Aparent contradiction - pressure and velocity

[Fanaticus]

So, I have in my mind an apparent contradiction that hopefully can be cleared up.

From early lessons we learn that an objects velocity in an x direction has no effect on its velocity in the y direction. So, the y component of a particles velocity is not effected when we change its x component of velocity. If we have a whole ensemble of particles, it should reason that the sum of their y components of velocity would also be unaffected by a change in their net x velocity. Now the average y velocities of an ensemble of particles determine what pressure is measured, so the pressure should not change.

But Bernoulli's equation says just the opposite... It says that if you increase the velocity in the x direction, the pressure drops. But the pressure comes from the particle's velocities in the y direction!

I think you can see my confusion at this apparent contradiction. Help me understand what is really going on!


Thx a lot

 

[uby]

i'm not an expert on fluid dynamics, but here are a couple starting points:

1) pressure is direction-less. it is not a vector quantity. pressure is a measure of the average kinetic energy particles/molecules in a fluid, or, equivalently, the average collision frequency of particles/molecules against a surface (all for a given temperature).

consider a cubic box containing a gas. the gas has a constant pressure throughout the box, and you would quantify this pressure by measuring the number of collisions of gas molecules with the box surfaces. the top/bottom surfaces would see the same number of collisions as the side surfaces per unit area for a given pressure. the same concept can apply to fluids as well.

2) bernoulli equation relates a dynamic pressure from a flow to the static (stagnation?) pressure at a point generated by the stoppage of the fluid flow. this does not speak towards the relationship in a fluid of pressure and velocity.

 

[Q_Goest]

But Bernoulli's equation says just the opposite... It says that if you increase the velocity in the x direction, the pressure drops. But the pressure comes from the particle's velocities in the y direction!

Bernoulli's says if velocity in x direction increases, pressure in x direction increases (dynamic pressure) and pressure in y direction decreases (static pressure).

 

[rcgldr]

That an objects velocity in an x direction has no effect on its velocity in the y direction. But Bernoulli's equation says just the opposite... It says that if you increase the velocity in the x direction, the pressure drops. But the pressure comes from the particle's velocities in the y direction!

For the first statement, if a force is applied to the object perpendicular to it's direction (a centripetal force), then no work is done, the kinetic energy remains the same, but the objects direction changes, and the components of velocity in the x, y, and z axis will change.

Bernoulli principle in it's simplest form ignores things like friction, viscosity or turbulence, and assumes a closed system, such as a pipe with varying diameters, which peforms no work on the fluid (or gas) (no friction, vicocity, or turbulent effects). Since the flow of mass across any cross section of the pipe is constant (else fluid or gas would be accumulating), then the speed of the flow is relative to the inverse of the cross sectional area. Since no work is done by the pipe, the only remaining cause for the changes in speed are pressure differentials, higher in the larger diameter sections of the pipe, lower in the narrower sections of the pipe, which result in the changes in net velocity in the direction of the flow. The average velocity and the total kinetic energy of the molecules in the fluid remains constant, but the net direction of these velocities will vary with the speed of the flow, and the component of kinetic energy calculated from the component of net velocity in the direction of flow will vary.

When the rate of flow increases in the Bernoulli pipe, the component of velocity in the direction of flow increases, and since the kinetic energy is constant, the component of velocity perpendicular to the direction of flow must decrease. What happens is that the nearly random collisions of the molecules become less random and more directional in during transitions to narrower sections of the pipe, resulting in a net component of acceleration during the transition to narrower sections and vice versa.

 

[Phlogistonian]

The particles collide with one another, exchanging energy in all directions.