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Meaning of a PDE expression

  1. Dec 27, 2009 #1
    Hi

    I have seen the expression for mass flow rate in one of the problems I am working on. I used to simply apply the expression for calculating the mass flow rate with respect to the position as (ρu + (∂u ∕∂x)dx) dy dz). ρ, u are density and velocity component respectively.

    I would like to understand the exact meaning of multiplication of (∂u ∕∂x)dx.
    I expect a good explanation as soon as possible.

    Thanks in advance.
    Madguy
     
  2. jcsd
  3. Dec 27, 2009 #2

    Astronuc

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    Staff: Mentor

    (ρu + (∂u ∕∂x)dx) should be ρ(u + (∂u ∕∂x)dx), if the fluid is incompressible or (ρu + (∂ρu ∕∂x)dx) if ρ = ρ(x), which would imply the fluid is compressible (or changing temperature).

    (∂u ∕∂x)dx = du, and ∂u ∕∂x is just the velocity gradient, i.e. u is not constant with x (position or displacement). This could be because the fluid is compressible or the cross-section (normal to flow) is changing.
     
  4. Dec 28, 2009 #3
    Hi Astronyc

    Thanks for your explanations!
     
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