= Solution
{c}
For a constant-<mass density> <incompressible flow> with no imposed <body force>, the two-dimensional <Navier-Stokes equation> is
$$
\boxed{\begin{aligned}
u_x+v_y&=0,\\
\rho(u_t+uu_x+vu_y)&=-p_x+\mu(u_{xx}+u_{yy}),\\
\rho(v_t+uv_x+vv_y)&=-p_y+\mu(v_{xx}+v_{yy}).
\end{aligned}}
$$
A conservative <body force>, if retained, may be incorporated into a modified <pressure>. Using scales $L,U_0,L/U_0$ and $\rho U_0^2$ gives
$$
\boxed{\mathrm{Re}=\frac{\rho U_0L}{\mu}=\frac{U_0L}{\nu},\qquad
\mathbf u_t+\mathbf u\cdot\nabla\mathbf u=-\nabla p+\mathrm{Re}^{-1}\nabla^2\mathbf u.}
$$
Here $\nu=\mu/\rho$ is the molecular <kinematic viscosity>. At high <Reynolds number>, inertia and the <pressure gradient> dominate on the outer length scale, while molecular viscous diffusion is small there. Thin <boundary layers>, narrow <shear flows> and fine turbulent scales can still have sufficiently large <velocity> gradients for viscous effects to matter. High <Reynolds number> permits instability and <turbulence>, but does not alone prove that every flow is turbulent. Molecular <viscous dissipation> remains essential at the small end of the turbulent cascade.
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