Interfacial dynamics and pinch-off singularities for axially symmetric Darcy flow

We study a model for the evolution of an axially symmetric bubble of inviscid fluid in a homogeneous porous medium otherwise saturated with a viscous fluid. The model is a moving boundary problem that is a higher-dimensional analogue of Hele-Shaw flow. Here we are concerned with the development of pinch-off singularities characterised by a blow-up of the interface curvature and the bubble subsequently breaking up into two; these singularities do not occur in the corresponding two-dimensional Hele-Shaw problem. By applying a novel numerical scheme based on the level set method, we show that solutions to our problem can undergo pinch-off in various geometries. A similarity analysis suggests that the minimum radius behaves as a power law in time with exponent $\alpha = 1/3$ just before and after pinch-off has occurred, regardless of the initial conditions; our numerical results support this prediction. Further, we apply our numerical scheme to simulate the time-dependent development and translation of axially symmetric Saffman-Taylor fingers and Taylor-Saffman bubbles in a cylindrical tube, highlighting key similarities and differences with the well-studied two-dimensional cases.Comment: 16 pages, 16 figure

Coupling Navier-Stokes and Cahn-Hilliard equations in a two-dimensional annular flow configuration

In this work, we present a novel isogeometric analysis discretization for the Navier-Stokes-Cahn-Hilliard equation, which uses divergence-conforming spaces. Basis functions generated with this method can have higher-order continuity, and allow to directly discretize the higherorder operators present in the equation. The discretization is implemented in PetIGA-MF, a high-performance framework for discrete differential forms. We present solutions in a twodimensional annulus, and model spinodal decomposition under shear flow

The mechanical design aspects of a small diameter vascular prosthesis

Bibliography: pages 81-86.Failure of medium to small diameter vascular grafts is believed to be in part due to the compliance mismatch between the native artery and the implanted graft. Consequently, designers are examining the use of more compliant materials for their manufacture. Ether free polyurethanes are currently amongst the most popular materials for use in biological implants although these materials are inherently too stiff for use in vascular prostheses. These materials can be made more compliant by introducing porosity. Apart from creating a more compliant overall material, under optimal biological conditions, the porosity may lead to cell in growth through the thickness of the graft allowing an endothelial cell layer to form on the inner flow surface. Compliance and cell ingrowth are both important characteristics that determine the successful functioning of the graft. The current work is part of a collaborative venture with the Cardiovascular Research Unit (CVRU) at the University of Cape Town to design and develop a new polyurethane graft. Finite element models are used to facilitate stress analyses and to evaluate the long-term behaviour and compliance of various graft designs made from a bio-inert thermoplastic polyurethane. Material properties of the polyurethane are determined from uniaxial tension tests, simple-shear tests and viscoelastic shear tests. The constitutive equations for a compressible, large strain hyper elastic material model with viscoelasticity are implemented in the finite element code using material constants calculated from the test data. The behaviour of the finite element model is verified by using a single element test and comparing results to the material data. The finite element model is validated for use m more sophisticated problems by comparing axi-symmetric models with in vitro experiments. An artery/graft anastomosis is then analysed by modelling the artery as an incompressible hyperplastic material. Further more complex graft designs are analysed with internal growth channels and spiral reinforcing winds. Viscoelastic effects are also examined. The modelling method is discussed and important results are noted