Laminar plume

A laminar plume is a narrow buoyancy-driven flow in which viscous diffusion balances inertia across the plume.

For a steady incompressible plume with localized vertical body force $b(x)\delta (y)$, decay of velocity and shear at transverse infinity gives

If $b(x)=B{x}^{-1/5}$, balancing the momentum flux and transverse viscosity gives

With $u=-{\psi }_y$, $v={\psi }_x$, $\eta =y/\Delta$, and $\psi =U\Delta f(\eta )$, the plume boundary-layer equation reduces to