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?