Analysis of Time-Periodic Navier-Stokes Equations in a Moving Domain and Numerical Computations with Radial Basis Neural Networks: Application to Artificial Hearts Blood Flow

Abstract

The dynamics of blood flow within an artificial heart (AH) chamber are governed by the time-periodic Navier-Stokes (NS) equations. These equations are coupled with a hyperbolic partial differential equation (PDE) that describes the dynamic behavior of the membrane (diaphragm). This coupled system of PDEs is subject to a moving boundary.

In this work, we decouple the system by assuming that the solution of the membrane equation is known within a well-defined Banach space. Subsequently, we solve only the NS problem and not the coupled fluid-structure problem. We conduct an analysis, specifically examining the existence and uniqueness of a time-periodic strong solution for the Navier-Stokes (NS) problem in dimensions 𝑁 = 2, 3 within a moving domain. The moving domain Ωₜ is supposed to be a 𝐿^∞(𝐖^{2,2̂*}) ∩ 𝐻^1(𝐇^2) ∩ 𝐻^2(𝐋^2) local perturbation of a 𝐶^{1,1} reference domain Ω₀ where 2̂* = 2 + {4 / 𝜀}, 𝜀 > 0 if 𝑁 = 2 and 2̂* = 3 + {𝜀 / (4 - 𝜀)}, 𝜀 ∈ (0, 3] if 𝑁 = 3. Typically, our moving domain Ωₜ is of class 𝐿^∞(𝐖^{2, 2̂*}) ∩ 𝐻^1(𝐇^2) ∩ 𝐻^2(𝐋^2), which is weaker that the 𝐶^3 in space regularity presented in the literature.

Subsequently, we proceed to numerically solve the problem in dimension 𝑁 = 2 using radial basis neural network (RBNN) functions. To validate our computational framework, we compare our results against existing literature, particularly by solving benchmark-driven cavity problems up to a Reynolds number 𝑅𝑒 = 10,000$. Finally, we solve numerically the time-periodic NS equations for different AH geometries. The results we obtain demonstrate the strong dependence of the blood flow behavior on the AH geometry and motivate the optimal shape design of AH, which we plan to address in future work.