A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis

We derive a Cahn–Hilliard–Darcy model to describe multiphase tumour growth taking interactions with multiple chemical species into account as well as the simultaneous occurrence of proliferating, quiescent and necrotic regions. A multitude of phenomena such as nutrient diffusion and consumption, angiogenesis, hypoxia, blood vessel growth, and inhibition by toxic agents, which are released for example by the necrotic cells, are included. A new feature of the modelling approach is that a volume-averaged velocity is used, which dramatically simplifies the resulting equations. With the help of formally matched asymptotic analysis we develop new sharp interface models. Finite element numerical computations are performed and in particular the effects of necrosis on tumour growth are investigated numerically. In particular, for certain modelling choices, we obtain some form of focal and patchy necrotic growth that have been observed in experiments.

On thin evaporating drops: When is the -law valid?

We study the evolution of a thin, axisymmetric, partially wetting drop as it evaporates. The effects of viscous dissipation, capillarity, slip and diffusion-dominated vapour transport are taken into account. A matched asymptotic analysis in the limit of small slip is used to derive a generalization of Tanner’s law that takes account of the effect of mass transfer. We find a criterion for when the contact-set radius close to extinction evolves as the square root of the time remaining until extinction – the famous $d^{2}$-law. However, for a sufficiently large rate of evaporation, our analysis predicts that a (slightly different) ‘$d^{13/7}$-law’ is more appropriate. Our asymptotic results are validated by comparison with numerical simulations.

On contact-line dynamics with mass transfer

We investigate the effect of mass transfer on the evolution of a thin, two-dimensional, partially wetting drop. While the effects of viscous dissipation, capillarity, slip and uniform mass transfer are taken into account, other effects, such as gravity, surface tension gradients, vapour transport and heat transport, are neglected in favour of mathematical tractability. Our focus is on a matched-asymptotic analysis in the small-slip limit, which reveals that the leading-order outer formulation and contact-line law depend delicately on both the sign and the size of the mass transfer flux. This leads, in particular, to novel generalisations of Tanner's law. We analyse the resulting evolution of the drop on the timescale of mass transfer and validate the leading-order predictions by comparison with preliminary numerical simulations. Finally, we outline the generalisation of the leading-order formulations to prescribed non-uniform rates of mass transfer and to three dimensions.

Emergent parabolic scaling of nano-faceting crystal growth

Nano-faceted crystals answer the call for self-assembled, physico-chemically tailored materials, with those arising from a kinetically mediated response to free-energy disequilibria ( thermokinetics ) holding the greatest promise. The dynamics of slightly undercooled crystal–melt interfaces possessing strongly anisotropic and curvature-dependent surface energy and evolving under attachment–detachment limited kinetics offer a model system for the study of thermokinetic effects. The fundamental non-equilibrium feature of this dynamics is explicated through our discovery of one-dimensional convex and concave translating fronts ( solitons ) whose constant asymptotic angles provably deviate from the thermodynamically expected Wulff angles in direct proportion to the degree of undercooling. These thermokinetic solitons induce a novel emergent facet dynamics, which is exactly characterized via an original geometric matched-asymptotic analysis. We thereby discover an emergent parabolic symmetry of its coarsening facet ensembles, which naturally implies the universal scaling law L ∼ t 1 / 2 for the growth in time t of the characteristic length L .

Second-order Wagner theory for two-dimensional water-entry problems at small deadrise angles

The theory of Wagner from 1932 for the normal symmetric impact of a two-dimensional body of small deadrise angle on a half-space of ideal and incompressible liquid is extended to derive the second-order corrections for the locations of the higher-pressure jet-root regions and for the upward force on the impactor using a systematic matched-asymptotic analysis. The second-order predictions for the upward force on an entering wedge and parabola are compared with numerical and experimental data, respectively, and it is concluded that a significant improvement in the predictive capability of Wagner's theory is afforded by proceeding to second order.

Drop fall-off from pendent rivulets

Drops fall off a viscous pendent rivulet on the underside of a plane when the inclination angle θ, measured with respect to the horizontal, is below a critical value θc. We estimate this θc by studying the existence of finite-amplitude drop solutions to a long-wave lubrication equation. Through a partial matched asymptotic analysis, we establish that fall-off occurs by two distinct mechanisms. For θ>ϕ, where ϕ is the static contact angle, a jet mechanism results when a mean-flow steepening effect cannot provide sufficient axial curvature to counter gravity. This fall-off mechanism occurs if the rivulet width B, which is normalized with respect to the capillary length H=(σ/ρg cosθ)1/2, exceeds a critical value defined by β=−cosB>1/4. For θ<ϕ, the normal azimuthal curvature is the dominant force against fall-off and the azimuthal capillary force. The corresponding critical condition is found to be 1.5β1/6>tanθ/tanϕ. Both criteria are in good agreement with our experimental data.

Initial motion of a viscous vortex ring

The behaviour of a viscous vortex ring is examined by a matched asymptotic analysis up to three orders. This study aims at investigating how much the location of maximum vorticity deviates from the centroid of the vortex ring, defined by P. G. Saffman (1970). All the results are presented in dimensionless form, as indicated in the following context. Let Γ be the initial circulation of the vortex ring, and R denote the ring radius normalized by its initial radius R i . For the asymptotic analysis, a small parameter ∊ = ( t / Re ) ½ is introduced, where t denotes time normalized by R 2 i / Γ , and Re = Γ/v is the Reynolds number defined with Γ and the kinematic viscosity v . Our analysis shows that the trajectory of maximum vorticity moves with the velocity (normalized by Γ/R i ) U m = – 1/4π R {ln 4 R /∊ + H m } + O (∊ ln ∊), where H m = H m ( Re, t ) depends on the Reynolds number Re and may change slightly with time t for the initial motion. For the centroid of the vortex ring, we obtain the velocity U c by merely replacing H m by H c , which is a constant –0.558 for all values of the Reynolds number. Only in the limit of Re → ∞, the values of H m and H c are found to coincide with each other, while the deviation of H m from the constant H c is getting significant with decreasing the Reynolds number. Also of interest is that the radial motion is shown to exist for the trajectory of maximum vorticity at finite Reynolds numbers. Furthermore, the present analysis clarifies the earlier discrepancy between Saffman’s result and that obtained by C. Tung and L. Ting (1967).