What Does the |x=0 = dy dz Mean in the Physical Interpretation of Divergence?

[Summer2442]

Hello,

I am new to calculus, and am having problems with divergence, I was reading something to explain the physical interpretation of divergence and i got stuck in the very first part.

it says that if we have a small volume dxdydz at the origin, and that a fluid flowing into this volume from the positive x-direction per unit time, the the rate of flow in is
= ρvx|x=0 = dy dz,
where ρ is the density at (x, y, z), and vx is the velocity of the fluid in the x-direction, what does "|x=0 = dy dz" part mean.

Thanks a lot.

 

[Erland]

"|x=0" probably means that the function, here ρv_x, should be evaluauted at x=0. So they imagine the volume element as a cuboid with one vertex at the origin and the sides as dx, dy, and dz in the posotive directions. One face of the cuboid is then contained in the plane x=0.

The flux into the volume through this face is then ρv_xdydz, so something seems wrong in what you wrote anyway.

Btw, you wrote "from the positive x-direction", but I assume that you meant "along the positive x-direction", so that positive v_x is directed to the right.

 

[Summer2442]

yes what i meant is "along the positive x-direction"

but about the flux into the volume, i agree that it should be 'ρvxdydz' but i am pretty sure this is what the book says 'ρvx|x=0 = dy dz', i think its a mistake.

I have another question please, it then says that the flux out of the opposing face is
ρvx|x=dx dydz
which i understand but then it equates this equation with the following
[ρvx + ∂ (ρvx)/∂x dx] dydz
which i do not understand, how was the partial derivative introduced and why?

thanks,

 

[Erland]

yes what i meant is "along the positive x-direction"

but about the flux into the volume, i agree that it should be 'ρvxdydz' but i am pretty sure this is what the book says 'ρvx|x=0 = dy dz', i think its a mistake.

Yes, it is a typo. It should be ρv_x|_x=0dxdy.

I have another question please, it then says that the flux out of the opposing face is
ρvx|x=dx dydz
which i understand but then it equates this equation with the following
[ρvx + ∂ (ρvx)/∂x dx] dydz
which i do not understand, how was the partial derivative introduced and why?

thanks,

The second expression is a linear approximation of the first one.

Recall that (ρv_x|_x=dx - ρv_x|_x=0)/dx ≈ ∂(ρv_x)/∂x|_x=0.

 

[engnr_arsalan]

i think it is talking about mass flow rate... density X velocity X area=flow rate..or more precisely mass flow rate..it says rho X V X x such that at x=0 area is dy dz..for differential element the area perpendicular to flow in x direction is dydz...
the second thing is taylor's theorem is being applied here..which means when fluid has flowed a length dx its mass flowrate has changed depnding upon dx..that is why patial derivative is introduced here..

 

[Summer2442]

Ok I get it now, thanks guys.