Shear dilatation/contraction of granular materials has long been recognized as an important process in granular flows but a comprehensive theoretical description of this process for a wide range of shear rates is not yet available. In this paper, a theoretical formulation of dilatation/contraction is proposed for continuum modelling of granular flows, in which the dilatation/contraction effects consist of a frictional component, which results from the rearrangement of enduring-contact force chains among particles, and a collisional component, which arises from inter-grain collisions. In this formulation, a frictional solid pressure, which considers the rearrangement of contact force chains under shear deformation, is proposed for the frictional dilatation/contraction, while well-established rheological laws are adopted for the collisional inter-grain pressure to account for the collisional dilatancy effect. The proposed formulation is first verified analytically by describing the shear-weakening behaviour of granular samples in a torsional shear rheometer and by capturing the incipient failure of both dry and immersed granular slopes. The proposed dilatation/contraction formulation is then further validated numerically by integrating it into a two-fluid continuum model and applying the model to study the collapse of submerged granular columns, in which the dilatation/contraction plays a critical role.

The evolution of the interfacial instabilities on a smooth and laminar liquid sheet formed by the oblique impingement of two liquid jets is investigated in the presence of external acoustic forcing. The forced liquid sheet exhibits different instability patterns, such as flapping flag, sinuous breakup and vibrating membrane structure, depending on the forcing frequency. The transition frequencies, where the instability pattern changes, are obtained using the linear theory of sheet instability and are shown to be increasing functions of the jet Weber number. These different instability patterns show distinct growth rate behaviours, which cannot be predicted using a unique theory such as the conventional linear theory of sheet breakup. The dynamics of liquid sheet breakup in the presence of forcing is further assessed with the help of established instability theories. Our measurements show that the sheet thickness distribution plays a major role in dictating the relative influence of different mechanisms on the instability characteristics. All the results are finally consolidated to arrive at the instability regime map on the Weber number versus frequency ($We$–$\tilde {\omega }$) space to serve as a guideline for adopting a suitable modelling approach.

We investigate the steady-state harmonic resonance of periodic interfacial gravity waves in a two-layer fluid with free surface. Two independent ‘external’ and ‘internal’ modes with separate linear dispersion relationships exist for this two-layer fluid. Exact harmonic resonance occurs when an external mode and an internal mode share the same phase speed and have an integer ratio of wavelengths. The singularity or small divisor caused by the exactly or nearly resonant component is successfully removed by the homotopy analysis method (HAM). Convergent series solutions are obtained of steady-state interfacial wave groups with harmonic resonance. It is found that steady-state resonant waves form a continuum in parameter space. For finite amplitude interfacial waves, the energy carried by surface waves mirrors that carried by interface waves as the water depth varies. As the upper layer depth increases, energy carried by both surface and interface waves transfers from the shorter resonant component to the longer primary one. The paper utilizes a HAM-based analytical approach to obtain a steady-state, periodic, interfacial wave system with exact- and near-resonant interactions between internal and external modes.

We consider nonlinear, directionally spread irregular wave fields in deep water and study the statistical properties of the total inline force that would be induced by the waves on a vertical circular cylinder. Starting from the two-dimensional Morison equation, we specifically investigate the effect of wave steepness and directionality on the probability density functions (PDFs) of the inertia and drag forces. To do so, we derive new analytical expressions for the PDFs of these forces based on first-order theory and compare them with the results of fully nonlinear numerical simulations. We show that the inertia force for the main direction ($x$) of the wave field is in general unaffected by nonlinear effects, while the inertia force for the direction perpendicular to the main direction ($y$) is subject to substantial third-order effects when the steepness is appreciable and the wave field becomes relatively long crested. Moreover, we show that the drag force for the $x$-direction is in general subject to substantial second-order effects. The drag force for the $y$-direction is also affected by second-order effects, but to a much smaller degree than the $x$-direction. It is, however, strongly affected by third-order effects under the same conditions as the inertia force in the $y$-direction. We conclude that the total force can be accurately approximated by first-order theory when the ratio $k_p D/\varepsilon$ is large (with $k_p$ the peak wavenumber, $D$ the cylinder's diameter and $\varepsilon$ the wave steepness), while first-order theory underestimates the probability of large forces considerably when this ratio is small.

The natural frequencies for sloshing without coupling with lateral tank motions differ from the natural sloshing frequencies with coupling. A consequence is that the nonlinear multimodal sloshing theory for prescribed tank motion should be revised when studying the liquid sloshing dynamics in connection with marine structures in ocean waves. The needed revisions are done for a rectangular rigid tank with finite liquid depth. Steady-state resonant solutions of the constructed nonlinear modal equations are derived to analytically describe the coupled resonant sloshing and sway of a floating rigid body in regular incident deep water waves with two-dimensional flow conditions at the lowest coupled sloshing–sway natural frequency. The steady-state theoretical results are validated by comparing them with the model tests by Rognebakke & Faltinsen (J. Ship Res., vol. 47, issue 3, 2003, pp. 208–221). The occurrence of an instability frequency range is theoretically justified.