Flow of mechanically incompressible, but thermally expansible viscous fluids A. Mikelic, A. Fasano, A. Farina Montecatini, Sept

LECTURE 1. Basic mathemathical modelling LECTURE 2. Mathematical problem LECTURE 3. Stability

Viscous fluids, such as molten glass, in many processes are mechanically incompressible but experience significant thermal induced volume change. In previous lectures we focus on models for viscous fluids that impose this behavior through a posited internal constraint, following the formalism which introduces constraint responses that do not produce entropy. The internal constraint approach yields simpler constitutive relations and therefore easier material characterization. The internal constraint theory preserves consistency with the thermodynamical balance laws, second law but there is no a priori guarantee that stability of the rest state is preserved. Indeed the rest state is not stable.

In this lecture we will analize the stability of the rest state. Stability of the rest state means that No energy or momentum sources

Following standard methods (see the original paper by Gibbs [1] and the recent paper by Bechtel [2]) there are three necessary and sufficient conditions for stability of the rest state: 1.Convexity of Gibbs free energy stability 2.Nonnegativity of C v 3.Nonnegativity of the of the so-called bulk modulus [1]. Gibbs, Collected Works, Am. J. Sci [2]. Bechtel, Roony, Forest, Int. J. Eng. Sci., 2004

The constitutive model we have developed is:

Let us analyze the rest state stability conditions: Condition 3: Nonnegativity of the of the so-called bulk modulus But in the developed theory there is no state equations that link p with the other variables. Such a condition is not clear !

But C v is not defined !! Only C p is defined. ….so ? ? Condition 2: Nonnegativity of C v

Condition 1: Convexity of Gibbs free energy stability Convexity conditions: OK Can be OK NO unless

….. But, may be,  depends also on pressure p !! Answer:  cannot depend on pressure. Indeed, assuming  p), the Clausius-Duhem inequality rewrites as Now, this inequality must hold true for all thermo-mechanical processes. Thus, it entails:

Remark According to classical thermodynamics the Gibbs free energy is the Legendre transformation of the Helmoltz free energy, i.e. with But in our case  does not depend on  and there is no state equation for p.  So also condition 1, which appeals to Gibbs free energy, seems not very clear !

Conclusions Although Condition 1 suggests that the rest state is instable, it is not clear if such a method is applicable to our model. Question: How do we proceed ? We try direct analysis

Rest state linear stability Summarizing, once stated the physical assumptions, the mathematical Model describing the dynamics of the fluid is the following 1 1 Viscosity  and thermal conductivity K have been considered not depending on temperature

Operating the scaling illustrated in lecture 2, the model can be rewritten as (1) Recall

Non-dimensional characteristic numbers We may write

When the model is used for simulating slow flows of very viscous heated fluids (melted glasses, melted polymer, etc.), often happens that K  is small and the Eckert number is very small (frequently smaller than ). Consequently, the terms containing the Eckert number are dropped in applications and the system reduces to (2) In the sequel we will analyze the linear stability of sistem (1) and (2)

Let us consider the fluid filling all the space and the rest state corresponding to the fluid at rest subject only to a uniform pressure and to isothermal conditions. We now study the linear stability of S, in absence of any body forces and heat sources.

Let us first linearize the equations considering small (dimensionless) perturbations of the state S, namely In particular, we have 2 2 As before, for simplifying our analysis we consider K  constant

The linearized form of the system (1) is

We now take the Fourier transform with respect to the spatial variable x of the dependent variable v *, p * and  . We recall that the Fourier transform, with respect to the spatial variable of a function f ( x; t) is defined by So,

Since

with

We have

Therefore we are left with with

We have to evaluate the eingenvalues of the matrix. Denoting them by i, i = 1; 2; 3 we have Now, since the secular equation is a third degree equation in ¸ whose coefficients are real, at least one eigenvalue is real. The above relations implies therefore that there exists at least a positive eigenvalue. We thus conclude that the state S is instable.

We now consider the linear stability of system (2). Proceeding as before

W In particular, and so Concerning p, we have p 0 as t

We therefore conclude that, if the dynamics is governed by system (2), then the state S is linearly stable.

Proposed remedies 1. Bechtel and coworkers proposed the constraint  (s). Actually such a constraint was studied by Scott [3] in the context of linear thermoelasticity. 2.Rajagopal [4] and coworkers propose to first develop the constitutive theories for unconstrained materials, eliminating, if necessary, pathological behavior by suitable restrictions on the functions (convexity, positivity definiteness, etc. ) and then obtaining the constrained for by a carefully chosen limiting process [3] Scott 1992, Q.J. Mech. Appl. Math. [4] Rajagopal et Al., Proc. R. Soc. Lond. (2004)