On the Solution of a Hyperbolic One-Dimensional Free Boundary Problem for a Maxwell Fluid
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 . The model describes the one-dimensional flow of a fluid which behaves as a nonlinear viscoelastic fluid if the stress is above a certain threshold and like a rate type fluid if the stress is below that threshold. The problem investigated here belongs to a series of extensions of the classical Bingham model we have proposed in recent years (see [2–5]).
In particular, in  we describe the one-dimensional flow of such a fluid in an infinite channel, assuming that in the outer part of the channel the material behaves as a viscoelastic upper convected Maxwell fluid, while in the inner core as a rate-type Oldroyd-B fluid. The general mathematical model is derived within the framework of the theory of natural configurations developed by Rajagopal and Srinivasa (see ). The constitutive equations are obtained imposing how the system stores and dissipates energy and exploiting the criterion of the maximization of the dissipation rate.
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 [7–9], it is clear that such materials have a viscoelastic behaviour (for instance, upper convected Maxwell fluid) which is observed if the applied stress is greater than a certain threshold (see, in particular, ).
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  for more details). The mathematical formulation turns out to be a hyperbolic free boundary problem which, in the authors knowledge, is new since it involves a telegrapher's equation coupled with an ODE describing the evolution of the interface.
The paper is structured as follows. In Section 2 we formulate the problem, namely, problem (2.1), and specify the basic assumptions. In Section 3 we give an equivalent formulation of the problem which leads to a nonlinear integrodifferential equation for the free boundary. We prove local existence and uniqueness for such an equation (see Theorem 3.3), under specific assumptions on the data.
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 4 we perform an asymptotic expansion in terms of a coefficient (representing the ration between the elastic characteristic time and the relaxation time of the viscoelastic material), which typically is of the order . This procedure allows to obtain approximations of the actual solution up to any order through an iterative procedure. We do not prove the convergence of the asymptotic approximations to the actual solution (whose existence is proved in Theorem 3.3), limiting ourselves to develop only the formal procedure. Indeed, the main advantage of this procedure is that, for each order of approximation, the governing equation for the velocity field is the “standard” wave equation, which is by far easier to handle than the telegrapher's equation. We end the paper with few conclusive remarks.
2. Mathematical Formulation
In this section we state the mathematical problem. We refer the reader to  for all the details describing how this simplified model was derived from the general one. In the general case, in the region , the fluid behaves as an Oldroyd-B type fluid. The problem we are studying here is a particular case of such a model, which stems from some specific assumptions on the physical parameters (fulfilled by materials like asphalt and bitumen). Under such assumptions, the inner core moves with uniform constant velocity .
We consider an orthogonal coordinate system and assume that the fluid is confined between two parallel plates placed at distance . We assume that the motion takes place along the -direction and that the velocity field has the form . We rescale the problem (in a nondimensional form) and, because of symmetry, we study the upper part of the layer (the space variable is rescaled by ). The geometry of the system we investigate is depicted in Figure 1.
The mathematical model is written for the velocity field in the viscoelastic region which is separated from the region with zero strain rate (uniform velocity) by the moving interface .
The nondimensional formulation is the following: where (i) is the material density, (ii) is the viscosity of the fluid, (iii) is the elastic modulus, (iv) is a positive parameter depending on the viscosity (see ), (v) is the Bingham number, (vi) is the velocity of the inner core, (vii), (viii) is the characteristic elastic time, (ix) is the relaxation time.
In the case of asphalt typical values are (see [8, 9])
Taking we get
Remark 2.1. In  we have proved that problem (2.1) admits a stationary solution provided and that the stationary solution is given by
3. An Equivalent Formulation
Before proceeding in proving analytical results of problem (2.1) we introduce the new coordinate system and the new variable With transformations (3.1)-(3.2), problem (2.1) becomes
where Notice that, by means of (3.2), the evolution equation for the new variable has become a nonhomogeneous Klein-Gordon equation .
