Fluid mechanics

Pressure in fluids and atmospheric pressure

Pressure in a stationary liquid rises with depth; atmospheric pressure is the force per area exerted by the weight of the air.

Water pushes on the bottom and walls of its container, and the pressure is greater farther below the surface. Air also has mass and weight, so it exerts atmospheric pressure around us.

p=p_0+\rho gh,\qquad p_{\mathrm{atm}}\approx1.01\times10^5\,\mathrm{Pa}

Definition: Pressure

Pressure $p=F/A$ is normal force $F$ per area $A$ , measured in pascals (Pa). In a uniform stationary liquid at depth $h$ , gauge pressure is $p-p_0=\rho gh$ , where $p_0$ is the surface pressure.

Vary depth and density to explore gauge pressure; the press shows pressure transmission to a larger piston.

Depth and vessel shape

Gauge pressure $\rho gh$ depends on depth, density, and $g$ , not the vessel’s shape. At the same depth in one connected stationary liquid, pressure is equal; absolute pressure also includes $p_0$ .

Atmospheric pressure

The air above exerts pressure on the liquid surface and on our bodies. Atmospheric pressure decreases with altitude because less air lies overhead; we usually do not feel its large value because internal pressures approximately balance it.

Example: Pressure at a lake bottom

At a depth of $3.0\,\mathrm m$ in water ( $\rho=1000\,\mathrm{kg/m^3}$ ), take $g=9.8\,\mathrm{m/s^2}$ . Find gauge pressure.

Solution

$p-p_0=\rho gh=1000\times9.8\times3.0=29{,}400\,\mathrm{Pa}=29.4\,\mathrm{kPa}$ .

Quick check

Example: Comparing pressure at two depths

A diver at $2.0,mathrm m$ depth experiences gauge pressure $ho gh=1000 imes9.8 imes2.0=19.6,mathrm{kPa}$ . At $5.0,mathrm m$ , gauge pressure is $49.0,mathrm{kPa}$ , an increase of $29.4,mathrm{kPa}$ . The difference depends only on the $3.0,mathrm m$ depth change; add the atmospheric pressure at the surface to obtain absolute pressure.

What is the gauge pressure in a stationary liquid at depth $h$ ?

At the same depth in the same stationary liquid, pressure primarily depends on:

References

David Halliday, Robert Resnick, Jearl Walker (2018). Fundamentals of Physics