Research problems

This section appears from time to time. Contributions are invited, and should be submitted to R. C. Thompson, Mathematics Department, University of California, Santa Barbara, CA 93106.     

Finite element computations of two-dimensional arterial flow in the presence of a transverse magnetic field

A finite element solution of the Navier-Stokes equations for steady flow under the magnetic effect through a double-branched two-dimensional section of a three-dimensional model of the canine aorta is discussed. The numerical scheme involves transforming the physical co-ordinates to a curvilinear boundary-fitted co-ordinate system. The shear stress at the wall is calculated for a Reynolds number of 1000 with the branch-to-main aortic flow rate ratio as a parameter. The results are compared with earlier works involving experimental data and found to be in reasonable qualitative agreement. The steady flow, shear stress and branch flow under the effect of a magnetic field have been discussed in detail.     

Numerical study of simultaneous natural convection heat transfer from both surfaces of a uniformly heated thin plate with arbitrary inclination

 Numerical studies were conducted to investigate the natural convection heat transfer around a uniformly heated thin plate with arbitrary inclination in an infinite space. The numerical approach was based on the finite volume technique with a nonstaggered grid arrangement. For handling the pressure–velocity coupling the SIMPLE-algorithm was used. QUICK scheme and first order upwind scheme were employed for discretization of the momentum and energy convective terms respectively. Plate width and heating rate were used to vary the modified Rayleigh number over the range of 4.8×10⁶ to 1.87×10⁸. Local and average heat transfer characteristics were compared with regarding to the inclination angle. The empirical expressions for local and average Nusselt number were correlated. It has been found that for inclination angle less than 10^∘, the flow and heat transfer characteristics are complicated and the average Nusselt number can not be correlated by one equation while for inclination angle larger than 10^∘, the average Nusselt number can be correlated into an elegant correlation. Received on 18 April 2001 / Published online: 29 November 2001     

Finite element method in the solution of the Euler and Navier-Stokes equations for internal flow

Finite element solutions of the Euler and Navier-Stokes equations are presented, using a simple dissipation model. The discretization is based on the weak-Galerkin weighted residual method and equal interpolation functions for all the unknowns are permitted. The nonlinearity is iterated upon using a Newton method and at each iteration the linear algebraic system is solved by a direct solver with all unknowns fully coupled. Results are presented for two-dimensional transonic inviscid flows and two- and three-dimensional incompressible viscous flows. Convergence of the algorithm is shown to be quadratic, reaching machine accuracy in very few iterations. The inviscid results demonstrate the existence of nonunique numerical solutions to the steady Euler equations.     

VECTOR FIELDS IN R2 WITH MAXIMAL INDEX

We consider the method of Poincaré to investigate thelocal index of vector fields in the plane. If m is the degreeof the first non-zero jet, X_m, of the vector field X at an isolatedzero, we explore the geometry of the pencil generated by thecoordinate functions of X_m when the absolute value of the indexof X, |ind (X)|, is m. We also find necessary and sufficientconditions for |ind (X)| to be m.     

Equilibrium distributions of stochastic networks without detailed balance

We present results for the determination of the equilibrium microstate probability distribution of a class of strongly interacting systems obeying stochastic dynamics but without the necessity of detailed balance. The specific case of Ising systems is highlighted, with applications to non-symmetric synaptic neural networks and more general recurrent Boolean networks.     

Substitution of calcium by strontium within selected calcium phosphates

Sr incorporation in the molecules of amorphous calcium phosphate, apatitic tricalcium phosphate, hydroxyapatite, octacalcium phosphate and dicalcium phosphate dihydrate was investigated. The concentration of Sr ranged from 225 to 1010 μ g / g, i.e. it overlapped with the physiological range of Sr concentrations in human bone. The leading experimental technique was extended X-ray absorption fine structure (EXAFS) at the Sr K edge. Results of these studies demonstrated the following: (1) Sr incorporation in the calcium phosphates is compound-dependent, (2) the coordination of incorporated Sr atoms in the Ca-P molecules is similar to that of Ca atoms, but interatomic distances are ≈0.015 nm larger, (3) in apatitic tricalcium phosphate, hydroxyapatite and octacalcium phosphate lattices Sr atoms may occupy selected Ca sites, which was not the case for dicalcium phosphate dihydrate, (4) in the apatite lattice Sr atoms are coordinated by 6 PO₄ tetrahedrals and (5) EXAFS spectra at the K edge of the incorporated Sr may be used to distinguish the structures of amorphous calcium phosphate, dicalcium phosphate dihydrate as well as apatite and its derivatives (apatitic tricalcium phosphate, octacalcium phosphate).     

A note on dephasing and pair-breaking

It is shown that the dephasing which suppresses the weak localization correction to the conductivity has the same physical origin than pair-breaking in superconductors and thus-following de Gennes-may be expressed in terms of the correlation function of the operator for time reversal.