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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

    olgranpappy

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    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.
     
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