We calculate the average swimming velocity and dispersion rate characterizing the transport of swimming gyrotactic micro-organisms suspended in homogeneous (simple) shear. These are requisite effective phenomenological coefficients for the macroscale continuum modelling of bioconvection and related collective-dynamics phenomena. The swimming cells are modelled as rigid axisymmetric dipolar particles subject to stochastic Brownian rotations. Calculations are effected via application of the generalized Taylor dispersion scheme. Attention is focused on finite (as opposed to weak) shear. Results indicate that the largest transverse average swimming velocities (essential to gyrotactic focusing) appear shortly after transition from the ‘tumbling’ mode of motion to cells swimming in the equilibrium direction. At sufficiently large shear rates, dispersivity is not monotonically decreasing with external-field intensity. Exceptional dispersion rates which are unique to non-spherical cells appear in the ‘intermediate domain’ of external fields. These are rationalized in terms of the corresponding deterministic problem (i.e. in the absence of diffusion) when cell rotary motion is governed by the simultaneous coexistence of multiple stable attractors.

We analyse the time response of a gas confined in a small-scale gap (of the order of or smaller than the mean free path) to an instantaneous jump in the temperature of its boundaries. The problem is formulated for a collisionless gas in the case where the relative temperature jump at each wall is small and independent of the other. An analytic solution for the probability density function is obtained and the respective hydrodynamic fields are calculated. It is found that the characteristic time scale for arriving at the new equilibrium state is of the order of several acoustic time scales (the ratio of the gap width to the most probable molecular speed of gas molecules). The results are compared with direct Monte Carlo simulations of the Boltzmann equation and good agreement is found for non-dimensional times (scaled by the acoustic time) not exceeding the system Knudsen number. Thus, the present analysis describes the early-time behaviour of systems of arbitrary size and may be useful for prescribing the initial system behaviour in counterpart continuum-limit analyses.

We study the flow-field generated in a one-dimensional wall-bounded gas layer due to an arbitrary small-amplitude time variation in the temperature of its boundaries. Using the Fourier transform technique, analytical results are obtained for the slip-flow/Navier-Stokes limit. These results are complemented by low-variance simulations of the Boltzmann equation, which are useful for establishing the limits of the slip-flow description, as well as for bridging the gap between the slip-flow analysis and previously developed free-molecular analytical predictions. Results are presented for both periodic ͑sinusoidal͒ and nonperiodic ͑step-jump͒ heating profiles. Our slip-flow solution is used to elucidate a singular limit reported in the literature for oscillatory heating of a dynamically incompressible fluid.

The transition to convection in the Rayleigh-Bénard problem at small Knudsen numbers is studied via a linear temporal stability analysis of the compressible "slip-flow" problem. No restrictions are imposed on the magnitudes of temperature difference and compressibility-induced density variations. The dispersion relation is calculated by means of a Chebyshev collocation method. The results indicate that occurrence of instability is limited to small Knudsen numbers ͑KnՇ 0.03͒ as a result of the combination of the variation with temperature of fluid properties and compressibility effects. Comparison with existing direct simulation Monte Carlo and continuum nonlinear simulations of the corresponding initial-value problem demonstrates that the present results correctly predict the boundaries of the convection domain. The linear analysis thus presents a useful alternative in studying the effects of various parameters on the onset of convection, particularly in the limit of arbitrarily small Knudsen numbers.

We consider the response of a gas in a microchannel to instantaneous (small-amplitude) non-periodic motion of its boundaries in the normal direction. The problem is formulated for an ideal monatomic gas using the Bhatnagar, Gross, and Krook (BGK) kinetic model, and solved for the entire range of Knudsen (Kn) numbers. Analysis combines analytical (collisionless and continuum-limit) solutions with numerical (low-variance Monte Carlo and linearized BGK) calculations. Gas flow, driven by motion of the boundaries, consists of a sequence of propagating and reflected pressure waves, decaying in time towards a final equilibrium state. Gas rarefaction is shown to have a “damping effect” on equilibration process, with the time required for equilibrium shortening with increasing Kn. Oscillations in hydrodynamic quantities, characterizing gas response in the continuum limit, vanish in collisionless conditions. The effect of having two moving boundaries, compared to only one considered in previous studies of time-periodic systems, is investigated. Comparison between analytical and numerical solutions indicates that the collisionless description predicts the system behavior exceptionally well for all systems of the size of the mean free path and somewhat larger, in cases where boundary actuation acts along times shorter than the ballistic time scale. The continuum-limit solution, however, should be considered with care at early times near the location of acoustic wavefronts, where relatively sharp flow-field variations result in effective increase in the value of local Knudsen number.

