# Understanding Pascal's Principle

What I understand by Pascal's principle is that the pressure that when there is an increase in pressure at any point in a confined fluid, there is an equal increase at every other point in the container. However, while this should make intuitive sense to me, the notion clashes in my head when I consider also the formula for pressure by a fluid; namely $$P = \rho gh$$. According to the formula, the deeper you go in the fluid, the greater the pressure. When considering an object in a confined fluid, the bottom of the object experiences a greater amount of pressure than the top of the object, hence a buoyant force arises. Could someone explain me what I am failing to grasp?

• Pascal's principle says that, at static equilibrium, at any given location in a fluid, pressure acts equally in all directions. That is, pressure is isotropic. Commented Apr 5, 2023 at 15:31

Pascal's principle is a special case of equation of continuity, $$$$\frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf{j}=\sigma$$$$ Where $$\rho$$, $$\mathbf{j}$$ and $$\sigma$$ are density, flux and source of some quantity respectively.

Now since in Pascal's principle we take that there is no source so that the equation becomes $$$$\frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf{j}=0$$$$

In case of an fluid $$\mathbf{j} = \rho\mathbf{v}$$ where $$\mathbf{v}$$ is flow velocity field (describing how the fluid is flowing)

And if the fluid is incompressible i.e density is not changing then it becomes $$$$\nabla\cdot\mathbf{v}=0$$$$

Means that $$\mathbf{v}$$ has no divergence. i.e volume is not diverging or it is constant.But it can happen the fluid is rotating inside i.e. curl need not vanish.

This means that if I try to compress the volume from one side other sides should compensate for it giving me a constant volume,

Now if I displace water $$\Delta V =0$$ from which we can arrive at Pascal's principle Basically if we displace water by some volume on one side other sides should be displaced too keeping the volume constant, now generalized force corresponding to displacement of volume is pressure, which implies that pressure must be equal to attain equilibrium (no displacement of volume).

Or Consider an uneven (different radius) cylinder with two ends connected to two piston, if I displace one piston say with $$\Delta x_1$$ the volume displacement would be $$A_1 \Delta x_1$$ which should be equal to other end $$A_2 \Delta x_2$$

$$A_1 \Delta x_1 =A_2 \Delta x_2$$

Now differentiate it with time twice and you'll get force giving you again that $$P_1 =P_2$$

Notice that in all this we never considered what is the pressure gradient inside the container, as long as the fluid is incompressible Pascal's principle is gonna work wether it's on earth or in space.

namely $$P=ρgh$$

it's actually the pressure difference and right thing to say would be $$\Delta P= \rho g \Delta h$$

Which implies as we go down pressure increases. But Pascal's principle doesn't depend on the pressure gradient or anything it's just a consequence of incompressiblity of fluid.

Both are correct , Pascals principle reflects the change in pressure observed at the liquid at any point throughout the entirety of the fluid. So if the pressure is observed at the deepest points of fluid where pressure will be more will be more as at any point than normal due to pascals principle. Now the pressure formula $$\rho g h$$ just says that when increase of height from surface is observed to have more pressure exerted by the fluid underneath(its weighed down by other molecules on top) so if the fluid at lets say point $$x$$ already exerts great pressure due to pascal's principle , the fluid is going to exert more pressure due to the pressure formula. Two of them work on the fluid at the same time

• Pascal's principle is defined with an external force on the system so this might help your intuition that both external force and pressure increase due to depth plays a part in the depth you are concerned with Commented Apr 4, 2023 at 5:38