# Pressure Propagation through Fluid

Tom79Tom
I have been searching for an explanation of how a change in pressure propagates thru a fluid to become isotropic. My question is that if the time scale measured is small enough ( much less than the distance divided speed of sound ) will we see the pressure change have an initial direction and most importantly does that direction require a hard surface (the container) to refract/reflect as suggested below ?

If it think of the average molecular kinetic action of the pressure increase as this pressure increase represents an increase in average energy and this will flow away from high to low( will have an initial direction) and the average increase in molecular energy will be transferred through the fluid until a hard surface is encountered at which point the kinetic energy will be dispersed (Reflected) in another direction of increasing randomness.

I have found the following blurb from https://www.princeton.edu/~asmits/Bicycle_web/pressure.html

Pressure: transmission through a fluid
An important property of pressure is that it is transmitted through the fluid. When an inflated bicycle tube is pressed at one point, for example, the pressure increases at every other point in the tube. Measurements show that the increase is the same at every point and equal to the applied pressure. For example, if an extra pressure of 5 psi were suddenly applied at the tube valve, the pressure would increase at every point of the tube by exactly this amount. This property of transmitting pressure undiminished is a well established experimental fact, and it is a property possessed by all fluids. The transmission does not occur instantaneously, but at a rate that depends on the speed of sound in the medium and the shape of the container. The speed of sound is important because it measures the rate at which pressure disturbances propagate (sound is just a pressure disturbance travelling through a medium). The shape of the container is important because pressure waves refract and reflect of the walls of the container and this increases the distance and time the pressure waves need to travel.

Mentor
Do you really mean isotropic, or do you mean homogeneous (uniform)? Isotropic refers to directionality at a given point in the fluid. Homogeneous refers to uniformity over a spatial extent.

Chet

Tom79Tom
Hi Chet I think I mean isotropic which I understood to mean equal in all directions it's the the term I see referred to in regards to pressure.
Does that it help ? I'm really most Interested in the process how/ before this state occurs
Do you really mean isotropic, or do you mean homogeneous (uniform)? Isotropic refers to directionality at a given point in the fluid. Homogeneous refers to uniformity over a spatial extent.

Chet
Chet

Mentor
Hi Chet I think I mean isotropic which I understood to mean equal in all directions it's the the term I see referred to in regards to pressure.
Does that it help ? I'm really most Interested in the process how/ before this state occurs

Chet
You are saying that you mean isotropic (identical pressure in all directions at a given location in the gas or liquid), but your description seems to be referring to homogeniety (variation with location in the gas or liquid), and what it takes for the pressure to become homogeneous again after a disturbance. I want to be absolutely sure that we are on the same page. The explanation for isotropy is very different from the explanation for how pressure inhomogenieties are dissipated.

Chet

Tom79Tom
You are saying that you mean isotropic (identical pressure in all directions at a given location in the gas or liquid), but your description seems to be referring to homogeniety (variation with location in the gas or liquid), and what it takes for the pressure to become homogeneous again after a disturbance. I want to be absolutely sure that we are on the same page. The explanation for isotropy is very different from the explanation for how pressure inhomogenieties are dissipated.

Chet
Hi Chet
I guess I am not sure on the terminology, it may help if I give a scenario,
The scenario I envisage is a tube completely enclosed and completely filled with a liquid at a given pressure . Within this tube is a turbine that when started will create a disturbance in a given direction. At the point before the turbine is started the pressure will be homogenous (?) but the turbine will increase average velocity in a given direction. I wish to understand (at very small timescales) how that increase in average velocity will be propagated through the fluid.
I propose while there is individual interactions of molecules that will redirect energy within the fluid if we look at the average increase in velocity I am thinking that this will require a hard surface to reflect and refract against ?
Is my assumption correct as this would be in line with the link explanation that shape of container is a factor in how this disturbance.

Note : I enclosed the tube because I want to get away from the conservation of mass / Bernoullis example to something more like wave theory. (Ie no net movement of molecules only velocity)

Mentor
Hi Chet
I guess I am not sure on the terminology, it may help if I give a scenario,
The scenario I envisage is a tube completely enclosed and completely filled with a liquid at a given pressure . Within this tube is a turbine that when started will create a disturbance in a given direction. At the point before the turbine is started the pressure will be homogenous (?) but the turbine will increase average velocity in a given direction. I wish to understand (at very small timescales) how that increase in average velocity will be propagated through the fluid.
I propose while there is individual interactions of molecules that will redirect energy within the fluid if we look at the average increase in velocity I am thinking that this will require a hard surface to reflect and refract against ?
Is my assumption correct as this would be in line with the link explanation that shape of container is a factor in how this disturbance.

Note : I enclosed the tube because I want to get away from the conservation of mass / Bernoullis example to something more like wave theory. (Ie no net movement of molecules only velocity)
OK. We're definitely on the same page now. You want to know how pressure disturbances propagate and dissipate in fluids and gases.

Do you want the answer to include equations (like the derivation of the wave equation), which is a partial differential equation, or does it have to be qualitative?

