. One is a mode-2 internal solitary wave propagating up the slope, from one three-layer system to another three-layer system. Depending on the height of the shelf, which determines the variation of the nonlinear coefficient of the vKdV equation, this can be classified into two cases. First, the case of a polarity change, in which the coefficient of the quadratic nonlinear term changes sign at a certain critical point on the slope, and second, the case with no such polarity change. In both these cases there is a small transfer of energy from the mode-2 wave to mode-1 waves. The other configuration is when the lower layer in the three-layer system goes to zero at a transition point on the slope, and beyond that point, there is a two-layer fluid system. A mode-2 internal solitary wave propagating up the slope cannot exist past this transition point. Instead it is extinguished and replaced by a mode-1 bore and trailing wave packet which moves onto the shelf.
High-Reynolds-number steady currents of relatively dense fluid propagating along a horizontal boundary become unstable and mix with the overlying fluid if the gradient Richardson number across the interface is less than 1/4. The process of entrainment produces a deepening mixing layer at the interface, which increases the gradient Richardson number of this layer and eventually may suppress further entrainment. The conservation of the vertically averaged buoyancy and momentum flux, as the current advances along the boundary, leads to two integral constraints relating the downstream flow with that upstream of the mixing zone. These constraints are equivalent to imposing a Froude number in the upstream flow. Using the ansatz that the dowstream velocity and buoyancy profiles in the current have a lower well-mixed region overlain by an interfacial layer of constant gradient, we can use these two constraints to quantify the total entrainment of ambient fluid into the flow as a function of the gradient Richardson number of the downstream flow. This leads to recognition that both subcritical and supercritical currents may develop downstream of the mixing zone. However, as the mixing increases and the interfacial layer gradually deepens, there is a critical point at which these two solution branches coincide. For each upstream Froude number, we can also determine the downstream flow with maximal entrainment. This maximal entrainment solution coincides with the convergence point of the supercritical and subcritical branches. We compare this with the entrainment predicted for those solutions with a gradient Richardson number of 1/4, which corresponds to the marginally stable case. As the upstream Froude number $Fr_{u}$ increases, the maximum depth of the interfacial mixing layer gradually increases until eventually, for $Fr_{u}>2.921$ , the whole current may become modified through entrainment. We discuss the relevance of these results for mixing in gravity-driven flows.
We explore the dynamics of turbulent bubble fountains produced when a descending stream of fresh water and air bubbles issues from a nozzle submerged in a tank of water. The bubbles have diameters of 2 to 5 mm and the fountains have source Froude numbers ranging from 10 to 240. The Reynolds numbers of the bubbly fountains range from 4000 to 24 000. The bubbles, carried into the tank by the downward jet of water, lead to a buoyancy force which reduces the downward momentum of the jet, thus producing a fountain. We find that $H_{F}$ , the downward penetration distance of the bubbles into the water reservoir, may be characterised by two parameters: $\unicode[STIX]{x1D6EC}$ , the ratio of the bubble rise speed to the characteristic fountain velocity, $u_{F}=f_{0}^{1/2}/m_{0}^{1/4}$ , and $Fr_{0}$ , the source Froude number, given by $m_{0}^{5/4}/(q_{W_{0}}f_{0}^{1/2})$ , where $q_{W_{0}}$ , $m_{0}$ and $f_{0}$ are the source volume, momentum and buoyancy fluxes. As $\unicode[STIX]{x1D6EC}$ increases, $H_{F}$ decreases, a result which is directly analogous to the height of rise of particles in a particle-laden fountain (Mingotti & Woods, J. Fluid Mech., vol. 793, 2016, R1). Also, we find that $H_{F}$ increases as $Fr_{0}$ increases, a result directly analogous to single-phase fountains (Turner, J. Fluid Mech., vol. 26, 1966, pp. 779–792). We present a model for the conservation of volume, momentum and buoyancy fluxes and use this to predict the penetration distance of the bubbles corresponding to that point at which the fountain liquid velocity equals the bubble rise speed. Using the best-fit value for the entrainment coefficient, $\unicode[STIX]{x1D6FC}=0.04\pm 0.004$ , we find that our experimental measurements of the bubble penetration distance are in good accord with the model predictions for $10<Fr_{0}<240$ and $2<\unicode[STIX]{x1D6EC}<15$ . In our experiments the bubble rise speed, $u_{slip}$ , is large compared to the entrainment velocity of the descending fountain. Thus, only a small fraction of the rising bubbles are re-entrained, and so the buoyancy flux of the fountain is approximately independent of depth. Flow-visualisation experiments also show that the liquid momentum flux is not exhausted at the point of bubble separation and so the liquid in the fountain continues to travel downward, separated from the bubbles. We use the new theoretical model to estimate the flux of air entrained into plunging water jets.
