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Applying Bernoulli's equation to magnetohydrodynamic flow

  1. May 20, 2010 #1
    1. The problem statement, all variables and given/known data
    A static electrically conducting fluid, in the presence of electric and magnetic fields, experiences a Lorentz force. Determine the fluid pressure at point (1,2,1) when the pressure LaTeX Code: p_{0} at origin (0,0,0) is under the effect of gravity and the electric and magnetic field are given by:

    i) [tex]E = 2\hat{i}, B = 4\hat{k}[/tex]
    ii) [tex]E = \hat{i}+3\hat{j}-\hat{k}, B = 2\hat{i}+\hat{j}+4\hat{k}[/tex]


    2. Relevant equations
    Lorentz force:
    [tex]F_{L} = E \times B[/tex]

    Bernoulli's equation:
    [tex]\frac{p}{\rho} + \frac{u \cdot u}{2} + gz = constant[/tex]

    3. The attempt at a solution
    i)
    Modify Bernoulli's equation to account for Lorentz's force:
    [tex]\frac{p}{\rho} + \frac{u \cdot u}{2} + gz + F_{L} = constant[/tex]
    Divide by g to find the heads
    [tex]\frac{p}{\rho g} + \frac{u \cdot u}{2 g} + z + \frac{F_{L}}{g} = constant[/tex]

    Apply modified Bernoulli's equation to the two points.
    [tex]\frac{p_{0}}{\rho g} + \frac{u \cdot u}{2 g} + z_{0} + \frac{F_{L}}{g} = \frac{p_{x}}{\rho g} + \frac{u \cdot u}{2 g} + z_{x} + \frac{F_{L}}{g}[/tex]
    [tex]\frac{p_{0}}{\rho g} + z_{0} = \frac{p_{x}}{\rho g} + z_{x}[/tex]
    [tex]\frac{p_{x}}{\rho g} = \frac{p_{0}}{\rho g} + z_{0} - z_{x}[/tex]
    [tex]p_{x} = p_{0} + (z_{0} - z_{x})\rho g[/tex]
    [tex]p_{x} = p_{0} - \rho g as (z_{0} - z_{x}) = 0 - 1[/tex]

    I'm pretty much stuck from here, I don't think I modified Bernoulli's equation properly because I don't end up using the Lorentz force in my calculation of the pressure at point (1,2,1).
     
    Last edited: May 20, 2010
  2. jcsd
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