1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Derivation of the viscosity term

  1. Jun 25, 2007 #1
    How good derivations are there for the viscosity term [tex]\nu \nabla^2 u_i[/tex], 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 [tex](x_1,x_2,x_3)[/tex], 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 [tex]u_i(x_1+\Delta x,x_2,x_3)=u_i+\Delta x \partial_1 u_i[/tex] would be canceling the friction from velocity [tex]u_i(x_1-\Delta x,x_2,x_3)=u_i-\Delta x\partial_1 u_i[/tex] in the linear approximation. So the second order approximation at least gives something, but a more rigor proof that it is precisly the [tex]\nabla^2[/tex] that suits this, would be nice.
  2. jcsd
  3. Jun 26, 2007 #2
    The force is proportional to the gradient.
    The term in the Navier-Stokes equations arises from a force balance on an elementary volume.
  4. Jun 26, 2007 #3


    User Avatar
    Homework Helper

    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 [tex]\nabla^2\vec u[/tex]or we can have a term[tex]\vec\nabla(\vec \nabla\cdot\vec u)[/tex]... but the latter is zero for an incompressable fluid so we can often ignore it and treat only the [tex]\nabla^2\vec u[/tex] term.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook