Buoyancy Effects Due to Accelerating Flow

Example 2: Force on a stationary sphere in a fluid that is accelerated against it.

$$\begin{align}\phi \left( {r,\theta ,t} \right) & = U\left( t \right)\left( {r + ... ...vert^2} \right\vert _{r = a} & = \frac{9}{8}U^2\sin^2\theta \notag \end{align}$$

Then,

$$\begin{align}F_x & = \left( { - \rho } \right)\left( {2\pi r^2} \right)\int\limi... ...row }\limits_{ = \rho \upsilon } + \frac{2}{3}\pi a^3\rho } \notag \end{align}$$

Part of F$_{x}$ is due to the pressure gradient which must be present to cause the fluid to accelerate:

$$\begin{array}{l} U\left( t \right):N.S.^\frac{\partial U}{\... ...mbox{ \ for uniform (1D) accelerated flow } \\ \end{array}$$

Force on the body due to the pressure field

$$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightha... ...ac{\partial p}{\partial x}d\upsilon } = \rho \forall \dot {U}$$

``Buoyancy'' force due to pressure gradient = $\rho \forall \dot {U}$

Analogue: Buoyancy force due to hydrostatic pressure gradient. g = gravitational acceleration $\leftrightarrow \quad \dot {U}$= fluid acceleration.

$$\begin{array}{l} p_s = - \rho gy \\ \nabla p_s = - \rho ... ...rall \hat {j} \mbox{\ Archimedes principle} \\ \end{array}$$

Summary: Total force on a fixed sphere in an accelerated flow

$$F_x = \dot {U}\left( \underset{\mbox{\tiny{Buoyancy}}}{\unde... ...style{3 \over 2}\rho \forall = \dot {U}3m_{\left( 1 \right)}$$

In general, for any body in an accelerated flow:

$$F_x = F_{\mbox{\tiny {buoyany}} }+\dot {U}m_{\left( 1 \right)},$$

where $m_{(1)}$ is the added mass in still water (from now on, m)

$$F_{x}= - \dot {U}m \mbox{\ for body acceleration with } \dot {U}$$

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