3.9.2 Forces

Total fluid force for $\underset{\begin{array}{c}\mbox{No} \\ \mbox{shear} \\ \mbox{stress}\end{array}}{\mbox{\underline{ideal}}}$ flow:

$$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightha... ... }}\over {F}} = \int\!\!\!\int\limits_B {p\hat {n}dS} \notag$$

For potential flow:

$$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightha... ...{\nabla \phi } \right\vert^2 + gy + c(t)} \right)\hat {n}dS}$$

For the hydrostatic $\left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$ }}\over {v}} \equiv \phi \equiv 0} \right)$case:

$$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightha... ...t\!\!\!\int}\limits_{\kern-5.5pt {\upsilon _B }} {d\upsilon }$$

Hydrodynamic Force:

$$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightha... ...vert {\nabla \phi } \right\vert^2} \right)\hat {n}dS} \notag$$

For steady $\left( {\frac{\partial \phi }{\partial t} \equiv 0} \right)$motion:

$$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightha... ...rac{1}{2}\rho \int\!\!\!\int\limits_B {v^2\hat {n}dS} \notag$$

Example: Hydrodynamic force on 2D cylinder in a steady uniform stream .

$$\begin{array}{l} \mathord{\buildrel{\lower3pt\hbox{$\scrip... ...\right\vert _{r = a}^2\cos \theta d\theta } \\ \end{array}$$

Potential function for the 2D cylinder:

$$\phi = U\cos \theta \left( {r + \frac{a^2}{r}} \right)$$

Velocity vector on the 2D cylinder surface:

$$\left. {\nabla \phi } \right\vert _{r = a} = \left( {\left. {... ...\theta }} \right\vert _{r = a} }_{ - 2U\sin \theta }} \right)$$

Square of the velocity vector on the 2D cylinder surface:

$$\left. {\left\vert {\nabla \phi } \right\vert^2} \right\vert _{r = a} = 4U^2\sin ^2\theta$$

Hydrodynamic force on the 2D cylinder:

$$F_x = \frac{\rho a}{2}\int\limits_0^{2\pi } {d\theta \left( {... ...frac{\pi }{2},\frac{3\pi }{2} \\ \end{array}}} }_{ \equiv 0}$$

Therefore, F$_{x}$= 0 no forces ( symmetry fore-aft of the streamlines). Similarly,

$$F_y = \left( {\frac{1}{2}\rho U^2} \right)(2a)2\int\limits_0^{2\pi } {d\theta \sin ^2\theta \sin \theta = 0} \notag$$

In fact, in general we find that $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$ }}\over {F}} \equiv 0$on any 2D or 3D body.
D'Alembert's ``paradox'':
No hydrodynamic force$^{\ast }$ acts on a body moving with constant translational velocity in an infinite, inviscid, irrotational fluid.
Note: The moment as measured in a local frame is not necessarily zero.

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