The propagation of pressure disturbances in fluids is related to the fact that fluids have both compressibility and mass. The dissipation of disturbances is related to the fact that fluids also exhibit the property of viscosity, which causes mechanical energy to dissipate to heat.

We will talk about the exact case that you alluded to in your post. First, we will consider the case where the cylinder is infinitely long so that no reflections at boundaries is possible yet.

How does this all should to you?

Chet

Tom79Tom
OK. We're definitely on the same page now. You want to know how pressure disturbances propagate and dissipate in fluids and gases.

Do you want the answer to include equations (like the derivation of the wave equation), which is a partial differential equation, or does it have to be qualitative?

The propagation of pressure disturbances in fluids is related to the fact that fluids have both compressibility and mass. The dissipation of disturbances is related to the fact that fluids also exhibit the property of viscosity, which causes mechanical energy to dissipate to heat.

We will talk about the exact case that you alluded to in your post. First, we will consider the case where the cylinder is infinitely long so that no reflections at boundaries is possible yet.

How does this all should to you?

Chet
Hi Chet
That's great, my calc is pretty rusty so if we could stick to the qualitative explanation that would help me greatly.
I'm interested why you would use an infinitely long example but i'm assuming that will help when you explain the example of a completely enclosed length at very small time scales ?
Quentin

klimatos
I may be completely misunderstanding this topic; but when I read the OP two objections sprang immediately to mind. Firstly, no fluid in a gravitational field experiences uniformity of pressure, even when the fluid is at net rest. Secondly, as soon as that turbine starts, nearly all equilibrium-based formulae become invalid and motion-induced pressure differences (Bernoulli effects) are created. If the turbine creates turbulence, then I believe that distribution of pressure within the fluid becomes almost impossible to predict. I am defining pressure as the force exerted against any exposed surface by the bombarding of fluid molecules against that surface.

Mentor
Hi Chet
That's great, my calc is pretty rusty so if we could stick to the qualitative explanation that would help me greatly.
I'm interested why you would use an infinitely long example but i'm assuming that will help when you explain the example of a completely enclosed length at very small time scales ?
Quentin
OK. Here goes.
Suppose you have an infinitely long cylinder with a liquid in it, and a massless piston is located at one end. The fluid is slightly compressible and it has mass (inertia). Now suppose that you suddenly apply a force to the piston, and subsequently maintain that force. The fluid immediately next to the piston will compress, but further out, the fluid will not even feel the effect yet (because the fluid has inertia). As time progresses, the size of the compression zone will get larger and larger as the compression wave propagates across the fluid. The leading edge of the compression zone will advance at the speed of sound in the fluid, given by ##\sqrt{\frac{B}{\rho}}##, where B represents the bulk modulus of the fluid and ρ is its density. The bulk modulus relates to the compressibility of the fluid, and the density relates to its inertia. Ahead of the compression zone, the pressure will still be equal to the pressure that existed before the piston force was applied. Within the compression zone, the pressure increase will be equal to the applied force on the piston divided by the cross sectional area.

There is more to the story, but I'll stop here for now. I want to make sure that you're OK with this part so far.

Chet

• Tom79Tom
Tom79Tom
OK. Here goes.
Suppose you have an infinitely long cylinder with a liquid in it, and a massless piston is located at one end. The fluid is slightly compressible and it has mass (inertia). Now suppose that you suddenly apply a force to the piston, and subsequently maintain that force. The fluid immediately next to the piston will compress, but further out, the fluid will not even feel the effect yet (because the fluid has inertia). As time progresses, the size of the compression zone will get larger and larger as the compression wave propagates across the fluid. The leading edge of the compression zone will advance at the speed of sound in the fluid, given by ##\sqrt{\frac{B}{\rho}}##, where B represents the bulk modulus of the fluid and ρ is its density. The bulk modulus relates to the compressibility of the fluid, and the density relates to its inertia. Ahead of the compression zone, the pressure will still be equal to the pressure that existed before the piston force was applied. Within the compression zone, the pressure increase will be equal to the applied force on the piston divided by the cross sectional area.

There is more to the story, but I'll stop here for now. I want to make sure that you're OK with this part so far.

Chet
Thanks Chet
Happy with that explanation so far go ahead.

Mentor
Now suppose we proceed a little differently. First we apply a sudden force to the piston, so that the compression wave starts to travel down the fluid in the cylinder (just as before). But, now, after a very short time (say the amount it takes for the compression wave to travel 1 ft), we suddenly remove the force. What do you think would happen in that situation?

Chet

Tom79Tom
Now suppose we proceed a little differently. First we apply a sudden force to the piston, so that the compression wave starts to travel down the fluid in the cylinder (just as before). But, now, after a very short time (say the amount it takes for the compression wave to travel 1 ft), we suddenly remove the force. What do you think would happen in that situation?

Chet
Hi Chet I guess when the pressure is released from the piston we would have a region of high pressure with two "fronts" to it, A low pressure zone in front of the wave and behind is that correct ?

