On the similarities between the simplified Phan-Thien–Tanner model and the finitely extensible nonlinear elastic dumbbell (Peterlin closure) model in simple and complex flows

Abstract

For many commonly used viscoelastic constitutive equations, it is well known that the limiting behavior is that of the Oldroyd-B model. Here, we compare the response of the simplified linear form of the Phan-Thien–Tanner model (“sPTT”) [Phan-Thien and Tanner, “A new constitutive equation derived from network theory,” J. Non-Newtonian Fluid Mech. 2, 353–365 (1977)] and the finitely extensible nonlinear elastic (“FENE”) dumbbell model that follows the Peterlin approximation (“FENE-P”) [Bird et al., “Polymer solution rheology based on a finitely extensible bead—Spring chain model,” J. Non-Newtonian Fluid Mech. 7, 213–235 (1980)]. We show that for steady homogeneous flows such as steady simple shear flow or pure extension, the response of both models is identical under precise conditions ([Formula: see text]). The similarity of the “spring” functions between the two models is shown to help understand this equivalence despite a different molecular origin of the two models. We then use a numerical approach to investigate the response of the two models when the flow is “complex” in a number of different definitions: first, when the applied deformation field is homogeneous in space but transient in time (so-called “start-up” shear and planar extensional flow), then, as an intermediate step, the start-up of the planar channel flow; and finally, “complex” flows (through a range of geometries), which, although being Eulerian steady, are unsteady in a Lagrangian sense. Although there can be significant differences in transient conditions, especially if the extensibility parameter is small [Formula: see text], under the limit that the flows remain Eulerian steady, we once again observe very close agreement between the FENE-P dumbbell and sPTT models in complex geometries.

Turbulent Taylor–Couette flow of dilute polymeric solutions: a 10-year retrospective

“…However, as demonstrated in Davoodi et al. [57], this ‘re-scaled‘ FENE-P dumbbell model offers two advantages: (i) the relaxation time can be directly determined from the experimentally measured value of the primary normal stress coefficient, and (ii) the analogy with the simplified Phan–Thien–Tanner model. The constitutive equation is solved in combination with the mass conservation equations, and the Navier–Stokes equation.…”

“…In this formulation, λ and Wi need to be adjusted as they do not correspond to the standard molecular definition for the relaxation time based on the Stokes drag coefficient on a sphere and the Hookean spring constant. However, as demonstrated in Davoodi et al [57], this 're-scaled' FENE-P dumbbell model offers two advantages: (i) the relaxation time can be directly determined from the experimentally measured value of the primary normal stress coefficient, and (ii) the analogy with the simplified Phan-Thien-Tanner model. The constitutive equation is solved in combination with the mass conservation equations, and the Navier-Stokes equation.…”

“…For example, with an increase of Re or decrease of El, the turbulent TC flow can be sequentially classified as ET, EDT and finally IDT. The coherent flow structures in these typical turbulent states Song et al [31,25] Zhu et al [56] Lin et al [54] Lee et al [33] Groisman & Steinberg [20] Dutcher & Muller [12] Song et al [26] Latrache et al [28,29] Rajappan & McKinley [34] Latrache & Mutabazi [30] Lacassagne et al [57] Lacassagne et al [58] Wi ~ 10 are displayed in figures 2 and 3. In order to provide a clear mechanistic picture of how variation in Re and Wi affects the flow transitions and identify the unique features of each flow regime, one must perform an in depth analysis of the drag modification, coherent flow structures, velocity and elastic stress statistics, mechanism of turbulent kinetic energy production, spectral features as well as the self-sustaining cycles of turbulence in each of these three distinct flow states.…”

A comparison between the FENE-P and sPTT constitutive models in large-amplitude oscillatory shear

“…Latreche et al (2021) also established analytical solutions for steady, fully developed, flows of the FENE-P and sPTT models in flat and circular ducts using the aforementioned substitution of parameter values. Davoodi et al (2022) then investigated the FENE-P and sPTT models for a range of steady and homogeneous flows, as well as unsteady and inhomogeneous flows. Due to the presence of the Lagrangian derivative term in the stress tensor form of the FENE-P model, significant differences were observed between the FENE-P and sPTT responses for the transient flows.…”

Flow of Viscoelastic Fluid with Microrotation at a Boundary Layer Flow of a Horizontal Circular Cylinder

“…Lastly, there is the BKBZ model which is the integral type of constitutive equations. Among the study that considers these models to represent the flow of viscoelastic fluid for the case of bluff body have been presented by several authors [5][6][7][8].…”