Coarse-grained description of thermo-capillary flow

A mesoscopic or coarse-grained approach is presented to study thermo-capillary induced flows. An order parameter representation of a two-phase binary fluid is used in which the interfacial region separating the phases naturally occupies a transition zone of small width. The order parameter satisfies the Cahn-Hilliard equation with advective transport. A modified Navier-Stokes equation that incorporates an explicit coupling to the order parameter field governs fluid flow. It reduces, in the limit of an infinitely thin interface, to the Navier-Stokes equation within the bulk phases and to two interfacial forces: a normal capillary force proportional to the surface tension and the mean curvature of the surface, and a tangential force proportional to the tangential derivative of the surface tension. The method is illustrated in two cases: thermo-capillary migration of drops and phase separation via spinodal decomposition, both in an externally imposed temperature gradient.Comment: To appear in Phys. Fluids. Also at http://www.scri.fsu.edu/~vinals/dj1.p

A thermodynamically consistent phase-field model for two-phase flows with thermocapillary effects

In this paper, we develop a phase-field model for binary incompressible (quasi-incompressible) fluid with thermocapillary effects, which allows for the different properties (densities, viscosities and heat conductivities) of each component while maintaining thermodynamic consistency. The governing equations of the model including the Navier-Stokes equations with additional stress term, Cahn-Hilliard equations and energy balance equation are derived within a thermodynamic framework based on entropy generation, which guarantees thermodynamic consistency. A sharp-interface limit analysis is carried out to show that the interfacial conditions of the classical sharp-interface models can be recovered from our phase-field model. Moreover, some numerical examples including thermocapillary convections in a two-layer fluid system and thermocapillary migration of a drop are computed using a continuous finite element method. The results are compared to the corresponding analytical solutions and the existing numerical results as validations for our model

On a non-isothermal diffuse interface model for two-phase flows of incompressible fluids

We introduce a diffuse interface model describing the evolution of a mixture of two different viscous incompressible fluids of equal density. The main novelty of the present contribution consists in the fact that the effects of temperature on the flow are taken into account. In the mathematical model, the evolution of the macroscopic velocity is ruled by the Navier-Stokes system with temperature-dependent viscosity, while the order parameter representing the concentration of one of the components of the fluid is assumed to satisfy a convective Cahn-Hilliard equation. The effects of the temperature are prescribed by a suitable form of the heat equation. However, due to quadratic forcing terms, this equation is replaced, in the weak formulation, by an equality representing energy conservation complemented with a differential inequality describing production of entropy. The main advantage of introducing this notion of solution is that, while the thermodynamical consistency is preserved, at the same time the energy-entropy formulation is more tractable mathematically. Indeed, global-in-time existence for the initial-boundary value problem associated to the weak formulation of the model is proved by deriving suitable a-priori estimates and showing weak sequential stability of families of approximating solutions.Comment: 26 page

Energy consistent discontinuous Galerkin methods for a quasi-incompressible diffuse two phase flow model

We design consistent discontinuous Galerkin finite element schemes for the approximation of a quasi-incompressible two phase flow model of Allen--Cahn/Cahn--Hilliard/Navier--Stokes--Korteweg type which allows for phase transitions. We show that the scheme is mass conservative and monotonically energy dissipative. In this case the dissipation is isolated to discrete equivalents of those effects already causing dissipation on the continuous level, that is, there is no artificial numerical dissipation added into the scheme. In this sense the methods are consistent with the energy dissipation of the continuous PDE system

Weak-strong uniqueness for the Navier-Stokes equation for two fluids with surface tension

In the present work, we consider the evolution of two fluids separated by a sharp interface in the presence of surface tension - like, for example, the evolution of oil bubbles in water. Our main result is a weak-strong uniqueness principle for the corresponding free boundary problem for the incompressible Navier-Stokes equation: As long as a strong solution exists, any varifold solution must coincide with it. In particular, in the absence of physical singularities the concept of varifold solutions - whose global in time existence has been shown by Abels [2] for general initial data - does not introduce a mechanism for non-uniqueness. The key ingredient of our approach is the construction of a relative entropy functional capable of controlling the interface error. If the viscosities of the two fluids do not coincide, even for classical (strong) solutions the gradient of the velocity field becomes discontinuous at the interface, introducing the need for a careful additional adaption of the relative entropy.Comment: 104 page

Study of basic physical processes in liquid rocket engines

Inconsistencies between analytical results and measurements for liquid rocket thrust chamber performance, which escape suitable explanations, have motivated the examination of the basic phys ical modeling formulations as to their unlimited application. The publication of Prof. D. Straub's book, 'Thermofluid-dynamics of Optimized Rocket Propulsions,' further stimulated the interest of understanding the gas dynamic relationships in chemically reacting mixtures. A review of other concepts proposed by Falk-Ruppel (Gibbsian Thermodynamics), Straub (Alternative Theory, AT), Prigogine (Non-Equilibrium Thermodynamics), Boltzmann (Kinetic Theory), and Truesdell (Rational Mechanism) has been made to obtain a better understanding of the Navier-Stokes equation, which is now used extensively for chemically reacting flow treatment in combustion chambers. In addition to the study of the different concepts, two workshops were conducted to clarify some of the issues. The first workshop centered on Falk-Ruppel's new 'dynamics' concept, while the second one concentrated on Straub's AT. In this report brief summaries of the reviewed philosophies are presented and compared with the classical Navier-Stokes formulation in a tabular arrangement. Also the highlights of both workshops are addressed