Mathematical Institute (University of Oxford)

Michael Negus

Modelling high-speed droplet impact onto an elastic membrane

The impact of a high-speed droplet onto an elastic membrane is a highly nonlinear process and poses a formidable modelling challenge due to both the multi-scale nature of the flow and the fluid-structure interaction between the droplet and the membrane. We present two modelling approaches for droplet impact onto elastic membranes: matched asymptotics and direct numerical simulations (DNS). Inviscid Wagner theory is used in the former to derive analytical expressions which approximate the behaviour of the droplet during the early stages of impact, while the DNS builds on the open-source volume-of-fluid code Basilisk. We demonstrate the strong influence that the thickness, tension and stiffness of the membrane have on the dynamics of the droplet and the membrane. We also quantitatively show that the speed the droplet spreads across the substrate is notably decreased when the membrane is more compliant, which is consistent with experimental findings that splashing can be inhibited by impacting onto a soft substrate. We conclude by showing how these methods are complementary, as a combination of both can lead to a thorough understanding of the droplet impact across timescales.

Helen Saville

Lubrication model of a valve-controlled, gravity-driven bioreactor

Hospitals sometimes experience shortages of donor blood platelet supplies, motivating research into in vitro production of platelets. We model a novel platelet bioreactor described in Shepherd et al. [1]. The bioreactor consists of an upper channel, a lower channel, and a cell-seeded porous collagen scaffold situated between the two. Flow is driven by gravity, and controlled by valves on the four inlets and outlets. The bioreactor is long relative to its width, a feature which we exploit to derive a lubrication reduction of Navier-Stokes flow coupled to Darcy. Models for two cases are considered: small amplitude valve oscillations, and order one amplitude valve oscillations. The former model is a systematic reduction; the latter incorporates a phenomenological approximation for the cross-sectional flow profile. As the shear stress experienced by cells influences platelet production, we use our model to quantify the effect of valve dynamics on shear stress.

1: Shepherd, J.H., Howard, D., Waller, A.K., Foster, H.R., Mueller, A., Moreau, T., Evans, A.L., Arumugam, M., Chalon, G.B., Vriend, E. and Davidenko, N., Biomaterials, 182, pp.135-144. (2018)

Several problems of great importance in the study of glaciers and ice sheets involve processes of attachment and reattachment of the ice from the bedrock. Consider, for example, an ice sheet sliding from the continent into the ocean, where it goes afloat. Another example is that of subglacial cavitation, a fundamental mechanism in glacial sliding where the ice detaches from the bedrock along the downstream area of an obstacle. Such problems are generally modelled as a viscous Stokes flow with a free boundary and contact boundary conditions. In this talk, I will present a framework for solving such problems numerically. I will start by introducing the mathematical formulation of these viscous contact problems and the challenges that arise when trying to approximate them numerically. I will then show how, given a stable scheme for the free boundary equation, one can build a penalty formulation for the viscous contact problem in such a way that the resulting algorithm remains stable and robust.

Organoids are three–dimensional multicellular tissue constructs. When cultured in vitro, they recapitulate the structure, heterogeneity, and function of their in vivo counterparts. As awareness of the multiple uses of organoids has grown, e.g. in drug discovery and personalised medicine, demand has increased for low–cost and efficient methods of producing them in a reproducible manner and at scale. We are working in collaboration with the biotechnology company Cellesce, who develop bioprocessing systems for the expansion of organoids at scale. Part of their technology includes a bioreactor, which utilises flow of culture media to enhance nutrient delivery to the organoids and facilitate the removal of waste metabolites. A key priority is ensuring uniformity in organoid size and reproducibility; qualities that depends on the bioreactor design and operating conditions. A complete understanding of the system requires knowledge of the spatial and temporal information regarding flow and the resulting oxygen and metabolite concentrations throughout the bioreactor. However, it is impractical to obtain this data empirically, due to the highly–controlled environment of the bioreactor posing difficulties for online real–time monitoring of the system. Thus, we exploit a mathematical modelling approach, to provide spatial as well as temporal information.

In the bioreactor, organoids are seeded as single cells in a layer of hydrogel. We present a general model for the nutrient and waste metabolite concentrations in the hydrogel and organoid regions of the bioreactor. Resolving for the millions of organoids within the hydrogel is computationally expensive and infeasible. Hence, we take a mathematical homogenisation approach to understand how the behaviour of the organoids on the microscale influences the macroscale behaviour in the hydrogel layer. We consider the case of growing organoids, with a temporally and spatially dependent radii, and exploit the separation of scales to systematically derive an effective macroscale model for metabolite transport. We explore some canonical problems to understand our homogenised system.

Many problems in engineering can be understood as controlling the bifurcation structure of a given device. For example, one may wish to delay the onset of instability, or bring forward a bifurcation to enable rapid switching between states. In this talk, we will describe a numerical technique for controlling the bifurcation diagram of a nonlinear partial differential equation by varying the shape of the domain. Our aim is to delay or advance a given branch point to a target parameter value. The algorithm consists of solving a shape optimization problem constrained by an augmented system of equations, called the Moore–Spence system, that characterize the location of the branch points. We will demonstrate the effectiveness of this technique on several numerical experiments on the Allen–Cahn, Navier–Stokes, and hyperelasticity equations.