PhD Seminar Session: Fluids and Waves

Hessah Almaaz; Jie Yang; Tanisha Kumari (University of Glasgow)

Thursday 29th October 14:00-15:00
Maths 311B

Abstract

Hessah Almaaz: Pulsatile fluid flow through axisymmetric porous membranes

In a range of physiological and industrial systems, particles are transported and filtered through permeable membranes. The net filtration depends on the form of the fluid delivery, including whether the flow is steady or pulsatile. In this study, we investigate the influence of fluid pulsation on particle transport through porous hollow fibres, which are typically long, thin, and axisymmetric. Although pulsatile flow has been suggested to enhance filtration efficiency, the mechanisms underlying it remain poorly understood.

To bridge this gap, we develop a fluid-mechanical model for pulsatile viscous flow in an axisymmetric porous fibre, explicitly accounting for inertial effects following rapid temporal variations in the fluid flux. The membrane is modelled as a homogeneous porous medium in terms of a depth-averaged version of Darcy’s law. We derive an evolution equation for the fluid flux by balancing viscous and inertial forces, which we analyse using a combination of numerical computation and asymptotic methods to assess the impact of inertial effects. Our theoretical framework is readily extendable to more complex geometries relevant to practical applications and gives insight into the extent to which fluid pulsation offers an efficient filtration mechanism

Jie Yang: Resonant triad interactions of gravity waves in variable-depth rectangular tanks

Controlling the side-to-side sloshing motion of a liquid in a container is crucial in many industrial processes, from transporting liquid cargo to managing aircraft fuel. Pronounced sloshing, however, may be sustained by two forms of resonant interaction: external resonances, induced by pitching or shaking the container at a natural frequency, or internal triad resonances, for which three wave modes correlated in space and time continually exchange energy via nonlinear interactions. Such three-wave spatiotemporal correlations are impossible for gravity waves in a rectangular tank of uniform depth, yet may be triggered at specific depths when the side walls are perturbed and the wave modes distorted. We demonstrate herein that smooth spatial variations in depth may likewise distort sloshing modes and excite a new family of triads in a rectangular tank. Our asymptotic analysis and numerical computation reveal that topography-induced triads may form either from three uncorrelated wave modes at uniform depth or from three correlated wave modes in the small-depth limit. The strength of the triad interaction depends strongly on the bottom profile, with triads generally being strongest for basins with shallow beaches and weakest for small-amplitude topographies. Our study highlights a new mechanism for exciting internal resonances in variable-depth, sloshing-prone basins, such as cargo tanks and harbours, and provides insight into practical resonance mitigation strategies.

Tanisha Kumari: The fluid mechanics of grounding zone wedges

We examine the deposition of a thin layer of viscous fluid beneath another as both fluids intrude towards an inviscid ocean, motivated by the formation of sedimentary grounding zone wedges beneath marine ice sheets. Over large time and length scales, such as those of the Antarctic ice sheet, both ice and the subglacial sediment, or till, over which it flows, exhibit fluid-like behaviour. We use this analogy to employ lubrication theory to model both ice and till as thin films of viscous fluid that spread over rigid bedrock and into an inviscid ocean. Upstream, the ice sheet is in contact with the bedrock. As it flows towards the ocean, the ice sheet detaches from the bedrock at the grounding line, beyond which it feeds into a freely floating ice shelf. We assume that vertical shear stress provides the dominant resistance to the flow of the ice and till and viscous extensional stress provides the dominant resistance to the flow of the shelf. While the rheology of ice and till is non-Newtonian, much of the underlying physical principles can be understood using a Newtonian rheology, which is where we begin whilst retaining the essential viscosplastic properties of till downstream of the grounding line. We demonstrate that a wedge necessarily forms at the grounding line and use a combination of asymptotic analysis and numerical simulations to characterise it across a range of key dimensionless parameter values.

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