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Anonymous Poster

Deriving Naviers Stokes Equations

06/16/2009 6:00 PM

Hello, I read some fluidmechanics and there was something I didn't understand.

The shear stress in a newtonian fluid is tau=viscosity*dV/dy, (no need to be dy, but dx and dz also can do.)

A shear component called tau(xx) came up, I have two questions about this component:

1. Shear is supposed to be parralell on a surface, so how does this shear component work? How can it point in the x-direction, when it is on the x-surface(yz-plane) and also is supposed to be in the yz-plane?

2. It is said that in a Newtonian fluid tau(xx)=2*viscosity*du/dx, where the velocity in the x-direction. Why is it this, why the number 2?, can you explain this if you look at the definition of viscosity in Newtonian fluids I posted first?

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Anonymous Poster
#1

Re: questions about deriving the naviers stokes equations

06/16/2009 6:02 PM

Second question:

My question is about the stress component tau(xy). It is said that it is viscositu*(du/dy+dv/dx). I can see out of the definition that it is supposed to be viscosity*du/dy, but why also the dv/dx part?(v is the y-compononent of the velocity).

These questions have been nagging me for ours now, I would appreciate some help.

PS: All the deriviatives are supposed the be partial deriviatives offcourse.

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#2

Re: questions about deriving the naviers stokes equations

06/16/2009 6:31 PM

Not going into the rest of it, but isn't there something pretty obvious going on with your Q2 and your post #2?

tau(xx) = viscosity*(du/dx+du/dx)

tau(xy) = viscosity*(dv/dx+du/dy).

Without any knowledge of this subject, would it be reasonable of me to suggest that

tau(xz) = viscosity*(dv/dx+dw/dz) ?

Feel free to kick me into touch - but if I'm wrong, could someone please explain?

(Symmetry rules, OK? )

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Anonymous Poster
#3
In reply to #2

Re: questions about deriving the naviers stokes equations

06/16/2009 7:40 PM

The tau(xx) you have written I didnt write. You have written the right tau(xy), but not the right tau(xz), there you need w which is the z-component of the velocity.

I don't really understand what you are trying to say, because you are just repeating me, but you mixed something up.

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#4
In reply to #3

Re: questions about deriving the naviers stokes equations

06/16/2009 11:49 PM

Review the basic theory and it will all become crystal clear.

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Anonymous Poster
#5
In reply to #4

Re: questions about deriving the naviers stokes equations

06/17/2009 5:21 AM

Say what!?? I've been reading this for hours now.

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