Let $b=h-\delta$ be the half-width of the cell-rich core. In fully developed pressure-driven flow the shear stress is fixed by momentum balance, independently of the local viscosity:
$$
\tau(y)=-Gy.
$$
With no slip at $y=h$,
$$
u(y)=\int_y^h\frac{Gs}{\mu(s)}\,ds.
$$
Interchanging the order of integration gives the total flux
$$
Q=2\int_0^h u(y)\,dy
=2G\int_0^h\frac{s^2}{\mu(s)}\,ds
=\frac{2G}{3}
\left[
\frac{b^3}{\mu}
+\frac{h^3-b^3}{\mu_w}
\right].
$$
By definition, the homogeneous effective fluid has
$$
Q=\frac{2Gh^3}{3\mu_{\rm eff}}.
$$
Writing $\varepsilon=\delta/h$ therefore gives
$$
\boxed{
\frac1{\mu_{\rm eff}}
=\frac{(1-\varepsilon)^3}{\mu}
+\frac{1-(1-\varepsilon)^3}{\mu_w}},
$$
or the reciprocal of the right-hand side.


Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 334 / 1 / b / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-334/1/b/ii/solution