Aerostatics

Aerostatics studies density allocation, especially in air. One of the applications of this is the barometric formula.

An aerostat is a lighter than air craft, such as an airship or balloon, which uses the principles of aerostatics to float.

Basic laws

Treatment of the equations of gaseous behaviour at rest is generally taken, as in hydrostatics, to begin with a consideration of the general equations of momentum for fluid flow, which can be expressed as:

${\displaystyle \rho [\frac{\mathrm{\partial }{U}_j}{\mathrm{\partial }t}+U_i\frac{\mathrm{\partial }{U}_j}{\mathrm{\partial }t}]=-\frac{\mathrm{\partial }P}{\mathrm{\partial }{x}_j}-\frac{\mathrm{\partial }{\tau }_{ij}}{\mathrm{\partial }{x}_i}+\rho g_j}$,

where ${\displaystyle \rho }$ is the mass density of the fluid, ${\displaystyle U_j}$ is the instantaneous velocity, ${\displaystyle P}$ is fluid pressure, ${\displaystyle g}$ are the external body forces acting on the fluid, and ${\displaystyle {\tau }_{ij}}$ is the momentum transport coefficient. As the fluid's static nature mandates that ${\displaystyle U_j=0}$, and that ${\displaystyle {\tau }_{ij}=0}$, the following set of partial differential equations representing the basic equations of aerostatics is found.

${\displaystyle \frac{\mathrm{\partial }P}{\mathrm{\partial }{x}_j}=\rho g_j}$

However, the presence of a non-constant density as is found in gaseous fluid systems (due to the compressibility of gases) requires the inclusion of the ideal gas law:

${\displaystyle \frac{P}{\rho }=RT}$,

where ${\displaystyle R}$ denotes the universal gas constant, and ${\displaystyle T}$ the temperature of the gas, in order to render the valid aerostatic partial differential equations:

${\displaystyle \frac{\mathrm{\partial }P}{\mathrm{\partial }{x}_j}=\rho \widehat{g_j}=\frac{P}{RT}\widehat{g_j}}$,

which can be employed to compute the pressure distribution in gases whose thermodynamic states are given by the equation of state for ideal gases.