The domain of problem (3.3) is depicted in Figure 2. We begin by considering the domain (see Figure 3). Here the solution has the representation formula (see ) where is the Riemann's function that solves the problem (see again Figure 3)
To determine the solution of problem (3.6) we set where
By means of (3.7) problem (3.6) becomes where (3.9)(1) is the modified Bessel equation of zero order. The solution of (3.9) is given by where is the modified Bessel function of zero order. It is easy to prove that the function defined by (3.10) satisfies problem (3.6). Moreover, since 
one can prove that where is the modified Bessel function of second order. Recalling that and, by (3.3)(3), (3.4), thatwe see thatand representation formula (3.5) can be rewritten as
Let us now write a representation formula for in the domain (see Figure 4). We once again make use of (3.6), where now has to be extended to the domain . Following , we extend imposing condition (3.3)(4), that is, . From the representation formula we get where is the coordinate of the intersection of the characteristic with . Relation (3.16) can be rewritten as From (3.18), the extended function , defined in , fulfills the following Volterra integral equation of second type: Equation (3.19) can be put in the more compact formwhere and
Due to the regularity of the kernel the function (which can be determined using the iterated kernels method, ) is a smooth function. Thus we extend as and the solution in the domain is given by
Remark 3.1. We notice that, considering the representation formulae (3.5) and (3.23), Moreover, taking the first derivatives (with respect to time and space ) of for the domains and it is easy to prove that, assuming the compatibility condition , where the derivatives in limits on the l.h.s. of (3.25) are evaluated using (3.5), while the ones on the r.h.s. using (3.23). This implies that the solution is across the characteristic , that is, the line that separates the domains and .
We now write the representation formula for in the domain . We proceed as in  assuming that the velocity of the free boundary is less than the velocity of the characteristics (i.e., ) and extending to the domain (see Figure 2) in a way such that (i.e., imposing the free boundary condition (3.3)(5)).
Given a point in the domain we define the point as the intersection of the characteristic (with negative slope) passing from and the free boundary (see Figure 5). It is easy to check that We consider once again the representation formula (3.5) and impose condition (3.3)(5), gettingFrom (3.28) we see that the extension to the domain is the solution of the following Volterra integral equation of second kind: Recalling (3.26) and proceeding as for the domain , the above can be rewritten aswhere and Once again the regularity of the kernel ensures the regularity of the solution . The function can thus be defined in the interval as The solution in the domain is thus given by where for simplicity of notation we have introducedand where is given by (3.32). Therefore for any fixed function with we have that the solution to problem (3.3)(1–5) is given by (3.5), (3.23), (3.33) with defined by (3.32). At this point we make use of (3.3)(6) to determine the evolution equation of the free boundary . We begin writing the derivatives and . To this aim we exploit formula (3.33) since and have to be evaluated on , which belongs to domain . Differentiating (3.33) with respect to we getwhile, differentiating (3.33) with respect to , we obtain Notice that Now we evaluate (3.35) and (3.36) on the free boundary , that is, At this point we insert (3.38), (3.3)(5) in (3.3)(6), obtaining which is a nonlinear integrodifferential equation of the first order and where is defined by (3.32). Equation (3.39) is the free boundary equation which, as we mentioned in the introduction, does no longer depend on the velocity field .
Next we remark that (3.39) can be further simplified. Indeed, recalling (3.10) and (3.12), so that, on , we have while, on Hence (3.39) reduces to
Remark 3.2. The function is continuous across the characteristic . Indeed where the limit is evaluated using (3.33) and the limit using (3.5) or (3.23). If we evaluate the derivatives and on the characteristic we get two different results depending on whether we are evaluating such derivatives in or . We can prove that where . It is easy to check that, imposing that (3.45) equals (3.46) and that (3.47) equals (3.48), we get the following condition: which is the condition that must be fulfilled if we want the first derivatives of to be continuous across the characteristic .
If we assume that the free boundary equation (3.3)(6) holds up to we get Moreover, from (3.43) we have that, when , We can therefore prove the following.
Theorem 3.3. If one assumes that compatibility condition and hypotheses (3.49), (3.50), (3.51) hold, then necessarily either or and problem (3.3) admits a unique local solution , such that . If one does not assume hypothesis (3.49) (meaning that the first derivatives of are not continuous along the characteristic ), then problem (3.3) admits a unique local solution , such that if and only if
Proof. If we suppose that (3.49), (3.50), (3.51) hold then we have The initial velocity is not physically acceptable since existence of a solution requires that . Therefore and, recalling (3.49), we have either or , since . If, on the other hand, we suppose that condition (3.49) does not hold, but we assume (3.52), then it is easy to show that so that for a sufficiently small time there exists a unique solution with . The existence of such a solution can be proved using classical tools like iterated kernels method (see ).
