# How does flow field density change with time?

#### Kirik

I was wondering what it means to have the density changing with time, does this reference the material derivitve (which seems to be at least partially with respect to time), or the straight partial derivitive of density WRT time?

Drho/Dt ~= 0 or partialrho/partialt ~=0?

The material derivative being D/Dt = partial/partialt + (Velocity dot gradient)
or with density: Drho/Dt = partialrho/partialt + (Velocity dot gradient)rho.

Given that the flow field may be examined with the conservation of mass, using the continuity equation:
parital rho/partial t + partial (u*rho)/partial x + partial (v*rho)/partial y + partial (w*rho)/partial z = 0

A possilble connection is that when the density is the same throughout the flow field, and unchanging with time (incompressible). Thus,
partial (u)/partial x + partial (v)/partial y + partial (w)/partial z will be equal to zero when rho(x,y,z,t) NEVER changes. Also, it is easy to obtain these partial derivitives when given a flowfield, like x=y^2-1, v=0, w=0.

However how can I determine if rho(x,y,z,t) changes with x,y,z but not with respect to t?

Expressions given for the conservation of mass including the material derivative are as follows:
Drho/Dt + rho*gradient dot velocity = 0
or
Drho/Dt + rho*(divergence of Velocity)= 0

Again, it seems that Drho/Dt will have to be zero when the divergence of Velocity is zero, which seems to be the same condition to show that density is not changing in an incompressible fluid.

However, does Drho/Dt =0 prove that density does not change WRT time, or that the sum of the local derivative (partialrho/partialt) just somehow cancels out the terms of the convective derivative (Velocity dot gradient rho)?

More importantly, may I claim that when div Velocity = gradient dot Velocity= 0 it follows that Velocity dot gradient = 0 as well (commutative property of the dot product). and thus the only term remaining when the velocity gradient is zero from
Drho/Dt + rho*gradient dot velocity = 0
is partialrho/partialt = 0, which would give me that rho(x,y,z,t) is not changing with time?

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#### Chestermiller

Mentor
If $\rho$ does not change with time at a specified spatial location, it means that the partial derivative of $\rho$ with respect to t is zero.

"How does flow field density change with time?"

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