Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 38D (Fluid Dynamics)

The incompressibility equation turns the advective term into a divergence:because $v_y=-u_x$. For a plume in an otherwise stationary environment, the boundary-layer pressure equals the ambient pressure and has no imposed $x$-gradient. Integrating the vertical momentum equation across $y$ therefore givesBoth boundary terms vanish as the velocity and shear decay away from the plume. Hence the integral momentum-flux balance for a two-dimensional plume is

Let $U(x)$ and $\Delta (x)$ denote the characteristic vertical speed and width. Integrating the result of part (a), with $b(x)=B{x}^{-1/5}$, gives the scalingThe boundary-layer scaling in the momentum equation balances vertical inertia against transverse viscosity:Substituting this into the momentum-flux relation givesand thereforeReturning to the viscous-inertial balance then givesThese are the similarity scaling of a two-dimensional laminar plume.

Choose the stream function conventionwhich satisfies continuity identically. Introduceso thatTake the similarity solutionThenandA direct differentiation shows that the terms containing $\eta f^{\prime }{f}^{\prime \prime }$ cancel:Also,while the delta scaling givesThe definitions of $D$ and $U_0$ implyAfter cancelling the common dimensional factor, the similarity equation for a two-dimensional laminar plume isThus