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Navier stokes equations

  1. Oct 13, 2013 #1


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    hey pf! can you tell me if this derivation sounds reasonable for the navier stokes equation, from newtons second law into a partial differential equation.

    i'm really just concerned with one part. specifically, i start the derivation with [itex]\Sigma F = ma[/itex]. I am comfortable with the force term but on the acceleration term I learned there are two ways to account for a change in momentum: boundary and volumetric change. i feel good on the volume term, but on the boundary term, i believe mass may leave through the boundary, yielding [tex]\iint_S (\rho \vec{V}) \cdot \vec{dS} \vec{V}[/tex] in order to maintain dimensionality through integration so we can obtain a partial differential equation it is necessary to convert this surface integral into a volume integral using the divergence theorem, yielding: [tex]\iiint_v \nabla \cdot (\rho \vec{V}){dv} \vec{V}[/tex] and thus if i shrink volume i obtain [tex]\nabla \cdot (\rho \vec{V}) \vec{V}[/tex] but the text has, in component form, [tex]\nabla \cdot (\rho u_i \vec{V})[/tex] where [itex]u_i[/itex] is the [itex]i^{th}[/itex] component of velocity [itex] \vec{V}[/itex] but this seems wrong. shouldn't it be written [tex]\nabla \cdot (\rho \vec{V})u_i[/tex] please help me out here! Thanks!!!
  2. jcsd
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