Mentor
Hi Chet I guess when the pressure is released from the piston we would have a region of high pressure with two "fronts" to it in front of the wave and behind is that correct ?
Yes. We would have an elevated pressure pulse region traveling down the cylinder at the speed of sound. The pressure ahead of the pulse would be the original undisturbed pressure, and the pressure behind the pulse would also be the original undisturbed pressure.

Are we together on this so far?

Chet

Tom79Tom
Yes. We would have an elevated pressure pulse region traveling down the cylinder at the speed of sound. The pressure ahead of the pulse would be the original undisturbed pressure, and the pressure behind the pulse would also be the original undisturbed pressure.

Are we together on this so far?

Chet

Mentor
Even though the leading and trailing edges of the pulse (region of elevated pressure) are traveling at the speed of sound, the portion of the fluid contained within the pulse at any instant of time is traveling at a much lower speed. Its velocity is only on the order of the compressive strain in the fluid times the speed of sound, and, since the fluid is very incompressible, the fluid velocity within the pulse might be only 0.1% of the speed of sound in the fluid. However, ahead of the pulse and behind the pulse, the fluid is essentially not moving, so that its speed is virtually zero. So at the leading edge of the pulse, even though the fluid velocity within the pulse is fairly moderate, there is a sharp change of fluid velocity from the value inside to pulse to zero up ahead of the pulse. Also, at the trailing edge of the pulse, there is a sharp fluid velocity change from the value of zero behind the pulse to the higher value inside the pulse. Associated with these sharp changes in fluid velocity are very rapid rates of fluid deformation at the leading and trailing edges of the pulse.

So far we haven't said anything about the property of the fluid called viscosity, and how that comes into play. The situation we have described up to now is that which would have prevailed if the viscosity of the fluid were zero. However, viscosity is a property with tends to dissipate mechanical energy. It works like a kind of damper, and the rate at which mechanical energy is dissipated is proportional to the viscosity times the square of the rate at which the fluid deforms. So we expect viscous damping to be significant at the leading and trailing edges of the traveling (pulse where the fluid deformation rate is high). What the viscous damping in the fluid tends to do is to disperse the sharp pressure pulse by smoothing its edges, lowering its peak, and spreading it out axially. So as the pressure pulse travels down our cylinder, it spreads out and decays. And, if we go far enough downstream, we will no longer be able to detect the pressure pulse. It will be completely dissipated by viscous damping.

Chet

• Tom79Tom
Tom79Tom
Hi Chet
Firstly thanks so much for your effort in answering my query. It is really appreciated.I do have a few queries

I think your explanation assists in how the velocity increase is spread (homogenized) from the initial disturbance but I am still unclear how the resultant pressure increase changes direction to become equal in all directions.

Doing some Wiki research I have found fluid deformation due to viscosity has two effects

"Dynamic (shear) viscosity

Viscosity is a property arising from collisions between neighbouring particles in a fluid that are moving at different velocities. When the fluid is forced through a tube, the particles which comprise the fluid generally move more quickly near the tube's axis and more slowly near its walls: therefore some stress, (such as a pressure difference between the two ends of the tube), is needed to overcome the friction between particle layers and keep the fluid moving.

Bulk viscosity

When a compressible fluid is compressed or expanded evenly, without shear, it may still exhibit a form of internal friction that resists its flow. These forces are related to the rate of compression or expansion by a factor σ, called the volume viscosity, bulk viscosity or second viscosity.

The bulk viscosity is important only when the fluid is being rapidly compressed or expanded, such as in sound and shock waves. Bulk viscosity explains the loss of energy in those waves, as described by Stokes' law of sound attenuation."

Can I assume that the deformation you are referring to at the leading and trailing edges to be caused by Bulk Viscosity effects ?

I really want to get back to viewing the disturbance as a mean velocity increase of the fluid particles motion in a given direction. How does this viscosity reduce this mean velocity and increase its randomness . I am interested in the molecular kinetic mechanism that causes this *

* Going back to the Princeton document this suggests that the shape of the container has an effect on how long this pressure difference takes to become equal in all directions.

I think I have sourced a clue from the No slip effect where it describes viscosity as the result of cohesive forces between layers travelling at different speeds having a friction like effect ?**

**I am still however unclear how this translate to random motion? To me won't this cohesion cause molecules in the slow layer to speed up and the faster layer to slow down . How does it cause the mean velocity to alter direction ?

Is this correct and as this is for referenced from Dynamic Viscosity how can I apply it to the velocity differences at the leading and trailing edges you describe in Bulk Viscosity.***

***To apply the same logic I have the same issue as wouldn’t the slow moving trail be accelerated by cohesion forces and the molecules ahead of the leading edge accelerated by repulsive forces but I don’t see how this changes direction ? What would cause this to become random ?

The last query I have is we have explored an example of a wave like disturbance travelling a great length until viscosity effects homogenize (?) the disturbance but what disturbance is continuous (we do not turn off the turbine/no trailing edge) and the pipe is very short so viscous effects are neglible before it encounters the end.What would happen to the increase in mean velocity at the end of the pipe would it reflect?

The question I really want to know is if a hard surface the only way that a mean velocity increase (Dynamic pressure) can change direction to become equal in all directions.