Thin Tube Structures Research Articles

NUMERICAL STUDY OF THE EQUATION ON THE GRAPH FOR THE STEADY STATE NON-NEWTONIAN FLOW IN THIN TUBE STRUCTURE

The dimension reduction for the viscous flows in thin tube structures leads to equations on the graph for the macroscopic pressure with Kirchhoff type junction conditions in the vertices. Non-Newtonian rheology of the flow generates nonlinear equations on the graph. A new numerical method for second order nonlinear differential equations on the graph is introduced and numerically tested.

Pressure boundary conditions for viscous flows in thin tube structures: Stokes equations with locally distributed Brinkman term

The steady state Stokes-Brinkman equations in a thin tube structure is considered. The Brinkman term differs from zero only in small balls near the ends of the tubes. The boundary conditions are: given pressure at the inflow and outflow of the tube structure and the no slip boundary condition on the lateral boundary. The complete asymptotic expansion of the problem is constructed. The error estimates are proved. The method of partial asymptotic dimension reduction is introduced for the Stokes-Brinkman equations and justified by an error estimate. This method approximates the main problem by a hybrid dimension problem for the Stokes-Brinkman equations in a reduced domain.

Partial Asymptotic Dimension Reduction for Steady State Non-Newtonian Flow with Strain Rate Dependent Viscosity in Thin Tube Structure

Partial Asymptotic Dimension Reduction for Steady State Non-Newtonian Flow with Strain Rate Dependent Viscosity in Thin Tube Structure

Steady state non-Newtonian flow in a thin tube structure: equation on the graph

The dimension reduction for the viscous flows in thin tube structures leads to equations on the graph for the macroscopic pressure with Kirchhoff type junction conditions at the vertices. Nonlinear equations on the graph generated by the non-Newtonian rheology are treated here. The existence and uniqueness of a solution of this problem is proved. This solution describes the leading term of an asymptotic analysis of the stationary non-Newtonian fluid motion in a thin tube structure with no-slip boundary condition on the lateral boundary.

Steady state non-Newtonian flow with strain rate dependent viscosity in thin tube structure with no slip boundary condition

The steady state non-Newtonian flow, with strain rate dependent viscosity in a thin tube structure, with no slip boundary condition, is considered. Applying the Banach fixed point theorem we prove the existence and uniqueness of a solution. An asymptotic approximation is constructed and justified by an error estimate.

Steady-State Navier–Stokes Equations in Thin Tube Structure with the Bernoulli Pressure Inflow Boundary Conditions: Asymptotic Analysis

Steady-state Navier–Stokes equations in a thin tube structure with the Bernoulli pressure inflow–outflow boundary conditions and no-slip boundary conditions at the lateral boundary are considered. Applying the Leray–Schauder fixed point theorem, we prove the existence and uniqueness of a weak solution. An asymptotic approximation of a weak solution is constructed and justified by an error estimate.

Time periodic Navier\u2013Stokes equations in a thin tube structure

The time periodic Navier–Stokes equations are considered in the three-dimensional and two-dimensional settings with Dirichlet boundary conditions in thin tube structures. These structures are finite union of thin cylinders (thin rectangles in the case of dimension two), where the small parameter ε is the ratio of the hight and the diameter of the cylinders. We consider the case of finite or big coefficient before the time derivative. This setting is motivated by hemodynamical applications. Theorems of existence and uniqueness of a solution are proved. Complete asymptotic expansion of a solution is constructed and justified by estimates of the difference of the exact solution and truncated series of the expansion in norms taking into account the first and second derivatives with respect to the space variables and the first derivative in time. The method of asymptotic partial decomposition of the domain is justified for the time periodic problem.

Non-Newtonian flows in domains with non-compact boundaries

Non-Newtonian flow of a viscous fluid in unbounded domains with cylindrical outlets is considered. The viscosity is assumed to be dependent on the shear rate. Applying the Banach fixed point theorem and the Hilbert spaces with exponential weight we prove the existence and uniqueness of solutions with an exponential stabilization to the quasi-Poiseuille flows at the outlets if the right hand side decays exponentially. These results may be used for the matching technique for flows in thin tube structures.

