Thermomechanical Equations Governing a Material With Prescribed Temperature-Dependent Density With Application to Nonisothermal Plane Poiseuille Flow

Abstract

The standard practice in the literature for modeling materials processing in which changes in temperature induce significant volume changes is based on the a posteriori substitution of a temperature-dependent expression for density into the governing equations for an incompressible material. In this paper we show this ad hoc approach misses important terms in the equations, and by example show the ad hoc equations fail to capture important physical effects. First we derive the three-dimensional equations which govern the deformation and heat transfer of materials with prescribed temperature-dependent density. Specification of density as a function of temperature translates to a thermomechanical constraint, in contrast to the purely mechanical incompressibility constraint, so that the constraint response function (“pressure”) enters into the energy equation as well as the momentum equation. Then we demonstrate the effect of the correct constraint response by comparing solutions of our thermomechanical theory with solutions of the ad hoc theory in plane Poiseuille flow. The differences are significant, both quantitatively and qualitatively. In particular, the observed phenomenon of expansion cooling is captured by the thermomechanically constrained theory, but not by the ad hoc theory.

Ill-Posedness in a Thermomechanically Consistent Constrained Theory for Materials With Prescribed Temperature-Dependent Density

“…A general proof can be found in (Rooney et al 1999) for thermomechanical processes satisfying the Gibbs conditions (Gibbs 1873(Gibbs , 1878. The local illposedness of the constrained theory (Cao et al 1996) found in this study certainly contradicts both the unconstrained theory and the common experience with nonisothermal viscous uids. Consequently, its direct use in transient ow phenomenon is not advised.…”

“…1 Introduction Cao et al (1996) developed a thermomechanically consistent constrained theory for materials with prescribed temperature-dependent density. This theory successfully captures the expansion cooling phenomenon observed in nonisothermal steady state Poiseuille ows of viscous uids (Winter 1977, Cao et al 1996 while the ad hoc theories (Kase and Matsuo 1965, Sabhapathy and Cheng 1986, Hayashi et al 1992, where a temperature-dependent density is a posteriori inserted into the classical theory for incompressible materials (Bird et al 1987), fail. In this regard, the constrained theory of Cao et al's marks a notable theoretical advance in modeling steady state nonisothermal ows.…”

“…Evolutionary equations exhibiting this behavior are often said to be locally illposed (Joseph 1990). The constrained theory of Cao et al (1996) is thus locally illposed.…”

Fluid Mechanics and Rheology of Liquid Fiber Flows: Fundamental Science and Technology Applications

“…This modeling advance, combined with the experimental advances described above, pushes our resolution of dynamic surface tension to fractions of a millisecond. An important result in [2] is the following: The rate and form of the decay can be inferred from downstream measurements, avoiding the need for measurements near the nozzle where rapid surface tension decay takes place. In [12] the equations for oscillating jets are extended to allow for shear-rate dependent viscosity.…”

“…Thermally induced shrinkage is an important feature of this process, and the state-of-the art modeling of the effect was a posteriori insertion of a temperature-dependent density function into the equations governing an incompressible fluid. In [2] we showed that this approach is thermodynamically inconsistent (it violates the second law), and derived, through the theory of a thermomechanically constrained material, a model which does not violate the second law. When applying this thermomechanically consistent model to pipe flows in that paper and [1], we predicted features qualitatively different than the commonly employed ad hoc model, in agreement with predictions based on the fully compressible theory.…”