Cilia and flagella are essential building blocks for biological fluid transport and locomotion at the micrometre scale. They often beat in synchrony and may transition between different synchronization modes in the same cell type. Here, we investigate the behaviour of elastic microfilaments, protruding from a surface and driven at their base by a configuration-dependent torque. We consider full hydrodynamic interactions among and within filaments and no slip at the surface. Isolated filaments exhibit periodic deformations, with increasing waviness and frequency as the magnitude of the driving torque increases. Two nearby but independently driven filaments synchronize their beating in-phase or anti-phase. This synchrony arises autonomously via the interplay between hydrodynamic coupling and filament elasticity. Importantly, in-phase and anti-phase synchronization modes are bistable and coexist for a range of driving torques and separation distances. These findings are consistent with experimental observations of in-phase and anti-phase synchronization in pairs of cilia and flagella and could have important implications on understanding the biophysical mechanisms underlying transitions between multiple synchronization modes.
In the present work, the role of diffusion and mixing in hot jet initiation and detonation propagation in a supersonic combustible hydrogen–oxygen mixture is investigated in a two-dimensional channel. A second-order accurate finite volume method solver combined with an adaptive mesh refinement method is deployed for both the reactive Euler and Navier–Stokes equations in combination with a one-step and two-species reaction model. The results show that the small-scale vortices resulting from the Kelvin–Helmholtz instability enhance the reactant consumption in the inviscid result through the mixing. However, the suppression of the growth of the Kelvin–Helmholtz instability and the subsequent formation of small-scale vortices imposed by the diffusion in the viscous case can result in the reduction of the mixing rate, hence slowing the consumption of the reactant. After full initiation in the whole channel, the mixing becomes insufficient to facilitate the reactant consumption. This applies to both the inviscid and viscous cases and is due to the absence of the unburned reactant far away from the detonation front. Nonetheless, the stronger diffusion effect in the Navier–Stokes results can contribute more significantly to the reactant consumption closely behind the detonation front. However, further downstream the mixing is expected to be stronger, which eventually results in a stronger viscous detonation than the corresponding inviscid one. At high grid resolutions it is vital to correctly consider physical viscosity to suppress intrinsic instabilities in the detonation front, which can also result in the generation of less triple points even with a larger overdrive degree. Numerical viscosity was minimized to such an extent that inviscid results remained intrinsically unstable while asymptotically converged results were only obtained when the Navier–Stokes model was applied, indicating that solving the reactive Navier–Stokes equations is expected to give more correct descriptions of detonations.
Since the speed of sound in water is much greater than that of the surface gravity waves, acoustic signals can be used for early warning of tsunamis. We simplify existing works by treating the sound wave alone without the much slower gravity wave, and derive a two-dimensional theory for signals emanating from a fault of finite length. Under the assumptions of a slender fault and constant sea depth, the asymptotic technique of multiple scales is applied to obtain analytical results. The modal envelopes of the two-dimensional sound waves are found to be governed by the Schrödinger equation and are solved explicitly. An approximate method is described for the inverse estimation of fault properties from the pressure record at a distant hydrophone.
