A numerical method for the solution of the Navier-Stokes equations is developed using an integral representation of the conservation equations. The velocity-vorticity formulation is employed, where the kinematics is given with the Poisson equation for a velocity vector, while the kinetics is represented with the vorticity transport equation. The corresponding boundary-domain integral equations are presented along with discussions of the kinetics and kinematics of the fluid flow problem. The boundary-domain integral formulation is developed and tested for natural convection flows in closed cavities with complex geometries.

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Dynamic thermography has been clinically proven to be a valuable diagnostic technique for skin tumor detection as well as for other medical applications, and shows many advantages over static thermography. Numerical modelling of heat transfer phenomena in biological tissue during dynamic thermography can aid the technique by improving process parameters or by estimating unknown tissue parameters based on measurement data. This paper presents a new non-linear numerical model of multilayer skin tissue containing a skin tumour together with thermoregulation response of the tissue during the cooling-rewarming process of dynamic thermography. The thermoregulation response is modelled by temperature-dependent blood perfusion rate and metabolic heat generation. The aim is to describe bioheat transfer more realistically. The model is based on the Pennes bioheat equation and solved numerically using a subdomain BEM approach treating the problem as axisymmetrical. The paper includes computational tests for Clark II and Clark IV tumours, comparing the models using constant and temperature-dependent properties which showed noticeable differences and highlighted the importance of using a local thermoregulation model. Results also show the advantage of using dynamic thermography for skin tumour screening and detection at an early stage. One of the contributions of this paper is a complete sensitivity analysis of 56 model parameters based on the gradient of the surface temperature difference between tumour and healthy skin. The analysis shows that size of the tumour, blood perfusion rate, thermoregulation coefficient of the tumour, body core temperature and density and specific heat of the skin layers in which the tumour is embedded are important for modelling the problem, and so have to be determined more accurately to reflect realistic skin response of the investigated tissue, while metabolic heat generation and its thermoregulation are not.

The solution of Navier-Stokes equations of time-dependent incompressible viscous fluid flow in planar geometry by the Boundary Domain Integral Method (BDIM) is discussed. The introduction of a subdomain technique to fluid flow problems is considered and improved in order to maintain the stability of BDIM. To avoid problems with flow kinematics computation in the sudomain mesh, a segmentation technique is proposed which combines the original BDIM with its subdomain variant and preserves its numerical stability. In order to reduce the computational cost of BDIM, which greatly depends on the solution of systems of linear equations, iterative methods are used. Conjugate gradient methods, conjugate gradients squared and an improved version of the biconjugate gradient method BiCGSTAB, together with the generalized minimal residual method, are used as iterative solvers. Different types of preconditioning, from simple Jacobi to incomplete LU factorization, are carried out and the performance of chosen iterative methods and preconditioners are reported. Test examples include backward facing step flow and flow through tubular heat exchangers. Test computation results show that BDIM is an accurate approximation technique which, together with the subdomain technique and powerful iterative solvers, can exhibit some significant savings in storage and CPU time requirements.

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