### A Preconditioning Mass Matrix to Avoid the Ill-Posed Two-Fluid Model

#### Abstract

Two-fluid models are central to the simulation of transport

processes in two-phase homogenized systems. Even though this physical

model has been widely accepted, an inherently non-hyperbolic and

non-conservative ill-posed problem arises from the mathematical point

of view. It has been demonstrated that this drawback occurs even for a

very simplified model, i.e., an inviscid model with no interfacial

terms. Lots of efforts have been made to remedy this anomaly and in

the literature two different types of approaches can be found. On one

hand, extra terms with physical origin are added to model the

interphase interaction, but even though this methodology seems to be

realistic, several extra parameters arise from each added term with

the associated difficulty in their estimation. On the other hand,

mathematical based-work has been done to find the way to remove the

complex eigenvalues obtained with two-fluid model

equations. Preconditioned systems, characterized as a projection of

the complex eigenvalues over the real axis, may be one of the choices.

The aim of this paper is to introduce a simple and novel mathematical

strategy based on the application of a preconditioning mass matrix

that circumvents the drawback caused by the non-hyperbolic behavior of

the original model. Although the mass and momentum conservation

equations are modified, the target of this methodology is to present

another way to reach a steady state solution (using a time marching

scheme), greatly valued by researchers in industrial process

design. Attaining this goal is possible because only the temporal term

is affected by the preconditioner. The obtained matrix has two

parameters that correct the non-hyperbolic behavior of the model: the

first one modifies the eigenvalues removing their imaginary part and

the second one recovers the real part of the original

eigenvalues. Besides the theoretical development of the

preconditioning matrix, several numerical results are presented to

show the validity of the method. [To appear in Journal of Applied Mechanics]

processes in two-phase homogenized systems. Even though this physical

model has been widely accepted, an inherently non-hyperbolic and

non-conservative ill-posed problem arises from the mathematical point

of view. It has been demonstrated that this drawback occurs even for a

very simplified model, i.e., an inviscid model with no interfacial

terms. Lots of efforts have been made to remedy this anomaly and in

the literature two different types of approaches can be found. On one

hand, extra terms with physical origin are added to model the

interphase interaction, but even though this methodology seems to be

realistic, several extra parameters arise from each added term with

the associated difficulty in their estimation. On the other hand,

mathematical based-work has been done to find the way to remove the

complex eigenvalues obtained with two-fluid model

equations. Preconditioned systems, characterized as a projection of

the complex eigenvalues over the real axis, may be one of the choices.

The aim of this paper is to introduce a simple and novel mathematical

strategy based on the application of a preconditioning mass matrix

that circumvents the drawback caused by the non-hyperbolic behavior of

the original model. Although the mass and momentum conservation

equations are modified, the target of this methodology is to present

another way to reach a steady state solution (using a time marching

scheme), greatly valued by researchers in industrial process

design. Attaining this goal is possible because only the temporal term

is affected by the preconditioner. The obtained matrix has two

parameters that correct the non-hyperbolic behavior of the model: the

first one modifies the eigenvalues removing their imaginary part and

the second one recovers the real part of the original

eigenvalues. Besides the theoretical development of the

preconditioning matrix, several numerical results are presented to

show the validity of the method. [To appear in Journal of Applied Mechanics]