Consider a gas with initial number density $n$ and temperature $T$. At the initial moment, the gas is spatially uniform, while the macroscopic velocity is parallel to the $x$-axis with a profile given by $v_x(x)=v_0 \sin⁡(kx)$. The molecules can be treated as rigid spheres of mass $m$ and radius $r$.

What condition for $v_0$ must be satisfied so that, at a certain moment and location, a multi-stream flow arises? It is sufficient to estimate the answer to an order of magnitude; exact numerical prefactors are not required.

A flow is considered “multi-stream” at a point xx if the local velocity distribution function $f(v_x)$ has at least two local maxima, where each maximum is at least twice the value of the local minimum between them.

Assume that $v_0 \gg \sqrt{k_B T/m}$