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They were first derived by Alexander Friedmann in 1922 from Einstein's field equations of gravitation for the Friedmann–Lemaître–Robertson–Walker metric and a perfect fluid with a given mass density ρ and pressure p. [1] The equations for negative spatial curvature were given by Friedmann in 1924. [2]
In fluid mechanics, the force density is the negative gradient of pressure. It has the physical dimensions of force per unit volume. Force density is a vector field representing the flux density of the hydrostatic force within the bulk of a fluid. Force density is represented by the symbol f, [1] and given by the following equation, where p is ...
A relatively simple version [1] of the vertical fluid pressure variation is simply that the pressure difference between two elevations is the product of elevation change, gravity, and density. The equation is as follows: =, where P is pressure, ρ is density, g is acceleration of gravity, and; h is height.
If mass density is ρ, the mass of the parcel is density multiplied by its volume m = ρA dx. The change in pressure over distance dx is dp and flow velocity v = dx / dt . Apply Newton's second law of motion (force = mass × acceleration) and recognizing that the effective force on the parcel of fluid is −A dp.
In fluid mechanics, the pressure-gradient force is the force that results when there is a difference in pressure across a surface. In general, a pressure is a force per unit area across a surface. A difference in pressure across a surface then implies a difference in force, which can result in an acceleration according to Newton's second law of ...
The equation to calculate the pressure inside a fluid in equilibrium is: ... so Φ = −ρ f gz where g is the gravitational acceleration, ρ f is the mass density of ...
ρ p is the mass density of the sphere [kg/m 3] ρ f is the mass density of the fluid [kg/m 3] g is the gravitational acceleration [m/s 2] Requiring the force balance F d = F e and solving for the velocity v gives the terminal velocity v s.
In this case the field is gravity, so Φ = −ρ f gz where g is the gravitational acceleration, ρ f is the mass density of the fluid. Taking the pressure as zero at the surface, where z is zero, the constant will be zero, so the pressure inside the fluid, when it is subject to gravity, is =. So pressure increases with depth below the surface ...