# Fluids Lecture 02

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### 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

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

...

Euler’s Equation:

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

where ${\displaystyle u}$ is a velocity field. We can use the vector identity:

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

where ${\displaystyle \nabla \times u\equiv \omega }$ describes vorticity, and ${\displaystyle {\frac {1}{2}}\nabla \left|u\right|^{2}}$ describes the energy of the field. Vorticity is a uniform shear flow, which is to say that ${\displaystyle v_{x}=\beta y}$, 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:

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

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 ${\displaystyle \left|\omega \right| \over 2}$.