Fluid Dynamics: Pressure Field - Vector or Scalar? | Homework Question

[shreddinglicks]

Homework Statement

This is not so much a homework problem but a question I have on the subject while studying. I have a terrible teacher who is contradicting my textbook constantly and I really want to learn this.

Is a pressure field vector or scalar? If given the velocity components, is it appropriate to get the magnitude and insert it into the Bernoulli eq to solve for the change in pressure? Which would give me a scalar answer. Or is the correct method to use Euler's eq to obtain the components of pressure changes? Which would be a vector. Or are both these methods ok?

Homework Equations

The Attempt at a Solution

 

[haruspex]

Pressure is a scalar; pressure gradient is a vector.
Beyond that, I am not clear on what methods you are proposing. How about you post the corresponding algebra?

 

[shreddinglicks]

Pressure is a scalar; pressure gradient is a vector.
Beyond that, I am not clear on what methods you are proposing. How about you post the corresponding algebra?

 

[haruspex]

I'll need more background.
State the actual problem.
Define your variables.
Identify where you are applying standard equations.

 

[shreddinglicks]

I'll need more background.
State the actual problem.
Define your variables.
Identify where you are applying standard equations.

u and v are the x and y components of velocity.

The 1st picture is applying Euler's eq. The 2nd picture is the Bernoulli eq.

 

[shreddinglicks]

this is Euler's eq

Bernoulli eq

 

[Chestermiller]

Homework Statement

This is not so much a homework problem but a question I have on the subject while studying. I have a terrible teacher who is contradicting my textbook constantly and I really want to learn this.

Is a pressure field vector or scalar?

Actually, it is neither. Pressure is the magnitude of the isotropic component of a second order tensor, we call the stress tensor. Since it is the magnitude of that component, we can regard it as a scalar. However, because of its tensorial character, it is accompanied by a certain type of directionality: for any arbitrarily oriented surface within a fluid (or at solid interface of a fluid), the pressure always acts perpendicular to the surface.

If given the velocity components, is it appropriate to get the magnitude and insert it into the Bernoulli eq to solve for the change in pressure?

Yes. You don't need the velocity components. All you need is the magnitude of the velocity vector for Bernoulli.

Which would give me a scalar answer.

Yes.

Or is the correct method to use Euler's eq to obtain the components of pressure changes? Which would be a vector.

No. Euler's eqns. are expressed in terms of the scalar magnitude of the pressure.

Or are both these methods ok?

Both methods are OK. But, Bernoulli is easier if you don't need the components of the velocity or the force components on solid boundaries.

 

[shreddinglicks]

Actually, it is neither. Pressure is the magnitude of the isotropic component of a second order tensor, we call the stress tensor. Since it is the magnitude of that component, we can regard it as a scalar. However, because of its tensorial character, it is accompanied by a certain type of directionality: for any arbitrarily oriented surface within a fluid (or at solid interface of a fluid), the pressure always acts perpendicular to the surface.Yes. You don't need the velocity components. All you need is the magnitude of the velocity vector for Bernoulli.

Yes.

No. Euler's eqns. are expressed in terms of the scalar magnitude of the pressure.

Both methods are OK. But, Bernoulli is easier if you don't need the components of the velocity or the force components on solid boundaries.

Thanks! That was very helpful. One last question though. You say Euler's equations are expressed in terms of the scalar magnitude of pressure. My textbook says the Euler's equations I posted in post #6 are the three component equations. Isn't that the vector components in x,y,z directions?

 

[Chestermiller]

Thanks! That was very helpful. One last question though. You say Euler's equations are expressed in terms of the scalar magnitude of pressure. My textbook says the Euler's equations I posted in post #6 are the three component equations. Isn't that the vector components in x,y,z directions?

As haruspex said, those are the components of the pressure gradient.

 

[shreddinglicks]

As haruspex said, those are the components of the pressure gradient.

Thanks again! I finally understand. Time for me to study some more.