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Applying Bernoulli's equation to magnetohydrodynamic flow

  • Thread starter Hendrick
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Homework Statement


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 [tex]p_{0}[/tex] 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]


Homework Equations


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

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

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[/tex] as [tex](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).
 

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