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Ablative therapies such as radio-frequency (RF) ablation are increasingly used for treatment of tumours in liver and other organs. Often large vessels limit the extent of the thermal lesion, and cancer cells close to the vessel survive resulting in local tumour recurrence. Accurate estimates of the heat convection coefficient h for large vessels will help improve ablation techniques, and are required for estimation of thermal lesion dimensions in simulations. Previous estimates of h did not consider that only part of the vessel is heated, and assumed uniform temperature distribution at the vessel wall. An analytical relationship between the heat convection coefficient, blood velocity and temperature is formulated. The heat convection coefficient evaluated will assist both simulations and design of proper protocols for in vivo measurements. The mathematical model developed in this work describes the exchange of heat between a solid surface and a moving fluid and it is based on energy and motion equations for Navier-Stokes fluids. A particular case of a laminar blood flow in the portal vein is studied when a portion of its surface is heated. The results show that heating a larger portion of the vessels reduces convective heat loss, which may result in more effective ablation strategies.

SUMMARYA modiÿed model for a binary uid is analysed mathematically. The governing equations of the motion consists of a Cahn-Hilliard equation coupled with a system describing a class of non-Newtonian incompressible uid with p-structure. The existence of weak solutions for the evolution problems is shown for the space dimension d = 2 with p ¿ 2 and for d = 3 with p ¿ 11=5. The existence of measure-valued solutions is obtained for d = 3 in the case 2 6 p ¡ 11=5. Similar existence results are obtained for the case of nondi erentiable free energy, corresponding to the density constraint | | 6 1. We also give regularity and uniqueness results for the solutions and characterize stable stationary solutions.

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We analyze the asymptotic behavior corresponding to the arbitrary high conductivity of the heat in the thermoelectric devices. This work deals with a steady-state multidimensional thermistor problem, considering the Joule effect and both spatial and temperature dependent transport coefficients under some real boundary conditions in accordance with the Seebeck-Peltier-Thomson cross-effects. Our first purpose is that the existence of a weak solution holds true under minimal assumptions on the data, as in particular nonsmooth domains. Two existence results are studied under different assumptions on the electrical conductivity. Their proofs are based on a fixed point argument, compactness methods, and existence and regularity theory for elliptic scalar equations. The second purpose is to show the existence of a limit model illustrating the asymptotic situation.Comment: 20 page

Key words Joule-Thomson effect, weak and classical solutions. MSC (2000) 35J55, 78A99, 80A20The transmission of an electric current in a conductor is a process in which some electrical energy is converted into heat (thermal energy). We deal with a nonlinear boundary value elliptic problem which describes the electrical heating of a solid conductor and the Joule-Thomson effect is taken into account. The existence of a weak solution is proved under both space and temperature dependence of the electrical and thermal conductivities. When the coefficients are only dependent on their temperature argument, some regularity results are stated.

The Boussinesq approximation to the Fourier-Navier-Stokes (F-N-S) flows under the electromagnetic field is considered. Such a model is the so-called Maxwell-Boussinesq approximation. We propose a new approach to the problem. We prove the existence and uniqueness of weak solutions to the variational formulation of the model. Some further regularity in W 1,2+δ , δ > 0, is obtained for the weak solutions. The shape sensitivity analysis by the boundary variations technique is performed for the weak solutions. As a result, the existence of the strong material derivatives for the weak solutions of the problem is shown. The result can be used to establish the shape differentiability for a broad class of shape functionals for the models of Fourier-Navier-Stokes flows under the electromagnetic field.