Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 74 / 4 / d / Solution

Seek a nonzero-buoyancy, separable similarity solution of the unsteady Boussinesq triangular-profile line-plume balances, using powers of distance and time. Let $\tau =t+t_{\ast }>0$, allowing a time origin, and initially take distance $z$ from the virtual origin. Balancing the powers in the momentum and buoyancy equations and the entrainment balance gives the formOne way to recover the exponents is to let $Q\propto z^a{\tau }^p$ and $M\propto z^b{\tau }^q$. Matching the spatial powers in ${\partial }_t(Q^2/M)$, $Q_z$ and $M/Q$ gives $a=2,b=3$; matching the time powers gives $p=-1,q=-2$. The momentum equation then gives the powers of $F$. In the resulting solution, $Q^2/M$ is time-independent, so the storage term in the first equation vanishes exactly.

Direct substitution yields the coefficient equationsFor a rising buoyant solution $C>0$, the last equation gives $B=2A/3$. The first then gives $A=\alpha {\rho }_0$, and the second gives $C=B$. The separable decaying line-plume similarity is thereforeReconstructing the physical fields confirmsFor example, ${\partial }_t(FQ/M)=-2\alpha {\rho }_0{z}^2/{\tau }^3$ and $F_z=2\alpha {\rho }_0{z}^2/{\tau }^3$ cancel; the momentum balance similarly gives $-\alpha {\rho }_0{z}^2/{\tau }^2+2\alpha {\rho }_0{z}^2/{\tau }^2=FQ/M$.

The unshifted form uses $t_{\ast }=0$ and is defined for $t>0$, with a singular zero-time limit. To give a finite solution for all $t\ge 0$, choose $t_{\ast }>0$. The equations also permit replacing $z$ by $z+z_{\ast }>0$, a plume virtual origin; at a finite physical source distance this family has flux histories proportional to ${\tau }^{-1},{\tau }^{-2},{\tau }^{-3}$. The Boussinesq approximation requires $(z+z_{\ast })/(g{\tau }^2)\ll 1$ in the region modelled. This is a decaying similarity family; its time and virtual origins require source or initial data. The PDF supplies neither, so a unique forced-startup history cannot be selected from it. In particular, the singular unshifted field should not be presented as a regular solution at $t=0$.