Stretching and breaking of particles in compressible random flows

TL;DR

The paper tackles how small-scale flow compressibility alters the extensional dynamics of tiny deformable particles in random flows. It develops a vector-based particle model and connects stationary size statistics to the full statistics of the strain rate via generalized Lyapunov exponents $L(q)$, deriving exact and scaling results for the delta-correlated BK flow and validating them with time-correlated renewing-flow simulations. A key finding is that increasing compressibility suppresses mean stretching but amplifies extreme stretching events, producing a power-law tail in the size distribution; notably, stiff particles break faster under higher compressibility, while highly elastic particles stretch less. The work highlights the central role of the entire strain-rate fluctuation spectrum (not just the mean) in predicting breakup statistics and provides a framework applicable to surface and bulk suspensions, with potential extensions to DNS and more realistic flows.

Abstract

A key feature of turbulent suspensions that involve floating particles on the surface or inertial particles in the bulk is the compressibility of the effective particle-phase velocity field. Little, however, is known about the effects of small-scale flow compressibility on the stretching and breaking of particles. Here, we gain insight into the nature of these effects by studying the deformation of tiny particles in model fluctuating flows. We consider a generic particle with extensional dynamics that are governed by a vector model, which accounts for elasticity, internal viscosity, and non-affine deformation. Applying the dynamical systems approach of Balkovsky, Fouxon & Lebedev (2000), we first obtain general results for the stationary statistics of particle extension in compressible chaotic flows. We then specialize to a time-decorrelated Gaussian random flow and derive an exact solution for the Batchelor regime of the compressible Kraichnan model. We also perform numerical simulations for a time-correlated renewing flow. While straining is suppressed on the average in compressible flows, our results show that large deviations of the strain rate strongly stretch particles and give rise to a power-law distribution of extensions. Extreme straining events are particularly important for stiff particles and, in the examples considered here, give rise to a counter-intuitive effect: stiff particles stretch more and break faster in flows of increasing compressibility. Highly-elastic particles, whose deformation is dictated by the mean straining, stretch less and break slower. Though based on specific random flows, our work shows how compressibility can affect the extensional dynamics of particles by altering the fluctuations of the strain rate, including its large deviations.

Figures (6)

• Figure 1: Graphical construction of the solution of \ref{['eq:alpha_l']} in ( a) the incompressible regime ($\wp=0$), ( b) the weakly compressible regime ($0<\wp<\wp_c$), and ( c) the strongly compressible regime ($\wp_c<\wp\leqslant 1$). The convex black curve is the graph of $L(\alpha)$, while the straight lines are $\alpha/(2\tau_p)$ for small (green line) and large (orange line) $\tau_p$.
• Figure 2: Plots of ( a) $L(q)/D$ as a function of $q$, ( b) $g \alpha /(1+\epsilon)$ as a function of $gD\tau_p$, and ( c) $g \alpha /(1+\epsilon)$ as a function of $g \lambda \tau_p$, for different values of $\wp$ in the Batchelor--Kraichnan flow. In all panels, $d=2$. The values of $\wp$ considered in ( a) and ( b) cover three regimes of compressibility: incompressible ($\wp=0$), weakly compressible ($0<\wp < d/4$), and strongly compressible $(d/4 < \wp \leqslant 1)$; only the incompressible and weakly compressible regimes are considered in ( c).
• Figure 3: Contour plot of the mean breakup time $T_{\mathrm{br}}$ as function of $D\tau_p$ and $\wp$ (note the logarithmically spaced contours) for ( a) small and ( b) large values of $D\tau_p$. In both panels, $d=2$, $\epsilon=0$, $g=1$, $R_{\mathrm{br}}/\rho=10$, and the elastic force is linear.
• Figure 4: $L(q)/\vert\lambda\vert$ vs $q$ for the two-dimensional Batchelor--Kraichnan model and the renewing flow with either $D\tau_f=0.01$ or $D\tau_f=0.2$. The degree of compressibility is ( a) $\wp=0$, ( b) $\wp=0.25$, and ( c) $\wp=1$. Panel (d) presents the variation of $\lambda/D$ with $\wp$ in the two flows.
• Figure 5: Plots of ( a) $L(q)/D$ as a function of $q$ and ( b) $g \alpha /(1+\epsilon)$ vs $gD\tau_p$ for different values of $\wp$ in the renewing flow with $\tau_f=0.2$. In panel ( b), the lines correspond to the numerical solution of \ref{['eq:alpha_l']}, whereas the symbols are estimates of the slope of $P_{\mathrm{st}}(R)$ for $R_{\mathrm{eq}}\ll R\ll R_{\mathrm{max}}$ obtained from numerical solutions of the stochastic differential equation for $\bm R(t)$. To obtain a nonzero equilibrium length and a finite maximum length, we modified \ref{['eq:orl']} by adding Brownian noise with an amplitude such that $R_{\mathrm{eq}}=1$ and by replacing the linear elastic force with a nonlinear force that enforces $R_{\mathrm{max}}=10^3$. For simplicity, we also took $\epsilon=0$ and $g=1$. The resulting stochastic differential equation for $\bm R(t)$ is given in \ref{['eq:dumbbell']} and \ref{['eq:fene']}. We integrated \ref{['eq:dumbbell']} numerically by using the Euler-Maruyama method with time step $dt=5\times 10^{-4}$ in combination with Öttinger's rejection algorithm, to preserve the constraint $R<R_{\mathrm{max}}$ (see o96, § 4.3.2). These simulations confirm the validity of \ref{['eq:alpha_l']} in the compressible case.