# Applying Bernoulli's equation to magnetohydrodynamic flow

## 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 $$p_{0}$$ at origin (0,0,0) is under the effect of gravity and the electric and magnetic field are given by:

i) $$E = 2\hat{i}, B = 4\hat{k}$$
ii) $$E = \hat{i}+3\hat{j}-\hat{k}, B = 2\hat{i}+\hat{j}+4\hat{k}$$

## Homework Equations

Lorentz force:
$$F_{L}$$=$$E \times B$$

Bernoulli's equation:
$$\frac{p}{\rho} + \frac{u \cdot u}{2} + gz = constant$$

## The Attempt at a Solution

i)
Modify Bernoulli's equation to account for Lorentz's force:
$$\frac{p}{\rho} + \frac{u \cdot u}{2} + gz + F_{L} = constant$$
Divide by g to find the heads
$$\frac{p}{\rho g} + \frac{u \cdot u}{2 g} + z + \frac{F_{L}}{g} = constant$$

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

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).