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I am trying to derive the equation for buoyancy frequency in a stratified fluid and am struggling with some of the equations. I have a limited background in fluid dynamics so I basically just need someone to break down the continuity equation for me, in terms that are easily understood.

From the set of notes I am looking at:

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This is the first step of the method for deriving the equation, can anyone explain this in simplistic terms?

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Related: physics.stackexchange.com/questions/54309/… –  Bernhard Feb 21 '13 at 21:54
In very simplistic terms, this equation represents the principle of conservation of mass (no creation, no destruction of matter). –  Christoph B. Apr 12 '13 at 10:22

2 Answers 2

The continuity equation for a compressible fluid is the following: $$\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\vec{v})=0$$ Divergence of the current $\rho\vec{v}$ contains the term $\rho\nabla\cdot\vec{v}$ which vanishes for incompressible fluid. The remainder is just $\frac{D\rho}{Dt}$, as joshphysics said.

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Recall the definition the the "material derivative" $$ \frac{D\rho}{Dt} := \frac{\partial\rho}{\partial t} + \vec v\cdot \nabla \rho $$ If $\nabla\cdot\vec v = 0$ (as for incompressible flow) then the continuity equation is $$ \frac{D\rho}{Dt} = 0 $$ and combining these results gives what is written.

I may have misunderstood your question though. Are you looking for the physical interpretation of these different quantities and steps?

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