= Solution
Let the upper plate in flow A move at speed $U$, with the lower plate fixed. The <Couette flow> profile is
$$
u_A(y)=\frac Uh\,y,
$$
so its mid-plane speed is $U/2$ and its total <viscous dissipation> per unit plate area is
$$
\dot Q_A=\int_0^h\mu\left(\frac Uh\right)^2dy
=\frac{\mu U^2}{h}.
$$
Write $G=-dp/dx>0$ for the constant pressure-gradient magnitude in flow B. The <plane Poiseuille flow> profile is
$$
u_B(y)=\frac{G}{2\mu}y(h-y).
$$
Equality of the mid-plane speeds gives
$$
\frac{Gh^2}{8\mu}=\frac U2,
\qquad
G=\frac{4\mu U}{h^2}.
$$
Since $u_B'(y)=G(h-2y)/(2\mu)$,
$$
\dot Q_B
=\frac{G^2}{4\mu}\int_0^h(h-2y)^2dy
=\frac{G^2h^3}{12\mu}
=\frac{4\mu U^2}{3h}.
$$
Hence
$$
\boxed{\frac{\dot Q_A}{\dot Q_B}=\frac34}.
$$
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