Applying Bernoulli's equation to magnetohydrodynamic flow

[Hendrick]

1. 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 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) $$E = 2\hat{i}, B = 4\hat{k}$$
ii) $$E = \hat{i}+3\hat{j}-\hat{k}, B = 2\hat{i}+\hat{j}+4\hat{k}$$2. Homework Equations
Lorentz force:
$$F_{L} = E \times B$$

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

3. 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).

 

[mark726]

Any help would be appreciated.

Hi there,

It seems like you are on the right track, but there are a few things that need to be clarified.

Firstly, in your attempt at a solution, you have used the modified Bernoulli's equation, which takes into account the Lorentz force. However, you have not included the Lorentz force in your calculation of the pressure at point (1,2,1). This is because the Lorentz force is already included in the equation, and you do not need to add it separately. The Lorentz force is included in the term $\frac{F_L}{g}$, which is already present in the equation.

Secondly, in your calculation of the pressure at point (1,2,1), you have used the equation $p_x = p_0 + (z_0 - z_x)\rho g$. This equation is incorrect, as it does not take into account the Lorentz force. The correct equation to use would be $\frac{p_x}{\rho g} = \frac{p_0}{\rho g} + z_0 - z_x + \frac{F_L}{g}$. This takes into account the Lorentz force and gives the correct pressure at point (1,2,1).

Finally, to solve this problem, you will need to use the given values for the electric and magnetic fields and solve for the Lorentz force. Once you have the Lorentz force, you can substitute it into the modified Bernoulli's equation and solve for the pressure at point (1,2,1).

Hope this helps!