Fluids Lecture 02

Summary

• Stresses in a fluid consist of body (long range) and surface (short range) forces.
• The stress in a fluid is described by a symmetric tensor of second order there are normal stresses and shear stresses, only the isotropic part survives in a fluid in equilibrium
• in hydrostatic equilibrium every fluid element must be in equilibrium

Dynamics of Ideal Fluids

The continuity equation reads:

$\int _{V}{{\frac {\partial \rho }{\partial t}}dV}+\int _{V}{\nabla \cdot (\rho u)dV}=0\,\!$

...

Euler’s Equation:

$\rho \left[{\frac {\partial u}{\partial t}}+(u\cdot \nabla )u\right]=-\nabla p-\rho \nabla \phi \,\!$

where $u$ is a velocity field. We can use the vector identity:

$(u\cdot \nabla )u\equiv (\nabla \times u)\times u+{\frac {1}{2}}\nabla \left|u\right|^{2}\,\!$

where $\nabla \times u\equiv \omega$ describes vorticity, and ${\frac {1}{2}}\nabla \left|u\right|^{2}$ describes the energy of the field. Vorticity is a uniform shear flow, which is to say that $v_{x}=\beta y$, or that in some direction, the velocity of the fluid tangential to that direction grows linearly with distance.

We may relate circulation and vorticity by stokes theorum:

$\Gamma =\int _{C}{udl}=\int _{S}{(\nabla \times u)\cdot ds}=\int _{s}{\omega \cdot ds}\,\!$

Vorticity is the circulation per unit area perpendicular to the surface. It is a measure of local rotation, and has nothing to do with global rotation. An individual speck of dust in a liquid spins with angular velocity $\left|\omega \right| \over 2$.