We present here an in-depth analysis of particle levitation stability and the role of the radial and axial forces exerted on fixed spherical and ellipsoidal particles levitated in an axisymmetric acoustic levitator, over a wide range of particle sizes and surrounding medium viscosities. We show that the stability behaviour of a levitated particle in an axisymmetric levitator is unequivocally connected to the radial forces: the loss of levitation stability is always due to the change of the radial force sign from positive to negative. It is found that the axial force exerted on a sphere of radius increases with increasing viscosity for ( is the acoustic wavelength), with the viscous contribution of this force scaling with the inverse of the sphere radius. The axial force decreases with increasing viscosity for spheres with . The radial force, on the other hand, decreases monotonically with increasing viscosity. The radial and axial forces exerted on an ellipsoidal particle are larger than those exerted on a volume-equivalent sphere, up to the point where the ellipsoid starts to act as an obstacle to the formation of the standing wave in the levitator chamber.

The energy decay of vortices in viscous fluids governed by the compressible Navier–Stokes equations is investigated. It is shown that the main reason for the slow decay is that zero eigenvalues exist in the matrix related to the dissipative terms. The theoretical analysis is purely mathematical and based on the energy method. To check the validity of the theoretical result in practice, numerical solutions to the Navier–Stokes equations are computed using a stable high-order finite difference method. The numerical computations corroborate the theoretical conclusion.

We investigate, through both asymptotic and numerical analysis, precessionally driven flow of a homogeneous fluid confined in a fluid-filled circular cylinder that rotates rapidly about its symmetry axis and precesses slowly about a different axis that is fixed in space. After demonstrating that the inviscid approximation is always divergent even far away from resonance, we derive a general asymptotic solution for an asymptotically small Ekman number in the rotating frame of reference describing the weakly precessing flow that satisfies the no-slip boundary condition and that is valid at or near or away from resonance. Numerical analysis of the same problem using the Galerkin method in terms of a Chebyshev polynomial expansion is also carried out, showing satisfactory agreement between the general asymptotic solution and the corresponding numerical solution at or near or away from resonance.

Irregular conduits, complex surfaces, and porous media often manifest more than one geometric wetting condition for spontaneous capillary flows. As a result, different regions of the flow exhibit different rates of flow, all the while sharing common dynamical capillary pressure boundary conditions. The classic problem of sudden capillary rise in tubes with interior corners is revisited from this perspective and solved numerically in the self-similar visco-capillary limit à la Lucas–Washburn. Useful closed-form analytical solutions are obtained in asymptotic limits appropriate for many practical flows in conduits containing one or more interior corner. The critically wetted corners imbibe fluid away from the bulk capillary rise, shortening the viscous column length and slightly increasing the overall flow rate. The extent of the corner flow is small for many closed conduits, but becomes significant for flows along open channels and the method is extended to approximate hemiwicking flows across triangular grooved surfaces. It is shown that an accurate application of the method depends on an accurate a priori assessment of the competing viscous cross-section length scales, and the expedient Laplacian scaling method is applied herein toward this effect.

The effect of the body on the lift force in hovering flight is studied here by including the effect of image vortex rings (IVRs) in the inviscid vortex ring model proposed by Rayner (J. Fluid Mech., vol. 91, 1979, pp. 697–730) and used by Wang & Wu (J. Fluid Mech., vol. 654, 2010, pp. 453–472) to study lift force due to wakes. The body is treated simply as an equivalent sphere following the data of Ellington (Phil. Trans. R. Soc. Lond. B, vol. 305, 1984a, pp. 17–40). It is shown that the body image reduces the lift by inducing a further downwash near the wing tip and an additional contraction to the real vortex rings (RVRs). The amount of force reduction due to body image is found to grow cubically with relative body size, defined by the equivalent radius relative to the wing span, and approximately linearly with the feathering parameter. For Apis and Bombus with large relative body size and large feathering parameter, the body images reduce lift by an amount near 8 % according to the present simplified analysis.

Following the Kolmogorov technique, an exact relation for a vector third-order moment is derived for three-dimensional incompressible stably stratified turbulence under the Boussinesq approximation. In the limit of a small Brunt–Väisälä frequency, isotropy may be assumed which allows us to find a generalized -law. For strong stratification, we make the ansatz that is directed along axisymmetric surfaces parameterized by a scaling law relating horizontal and vertical coordinates. An integration of the exact relation under this hypothesis leads to a generalized Kolmogorov law which depends on the intensity of anisotropy parameterized by a single coefficient. By using a scaling relation between large horizontal and vertical length scales we fix this coefficient and propose a unique law.