Pressure in a moving fluid

Question

I’m having some quite large conceptual difficulties which I hope can be resolved by considering this example.

When considering the pressure in the sheet very far away from the impact zone the argument is “the streamlines are parallel, therefore it’s atmospheric pressure”. This brings a few questions:

1) in a moving fluid, is the pressure exerted perpendicular to the streamlines not greater than on surfaces parallel?

2) Why is the pressure atmospheric?

3)What is the pressure term in Bernoulli for a moving fluid? Is it the pressure in the direction of motion, or the pressure being exerted in it by the surroundings? (This is obviously gonna be easier to answer eight rh help of 1).

4) why is the pressure very far away from the point of impact the same as the initial pressure (and hence the velocity the same)?

Answer 1

1. Pressre in a fluid at a point has the same magnitude in every direction. If the "pressure" was different in two perpendicular directions, then there would be a shear force acting on the fluid in directions in between those two. A fluid can not support shear stresses, by definition - that is the difference between a fluid and a solid. (See any textbook on solid mechanics or strength of materials to understand why there would be a shear force, or google for "Mohr's circle".)

2. It is atmospheric at the surface of the water stream because of equilibrium. If it was different, there would be a net force on the surface and the shape of the stream would change. It is not necessarily atmospheric at every point inside the water jet. If the water is accelerating (which is must be in your example, since its velocity changes direction from vertical to horizontal) the acceleration is caused by the pressure gradient, consistent with Newton's second law of motion.

   For example, in the vertical part of the flow, the downwards velocity will be increasing because of gravity. If the fluid is incompressible, the cross section of the stream will therefore get smaller as the downward velocity increases. Therefore, the fluid particles are also moving sideways towards the center of the stream, and that motion is produced by a change in pressure between the center and the edge of the stream.

   Applying Bernoulli's equation to this type of flow is complicated. It is much simpler if the flow is in a pipe, where the fluid pressure at the boundary is not constant (it is balanced by changes in the stress in the pipe).

   You can apply Newton's second law in the form of "force = rate of change of momentum" to find the force on the plate, without knowing the detailed pressure distribution in the fluid.

3. Since there is only one value of "pressure" at each point in the fluid, this question doesn't arise.

Answer 2

For a non-viscous fluid, the stress tensor is diagonal and isotropic. There is therefore only one force per unit area normal to the surface and which is the pressure. It does not depend on the orientation of the surface considered.

On the other hand, if you insert in a fluid a material small surface normal to the flow, you will find a different force because you modify the flow.

Finally, if the flow is unidirectional, then a fluid particle does not accelerate in directions perpendicular to the direction of flow. If we denote Oz this direction, Newton's law projected according to the normal is $\left( -\overrightarrow{\nabla }p+\mu \overrightarrow{g} \right)\cdot \overrightarrow{{{e}_{y}}}=0$ and $\left( -\overrightarrow{\nabla }p+\mu \overrightarrow{g} \right)\cdot \overrightarrow{{{e}_{x}}}=0$ : the pressure varies as in static in a plane perpendicular to the flow. This is happening quite far from the point of impact, with almost parallel flow lines.

Sorry for my poor english !