Hydrostatic equilibrium

This is a whole presentation on hydrostatic equilibrium a topic in fluid flow operations for chemical engineer made by me as a part of term work for our vgec college . Hope you will find its useful

1. Topic :- Hydrostatic Equilibrium Term Work by:- Aniket Jha (200170105005) Adarsh prajapati(200170105001) Harmish Kotadiya(200170105002) Prince Bhalala(200170105004)

2. Definition Hydrostatic equilibrium is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force.In the planetary physics of the Earth, the pressure-gradient force prevents gravity from collapsing the planetary atmosphere into a thin, dense shell, whereas gravity prevents the pressure-gradient force from diffusing the atmosphere into outer space .

3. Application of hydrostatic equilibrium In Fluids (Fluid Mechanics ) In planets and stars (planetary geology )

4. Atmospheric hydrostatic equilibrium

5. The Atmosphere’s vertical pressure structure plays a critical role in weather and climate The Atmosphere’s basic pressure structure is determined by the hydrostatic balance of forces To a good approximation, every air parcel is acted on by three forces that are in balance, leading to no net force . There are  3 forces   that determine hydrostatic balance : One force is downwards (negative) onto the top of the cuboid from the pressure,  p , of the fluid above it. It is, from the definition of  pressure Similarly, the force on the volume element from the pressure of the fluid below pushing upwards (positive) is: Fbottom= pbottomA   Finally, the weight of the volume element causes a force downwards. If the  density is ρ, the volume is  V , which is simply the horizontal area  A  times the vertical height,  Δz , and  g  the standard gravity, then:

6. Fweight=−ρVg=−ρgAΔz By balancing these forces, the  total force on the fluid  is: ∑ F = Fbottom + Ftop + Fweight = pbottom A −ptop A − ρgAΔ z This sum equals zero if the air's velocity is constant or zero. Dividing by  A , p top− pbottom =− ρgΔz P top  − P bottom  is a change in pressure , and Δz is the height of the volume element – a change in the distance above the ground. By saying these changes are infinitesimally small, the equation can be written in differential form, where  dp  is top pressure minus bottom pressure just as dz is top altitude minus bottom altitude. d p = − ρ gdz The result is the equation: dpdz =− ρg This equation is called the  Hydrostatic Equation

7.   The A tmospheric pressure falls off exponentially with height at a rate given by the scale height. Thus, for every 7 km increase in altitude, the pressure drops by about 2/3. At 40 km, the pressure is only a few tenths of a percent of the surface pressure. Similarly, the concentration of molecules is only a few tenths of a percent, and since molecules scatter sunlight, you can see in the picture below that the scattering is much greater near Earth's surface than it is high in the atmosphere

8. Hydrostatic equilibrium in fluids Hydrostatic equilibrium in fluids Incompressible fluids Compressible fluids

9. Incompressible fluids If the fluid is incompressible, so that the density is independent of the pressure, the weight of a column of liquid is just proportional to the height of the liquid above the level where the pressure is measured.

10. Compressible fluids Hydrostatic equilibrium is a little more complicated to apply to air, because air is very compressible. The same principle still applies, but we now have to deal with a density that varies with pressure and temperature For an ideal gas the density and pressure are related by the equation

11. Application Application of Fluid Mechanics Hydrostatic Balance A strophysics Planetary Geology Atmosphere modelling

12. Hydrostatic Balance The hydrostatic equilibrium pertains to  hydrostatics  and the  principles of equilibrium  of  fluids . A hydrostatic balance is a particular balance for weighing substances in water. Hydrostatic balance allows the  discovery  of their  specific gravities . This equilibrium is strictly applicable when an ideal fluid is in steady horizontal laminar flow, and when any fluid is at rest or in vertical motion at constant speed. It can also be a satisfactory approximation when flow speeds are low enough that acceleration is negligible

13. Astrophysics In any given layer of a  star , there is a hydrostatic equilibrium between the outward thermal pressure from below and the weight of the material above pressing inward. The  isotropic  gravitational field compresses the star into the most compact shape possible. A rotating star in hydrostatic equilibrium is an  oblate spheroid  up to a certain (critical) angular velocity. An extreme example of this phenomenon is the star  Vega , which has a rotation period of 12.5 hours. Consequently, Vega is about 20% larger at the equator than at the poles. A star with an angular velocity above the critical angular velocity becomes a  Jacobi (scalene) ellipsoid , and at still faster rotation it is no longer ellipsoidal but  piriform  or  oviform , with yet other shapes beyond that, though shapes beyond scalene are not stable. [5] If the star has a massive nearby companion object then  tidal forces  come into play as well, distorting the star into a scalene shape when rotation alone would make it a spheroid. An example of this is  Beta Lyrae . Hydrostatic equilibrium is also important for the  intracluster medium , where it restricts the amount of fluid that can be present in the core of a  cluster of galaxies . We can also use the principle of hydrostatic equilibrium to estimate the  velocity dispersion  of  dark matter  in clusters of galaxies

14. Planetary Geology The concept of hydrostatic equilibrium has also become important in determining whether an astronomical object is a  planet ,  dwarf planet , or  small Solar System body . According to the  definition of planet  adopted by the  International Astronomical Union  in 2006, one defining characteristic of planets and dwarf planets is that they are objects that have sufficient gravity to overcome their own rigidity and assume hydrostatic equilibrium. Such a body will often have the differentiated interior and geology of a world (a  planemo ), though near-hydrostatic or formerly hydrostatic bodies such as the proto-planet  4 Vesta  may also be differentiated and some hydrostatic bodies (notably  Callisto ) have not thoroughly differentiated since their formation. Often the equilibrium shape is an  oblate spheroid , as is the case with Earth. However, in the cases of moons in synchronous orbit, nearly unidirectional tidal forces create a  scalene ellipsoid . Also, the purported dwarf planet  Haumea  is scalene due to its rapid rotation, though it may not currently be in equilibrium.

15. Atmospheric Modelling In the atmosphere, the pressure of the air decreases with increasing altitude. This pressure difference causes an upward force called the  pressure-gradient force . The force of gravity balances this out, keeping the atmosphere bound to Earth and maintaining pressure differences with altitude