Solution Formula of Korteweg Type by Using Partial Fourier Transform Methods in Half-Space without Surface Tension

Sharp-interface models and diffuse-interface models are the two basic types of models that describe liquid-vapour flow for compressible fluids. Their depictions of the line dividing liquid from vapour are different. The interface is modeled as a hypersurface in sharp-interface models. Sharp-interface models are free-boundary problems from a mathematical perspective since the position of the interface is a priori unknown and therefore a component of the solution to the free-boundary problem. A unique system of partial differential equations describes the motion of the fluid in the liquid and vapour phases, respectively. At the interface, boundary conditions between these systems are connected.. A mathematical model for liquid-vapour flows including phase transition known as the Navier-Stokes-Korteweg system which is the extension of the compressible Navier-Stokes equations. The purpose of Ihis article, we consider the soluton formula of Korteweg fluid model in half-space without surface tension. Since we consider in half-space case, Partial Fourier transform become appropriate method to find the formula of velocity and density for Korteweg type. The solution formula of the model problem for the velocity (u) and the (φ) are obtained by using the invers of partial Fourier transform. It consist multipliers. For the future research, we can investigate the estimation of the multiplier. Furthermore, by using Weis’s multiplier theorem we can find not only maximal Lp-Lq regularity class, but also we can consider the local well-posedness of the model problem.

Weak solutions to problems involving inviscid fluids

We consider an abstract functional-differential equation derived from the pressure-less Euler system with variable coefficients that includes several systems of partial differential equations arising in the fluid mechanics. Using the method of convex integration we show the existence of infinitely many weak solutions for prescribed initial data and kinetic energy

Thermodynamics of viscoelastic rate-type fluids with stress diffusion

We propose thermodynamically consistent models for viscoelastic fluids with a stress diffusion term. In particular, we derive variants of compressible/incompressible Maxwell/Oldroyd-B models with a stress diffusion term in the evolution equation for the extra stress tensor. It is shown that the stress diffusion term can be interpreted either as a consequence of a nonlocal energy storage mechanism or as a consequence of a nonlocal entropy production mechanism, while different interpretations of the stress diffusion mechanism lead to different evolution equations for the temperature. The benefits of the knowledge of the thermodynamical background of the derived models are documented in the study of nonlinear stability of equilibrium rest states. The derived models open up the possibility to study fully coupled thermomechanical problems involving viscoelastic rate-type fluids with stress diffusion.Comment: The benefits of the knowledge of the thermodynamical background of the derived models are now documented in the study of nonlinear stability of equilibrium rest state

Numerical simulation methods for phase-transitional flow

The object of the present dissertation is a numerical study of multiphase flow of one fluid component. In particular, the research described in this thesis focuses on the development of numerical methods that are based on a diffuse-interface model (DIM). With this approach, the modeling problem posed by the presence of moving boundaries in the flow domain, namely the interfaces between different phases, can be solved in a way that preserves the characteristic physical features related to the interfaces, such as surface tension and phase transitions. The first, largest part of the dissertation describes how to apply the DIM formulation that has been adopted, commonly identified as Korteweg formulation, in numerical simulations, without altering the physical parameters of the fluid. The issues of stability and accuracy of the solution, which can be severely compromised by the elliptical and dispersive nature of the set of governing equations, are extensively discussed. Therefore, before discretizing the governing equations a transformation of variables is performed, which removes the most important dispersive terms and greatly increases the stability of the numerical method. The latter is tested on several benchmark two-phase flow problems and for various grid refinements, when a Van der Waals equation of state is used and the temperature is in the vicinity of the critical value. To study the behavior of the flow when the temperature and the velocity fields are coupled, not only isothermal but also non-isothermal simulations are performed and analyzed. This includes a phasetransitional flow where the initial temperature field is such that latent heat plays a major role. Next, the feasibility of a combination of the DIM formulation with Large Eddy Simulation (LES) for turbulent multiphase flow, which is typical in several industrial applications, is explored and tested on one of the isothermal flow simulations. First the various subgrid terms resulting from filtering the governing equations are assessed in an a priori analysis, and different models for the most important subgrid terms are evaluated. Subsequently, a real LES is performed with the best subgrid model based on this analysis and its results are compared with filtered results from a direct numerical simulation. The research carried out for DIM and DIM-LES simulations is intended as the first step towards the development of models for interface mass and heat transfer that can be applied in commercial flow solvers for turbulent phase-transitional flow on industrial problems. Therefore, this research represents an ideal bridge towards the last part of the dissertation, in which a CFD (Computational Fluid Dynamics) model is developed and tested for an industrial application of turbulent phase-transitional flow: the direct-contact condensation of superheated steam injected in water. This model is implemented in the commercial CFD software package ANSYS Fluent. The purpose of this work is twofold. On the one hand, a condensation model for the mass transfer rate at the steam–water interface, based on kinetic gas theory, is tested by comparison of the results with experiments conducted at the Department of Mechanical Engineering of TU/e within the scope of the same research project. By testing the phase change model, useful information can be obtained on the grid requirements and the turbulence model. On the other hand, comparison with experiments, also conducted at TU/e, can be made for the case of steam injected in a fully developed turbulent cross-flow of water in a square duct. To this purpose, results are shown for a three-dimensional simulation performed for the assigned geometry of the experimental setup and for one set of operating conditions used in the experiments. All simulations performed with Fluent are based on a Volume-of-Fluid (VOF) multiphase formulation and on the Reynolds-averaged Navier-Stokes (RANS) equations approach for turbulent flow. Both are typically adopted in the industrial two-phase flow CFD

Pore-scale modeling of phase change in porous media

The combination of high-resolution visualization techniques and pore-scale flow modeling is a powerful tool used to understand multiphase flow mechanisms in porous media and their impact on reservoir-scale processes. One of the main open challenges in pore-scale modeling is the direct simulation of flows involving multicomponent mixtures with complex phase behavior. Reservoir fluid mixtures are often described through cubic equations of state, which makes diffuse-interface, or phase-field, theories particularly appealing as a modeling framework. What is still unclear is whether equation-of-state-driven diffuse-interface models can adequately describe processes where surface tension and wetting phenomena play important roles. Here we present a diffuse-interface model of single-component two-phase flow (a van der Waals fluid) in a porous medium under different wetting conditions. We propose a simplified Darcy-Korteweg model that is appropriate to describe flow in a Hele-Shaw cell or a micromodel, with a gap-averaged velocity. We study the ability of the diffuse-interface model to capture capillary pressure and the dynamics of vaporization-condensation fronts and show that the model reproduces pressure fluctuations that emerge from abrupt interface displacements (Haines jumps) and from the breakup of wetting films

The sharp-interface limit for the Navier--Stokes--Korteweg equations

We investigate the sharp-interface limit for the Navier--Stokes--Korteweg model, which is an extension of the compressible Navier--Stokes equations. By means of compactness arguments, we show that solutions of the Navier--Stokes--Korteweg equations converge to solutions of a physically meaningful free-boundary problem. Assuming that an associated energy functional converges in a suitable sense, we obtain the sharp-interface limit at the level of weak solutions