Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 38C / c / Solution

For lower speed $U$, stationary upper tangential speed, upper normal speed $-V$, and equal endpoint pressures, define $\langle f\rangle=L^{-1}\int_0^Lf(x)\,dx$. Then
$$
p_x=\frac{6\mu U}{h^2}-\frac{12\mu(Vx+C)}{h^3},
$$
where
$$
C=\frac U2\frac{\langle h^{-2}\rangle}{\langle h^{-3}\rangle}
-V\frac{\langle xh^{-3}\rangle}{\langle h^{-3}\rangle}.
$$


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