Change to the frame in which the fluid is at rest far away and a sphere of radius…

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow · Kinematics, conservation laws and ideal flow

Fluid Dynamics, Ch Potential flow, conformal mapping and the finite wing

Problem

Change to the frame in which the fluid is at rest far away and a sphere of radius

a

moves through it at speed

U

. The velocity potential of this flow is the dipole

\phi = -\dfrac{Ua^3\cos\theta}{2r^2}

, chosen so that

u_r(a,\theta) = U\cos\theta

matches the normal velocity of the moving sphere's surface. ((a)) Find

u_r = \partial\phi/\partial r

and

u_\theta = r^{-1}\partial\phi/\partial\theta

, and hence show that the kinetic energy of the fluid,

\tfrac12\rho\int_{r \ge a}(u_r^2+u_\theta^2)\,\mathrm{d}V

with

\mathrm{d}V = 2\pi r^2\sin\theta\,\mathrm{d}r\,\mathrm{d}\theta

, equals

\tfrac12m_aU^2

with

m_a = \tfrac12\rho\cdot\tfrac43\pi a^3

, the added mass. ((b)) A hollow plastic ball of radius

a = 2.0\,\mathrm{cm}

and uniform density

100\,\mathrm{kg\,m^{-3}}

is released from rest, fully submerged, in still water (

\rho = 998\,\mathrm{kg\,m^{-3}}

). Using

(m_{\text{ball}}+m_a)\dfrac{\mathrm{d}U}{\mathrm{d}t} = (\rho_{\text{water}}-\rho_{\text{ball}})Vg

at the instant of release, find its initial acceleration as a multiple of

g

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