A comprehensive experimental study is presented to analyse the instabilities of a magnetic fluid drop surrounded by miscible fluid confined in a Hele-Shaw cell. The experimental conditions include different magnetic fields (by varying the maximum pre-set magnetic field strengths, $H$ , and sweep rates, $SR=\text{d}H_{t}/\text{d}t$ , where $H_{t}$ is the instant magnetic field strength), gap spans, $h$ , and magnetic fluid samples, and are further coupled into a modified Péclect number $Pe^{\prime }$ to evaluate the instabilities. Two distinct instabilities are induced by the external magnetic fields with different sweep rates: (i) a labyrinthine fingering instability, where small fingerings emerge around the initial circular interface in the early period, and (ii) secondary waves in the later period. Based on 81 sets of experimental conditions, the initial growth rate of the interfacial length, $\unicode[STIX]{x1D6FC}$ , of the magnetic drop is found to increase linearly with $Pe^{\prime }$ , indicating that $\unicode[STIX]{x1D6FC}$ is proportional to the square root of the $SR$ and $h^{3/2}$ at the onset of the labyrinthine instability. In addition, secondary waves, which are characterised by the dimensionless wavelength $\unicode[STIX]{x1D6EC}=\unicode[STIX]{x1D706}/h$ , can only be triggered when the three-dimensional magnetic microconvection is strong enough to make $Pe^{\prime }$ exceed a critical value, i.e. $Pe^{\prime }>19\,000$ , where $\unicode[STIX]{x1D706}$ is the wavelength of the secondary wave. In this flow regime of high $Pe^{\prime }$ , the length scale of the secondary wave instability is found to be $\unicode[STIX]{x1D6EC}=7\pm 1$ , corresponding to the Stokes regime; meanwhile, in the flow regime of low $Pe^{\prime }$ , the flow corresponds to the Hele-Shaw regime introduced by Fernandez et al. (J. Fluid Mech., vol. 451, 2002, pp. 239–260).
We provide experimental measurements for the effective scaling of the Taylor–Reynolds number within the bulk $\mathit{Re}_{\unicode[STIX]{x1D706},\mathit{bulk}}$ , based on local flow quantities as a function of the driving strength (expressed as the Taylor number $\mathit{Ta}$ ), in the ultimate regime of Taylor–Couette flow. We define $Re_{\unicode[STIX]{x1D706},bulk}=(\unicode[STIX]{x1D70E}_{bulk}(u_{\unicode[STIX]{x1D703}}))^{2}(15/(\unicode[STIX]{x1D708}\unicode[STIX]{x1D716}_{bulk}))^{1/2}$ , where $\unicode[STIX]{x1D70E}_{bulk}(u_{\unicode[STIX]{x1D703}})$ is the bulk-averaged standard deviation of the azimuthal velocity, $\unicode[STIX]{x1D716}_{bulk}$ is the bulk-averaged local dissipation rate and $\unicode[STIX]{x1D708}$ is the liquid kinematic viscosity. The data are obtained through flow velocity field measurements using particle image velocimetry. We estimate the value of the local dissipation rate $\unicode[STIX]{x1D716}(r)$ using the scaling of the second-order velocity structure functions in the longitudinal and transverse directions within the inertial range – without invoking Taylor’s hypothesis. We find an effective scaling of $\unicode[STIX]{x1D716}_{\mathit{bulk}}/(\unicode[STIX]{x1D708}^{3}d^{-4})\sim \mathit{Ta}^{1.40}$ , (corresponding to $\mathit{Nu}_{\unicode[STIX]{x1D714},\mathit{bulk}}\sim \mathit{Ta}^{0.40}$ for the dimensionless local angular velocity transfer), which is nearly the same as for the global energy dissipation rate obtained from both torque measurements ( $\mathit{Nu}_{\unicode[STIX]{x1D714}}\sim \mathit{Ta}^{0.40}$ ) and direct numerical simulations ( $\mathit{Nu}_{\unicode[STIX]{x1D714}}\sim \mathit{Ta}^{0.38}$ ). The resulting Kolmogorov length scale is then found to scale as $\unicode[STIX]{x1D702}_{\mathit{bulk}}/d\sim \mathit{Ta}^{-0.35}$ and the turbulence intensity as $I_{\unicode[STIX]{x1D703},\mathit{bulk}}\sim \mathit{Ta}^{-0.061}$ . With both the local dissipation rate and the local fluctuations available we finally find that the Taylor–Reynolds number effectively scales as $\mathit{Re}_{\unicode[STIX]{x1D706},\mathit{bulk}}\sim \mathit{Ta}^{0.18}$ in the present parameter regime of $4.0\times 10^{8}<\mathit{Ta}<9.0\times 10^{10}$ .