Existing studies on sound wave propagation in rarefied gases examine sound generation by actuated boundaries subject to isothermal boundary conditions. While these conditions are simple to analyze theoretically, they are more challenging to apply in practice compared to heat-flux conditions. To study the effect of modifying the thermal boundary conditions, the present work investigates the impact of replacing the isothermal with heat-flux conditions on propagation of acoustic waves in a microchannel. The linearized problem is formulated for an ideal hard-sphere gas, and the effect of heat-flux prescription is demonstrated through comparison with counterpart results for isothermal boundaries. Analytical solutions are obtained for a gas at collisionless (highly rarefied) and continuum-limit conditions, and validated through comparison with direct simulation Mote Carlo predictions. Remarkably, it is found that prescription of heat flux at the walls, altering the energy balance within the medium, has a significant effect on acoustic wave propagation in the gas. In particular, when optimized with respect to the boundary acoustic signal applied, the heat flux condition may be used to achieve “acoustic cloaking” of the moving wall, a much desired property in classical acoustics.

We consider the linear temporal stability of a Couette flow of a Maxwell gas within the gap between a rotating inner cylinder and a concentric stationary outer cylinder both maintained at the same temperature. The neutral curve is obtained for arbitrary Mach (Ma) and arbitrarily small Knudsen (Kn) numbers by use of a ‘slip-flow’ continuum model and is verified via comparison to direct simulation Monte Carlo results. At subsonic rotation speeds we find, for the radial ratios considered here, that the neutral curve nearly coincides with the constant-Reynolds-number curve pertaining to the critical value for the onset of instability in the corresponding incompressible-flow problem. With increasing Mach number, transition is deferred to larger Reynolds numbers. It is remarkable that for a fixed Reynolds number, instability is always eventually suppressed beyond some supersonic rotation speed. To clarify this we examine the variation with increasing (Ma) of the reference Couette flow and analyse the narrow-gap limit of the compressible TC problem. The results of these suggest that, as in the incompressible problem, the onset of instability at supersonic speeds is still essentially determined through the balance of inertial and viscous-dissipative effects. Suppression of instability is brought about by increased rates of dissipation associated with the elevated bulk-fluid temperatures occurring at supersonic speeds. A useful approximation is obtained for the neutral curve throughout the entire range of Mach numbers by an adaptation of the familiar incompressible stability criteria with the critical Reynolds (or Taylor) numbers now based on average fluid properties. The narrow-gap analysis further indicates that the resulting approximate neutral curve obtained in the (Ma, Kn) plane consists of two branches: (i) the subsonic part corresponding to a constant ratio (Ma/Kn) (i.e. a constant critical Reynolds number) and (ii) a supersonic branch which at large Ma values corresponds to a constant product Ma Kn. Finally, our analysis helps to resolve some conflicting views in the literature regarding apparently destabilizing compressibility effects.

The prevailing view of the dynamics of flapping flags is that the onset of motion is caused by temporal instability of the initial planar state. This view is re-examined by considering the linearized two-dimensional motion of a flag immersed in a high-Reynolds-number flow and taking account of forcing by a ‘street’ of vortices shed periodically from its cylindrical pole. The zone of nominal instability is determined by analysis of the self-induced motion in the absence of shed vorticity, including the balance between flag inertia, bending rigidity, varying tension and fluid loading. Forced motion is then investigated by separating the flag deflection into ‘vortex-induced’ and ‘self’ components. The former is related directly to the motion that would be generated by the shed vortices if the flag were absent. This component serves as an inhomogeneous forcing term in the equation satisfied by the ‘self’ motion. It is found that forced flapping is possible whenever the Reynolds number based on the pole diameter ReD ≳ 100, such that a wake of distinct vortex structures is established behind the pole. Such conditions typically prevail at mean flow velocities significantly lower than the critical threshold values predicted by the linear theory. It is therefore argued that analyses of the onset of flag motion that are based on ideal, homogeneous flag theory are incomplete and that consideration of the pole-induced fluid flow is essential at all relevant wind speeds.