Fluid Mechanics seventh edition by Frank M. White · Problem 2.25

Pressure Distribution in a Fluid — Frank M. White

Physics — Fluid Mechanics · the-atmosphere

Problem

As measured by NASA's Viking landers, the atmosphere of Mars, where g≈3.71g \approx 3.71 m/s2^2, is almost entirely carbon dioxide, and the surface pressure averages 700 Pa. The temperature is cold and drops off exponentially: T≈Toe−CzT \approx T_o e^{-Cz}, where C=1.3E-5C = 1.3\text{E-5} m−1^{-1} and To=250T_o = 250 K. For example, at 20,000 m altitude, T≈193T \approx 193 K. (a)(a) Find an analytic formula for the variation of pressure with altitude. (b)(b) Find the altitude where pressure on Mars has dropped to 1 pascal.
Intuitive explanation

The key is recognizing that the barometric formula—which relates pressure to altitude—depends on how density changes with height. Since temperature varies exponentially on Mars, you can't use the constant-temperature approximation; instead, substitute the temperature profile into the hydrostatic equation and the ideal gas law, then separate and integrate. This transforms an exponential temperature decay into a different pressure profile that you can solve analytically.

Intuitive explanation

The key is recognizing that the barometric formula—which relates pressure to altitude—depends on how density changes with height. Since temperature varies exponentially on Mars, you can't use the constant-temperature approximation; instead, substitute the temperature profile into the hydrostatic equation and the ideal gas law, then separate and integrate. This transforms an exponential temperature decay into a different pressure profile that you can solve analytically.

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