The short- and long-time equilibrium transport properties of a hydrodynamically interacting suspension confined by a spherical cavity are studied via Stokesian dynamics simulations for a wide range of particle-to-cavity size ratios and particle concentrations. Many-body hydrodynamic and lubrication interactions between particles and with the cavity are accounted for utilizing recently developed mobility and resistance tensors for spherically confined suspensions (Aponte-Rivera & Zia, Phys. Rev. Fluids, vol. 1(2), 2016, 023301). Study of particle volume fractions in the range $0.05\leqslant \unicode[STIX]{x1D719}\leqslant 0.40$ reveals that confinement exerts a qualitative influence on particle diffusion. First, the mean-square displacement over all time scales depends on the position in the cavity. Additionally, at short times, the diffusivity is anisotropic, with diffusion along the cavity radius slower than diffusion tangential to the cavity wall, due to the anisotropy of hydrodynamic coupling and to confinement-induced spatial heterogeneity in particle concentration. The mean-square displacement is anisotropic at intermediate times as well and, surprisingly, exhibits superdiffusive and subdiffusive behaviours for motion along and perpendicular to the cavity radius respectively, depending on the suspension volume fraction and the particle-to-cavity size ratio. No long-time self-diffusive regime exists; instead, the mean-square displacement reaches a long-time plateau, a result of entropic restriction to a finite volume. In this long-time limit, the higher the volume fraction is, the longer the particles take to reach the long-time plateau, as cooperative rearrangements are required as the cavity becomes crowded. The ordered dynamical heterogeneity seen here promotes self-organization of particles based on their size and self-mobility, which may be of particular relevance in biophysical systems.
Grad’s method of moments is employed to develop higher-order Grad moment equations – up to the first 26 moments – for dilute granular gases within the framework of the (inelastic) Boltzmann equation. The homogeneous cooling state of a freely cooling granular gas is investigated with the Grad 26-moment equations in a semi-linearized setting and it is shown that the granular temperature in the homogeneous cooling state still decays according to Haff’s law while the other higher-order moments decay on a faster time scale. The nonlinear terms of the fully contracted fourth moment are also considered and, by exploiting the stability analysis of fixed points, it is shown that these nonlinear terms have a negligible effect on Haff’s law. Furthermore, an even larger Grad moment system, which includes the fully contracted sixth moment, is also scrutinized and the stability analysis of fixed points is again exploited to conclude that even the inclusion of the scalar sixth-order moment into the Grad moment system has a negligible effect on Haff’s law. The constitutive relations for the stress and heat flux (i.e. the Navier–Stokes and Fourier relations) are derived through the Grad 26-moment equations and compared with those obtained via the Chapman–Enskog expansion and via computer simulations. The linear stability of the homogeneous cooling state is analysed through the Grad 26-moment system and various subsystems by decomposing them into longitudinal and transverse systems. It is found that one eigenmode in both longitudinal and transverse systems in the case of inelastic gases is unstable. By comparing the eigenmodes from various theories, it is established that the 13-moment eigenmode theory predicts that the unstable heat mode of the longitudinal system remains unstable for all wavenumbers below a certain coefficient of restitution, while any other higher-order moment theory shows that this mode becomes stable above some critical wavenumber for all values of the coefficient of restitution. In particular, the Grad 26-moment theory leads to a smooth profile for the critical wavenumber, in contrast to the other considered theories. Furthermore, the critical system size obtained through the Grad 26-moment theory is in excellent agreement with that obtained through existing theories.
This paper deals with the hydrodynamics of a viscous liquid passing through the hole in a deflating hollow sphere. I employ the method of complementary integrals and calculate in closed form the pressure and streamfunction for the axisymmetric, creeping motion coming from changes in radius. The resulting flow fields describe the motion of a deformable spherical cap in a viscous environment, where the deformations include changes in the size of the spherical cap, the size of the hole and translation of the body along the axis of symmetry. The calculations yield explicit expressions for the jumps in pressure and resistance coefficients for the combined deformations. The equation for the translation force shows that a freely suspended spherical cap is able to propel as an active swimmer. The expression for pressure contains the classic Sampson flow rate equation as a limiting case, but simulations show that the pressure must also account for the velocity of hole widening to correctly predict outflow rates in physiology. Movies based on the closed-form solutions visualize the flow fields and pressures as part of physical processes.