Vertical pressure variation
Depending on the fluid in question and the context being referred to, it may also vary significantly in dimensions perpendicular to elevation as well, and these variations have relevance in the context of pressure gradient force and its effects.
Since g is negative, an increase in height will correspond to a decrease in pressure, which fits with the previously mentioned reasoning about the weight of a column of fluid.
When density and gravity are approximately constant (that is, for relatively small changes in height), simply multiplying height difference, gravity, and density will yield a good approximation of pressure difference.
If the density of the fluid varies with height, mathematical integration would be required.
Whether or not density and gravity can be reasonably approximated as constant depends on the level of accuracy needed, but also on the length scale of height difference, as gravity and density also decrease with higher elevation.
The barometric formula depends only on the height of the fluid chamber, and not on its width or length.
As expressed by W. H. Besant,[3] The Flemish scientist Simon Stevin was the first to explain the paradox mathematically.
[4] In 1916 Richard Glazebrook mentioned the hydrostatic paradox as he described an arrangement he attributed to Pascal: a heavy weight W rests on a board with area A resting on a fluid bladder connected to a vertical tube with cross-sectional area α. Pouring water of weight w down the tube will eventually raise the heavy weight.
Balance of forces leads to the equation Glazebrook says, "By making the area of the board considerable and that of the tube small, a large weight W can be supported by a small weight w of water.
"[5] Hydraulic machinery employs this phenomenon to multiply force or torque.
[6][7] If one is to analyze the vertical pressure variation of the atmosphere of Earth, the length scale is very significant (troposphere alone being several kilometres tall; thermosphere being several hundred kilometres) and the involved fluid (air) is compressible.
Gravity can still be reasonably approximated as constant, because length scales on the order of kilometres are still small in comparison to Earth's radius, which is on average about 6371 km,[8] and gravity is a function of distance from Earth's core.
For smaller height differences, including those from top to bottom of even the tallest of buildings, (like the CN Tower) or for mountains of comparable size, the temperature variation will easily be within the single-digits.
An alternative derivation, shown by the Portland State Aerospace Society,[10] is used to give height as a function of pressure instead.
where (with values used in the article) A more general formula derived in the same article accounts for a linear change in temperature as a function of height (lapse rate), and reduces to above when the temperature is constant:
