 Solution

ID: past-exam-of-the-mathematical-mechanical-biology-course-of-the-university-of-oxford/2023/2/a/solution

e_{r}, e_{φ}, e_{z}

. The meridional unit tangent vector and outward unit normal vector consistent with the given geometry are

e_{s} = cos θ e_{r} - sin θ e_{z}, n = sin θ e_{r} + cos θ e_{z} .

θ

independent of

φ

. Consequently

\partial_{s} e_{s} = - θ_{s} n, \partial_{φ} e_{φ} = - e_{r}, e_{r} = cos θ e_{s} + sin θ n .

\partial_{s} (r t_{s} e_{s}) \partial_{φ} (t_{φ} e_{φ}) = (r_{s} t_{s} + r \partial_{s} t_{s}) e_{s} - r t_{s} θ_{s} n, = (\partial_{φ} t_{φ}) e_{φ} - t_{φ} cos θ e_{s} - t_{φ} sin θ n .

Equating the three independent components to zero, and using

r_{s} = cos θ

, gives

e_{s} e_{φ} n : r \partial_{s} t_{s} + (t_{s} - t_{φ}) cos θ = 0, : \partial_{φ} t_{φ} = 0, : r P - r t_{s} θ_{s} - t_{φ} sin θ = 0.

Thus, away from the rotation axis where

r > 0

P = t_{s} θ_{s} + t_{φ} \frac{sin θ}{r}, \partial_{s} t_{s} = \frac{cos θ}{r} (t_{φ} - t_{s}), \partial_{φ} t_{φ} = 0.

The two geometric factors

θ_{s}

and

sin θ / r

are the signed meridional and azimuthal principal curvatures in this convention. The normal equation is the Young–Laplace equation generalized to unequal principal membrane tensions.

 New to topics? Read the docs here!