# Questions about deriving the naviers stokes equations

1. Jun 16, 2009

### hyper

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?

Then 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.

2. Jun 16, 2009

### Andy Resnick

There's a few questions here, let me try sorting it out:

1) the shear is a tensor quantity, each component is defined as $$\tau_{ij} = \mu V_{i,j}$$, where I assumed a linear homogeneous medium (the viscosity is a scalar) and V_i,j means the j'th partial derivative of i'th component of V, for example $$\tau_{xy} = \mu \frac{\partial V_{x}}{\partial y}$$

2) I visualize tensors as the surface of a cube; each face has three directions associated with it (1 normal and 2 in-plane). The normal components, tau_ii, correspond to pressure- the action on the cube is to expand or contract the cube. The off-diagonal components are shear, and act to deform the cube into a rhombohedron.

3) Your other questions seem to be matters of notation; factors of '2' and '1/2' sometimes appear since the shear stress is symmetric... or am I missing something?

3. Jun 16, 2009

### hyper

I dont see how tau(xx) can be the preassure, since the preassure is another part in the equations in my book, and it is also another part in the anvier stokes equations.

4. Jun 16, 2009

### Andy Resnick

I guess I need to have a better idea of what your book is presenting- can you be more specific?

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