The Simplest Viscous Flow

TL;DR

This paper presents a minimalistic model of viscous shear flow using just two particles, demonstrating steady shear, Hamiltonian dynamics, and multifractal distributions with irreversible features.

Contribution

It introduces the simplest possible example of atomistic shear flow with a novel Hamiltonian approach and explores its steady-state and multifractal properties.

Findings

Steady shear flow modeled with two particles and a realistic pair potential.

Hamiltonian dynamics with periodic boundary conditions and shear strain rate.

Multifractal distributions with irreversible characteristics from time-reversible equations.

Abstract

We illustrate an atomistic periodic two-dimensional stationary shear flow, $u_x = \langle \ \dot x \ \rangle = \dot \epsilon y$, using the simplest possible example, the periodic shear of just two particles ! We use a short-ranged "realistic" pair potential, $\phi(r<2) = (2-r)^6 - 2(2-r)^3$. Many body simulations with it are capable of modelling the gas, liquid, and solid states of matter. A useful mechanics generating steady shear follows from a special ("Kewpie-Doll" $\sim$ "$qp$-Doll") Hamiltonian based on the Hamiltonian coordinates $\{ q \}$ and momenta $\{ p \}$ : ${\cal H}(q,p) \equiv K(p) + \Phi(q) + \dot \epsilon \sum qp$. Choosing $qp \rightarrow yp_x$ the resulting motion equations are consistent with steadily shearing periodic boundaries with a strain rate $(du_x/dy) = \dot \epsilon$. The occasional $x$ coordinate jumps associated with periodic boundary crossings in the $y$ direction provide a Hamiltonian that is a piecewise-continuous function of time. A time-periodic isothermal steady state results when the Hamiltonian motion equations are augmented with a continuously variable thermostat generalizing Shuichi Nos\'e's revolutionary ideas from 1984. The resulting distributions of coordinates and momenta are interesting multifractals, with surprising irreversible consequences from strictly time-reversible motion equations.