The Simplest Viscous Flow

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.