Phase-field model for Hele-Shaw flows with arbitrary viscosity contrast. I. Theoretical approach

We present a phase-field model for the dynamics of the interface between two inmiscible fluids with arbitrary viscosity contrast in a rectangular Hele-Shaw cell. With asymptotic matching techniques we check the model to yield the right Hele-Shaw equations in the sharp-interface limit and compute the corrections to these equations to first order in the interface thickness. We also compute the effect of such corrections on the linear dispersion relation of the planar interface. We discuss in detail the conditions on the interface thickness to control the accuracy and convergence of the phase-field model to the limiting Hele-Shaw dynamics. In particular, the convergence appears to be slower for high viscosity contrasts.Comment: 17 pages in revtex. changes: 1 reference adde

Derivation of a Hele-Shaw type system from a cell model with active motion

We formulate a Hele-Shaw type free boundary problem for a tumor growing under the combined effects of pressure forces, cell multiplication and active motion, the latter being the novelty of the present paper. This new ingredient is considered here as a standard diffusion process. The free boundary model is derived from a description at the cell level using the asymptotic of a stiff pressure limit. Compared to the case when active motion is neglected, the pressure satisfies the same complementarity Hele-Shaw type formula. However, the cell density is smoother (Lipschitz continuous), while there is a deep change in the free boundary velocity, which is no longer given by the gradient of the pressure, because some kind of 'mushy region' prepares the tumor invasion

Energy balance simulation of a wheat canopy using the RZ-SHAW (RZWQM-SHAW) model

RZ-SHAW is a new hybrid model coupling the Root Zone Water Quality Model (RZWQM) and the Simultaneous Heat and Water (SHAW) model to extend RZWQM applications to conditions of frozen soil and crop residue cover. RZ-SHAW offers the comprehensive land management options of RZWQM with the additional capability to simulate diurnal changes in energy balance needed for simulating the near-surface microclimate and leaf temperature. The objective of this study was to evaluate RZ-SHAW for simulations of radiation balance and sensible and latent heat fluxes over plant canopies. Canopy energy balance data were collected at various growing stages of winter wheat in the North China Plain (36° 57'N, 116° 6'E, 28 m above sea level). RZ-SHAW and SHAW simulations using hourly meteorological data were compared with measured net radiation, latent heat flux, sensible heat flux, and soil heat flux. RZ-SHAW provided similar goodness-of-prediction statistics as the original SHAW model for all the energy balance components when using observed plant growth input data. The root mean square error (RMSE) for simulated net radiation, latent heat, sensible heat, and soil heat fluxes was 29.7, 30.7, 29.9, and 25.9 W m -2 for SHAW and 30.6, 32.9, 34.2, and 30.6 W m -2 for RZ-SHAW, respectively. Nash-Sutcliffe R 2 ranged from 0.67 for sensible heat flux to 0.98 for net radiation. Subsequently, an analysis was performed using the plant growth component of RZ-SHAW instead of inputting LAI and plant height. The model simulation results agreed with measured plant height, yield, and LAI very well. As a result, RMSE for the energy balance components were very similar to the original RZ-SHAW simulation, and latent, sensible, and soil heat fluxes were actually simulated slightly better. RMSE for simulated net radiation, latent heat, sensible heat, and soil heat fluxes was 31.5, 30.4, 30.2, and 27.6 W m -2, respectively. Overall, the results demonstrated a successful coupling of RZWQM and SHAW in terms of canopy energy balance simulation, which has important implications for prediction of crop growth, crop water stress, and irrigation scheduling

A phase-field model of Hele-Shaw flows in the high viscosity contrast regime

A one-sided phase-field model is proposed to study the dynamics of unstable interfaces of Hele-Shaw flows in the high viscosity contrast regime. The corresponding macroscopic equations are obtained by means of an asymptotic expansion from the phase-field model. Numerical integrations of the phase-field model in a rectangular Hele-Shaw cell reproduce finger competition with the final evolution to a steady state finger the width of which goes to one half of the channel width as the velocity increases

The Hele-Shaw asymptotics for mechanical models of tumor growth

Models of tumor growth, now commonly used, present several levels of complexity, both in terms of the biomedical ingredients and the mathematical description. The simplest ones contain competition for space using purely fluid mechanical concepts. Another possible ingredient is the supply of nutrients through vasculature. The models can describe the tissue either at the level of cell densities, or at the scale of the solid tumor, in this latter case by means of a free boundary problem. Our first goal here is to formulate a free boundary model of Hele-Shaw type, a variant including growth terms, starting from the description at the cell level and passing to a certain limit. A detailed mathematical analysis of this purely mechanical model is performed. Indeed, we are able to prove strong convergence in passing to the limit, with various uniform gradient estimates; we also prove uniqueness for the asymptotic Hele-Shaw type problem. The main tools are nonlinear regularizing effects for certain porous medium type equations, regularization techniques \`a la Steklov, and a Hilbert duality method for uniqueness. At variance with the classical Hele-Shaw problem, here the geometric motion governed by the pressure is not sufficient to completely describe the dynamics. A complete description requires the equation on the cell number density. Using this theory as a basis, we go on to consider the more complex model including nutrients. We obtain the equation for the limit of the coupled system; the method relies on some BV bounds and space/time a priori estimates. Here, new technical difficulties appear, and they reduce the generality of the results in terms of the initial data. Finally, we prove uniqueness for the system, a main mathematical difficulty.Comment: 34 pages, 3 figure