Derivation of the viscosity term

[jostpuur]

How good derivations are there for the viscosity term $$\nu \nabla^2 u_i$$, that contributes to the acceleration of fluid, in Navier-Stokes equations? I can see that this term is intuitively reasonable. If I wanted to approximate the velocity of the fluid in an environment of some point $$(x_1,x_2,x_3)$$, in attempt to solve a friction that drags the fluid with it in this point, I couldn't use the first derivatives, because for example the friction from velocity $$u_i(x_1+\Delta x,x_2,x_3)=u_i+\Delta x \partial_1 u_i$$ would be canceling the friction from velocity $$u_i(x_1-\Delta x,x_2,x_3)=u_i-\Delta x\partial_1 u_i$$ in the linear approximation. So the second order approximation at least gives something, but a more rigor proof that it is precisly the $$\nabla^2$$ that suits this, would be nice.

 

[lalbatros]

The force is proportional to the gradient.
The term in the Navier-Stokes equations arises from a force balance on an elementary volume.

 

[olgranpappy]

How good derivations are there for the viscosity term $$\nu \nabla^2 u_i$$

Landau (vol 6) derives this term starting from a "most general" form of the stress tensor linear in the velocity 'u'.
I personally prefer Batchelor "An introduction to fluid dynamics." check it out.

Basically we have a vector du/dt on the LHS of our hydrodynamic equation and we want terms on the RHS that are also vectors and linear in u. We can have your viscosity term $$\nabla^2\vec u$$or we can have a term$$\vec\nabla(\vec \nabla\cdot\vec u)$$... but the latter is zero for an incompressable fluid so we can often ignore it and treat only the $$\nabla^2\vec u$$ term.