Constitutive Equations Of Differential Type Research Articles

On the similarities of the sPTT and FENE-P models for polymeric fluids

<h2>Abstract</h2> For many commonly used single mode viscoelastic constitutive equations of differential type, it is well known that they share many features. For example, in certain parameter limits the models due to Giesekus, Phan-Thien Tanner and FENE-type models approach the Oldroyd-B model. In this talk, I'll compare the response of the linear form of the simplified Phan-Thien Tanner model [due to Phan-Thien and Tanner, Journal of Non-Newtonian Fluid Mechanics, 2 (1977) 353–365] (the "sPTT") and the Finitely Extensible Nonlinear Elastic model that follows the Peterlin approximation [due to Bird et al., Journal of Non-Newtonian Fluid Mechanics, 7 (1980) 213–235] (the "FENE-P"). I'll show that for steady homogeneous flows such as steady simple shear flow or pure extension the response of both models is identical under certain conditions. For more general flows we show analytically that the results from the two models only formally approach each other when both the polymer concentration and Weissenberg number is small. We then use a numerical approach to investigate the response of the two models when the flow is "complex" under two different definitions: firstly, when the applied deformation field is homogenous in space but transient in time (so called "start-up" shear and planar extensional flow) and then for "complex" flows (through a range of geometries) which, although Eulerian steady, are unsteady in a Lagrangian sense. Under the limit that the flows remain Eulerian steady (so the Weissenberg number is typically small), we see once again a very close agreement between the FENE-P and sPTT models.

Modeling contraction flows and swell of polymer melts extruded from dies of various lengths

AbstractThis article describes isothermal contraction flows and extrudate swell, using our recent model for fast (high Deborah number) contraction flows of polymers melts. The model analyzes the polymer flow in several regions as one continuous process. This approach makes possible to evaluate the extrudate swell as a result of complex contraction polymer flow in dies of various length. Using the asymptotic matching conditions for the change in flow type at the die exit allowed us to calculate the swelling profile for extrudate in the flow direction. The present calculations performed using a multi‐mode viscoelastic constitutive equation of differential type, are compared with the experimental/direct numerical data including basic rheological tests. The presented complex model for contraction flow and swelling consists of no fitting parameter and is applicable for calculations using any viscoelastic constitutive equation. POLYM. ENG. SCI., 2008. © 2008 Society of Plastics Engineers

Bubble oscillation and inertial cavitation in viscoelastic fluids

Non-linear acoustic oscillations of gas bubbles immersed in viscoelastic fluids are theoretically studied. The problem is formulated by considering a constitutive equation of differential type with an interpolated time derivative. With the aid of this rheological model, fluid elasticity, shear thinning viscosity and extensional viscosity effects may be taken into account. Bubble radius evolution in time is analyzed and it is found that the amplitude of the bubble oscillations grows drastically as the Deborah number (the ratio between the relaxation time of the fluid and the characteristic time of the flow) increases, so that, even for moderate values of the external pressure amplitude, the behavior may become chaotic. The quantitative influence of the rheological fluid properties on the pressure thresholds for inertial cavitation is investigated. Pressure thresholds values in terms of the Deborah number for systems of interest in ultrasonic biomedical applications, are provided. It is found that these critical pressure amplitudes are clearly reduced as the Deborah number is increased.

Simulation of linear polymer melts in transient complex flow

Recently, much progress has been made in improving the modelling of linear polymer melts with the aid of reptation theory. In simple shear flows, this has resulted in a much better prediction of the shear viscosity and normal stress ratio. Here, we evaluate in complex flow the transient and steady-state behaviour of a recently proposed reptation model, the Marrucci–Greco–Ianniruberto model [G. Marrucci, F. Greco, G. Ianniruberto, Rheol. Acta, 2000, submitted for publication], that includes convective constraint release and a force balance on the entanglement nodes. To incorporate integral type models into the numerical framework of Lagrangian particle methods, developed previously to simulate dilute polymer solutions, we have included the so-called deformation field method. For the contraction/expansion flow that we consider, we find that a correction of the convective constraint release contribution to the relaxation time is necessary to avoid the unphysical situation of negative relaxation times. With this correction, we could obtain mesh and time convergence for high Weissenberg numbers without adding any solvent viscosity. We find that in complex flow also, both the steady-state and transient response of the integral model can be very well approximated by a constitutive equation of differential type. Due to the dominance of the strong thinning in both shear and elongational flows for the model, however, the inelastic Carreau–Yasuda model reproduces the steady-state kinematics and pressure drop as well.