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Problems with divergenceby Summer2442
Tags: divergence 
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#1
Jan1613, 11:58 AM

P: 8

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 xdirection per unit time, the the rate of flow in is = ρvxx=0 = dy dz, where ρ is the density at (x, y, z), and vx is the velocity of the fluid in the xdirection, what does "x=0 = dy dz" part mean. Thanks Alot. 


#2
Jan1613, 04:59 PM

P: 345

"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_{x}dydz, so something seems wrong in what you wrote anyway. Btw, you wrote "from the positive xdirection", but I assume that you meant "along the positive xdirection", so that positive v_{x} is directed to the right. 


#3
Jan1613, 05:57 PM

P: 8

yes what i meant is "along the positive xdirection"
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 'ρvxx=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 ρvxx=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, 


#4
Jan1713, 12:56 PM

P: 345

Problems with divergence
Recall that (ρv_{x}_{x=dx}  ρv_{x}_{x=0})/dx ≈ ∂(ρv_{x})/∂x_{x=0}. 


#5
Jan1713, 01:57 PM

P: 41

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.. 


#6
Jan1713, 05:24 PM

P: 8

Ok I get it now, thanks guys.



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