An improved correlation of the pressure drop in stenotic vessels using Lorentz's reciprocal theorem

A mathematical model of the human cardiovascu lar system in conjunction with an accurate lumped model for a stenosis can provide better insights into the pressure wave propagation at pathological conditions. In this study, a the oretical relation between pressure drop and flow rate based on Lorentz＇s reciprocal theorem is derived, which offers an identity to describe the relevance of the geometry and the convective momentum transport to the drag force. A voxel based simulator VFLOW VOF3D, where the vessel geome try is expressed by using volume of fluid （VOF） functions, is employed to find the flow distribution in an idealized steno sis vessel and the identity was validated numerically. It is revealed from the correlation flow in a stenosis vessel can that the pressure drop of NS be decomposed into a linear term caused by Stokes flow with the same boundary condi tions, and two nonlinear terms. Furthermore, the linear term for the pressure drop of Stokes flow can be summarized as a correlation by using a modified equation of lubrication the ory, which gives favorable results compared to the numerical ones. The contribution of the nonlinear terms to the pressure drop was analyzed numerically, and it is found that geomet ric shape and momentum transport are the primary factors for the enhancement of drag force. This work paves a way to simulate the blood flow and pressure propagation under dif ferent stenosis conditions by using 1D mathematical model.

On the Four Conjectures of G G Lorentz Type in L_ρ~2[-a,a]

In the space C[-1,1],G G Lorentz proposed four conjectures on the properties of the polynomials of the best approximation in 1977,1978 and 1980.The present paper transplants the four conjectures in the space Lρ2[-a,a] and proves them being all right in only one theorem under the corresponding conditions,although each of the original conjectures is very difficulty.