# Pascal's Principle vs. Bernoulli's Principle

Hi,

There is a basic problem I am having with fluid dynamics that has been really confusing me.

I have been told that as a result of conservation of energy and Pascal's principle, for an incompressible fluid Pin=Pout, or pressure is constant.

However, pressure is not necessarily constant in Bernoulli's equation (as if it were, pressure wouldn't even be in the equation).

Is it that pressure is constant if the fluid is static and there is no height difference? If so, textbooks say that it applies in that weird U-shaped container where there is definitely a height difference. How does this work?

sophiecentaur
Gold Member
2020 Award
Pascal's principle doesn't involve acceleration of the fluid; it describes a static, equilibrium situation. Bernoulli's equation is based on the fact that the same volume of fluid has to leave as the volume leaving. If there's a change in cross sectional area then liquid has to change velocity, which requires a force imbalance to accelerate the fluid.

Abhishek kumar
Hi,

There is a basic problem I am having with fluid dynamics that has been really confusing me.

I have been told that as a result of conservation of energy and Pascal's principle, for an incompressible fluid Pin=Pout, or pressure is constant.

However, pressure is not necessarily constant in Bernoulli's equation (as if it were, pressure wouldn't even be in the equation).

Is it that pressure is constant if the fluid is static and there is no height difference? If so, textbooks say that it applies in that weird U-shaped container where there is definitely a height difference. How does this work?

Pascal law state that if pressure in a liquid is changed at aparticular point,the change is transmitted to the entire liquid without being diminished in magnitude
And u shape has always same height on either side if there is no any external pressure and difference in cross sectional areas

Chestermiller
Mentor
Pascal's law just says that, at a given location in a fluid (that is either static or flowing), the pressure acts equally in all directions at that point. This says nothing about how the pressure changes from location to location. Bernoulli's equation is a conservation of energy equation for the fluid, and describes how the variations of pressure, velocity, and elevation of the fluid vary with location.

So, Sal Kahn said that Pin=Pout--when is that applicable?

Chestermiller
Mentor
So, Sal Kahn said that Pin=Pout--when is that applicable?
Who’s Sal Khan?

cjl
So, Sal Kahn said that Pin=Pout--when is that applicable?

Basically only when talking about static (non-moving) fluids where change in pressure due to height effects is negligible.

Chestermiller
Mentor
Basically only when talking about static (non-moving) fluids where change in pressure due to height effects is negligible.
This is not correct. Certainly, in the standardTorricelli problem, the inlet and outlet pressures are atmospheric.

Who’s Sal Khan?

Chestermiller
Mentor
Do you have a specific problem that you are having trouble understanding and that you would like to focus on? If so, please state it.

Do you have a specific problem that you are having trouble understanding and that you would like to focus on? If so, please state it.
No, I'm just confused, as it seems as though the equation Sal gave is inconsistent with Bernoulli's equation.

Chestermiller
Mentor
No, I'm just confused, as it seems as though the equation Sal gave is inconsistent with Bernoulli's equation.
Please write out the equation for us that is causing you confusion and how you feel that it is not consistent with the Bernoulli equation.

That's what I had in the OP; Sal gives the equation Pin=Pout, but in the Bernoulli equation P is not necessarily constant.

cjl
That's because, despite Chestermiller confusingly trying to assert otherwise, it is a principle of fluid statics, and thus is only meant to be applied to unmoving fluids.

cjl
This is not correct. Certainly, in the standardTorricelli problem, the inlet and outlet pressures are atmospheric.

Sure, problems exist where the pressure at two points is identical despite fluid moving, but the principle in question is a principle of fluid statics, and does not apply in general unless the fluid is not moving.

Chestermiller
Mentor
Sure, problems exist where the pressure at two points is identical despite fluid moving, but the principle in question is a principle of fluid statics, and does not apply in general unless the fluid is not moving.
The Bernoulli equation does not only apply to fluid statics.

cjl
The Bernoulli equation does not only apply to fluid statics.

Of course not. Pascal's principle is the principle at question here though, and it only applies to statics. Bernoulli only really is meaningful with fluid dynamics, hence the apparent contradiction between the two.

Chestermiller
Mentor
I looked over the Khan video, and, as far as I can tell, the analysis (although poorly presented) is essentially consistant with the Bernoulli equation. Which part don't you feel is consistent with Bernoulli?

Applying Bernoulli, $$P_1 + \rho g z_1=P_2 + \rho g z_2$$where ##P_1=F_1/A_1## and ##P_2=F_2/A_2##

Now, geometrically, $$z_2-z_1=D_2+D_1$$

So $$P_1=P_2+\rho g (D_1+D_2)$$

Now, if ##D_1## is differentially small, then so is ##D_2##, and the Bernoulli equation approaches:

$$P_1\approx P_2$$

Now do you see why I think the presentation in the Khan video was poor.

Isaac0427
Now do you see why I think the presentation in the Khan video was poor.
Very much so. So, Pascal's principle is just for static situations where there is no difference in height. Would it also apply when the system is in a non-static equilibrium, still with no difference in height?

Thank you very much.

cjl
No, that's when you would use Bernoulli (for non-static cases that are still at a steady state and where you can ignore viscosity), or delve into increasingly complex fluid dynamics equations depending on what assumptions you make.

Isaac0427
Chestermiller
Mentor
Very much so. So, Pascal's principle is just for static situations where there is no difference in height. Would it also apply when the system is in a non-static equilibrium, still with no difference in height?

Thank you very much.
Pascal's law (often incorrectly stated) refers to conditions at a single point in a fluid, not to multiple locations, or to locations throughout the fluid. It applies whether of not the fluid is flowing. It says that the pressure at a point acts equally in all directions at the point. In fluid mechanics parlance, fluid pressure is "isotropic."

Isaac0427
Okay, thank you. I understand this now.