Influence of a Given Field of Temperature on the Blood Pressure Variation: Variational Analysis, Numerical Algorithms and Simulations

This article presents a fluid–elastic structure interaction (FSI) problem when the temperature variation of the two media is also taken into account. We introduced the mathematical description of this interaction in a recent article. Our model includes the coupling between the fluid and the elastic medium equations and, in addition, the coupling with the temperature equations. The novelty of this approach is that we succeed in analyzing a complicated double-coupled problem that allows us to describe more complex physical phenomena both from the theoretical and numerical points of view. Since the main goal of this article is to analyze the influence of an exterior field of temperature on fluid pressure variations, the theoretical results obtained in our previous article are completed with qualitative properties concerning the fluid pressure, such as existence, regularity and uniqueness. Our study continues with approximation schemes: in order to improve the unknowns regularity, we introduce the pressure approximation by a sequence of viscoelastic pressure functions and we prove the weak convergence of this sequence to the pressure; then, we present a numerical approximation scheme with stability and convergence results and Uzawa’s algorithm. The last part of the article is devoted to numerical simulations that rely on the numerical schemes introduced and studied before and highlight some physical phenomena related to the considered problem.