Stretching of Polymers in Isotropic Turbulence: A Statistical Closure

We present a new closure for the mean rate of stretching of a dissolved polymer by homogeneous isotropic turbulence. The polymer is modeled by a bead-spring-type model (e.g., Oldroyd B, FENE-P, Giesekus) and the analytical closure is obtained assuming the Lagrangian velocity gradient can be modeled as a Gaussian, white-noise stochastic process. The resulting closure for the mean stretching depends upon the ratio of the correlation time for strain and rotation. Additionally, we derived a second-order expression for circumstances when strain and rotation have a finite correlation time. Finally, the base level closure is shown to reproduce results from direct numerical simulations by simply modifying the coefficients.

Figure 1

Plot of $\mathrm{Tr}(\mathbf{\Lambda })/[\mathrm{Tr}(\overline{\mathsf{C}})\mathrm{W}{\mathrm{e}}_{\Omega }]$ vs $\Omega$ at fixed ${\mathrm{We}}_{\Omega }=10$. Straight horizontal line is white-noise prediction: $3{\alpha }_1+{\beta }_1=2/3$, and the dashed line corresponds to Eq. (9). Symbols are Brownian dynamics simulations; error bars indicate 1 standard deviation, except where they are smaller than the symbol.

Figure 2

Plot of $Y\equiv \mathrm{Tr}(\mathbf{\Lambda })/[\mathrm{Tr}(\overline{\mathsf{C}})]$ vs We from direct numerical simulations. The solid line is a fit through the data ($Y=0.1822+0.3952\mathrm{We}$).

Figure 3

Plot of $\mathrm{Tr}(\mathbf{\Lambda })$ vs $t/{\tau }_{\eta }$, where ${\tau }_{\eta }$ is the Kolmogorov time scale. The lines are from direct numerical simulations and the points are Eq. (10) with the tuned coefficients.