Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 352 / 3 / b / ii / Solution

The volume flux carried with the particle skeleton is

Q_{0} = 2 π \int_{0}^{R} w_{s} (r) r d r = \frac{π G ( R - a ) ^{2} ( 3 R ^{2} + 2 R a + a ^{2} )}{24 η} .

By Darcy law, the fluid moves relative to the solids at

k G / η

. The total mixture-volume flux is therefore

Q_{T} = Q_{0} + \frac{2 πk G}{η} \int_{0}^{R} [1 - ϕ (r)] r d r .

This is explicit because

ϕ (r)

is given above. For example, if

c = \frac{G}{2 p _{s}}, T = c (R - a),

then

\int_{0}^{R} ϕ r d r = ϕ_{m} {\frac{a ^{2}}{2} + \frac{2 a}{c} [T - lo g (1 + T)] + \frac{2}{c ^{2}} [\frac{T ^{3}}{3} - \frac{T ^{2}}{2} + T - lo g (1 + T)]} .

 Ancestors (12)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version