Thermodynamically Consistent Modeling for Dissolution/Growth of Bubbles in an Incompressible Solvent

Abstract

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.

Well-posedness analysis of multicomponent incompressible flow models

“…Theorem 2. In the situation of Theorem 1 we assume, in addition, that s > 5 and that k is the function defined in (7). We define a matrix B i, j (ρ) := M i, j (ρ)/ρ j for i, j = 1, .…”

“…Of course, this does not mean that = const cannot be a good approximation under special circumstances. In [7] for instance, a class of multicomponent mixtures has been introduced for which the use of the incompressible Navier-Stokes equation is realistic: Incompressibility is assumed for the solvent only, and diffusion is considered against the solvent velocity. See also the discussion in the paragraph 4.8 of [27] on incompressible mixtures.…”

“…, m N > 0, y i = n i / N j=1 n j are the number fractions, and μ ref i are reference values of the chemical potentials. For the mathematical theory in this paper, more general structures in (7) will however be admitted. The isothermal Gibbs-Duhem equation: dp = N i=1 ρ i dμ i defines the intrinsic relationship between (1), (6), and the pressure field.…”

“…In the special case that the gradient of k is explicitly invertible on S 1 (see (14)), the statements can also be proved by direct algebraic computations yielding in many cases explicit formulae; see Sect. 4 in [14] for a complete characterisation of the example (7). Lemma 4.…”

“…We adopt the assumptions of Theorem 1 for the tensor M : R N + → R N ×N , and we assume that k : R N + → R is given by (7). We assume moreover that there is a continuous function C = C(|ρ|), bounded on compact subsets of R N \{0}, such that B i, j (ρ) := M i, j (ρ)/ρ j , with entries belonging to C 1 (R N + ), satisfies for all ρ ∈ R N + the conditions We next investigate the derivatives.…”