For
$$
\mathbf U=z\widehat{\mathbf x},
\qquad
\Theta=z-\frac{\operatorname{Ro}}B y,
$$
both material derivatives vanish because the fields are independent of $x$. Since $\widehat{\mathbf z}\times\mathbf U=z\widehat{\mathbf y}$, steady momentum balance requires
$$
P_y=-z,
\qquad
P_z=\frac B{\operatorname{Ro}}z-y.
$$
These derivatives are compatible and integrate to
$$
\boxed{P=-yz+\frac{B}{2\operatorname{Ro}}z^2+\text{constant}}.
$$
The cross-stream temperature gradient and vertical stratification are in <thermal-wind balance> with the vertical shear.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 331 / 3 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-331/3/b/solution