Write $\widehat\rho(z)$ for the ambient density far from the wall and $\rho(z)$ for the density at the wall. Across the plume, let $y=x/b$ and use the prescribed triangular profiles
$$
w=W(1-y),
\qquad
\rho_p=\widehat\rho-(\widehat\rho-\rho)(1-y),
\qquad 0\leq y\leq1.
$$
Direct integration gives the <volume flux>, <mass flux>, <momentum flux>, and density-weighted <buoyancy flux>, all per unit radiator length:
$$
\boxed{V=\int_0^b w\,dx=\frac{bW}{2},}
$$
$$
\boxed{Q=\int_0^b\rho_pw\,dx
=\frac{bW}{6}(\widehat\rho+2\rho),}
$$
$$
\boxed{M=\int_0^b\rho_pw^2\,dx
=\frac{bW^2}{12}(\widehat\rho+3\rho),}
$$
$$
\boxed{F=\int_0^b g(\widehat\rho-\rho_p)w\,dx
=\frac{gbW}{3}(\widehat\rho-\rho).}
$$
These coefficients distinguish the <triangular-profile wall line plume> from a <top-hat plume model>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 345 / 3 / a / Solution

In a triangular-profile wall line plume, axial velocity and <reduced gravity> decrease linearly from their wall values to zero across the plume width $b$. If the wall speed is $W$, its volume and kinematic momentum fluxes per unit span are $V=bW/2$ and $J=bW^2/3$.

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 Incoming links: Triangular-profile wall line plume