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Airflow in a porous wall

  1. Oct 4, 2011 #1
    1. The problem statement, all variables and given/known data

    What we know is that we have two impermeable walls with a porous material inside (the porous material gives a simply restraining force proportional to the movement itself). we have a stationaly flow, so that dw/dt=du/dt=0. u and w are the velocities in the x (horizontal) and z (vertical) directions, respectively. This is a 2-d problem, so we don't bother with the y-axis.


    [PLAIN]http://img135.imageshack.us/img135/4820/wallo.png [Broken]

    The temperature is a simple linear function of x, where Ty is the temperature of the outside wall and Ti is the temperature of the inside, like shown on the picture.

    So what I'm trying to find is an expression of the vertical velocity w as a function of x, that is, w(x). A hint is given that ∂P/∂x=0 when P is pressure. Also, I need an expression for P.


    2. Relevant equations

    it is given that (stationary solution and Darcy's law for flow through porous material).
    0= - ∂P/∂z - ρg - μ/kw

    the temperature as a function of x is,
    T(x)=1/2(Ti+Ty+(Ti-Ty)/L * x)

    the density ρ of the fluid is
    ρ=ρ0(1-αT) , where

    as a boundary condition or whatever, we know that
    w=0 @ x=0.

    3. The attempt at a solution

    to find and expression for w, it seems logical to solve the first equation for w and get
    w=-k/μ(∂P/∂z+ρg), and then subtitute for ρ and then for T(x). But then I get this large awkward equation, and i have not used the hint that ∂P/∂x=0.

    perhaps one should consider the continuity equation too, ∂u/∂x=-∂w/z ?

    any insightful ideas? :)
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
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