Remark 3.4. Let us consider the limit case in which and . In this particular situation the Riemann's function and the solution is given by and the free boundary equation is the characteristic with positive slope passing through , that is, namely, So, setting , for , the region with uniform velocity (the inner core) has disappeared. For , the solution is thus found solving where and are determined evaluating (3.55) at time . To solve problem (3.58) we introduce the new variable and rescale time with Problem (3.58) becomes whose solution is  where
4. Asymptotic Expansion
In this section we look for a solution to problem (3.3)(1) in the following form: This allows to obtain a sequence of problems for each with the free boundary being given by . (We remark that the sequence is not, in general, an asymptotic sequence.) Such an analysis is motivated by the fact that, in practical cases (asphalt and bitumen), (see (2.3)). Hence, it makes sense to look for a “perturbative” approach for the system (3.3).
We do not discuss the issue of the convergence of series (4.1) and of the sequence , which is beyond the scope of the present paper. We limit ourselves to a formal derivation of the free boundary problems that can be obtained plugging (4.1)(1) into (3.3): Hence, for each , we have
and the following free boundary problems We immediately remark that, in each problem, the governing equation is no longer a telegrapher's equation, but a nonhomogeneous wave equation. Hence, using classical d'Alembert formula, we can write the representation formula for each domain , , and for each order of approximation . In particular, in we have while in and in Proceeding as in Section 3 we can show that the evolution equations of the free boundary at each step are given by At the zero order, assuming the compatibility condition (see problem (4.4)), we have
At the first order (see problem (4.5)), we assume that the compatibility condition of second order holds in the corner . This means that we can differentiate the free boundary equation (4.5)(6) and take the limit for . We have
which, when , reduces to
For the generic th order (see problem (4.6)), we assume that the compatibility conditions in the corner hold up to order . Therefore we can take the derivative of (4.6)(6), obtaining
which, in the limit , reduces to We therefore conclude that, assuming enough regularity for each problem , (4.10)–(4.12) posses a unique local solution with and .
Before proceeding further we suppose that has the following properties:
(H2) , ,
(H3) and ,
(H4) satisfies all the compatibility conditions up to any order in the corner .
4.1. Zero-Order Approximation
We introduce the new variable , so that (4.10) can be rewritten as
with . Then we look for the solution which fulfills the following Cauchy problem: that is, Recalling that , that is, from (4.19)(1) we realize that (4.21) is fulfilled if Therefore, under hypothesis (4.22), local existence of a classical solution is guaranteed. Such a solution is given by , where is determined inverting (4.20).
4.2. First-Order Approximation
We now have to solve the problem with . Proceeding as in Section 4.1 we introduce the new variable , so that (4.23) becomes
and we have to solve the following Cauchy problem: We notice that (4.25)(1) is a Bernoulli equation. Therefore, setting , problem (4.25) becomes whose solution is given by which make sense only if . Integrating (4.27) by parts we get We recall from the previous section that the condition is guaranteed if
which, by virtue of(4.25)(1), is equivalent to require that Hence, under assumption , the discriminant , and (4.30) is fulfilled when Therefore, in order to have a unique local solution, we must require that which, exploiting (4.28), becomes The latter is automatically satisfied, under assumption , recalling that . So, also for the first order we have local uniqueness and existence of the solution , where is obtained inverting (4.28).
4.3. th-Order Approximation
We now consider here the th-order approximation. The evolution equation of the free boundary is given by (4.12). Proceeding as in the previous sections we set , so that (4.12) can be rewritten as with . Once again we look for , solving this Cauchy problem Now, hypothesis and (4.17) entailTherefore So for sufficiently small, we can approximate the integral on the r.h.s. of (4.37) in the following way: where is a smooth function of , determined exploiting (4.7), (4.8), and (4.9). In particular,