# Preprint 2013-002

# A two-fluid four-equation model with instantaneous thermodynamical equilibrium

## Alexandre Morin and Tore Flåtten

**Abstract:**
We analyse a four-equation version of a common two-fluid model for pipe flow,
containing one mixture mass equation and one mixture energy equation.
The motivation is to obtain a fluid-dynamical model
where the mixture is in thermodynamical equilibrium at all time.
We start from a five-equation model with instantaneous thermal equilibrium,
to which we add phase relaxation terms.
An interfacial velocity appears, for which we give an expression
based on the second law of thermodynamics.
We then derive the limit of this model
when the relaxation becomes instantaneous.
The time derivatives appearing in this process are subsequently transformed
into spatial derivatives to be able to use
numerical methods for conservation laws.
The Jacobian matrix of the fluxes can then be evaluated,
and the system be put into quasilinear form.
From the Jacobian matrix, we are able to extract
the sound speed intrinsic to the model.
By comparison to the sound speed in other two-phase flow models,
we extend some previous results showing that
the effect of relaxation on sound speed
is independent of the order in which the variables are relaxed.
We also check the subcharacteristic condition
and place the model in a hierarchy of two-phase flow models.
Finally, this model requires a regularisation term to be hyperbolic.
With the help of a perturbation method,
we find an expression for this term
that makes the model conditionally hyperbolic.