Applying Bernoulli's equation to magnetohydrodynamic flow

In summary, the problem involves determining the fluid pressure at a given point in the presence of electric and magnetic fields. Using Bernoulli's equation, the calculation is modified to account for the Lorentz force and the pressure is expressed in terms of the pressure at the origin and the difference in heights between the two points. However, it is unclear how to incorporate the Lorentz force into the final calculation. For the second part of the problem, there is uncertainty on how to proceed with the given information.
  • #1
Hendrick
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0

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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  • #2
How do I use the Lorentz force in my calculation?ii)I'm not sure how to approach this question at all.
 

What is Bernoulli's equation and how is it used in magnetohydrodynamic flow?

Bernoulli's equation is a fundamental principle in fluid mechanics that relates the pressure, velocity, and elevation of a fluid. It is commonly used in magnetohydrodynamic flow, which is the study of the behavior of electrically conductive fluids in the presence of a magnetic field. In this context, Bernoulli's equation is used to describe the relationship between the pressure, velocity, and magnetic field strength in the fluid.

What is the significance of magnetohydrodynamic flow and why is it important to study?

Magnetohydrodynamic flow has many practical applications, such as in the design of electric generators, plasma confinement devices, and magnetohydrodynamic pumps. It is also important to study because it can help us better understand natural phenomena such as the Earth's magnetic field and the behavior of gases in stars and other astronomical bodies.

Can Bernoulli's equation be applied to all types of magnetohydrodynamic flow?

Bernoulli's equation can be applied to most types of magnetohydrodynamic flow, but there are some cases where it may not be accurate, such as when the fluid is highly compressible or when the magnetic field is very strong. In these cases, more complex equations may be needed to accurately describe the flow.

What are the assumptions made in applying Bernoulli's equation to magnetohydrodynamic flow?

The main assumptions made in applying Bernoulli's equation to magnetohydrodynamic flow are that the fluid is incompressible, the flow is steady, and there is no friction or energy loss in the system. Additionally, the magnetic field must be constant and not affected by the fluid flow.

How does including the magnetic field in Bernoulli's equation affect the results of magnetohydrodynamic flow?

Including the magnetic field in Bernoulli's equation can significantly affect the results of magnetohydrodynamic flow. The magnetic field can alter the pressure and velocity distribution in the fluid, leading to changes in the flow patterns and overall behavior. It can also affect the amount of energy required to move the fluid, as the magnetic field exerts a force on the conductive particles in the fluid.

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