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Because the energy per unit mass of liquid in a well-mixed reservoir is uniform throughout, Bernoulli's equation can be used to analyze the fluid flow everywhere in that reservoir (including pipes or flow fields that the reservoir feeds) except where viscous forces dominate and erode the energy per unit mass. [6]: Example 3.5 and p.116
In compressible flow, however, the gas density and temperature also become variables. This requires two more equations in order to solve compressible-flow problems: an equation of state for the gas and a conservation of energy equation. For the majority of gas-dynamic problems, the simple ideal gas law is the appropriate state equation.
The equation above is a vector equation in a three-dimensional flow, but it can be expressed as three scalar equations in three coordinate directions. The conservation of momentum equations for the compressible, viscous flow case is called the Navier–Stokes equations. [2] Conservation of energy
Then for an ideal gas the compressible Euler equations can be simply expressed in the mechanical or primitive variables specific volume, flow velocity and pressure, by taking the set of the equations for a thermodynamic system and modifying the energy equation into a pressure equation through this mechanical equation of state. At last, in ...
Dynamic pressure is one of the terms of Bernoulli's equation, which can be derived from the conservation of energy for a fluid in motion. [1] At a stagnation point the dynamic pressure is equal to the difference between the stagnation pressure and the static pressure, so the dynamic pressure in a flow field can be measured at a stagnation point ...
Flux F through a surface, dS is the differential vector area element, n is the unit normal to the surface. Left: No flux passes in the surface, the maximum amount flows normal to the surface.
The Bernoulli equation applicable to incompressible flow shows that the stagnation pressure is equal to the dynamic pressure and static pressure combined. [1]: § 3.5 In compressible flows, stagnation pressure is also equal to total pressure as well, provided that the fluid entering the stagnation point is brought to rest isentropically.
is the flow velocity. and is the heat flux vector. