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Math Help - Integrating a function of compressibility

  1. #1
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    Integrating a function of compressibility

    Hi, i have a differential equation I am trying to integrate, but I do not know how to do it. The equation is:

    dp = rho*g*dx

    rho = rho_0 + (p(h)-p_0)/c^2

    dp = (rho_0 + (p(h)-p_0)/c^2)*g*dh

    This is a linear equation for the compressibility in a fluid. p(h) is the pressure at a given height below the surface of the fluid, p_0 is a reference pressure (above the fluid, atmospheric), rho_0 is the density of this fluid at that pressure, c is a constant and g is the gravity constant.

    Anyone knows what the expression for p(h) will be?
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  2. #2
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    Quote Originally Posted by balloo View Post
    Hi, i have a differential equation I am trying to integrate, but I do not know how to do it. The equation is:

    dp = rho*g*dx

    rho = rho_0 + (p(h)-p_0)/c^2

    dp = (rho_0 + (p(h)-p_0)/c^2)*g*dh

    This is a linear equation for the compressibility in a fluid. p(h) is the pressure at a given height below the surface of the fluid, p_0 is a reference pressure (above the fluid, atmospheric), rho_0 is the density of this fluid at that pressure, c is a constant and g is the gravity constant.

    Anyone knows what the expression for p(h) will be?
    Is this part of a bigger question? Could you post the question exactly as it was?
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  3. #3
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    No, not really. It is only this function I need to integrate for it self.
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  4. #4
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    Quote Originally Posted by balloo View Post
    Hi, i have a differential equation I am trying to integrate, but I do not know how to do it. The equation is:

    dp = rho*g*dx

    rho = rho_0 + (p(h)-p_0)/c^2

    dp = (rho_0 + (p(h)-p_0)/c^2)*g*dh

    This is a linear equation for the compressibility in a fluid. p(h) is the pressure at a given height below the surface of the fluid, p_0 is a reference pressure (above the fluid, atmospheric), rho_0 is the density of this fluid at that pressure, c is a constant and g is the gravity constant.

    Anyone knows what the expression for p(h) will be?
     dp = g\bigg(\rho_0 + \frac{p(h)-p_0}{c^2}\bigg)dh

     dp =  \bigg(g \rho_0 + \frac{g}{c^2}(p(h)-p_0)\bigg)dh

     dp =  \bigg(g \rho_0 + \frac{g}{c^2}(p(h))-\frac{g p_0}{c^2}\bigg)dh

     \int dp =  \int \bigg(g \rho_0 + \frac{g}{c^2}(p(h))-\frac{g p_0}{c^2}\bigg)dh


     \int dp =  \int g \rho_0 dh + \int \frac{g}{c^2}(p(h))dh- \int \frac{g p_0}{c^2}dh

     p = \rho_0 g h  + \frac{g}{c^2}\int (p(h))dh-  \frac{gh p_0}{c^2}

     p = g h\bigg(\rho_0 - \frac{p_0}{c^2}\bigg)  + \frac{g}{c^2}\int (p(h))dh

    Can't really do much more than this.
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