211 research outputs found
Wellposedness of the discontinuous ODE associated with two-phase flows
We consider the initial value problem \dot x (t) = v(t,x(t)) \;\mbox{ for
} t\in (a,b), \;\; x(t_0)=x_0 which determines the pathlines of a two-phase
flow, i.e.\ is a given velocity field of the type with denoting the bulk phases
of the two-phase fluid system under consideration. The bulk phases are
separated by a moving and deforming interface . Since we allow for
flows with phase change, these pathlines are allowed to cross or touch the
interface. Imposing a kind of transversality condition at , which
is intimately related to the mass balance in such systems, we show existence
and uniqueness of absolutely continuous solutions of the above ODE in case the
one-sided velocity fields are continuous in and locally Lipschitz continuous in
. Note that this is a necessary prerequisite for the existence of
well-defined co-moving control volumes for two-phase flows, a basic concept for
mathematical modeling of two-phase continua
Thermodynamically consistent modeling for dissolution/growth of bubbles in an incompressible solvent
We derive mathematical models of the elementary process of dissolution/growth
of bubbles in a liquid under pressure control. The modeling starts with a fully
compressible version, both for the liquid and the gas phase so that the entropy
principle can be easily evaluated. This yields a full PDE system for a
compressible two-phase fluid with mass transfer of the gaseous species. Then
the passage to an incompressible solvent in the liquid phase is discussed,
where a carefully chosen equation of state for the liquid mixture pressure
allows for a limit in which the solvent density is constant. We finally provide
a simplification of the PDE system in case of a dilute solution
Continuum thermodynamics of chemically reacting fluid mixtures
We consider viscous, heat conducting mixtures of molecularly miscible
chemical species forming a fluid in which the constituents can undergo chemical
reactions. Assuming a common temperature for all components, we derive a closed
system of partial mass and partial momentum balances plus a mixture balance of
internal energy. This is achieved by careful exploitation of the entropy
principle and requires appropriate definitions of absolute temperature and
chemical potentials, based on an adequate definition of thermal energy
excluding diffusive contributions. The resulting interaction forces split into
a thermo-mechanical and a chemical part, where the former turns out to be
symmetric in case of binary interactions. For chemically reacting systems and
as a new result, the chemical interaction force is a contribution being
non-symmetric outside of chemical equilibrium. The theory also provides a
rigorous derivation of the so-called generalized thermodynamic driving forces,
avoiding the use of approximate solutions to the Boltzmann equations. Moreover,
using an appropriately extended version of the entropy principle and
introducing cross-effects already before closure as entropy invariant couplings
between principal dissipative mechanisms, the Onsager symmetry relations become
a strict consequence. With a classification of the factors in the binary
products of the entropy production according to their parity--instead of the
classical partition into so-called fluxes and driving forces--the apparent
anti-symmetry of certain couplings is thereby also revealed. If the diffusion
velocities are small compared to the speed of sound, the Maxwell-Stefan
equations follow in the case without chemistry, thereby neglecting wave
phenomena in the diffusive motion. This results in a reduced model with only
mass being balanced individually. In the reactive case ..
Global existence for a class of reaction-diffusion systems with mass action kinetics and concentration-dependent diffusivities
In this work we study the existence of classical solutions for a class of
reaction-diffusion systems with quadratic growth naturally arising in mass
action chemistry when studying networks of reactions of the type with Fickian diffusion, where the diffusion
coefficients might depend on time, space and on all the concentrations of
the chemical species. In the case of one single reaction, we prove global
existence for space dimensions . In the more restrictive case of
diffusion coefficients of the type , we use an -approach to
prove global existence for . In the general case of networks of such
reactions we extend the previous method to get global solutions for general
diffusivities if and for diffusion of type if .
In the latter quasi-linear case of and for space dimensions
and , global existence holds for more than quadratic reactions. We can
actually allow for more general rate functions including fractional power
terms, important in applications. We obtain global existence under appropriate
growth restrictions with an explicit dependence on the space dimension
Highly accurate numerical computation of implicitly defined volumes using the Laplace-Beltrami operator
This paper introduces a novel method for the efficient and accurate
computation of the volume of a domain whose boundary is given by an orientable
hypersurface which is implicitly given as the iso-contour of a sufficiently
smooth level-set function. After spatial discretization, local approximation of
the hypersurface and application of the Gaussian divergence theorem, the volume
integrals are transformed to surface integrals. Application of the surface
divergence theorem allows for a further reduction to line integrals which are
advantageous for numerical quadrature. We discuss the theoretical foundations
and provide details of the numerical algorithm. Finally, we present numerical
results for convex and non-convex hypersurfaces embedded in cuboidal domains,
showing both high accuracy and thrid- to fourth-order convergence in space.Comment: 25 pages, 17 figures, 3 table
Well-posedness analysis of multicomponent incompressible flow models
In this paper, we extend our study of mass transport in multicomponent
isothermal fluids to the incompressible case. For a mixture, incompressibility
is defined as the independence of average volume on pressure, and a weighted
sum of the partial mass densities stays constant. In this type of models, the
velocity field in the Navier-Stokes equations is not solenoidal and, due to
different specific volumes of the species, the pressure remains connected to
the densities by algebraic formula. By means of a change of variables in the
transport problem, we equivalently reformulate the PDE system as to eliminate
positivity and incompressibility constraints affecting the density, and prove
two type of results: the local-in-time well-posedness in classes of strong
solutions, and the global-in-time existence of solutions for initial data
sufficiently close to a smooth equilibrium solution
A Kinematic Evolution Equation for the Dynamic Contact Angle and some Consequences
We investigate the moving contact line problem for two-phase incompressible
flows with a kinematic approach. The key idea is to derive an evolution
equation for the contact angle in terms of the transporting velocity field. It
turns out that the resulting equation has a simple structure and expresses the
time derivative of the contact angle in terms of the velocity gradient at the
solid wall. Together with the additionally imposed boundary conditions for the
velocity, it yields a more specific form of the contact angle evolution. Thus,
the kinematic evolution equation is a tool to analyze the evolution of the
contact angle. Since the transporting velocity field is required only on the
moving interface, the kinematic evolution equation also applies when the
interface moves with its own velocity independent of the fluid velocity. We
apply the developed tool to a class of moving contact line models which employ
the Navier slip boundary condition. We derive an explicit form of the contact
angle evolution for sufficiently regular solutions, showing that such solutions
are unphysical. Within the simplest model, this rigorously shows that the
contact angle can only relax to equilibrium if some kind of singularity is
present at the contact line. Moreover, we analyze more general models including
surface tension gradients at the contact line, slip at the fluid-fluid
interface and mass transfer across the fluid-fluid interface.Comment: 25 pages, 6 figures; accepted manuscript
- …