Questions about deriving the naviers stokes equations

[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.

 

[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?

 

[hyper]

I don't 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.

 

[Andy Resnick]

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