Dissipation effects in hydrodynamic stability of viscoelastic fluids

Abstract

PIE plo = mole fraction of component I = characteristic time of flow process = characteristic or natural time of fluid = viscosity at zero shear rate = final or equilibrium mass density of solvent in = initial mass density of solvent in sample sample In this paper an analysis is made of the hydrodynamic stability of a Boussinesq viscoelastic fluid undergoing plane Couette flow with a superposed temperature gradient. Of special interest is the effect of including the dissipation term in the energy equation. This term is shown to destabilize the fluid for most values of disturbance wave number and material parameters and to cause overstability for all values of the Brinkman number.At a critical Weissenberg number of 1, a rheological instability is developed which is essentially independent of the Reynolds, Prandtl, and Brinkman numbers. SCOPEOne of the most common and important processing operations in the polymer industry is fiber spinning. This operation is depicted schematically in Figure 1 and consists of many sequential flows, each of which is important in the production of an end product of satisfactory material properties. For the purposes of the research described, the process will be divided heuristically into four regimes:Correspondence concerning this paper should be addressed to Larry V. McIntire. 1.The die entry. Here the polymer melt is forced (by an extruder or other pressure source) from the reservoir into a capillary of very small diameter (relative to reservoir dimensions). The entry flow is very complicated and combines aspects of a complicated shear flow with an accelerated elongational flow. The capillary.Here an approximation to simple shear flow is developed. Viscous dissipation induced temperature effects may be important.3. Die exit. The complex phenomenon of die swell is encountered in this region.

Instabilities in polymer processing

“…The calculations shown in Figure 11 for instability of a Maxwell fluid in a long slit suggest that there should be a breakdown of laminar flow at low Reynolds number for a sufficiently elastic liquid, Such a result is predicted by Gorodstov and Leonov (1967) for plane Couette flow, but critical parameters were not calculated. McIntire ( 1972) and Bonnett and McIntire (1975) have computed a low Reynolds number instability in this flow at a recoverable shear of about one half. The result is independent of heat transfer considerations in the papers, which are discussed in a later section, and appears to be relevant to melt fracture.…”

“…We have already mentioned the work of Bonnett and McIntire ( 1975), who consider the interaction of viscoelasticity, viscous heat generation, and an imposed temperature gradient in plane Couette flow, with the density the only physical property that varies with temperature. This follows a series of papers dealing with simpler aspects of the same problem (McIntire andSchowalter, 1970, 1972;McIntire, 1972) which all involve viscoelasticity and gravity driven convection.…”

Numerical Study of Natural Convection of Power Law Fluid in a Square Cavity Fitted with a Uniformly Heated T-Fin

“…The test spaces for field variables like pressure, velocity and temperature are defined by considering W = H 1 (Ω) 3 to be the test subspaces for u, v, θ and Q = L 2 (Ω) to be the test space for pressure. The weak form of the momentum equation in the x-direction for the whole domain Ω is given as…”

“…A two-dimensional free convective isothermal flow of air enclosed in a horizontal cavity by prescribing temperature flux conditions at the wall was investigated computationally by Newel and Schdmit [2]. Bonnett and Mclntire [3] tested flow stability in the convective flow of the power law and Ellis models by employing the computational technique. They calculated the critical magnitude of Rayleigh numbers for which free convection converts to forced convection.…”