2.1 Variation of Pressure with Height

\[\frac{dP}{dz} = -\rho g\] Hydrostatic pressure is due to the weight of the fluid element.

  • For a fluid at rest, pressure is the same at any horizontal level of connected fluid.

  • According to Pascal’s law pressure at a point in a static fluid is equal in all directions.

  • Pressure variation due to depth for a incompressible fluid is given by \[\Delta P = \rho g h\]

  • Considering atmospheric air as an ideal gas, \[\rho = \frac{PM}{RT} \qquad \text{(where $M$ is molecular weight)}\] For variation of \(T\) with \(z\), we get \[\int_{P_0}^P \frac{dP}{P} = -\frac{Mg}{R}\int_0^z\frac{dz}{T(z)}\] If the temperature of atmosphere can be considered as constant (i.e., isothermal atmosphere), then, by integrating the above equation, we get \[P = P_0\exp\left(-\frac{Mg}{RT}z\right)\]