question archive Consider a slab of gas in hydrostatic equilibrium in an external gravitation potential ($) linearly dependent on the vertical coordinate: P(z) = gz, with g a positive constant
Consider a slab of gas in hydrostatic equilibrium in an external gravitation potential ($) linearly dependent on the vertical coordinate: P(z) = gz, with g a positive constant
Subject:Mechanical EngineeringPrice: Bought3
Consider a slab of gas in hydrostatic equilibrium in an external gravitation potential ($) linearly dependent on the vertical coordinate: P(z) = gz, with g a positive constant. 1. Use the Euler equation to derive a formula for the vertical hydrostatic equilibrium. Hint: In the equilibrium, there is no acceleration of fluid elements. 2. Assuming that the medium is also in thermal equilibrium, derive the vertical density profile p(z). Give an expression for the scaleheight h. 3. Consider the case of the Earth's atmosphere as a molecular gas of only Nitrogen (with g =9.8 x 10' cms 2 and T = 280 K) and estimate the scaleheight. 4. Assume then that the medium does, instead of isothermal, behave adiabatically (P = Kp?) and find the vertical density profile in this case. Tip: in the final equation it may be useful to define the scale height as 1 h = (1) y-1g where cs is the adiabatic sound speed.
