Stochastics in Fluids

Table of Contents

1. Frontmatter

2. On Wild Solutions to the Compressible Euler System

   Ondřej Kreml

   Abstract

   In this chapter, we survey recent results concerning the existence and properties of so-called wild solutions to the compressible Euler system in two spatial dimensions. These solutions are constructed using the convex integration method developed in this context by De Lellis and Székelyhidi (Ann Math 170:1417–1436;2009, Arch Rational Mech Anal 195:225–260;2010). We introduce the Riemann problem for the compressible Euler system, classify its one-dimensional self-similar solutions, and summarize results related to the uniqueness and nonuniqueness of these solutions within the class of admissible weak solutions. We also discuss the existence of wild solutions for regular initial data and the failure of criteria based on maximal dissipation of energy to select physically relevant solutions.

3. Comparative Study of 2D-2D FEM-FSI and 2D-1D Reduced FD Models for Respiratory Flow Simulation

   Anna Lancmanová, Alexander Drobny, Elfriede Friedmann, Tomáš Bodnár

   Abstract

   This contribution presents a computational comparison and cross-validation of a coupled fluid-structure interaction (FSI) model considering an oscillatory (direction reversing) flow of an incompressible fluid in a partially elastic channel. The solution of a full 2D FSI model solved by a finite-element method (FEM) is compared with a simplified coupled 2D-1D model solved by finite difference method (FDM). Besides providing a direct comparison of instantaneous velocity profiles with the Womersley analytical solution in the rigid part of the channel, the data from the elastic part considering fluid-structure interaction are compared as well. The results show a good mutual agreement of both models, with obvious limitations in the elastic region, where the computational setups for both models can’t be exactly identical.

4. Non-Newtonian Compressible Fluids with Stochastic Right-Hand Side

   Pavel Ludvík, Václav Mácha

   Abstract

   Fluids with shear rate-dependent viscosity form a special class of non-Newtonian fluids that play a crucial role in continuum dynamics. We consider a compressible barotropic flow of such fluids, governed by a generalized Navier-Stokes system. Due to the nonlinearity in the elliptic term, obtaining the existence of weak solutions is challenging. To address this, we introduce the concept of a measure-valued solution, whose existence is established in a very general setting using an abstract theorem on the existence of a Young measure. We also discuss the proof of this key result.

5. Weak-Strong Uniqueness for a Compressible Fluid-Rigid Body Interaction Problem with General Inflow-Outflow Boundary Data

   Šárka Nečasová, Ana Radošević

   Abstract

   The chapter deals with the problem of the motion of a rigid body in a domain filled by the compressible fluid. We consider the nonhomogeneous boundary condition of the velocity and inflow boundary condition for the density. We study the weak-strong uniqueness property for such problem. The novelty lies in the fact we are considering the general inflow-outflow boundary data which were not considered before in the context of fluid-structure interaction of weak-strong uniqueness.

6. Calculation of Pressure and Forces in Stochastic Vortex Methods

   Andrzej F. Nowakowski

   Abstract

   This chapter discusses pressure calculations and aerodynamic forces in external flows. The dynamic variables are recovered from the kinematic velocity-vorticity formulation of the Navier-Stokes equations. Both velocity and vorticity fields may result from vortex methods. In the example presented, the kinematic variables are generated from the stochastic vortex blob simulation of the viscous incompressible flow. The vorticity field is a discontinuous function. This contribution demonstrates how to find pressure as a solution to the variational problem which forms a weak version of the momentum equation. It does not require any explicit boundary conditions for pressure, and no derivatives of vorticity have to be calculated in this approach. The correctness of the calculation of the pressure field and consequently of the aerodynamic forces is verified by an alternative method of calculating the aerodynamic forces which directly applies formulas involving the velocity and vorticity fields.

7. On the Steady Motion of a Navier-Stokes Flow Across a Sieve with Prescribed Pressure Drop in a Finite Pipe

   Gianmarco Sperone

   Abstract

   The steady motion of a viscous incompressible fluid through a sieve (i.e., a wall perforated with a large number of small holes), in a pipe of finite length, is modeled through the Navier-Stokes equations under mixed boundary conditions involving the Bernoulli pressure and the tangential velocity on the inlet and outlet of the tube, while the pressure drop is prescribed along the pipe. Applying the classical energy method in homogenization theory, we study the asymptotic behavior of the solutions to this system, without any restriction on the magnitude of the data, as the diameters of the perforations vanish. Regardless of the initial scaling and distribution of the holes, we show that the sieve asymptotically becomes a wall, meaning that the effective equations are two, independent, stationary Navier-Stokes systems with a no-slip boundary condition on the wall. In the absence of external forces, we prove, furthermore, that the fluid motion becomes quiescent in the homogenization limit.

8. On the Derivation of the Compressible Primitive Equations

   Tong Tang, Šárka Nečasová

   Abstract

   The chapter deals with the problem of the derivation of the compressible primitive equation (viscous and inviscid) rigorously from compressible anisotropic Navier-Stokes equations and Euler equations, respectively.