The Simplest Viscous Flow

Abstract

We illustrate an atomistic periodic two-dimensional stationary shear flow, $u_x=⟨\dot{x}⟩=\dot{\epsilon }y$, using the simplest possible example, the periodic shear of just two particles ! We use a short-ranged "realistic" pair potential, $\varphi (r<2)=(2-r{)}^6-2(2-r{)}^3$. 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 $\{p\}$ : $\mathcal{H}(q,p)\equiv K(p)+\Phi (q)+\dot{\epsilon }\sum qp$. Choosing $qp\to y{p}_x$ the resulting motion equations are consistent with steadily shearing periodic boundaries with a strain rate $(d{u}_x/dy)=\dot{\epsilon }$. 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 Nosé'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.

26 pages with 9 figures destined for CMST. Expanded Acknowledgement and Second-Order Runge-Kutta Addendum