Turbulent stretching of dumbbells with hydrodynamic interactions: an analytical study

TL;DR

This work analyzes how hydrodynamic interactions (HI) between beads affect the stretching of a dumbbell in a fluctuating flow. Using the Batchelor-Kraichnan model, the authors derive and solve a Fokker-Planck equation for the end-to-end separation, obtaining the stationary extension PDF and polymer stress, and they compare different HI tensors (OB, RPY, ZO). A key finding is that HI preferentially enhance stretching for stiff dumbbells (low $\mathrm{Wi}$) and smooth the coil-stretch transition when HI are treated with the consistently averaged approximation, with quantitative agreement to Brownian dynamics simulations in direct numerical simulations of turbulence. The results suggest HI have only modest effects in turbulent stretching and can often be neglected in multiscale simulations, while providing a principled framework for incorporating HI through averaged theories. The study also demonstrates that the consistently averaged HI approach captures essential Wi- and $b$-dependent behaviors, offering practical guidance for modeling polymer solutions in chaotic flows.

Abstract

We study the stretching of an elastic dumbbell in a turbulent flow, with the aim of understanding and quantifying the effect of hydrodynamic interactions (HI) between the beads of the dumbbell. Adopting the Batchelor-Kraichnan model for the flow, we derive a Fokker-Planck equation and solve it analytically to obtain the probability distribution of the dumbbell's extension. Using different formulations of the HI tensor, we find that HI preferentially enhances the stretching of stiff dumbbells, i.e., those with a small Weissenberg number. We also evaluate the averaging approximations commonly used to simplify the description of HI effects; the consistently-averaged approximation shows that HI result in a less-pronounced coil-stretch transition in chaotic flows. Finally, we confirm the relevance of our analytical results by a comparison with Brownian dynamics simulations of dumbbells transported in a direct numerical simulation of homogeneous isotropic turbulence.

Figures (6)

• Figure 1: (a) Increase of the polymer mean square extension as a function of the Weissenberg number, for different values of the hydrodynamic-interaction parameter. The inset shows the mean square extension rescaled with its value for $h^*=0$, denoted as $\langle R^2\rangle_0$; the color code is the same as in the main plot. (b) Increase of the mean polymer stress as a function of the Weissenberg number, for different values of the hydrodynamic-interaction parameter. The RPY hydrodynamic-interaction tensor is used for both panels and the extensibility parameter is $b=1200$. The scaling factors $R^2_\mathrm{eq}$ and $\overline{\mathsf{T}}_\mathrm{eq}$ are the equilibrium values of the mean square extension and the polymeric stress.
• Figure 2: (a) Stationary PDF of the polymer extension for different values of $h^*$ and $\mathrm{Wi}=0.3$ (main plot) and $\mathrm{Wi}=3.0$ (inset). The RPY hydrodynamic-interaction tensor is used to model HI and the extensibility parameter is $b=1200$. To facilitate a comparison, the PDFs are plotted over a limited range of $R$. (b) The same PDFs as in the main plot of panel (a) in double logarithmic scale. The dashed lines indicate the slope of the PDF for $\mathrm{Wi}=0.3$ and $h^*=0$ as given by Eq. \ref{['eq:pdf_hstar_0']}.
• Figure 3: (a) Stationary PDF of the extension for $b=1200$, $h^*=0.25$, $\mathrm{Wi}=0.1,0.3$, and different forms of the hydrodynamic-interaction tensor. For the OB tensor, $3a/2 \leq R \leq R_\mathrm{max}$. (b) Stationary PDF of the extension for the OB and RPY tensors, when reflecting boundary conditions are applied at $R=3a/2$ for both cases. The parameters for this panel are $b=1200$, $h^*=0.25$, $\mathrm{Wi}=0.1$.
• Figure 4: (a) Variation of the prefactor in Eq. \ref{['eqn:effWi']} with Wi for different values of $b$ and $h^*$. (b) Increase of the polymer mean square extension as a function of Wi for the RPY tensor, and the preaveraged and consistently averaged versions of the OB tensor. Here $b=1200$ and $h^*=0.25$. The inset shows the mean polymer stress vs Wi for the same parameters and hydrodynamic-interaction tensors.
• Figure 5: (a) PDF of the rescaled polymer extension for the RPY tensor and the preaveraged and consistently averaged versions of the OB tensor. The parameters are $b=1200$, $h^*=0.25$, and $\mathrm{Wi}=0.3$. The inset shows the same PDFs in double logarithmic scale; the dashed line indicates the slope of the PDF for same $b$ and $\mathrm{Wi}$, but $h^*=0$, as given by Eq. \ref{['eq:pdf_hstar_0']}. (b) The same as panel (a) with $\mathrm{Wi}=3$.