Existence of nonstationary Poiseuille-type solutions under minimal regularity assumptions
- PDF / 424,566 Bytes
- 17 Pages / 547.087 x 737.008 pts Page_size
- 97 Downloads / 197 Views
Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Existence of nonstationary Poiseuille-type solutions under minimal regularity assumptions ˇ K. Pileckas and R. Ciegis Abstract. Existence and uniqueness of a solution to the nonstationary Navier–Stokes equations having a prescribed flow rate (flux) in the infinite cylinder Π = {x = (x , xn ) ∈ Rn : x ∈ σ ⊂ Rn−1 , −∞ < xn < ∞, n = 2, 3} are proved. It is assumed that the flow rate F ∈ L2 (0, T ) and the initial data u0 = 0, . . . , 0, u0n ∈ L2 (σ). The nonstationary Poiseuille solution has the form u(x, t) = 0, . . . , 0, U (x , t) , p(x, t) = −q(t)xn + p0 (t), where (U (x , t), q(t)) is a solution of an inverse problem for the heat equation with a specific over-determination condition. Mathematics Subject Classification. 35Q30, 76D03, 76D05. Keywords. Nonstationary Navier–Stokes equations, Cylindrical domain, Nonstationary Poiseuille-type solution, Inverse problem, Heat equation, Minimal regularity.
1. Introduction Nowadays, mathematical modeling extensively expands in physiology and medicine all over the world in order to help in the choice of the optimal strategy of medical treatment. The important topic in such modeling deals with multiscale mathematical models of the blood circulation in networks of vessels. However, the full three-dimensional computations are currently very time consuming and can be applied only for small parts of the blood circulation system. That is why a new trend is related to the creation of hybrid dimension models, combining the one-dimensional reduction in the regular zones with three-dimensional zooms in small zones of singular behavior. Starting from the Navier–Stokes equations written everywhere in the blood flow area, it derives the one-dimensional Poiseuille-type nonstationary solutions in the main part of the domain (in straight vessels) with three-dimensional zooms in small parts near the bifurcations of vessels and clot formation zones. It prescribes mathematically justified size of the zoomed areas and asymptotically exact junction conditions. This approach uses the method of asymptotic partial decomposition of domain proposed by Panasenko [14] and developed in [15–18]. These new mathematically justified hybrid dimension models require essentially smaller computational resources. The one-dimensional Poiseuille-type flows in straight vessels play a very important part in these methods. Poiseuille flows are also important in the study of motion in “bent” pipes or in pipes of varying cross sections that end up in the shape of semi-infinite straight pipes Π+ , etc. In such problems, Poiseuille flows are attained at very large distances in Π+ (see [22–24,27]). The steady-state Poiseuille flow in an infinite straight pipe Π = {x = (x , xn ) ∈ Rn : x ∈ σ ⊂ Rn−1 , −∞ < xn < ∞, n = 2, 3} of constant cross-sectional σ was described by Jean Louis Poiseuille in 1841 (see [25]). Today this classical solution of the Navier–Stokes equation seems to be trivial although it is used in numerous studies of fluid motion. The Poiseuille flow is c
Data Loading...
