We study a hyperbolic (telegrapher's equation) free boundary problem describing the pressure-driven channel flow of a Bingham-type fluid whose constitutive model was derived in the work of Fusi and Farina (2011). The free boundary is the surface that separates the inner core (where the velocity is uniform) from the external layer where the fluid behaves as an upper convected Maxwell fluid. We present a procedure to obtain an explicit representation formula for the solution. We then exploit such a representation to write the free boundary equation in terms of the initial and boundary data only. We also perform an asymptotic expansion in terms of a parameter tied to the rheological properties of the Maxwell fluid. Explicit formulas of the solutions for the various order of approximation are provided.

In this paper we study the well posedness of a hyperbolic free boundary problem arisen from a one-dimensional model for the channel flow of a rate-type fluid with stress threshold presented in [

In particular, in [

The main practical motivation behind this study comes from the analysis of materials like asphalt or bitumen which exhibit a stress threshold beyond which they change its rheological properties. Indeed from the papers [

The mathematical formulation for the channel flow driven by a constant pressure gradient consists in a free boundary problem involving a hyperbolic telegrapher's equation (Maxwell fluid) and a third-order equation (Oldroyd-B fluid). The free boundary is the surface dividing the two domains: the inner channel core and the external layer. Due to the high complexity of the general problem, here we have considered a simplified version which arises when the order of magnitude of some physical parameters involved in the general model ranges around particular values. In such a case we have that the velocity of the inner core is constant in space and time, while the outer part behaves as a viscoelastic upper convected Maxwell fluid (see [

The paper is structured as follows. In Section

The interesting aspect of the mathematical analysis lies on the technique we employ to reduce the complete problem to a single integrodifferential equation from which some mathematical properties can be derived (the free boundary equation can be solved autonomously from the governing equation of the velocity field). Such a methodology is a generalization of a technique already introduced in [

In Section

In this section we state the mathematical problem. We refer the reader to [

We consider an orthogonal coordinate system

Upper part of the channel.

The mathematical model is written for the velocity field

The nondimensional formulation is the following:

In the case of asphalt typical values are (see [

Taking

In [

Before proceeding in proving analytical results of problem (

where

The domain of problem (

Sketch of the domain.

The domain

To determine the solution of problem (

By means of (_{(1)} is the modified Bessel equation of zero order. The solution of (

one can prove that_{(3)}, (

Let us now write a representation formula for _{(4)}, that is,

The domain

Due to the regularity of the kernel

We notice that, considering the representation formulae (

We now write the representation formula for _{(5)}).

Given a point _{(5)}, getting_{(1–5)} is given by (_{(6)} to determine the evolution equation of the free boundary _{(5)} in (_{(6)}, obtaining

The domain

Next we remark that (

The function

If we assume that the free boundary equation (_{(6)} holds up to

If one assumes that compatibility condition

If we suppose that (

Let us consider the limit case in which

In this section we look for a solution to problem (_{(1)} in the following form:

We do not discuss the issue of the convergence of series (_{(1)} into (

and the following free boundary problems

At the first order (see problem (_{(6)} and take the limit for

which, when

For the generic _{(6)}, obtaining

which, in the limit

Before proceeding further we suppose that

(H1)

(H2)

(H3)

(H4)

We introduce the new variable

with _{(1)} we realize that (

We now have to solve the problem

and we have to solve the following Cauchy problem:_{(1)} is a Bernoulli equation. Therefore, setting

which, by virtue of(_{(1)}, is equivalent to require that

We now consider here the _{(1)} is once again a Bernoulli equation which can be integrated providing

Proceeding as before we derive the conditions ensuring

We have studied a hyperbolic (telegrapher's equation) free boundary problem derived from the model for a pressure-driven channel flow of a particular Bingham-like fluid described in [

We have shown that local existence and uniqueness is guaranteed under some appropriate assumptions on the initial and boundary data (Theorem

A further extension of the analysis we have performed in this paper (which is a limit case of the model presented in [