Continuity equation, partial derivative and differential operators

[pisgirl]

Hi all!
I have the following slide, and whilst I understand that the original point is "the rate of density, ρ, in each volume element is equal to the mass flux"...i am totally lost on the mathematics! (And I am meant to be teaching this tomorrow). I do not have any information on what the indivudual symbols refer to, I guess A is area and t is time etc. Can anyone understand this:

-∇(ρv)=$\frac{\partial}{\partial t}$(ρd$\tau$) where d$\tau$=Adx
-∇.v = $\frac{\partial}{\partial t}$ (Adx)
-$\frac{\partial v}{\partial x}$ Adx = $\frac{\partial}{\partial t}$(Adx)
-$\frac{\partial v}{\partial x}$=$\frac{1}{A}$ $\frac{\partial A}{\partial T}$

Unfortunately I am not sure how to even get from line 1 to 2

and how t combine partial with full!

Argh! Thank you in advance!

 

[I like Serena]

Welcome to PF, pisgirl!

Hi all!
I have the following slide, and whilst I understand that the original point is "the rate of density, ρ, in each volume element is equal to the mass flux"...i am totally lost on the mathematics! (And I am meant to be teaching this tomorrow). I do not have any information on what the indivudual symbols refer to, I guess A is area and t is time etc. Can anyone understand this:

-∇(ρv)=$\frac{\partial}{\partial t}$(ρd$\tau$) where d$\tau$=Adx
-∇.v = $\frac{\partial}{\partial t}$ (Adx)
-$\frac{\partial v}{\partial x}$ Adx = $\frac{\partial}{\partial t}$(Adx)
-$\frac{\partial v}{\partial x}$=$\frac{1}{A}$ $\frac{\partial A}{\partial T}$

Unfortunately I am not sure how to even get from line 1 to 2

and how t combine partial with full!

Argh! Thank you in advance!

I also have difficulty to make sense of these formulas.

I think you can only get from line 1 to line 2 if ρ is independent from both time and location.
Then ρ can be moved outside the differentiation and get canceled.
But I presume ρ is not supposed to be constant?
Would it be dependent on time?

Can you indicate which quantities are supposed to be vectors and which scalar?
And which quantities depend on place and/or time?

Line 2 contains ∇.v making it ambiguous what v represents.
What does it represent?
Volume? Velocity? Specific volume?
Note the difference between ∇v and ∇.v.
The first is a gradient which requires a scalar function and yields a vector.
The second is the divergence which requires a vector function and yields a scalar.

In line 3 we see that it was either
$∇v = (\frac{dv}{dx}, \frac{dv}{dy}, \frac{dv}{dz})$, but where did the other components go then? And where did Adx come from?
$∇ \cdot \mathbf v=\frac{dv_x}{dx} + \frac{dv_y}{dy} + \frac{dv_z}{dz}$.
If we assume $v_y = v_z = 0$, we're still left with an Adx that comes out of nowhere.

In line 4 suddenly a T pops up.
Temperature?
Or a typo that should have been $\tau$?