I Small scale homogeneity

  • Thread starter Apashanka
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From the continuity equation ##\frac{\partial \rho}{\partial t}+\rho (\nabla • u)=0## where ##\rho## is the mass density and is homogeneous and ##u## is the velocity of expansion or contraction.
For an expanding volume this becomes ##\nabla•u=\frac{\dot v(t)}{v(t)}=\Theta## which gives the rate of expansion ,##v(t)## is the volume.
Now if we consider our volume ##V(t)## to be consist of small patches of volume ##v(t)## where ##\rho## is homogeneous in ##v(t)## ,local expansion rate of volume ##V(t)## is therefore ##\Theta## ,now if the variance of this ##\Theta## is taken over ##V(t)## it will be ##\sigma_\Theta=[\bar{\Theta^2}-(\bar \Theta)^2]##.
Now my question is it small scale homogeneity is sufficient for this non-zero variance of volume expansion rate??
 
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Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
 

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