Vertical pressure variation
Pressure varies with elevation due to gravity and fluid density.
Vertical pressure variation is the variation in pressure as a function of elevation. Depending on the fluid in question and the context, it may also vary significantly in dimensions perpendicular to elevation, and these variations have relevance in the context of pressure gradient force and its effects. The vertical variation is especially significant, as it results from the pull of gravity on the fluid; for the same given fluid, a decrease in elevation within it corresponds to a taller column of fluid weighing down on that point.
- basic_formula
- dP/dh = -ρg
- key_variables
- P (pressure), ρ (density), g (gravity), h (height)
- hydrostatic_paradox
- Any quantity of liquid, however small, may be made to support any weight, however large.
- atmospheric_model
- Ph = P0 * e^(-mgh/kT)
- height_from_pressure
- z = -(RT/g) * ln(P/P0)
Lore & Background
A relatively simple version of vertical fluid pressure variation is that the pressure difference between two elevations is the product of elevation change, gravity, and density. The equation dP/dh = -ρg shows that an increase in height corresponds to a decrease in pressure. When density and gravity are approximately constant, multiplying height difference, gravity, and density yields a good approximation of pressure difference. For a liquid with uniform density, pressure at another point is given by P1 = P0 - ρg(h1 - h0). Where different fluids are layered, the total pressure difference is obtained by adding the pressure differences for each fluid.
Reader's Guide
The barometric formula depends only on the height of the fluid chamber, not on its width or length. Given a large enough height, any pressure may be attained, a feature called the hydrostatic paradox. The Flemish scientist Simon Stevin was the first to explain the paradox mathematically. Pouring water of weight w down the tube will eventually raise the heavy weight, leading to the equation W = wA/α. Hydraulic machinery employs this phenomenon to multiply force or torque. In the context of Earth's atmosphere, air is compressible and density varies significantly with height. Using the ideal gas law, a more accurate formula yields pressure as an exponential function of height: Ph = P0 e^(-mgh/kT). An alternative derivation gives height as a function of pressure: z = -(RT/g) ln(P/P0).
Did You Know?
- The hydrostatic paradox states that any quantity of liquid, however small, may be made to support any weight, however large.
- Simon Stevin was the first to explain the hydrostatic paradox mathematically.
- For Earth's atmosphere, pressure decreases exponentially with height, not linearly.
More in Classical And Continuum Mechanics 1-19
Spotted an error? Know more?
This is a living reference — every entry is fact-audited, and reader corrections feed straight into our audit queue. Suggest an edit · See this site's audit record