Asymptotic analysis of the non-steady Navier–Stokes equations in a tube structure. II. General case

The non-steady Navier–Stokes equations with Dirichlet boundary conditions are considered in thin tube structures. These domains are connected finite unions of thin finite cylinders (in the 2D case respectively thin rectangles). The complete asymptotic expansion of the solution is constructed. It contains a regular part and three types of the boundary layer correctors: “in-space”, “in-time” and “in-space-and-in-time”. The estimates for the difference of the exact solution and its Jth asymptotic approximation are proved.

Asymptotic analysis of the non-steady Navier–Stokes equations in a tube structure. I. The case without boundary-layer-in-time

The non-steady Navier–Stokes equations with Dirichlet boundary conditions are considered in thin tube structures. These domains are connected finite unions of thin finite cylinders (in the 2D case respectively thin rectangles). The complete asymptotic expansion of the solution is constructed in the case without boundary-layer-in-time. The estimates for the difference of the exact solution and its J-th asymptotic approximation is proved. The method of asymptotic partial domain decomposition is formulated and justified for the non-steady Navier–Stokes equations in a tube structure. It gives the asymptotically exact interface conditions of coupling of the 1D and 3D models of the flow. Note that the obtained results hold true for the important in applications case of time periodic flows.

Divergence equation in thin-tube structures

Divergence equation is considered in a thin-tube structure, connected finite union of thin finite cylinders (in the case of two dimensions, respectively, thin rectangles) where the ratio of the diameter and the heights of the cylinders is the small parameter . Various norms of the solution of divergence equation are estimated over the norms of the right-hand side. The exact dependence of constants in these estimates on the small parameter is obtained.

ASYMPTOTIC PARTIAL DOMAIN DECOMPOSITION IN THIN TUBE STRUCTURES: NUMERICAL EXPERIMENTS

The method of asymptotic partial domain decomposition for thin tube structures (finite unions of thin cylinders) is revisited. Its application to the Newtonian and non-Newtonian flows in large systems of vessels is considered. The possibility of a parallelization of its algorithm is discussed for linear and nonlinear models.

Asymptotic expansion of the solution of the steady Stokes equation with variable viscosity in a two-dimensional tube structure

The Stokes equation with the varying viscosity is considered in a thin tube structure, i.e., in a connected union of thin rectangles with heights of order ɛ < < 1 and with bases of order 1 with smoothened boundary. An asymptotic expansion of the solution is constructed: it contains some Poiseuille type flows in the channels (rectangles) with some boundary layers correctors in the neighborhoods of the bifurcations of the channels. The estimates for the difference of the exact solution and its asymptotic approximation are proved.

A Viscous Fluid Flow through a Thin Channel with Mixed Rigid‐Elastic Boundary: Variational and Asymptotic Analysis

We study the nonsteady Stokes flow in a thin tube structure composed by two thin rectangles with lateral elastic boundaries which are connected by a domain with rigid boundaries. After a variational approach of the problem which gives us existence, uniqueness, regularity results, and some a priori estimates, we construct an asymptotic solution. The existence of a junction region between the two rectangles imposes to consider, as part of the asymptotic solution, some boundary layer correctors that correspond to this region. We present and solve the problems for all the terms of the asymptotic expansion. For two different cases, we describe the order of steps of the algorithm of solving the problem and we construct the main term of the asymptotic expansion. By means of the a priori estimates, we justify our asymptotic construction, by obtaining a small error between the exact and the asymptotic solutions.

Asymptotic analysis of the steady stokes equation with randomly perturbed viscosity in a thin tube structure

The Stokes equation with the nonconstant viscosity is considered in a thin tube structure, i.e., in a connected union of thin rectangles with heights of order e � 1 and bases of order 1 with smoothened boundary. An asymptotic expansion of the solution is constructed. In the case of random perturbations of the constant viscosity, we prove that the leading term for the velocity is deterministic, while for the pressure it is random, but the expectations of the pressure satisfies the deterministic Darcy equation. Estimates for the difference between the exact solution and its asymptotic approximation are proved. Bibliography: 11 titles. Illustrations: 3 figures.

Parallelization of the algorithm of asymptotic partial domain decomposition in thin tube structures

The method of asymptotic partial domain decomposition for thin tube structures (finite unions of thin cylinders) is revisited. Its application to the Newtonian and non-Newtonian flows in great systems of vessels is considered. The possibility of a parallelization of its algorithm is discussed for linear and non-linear models.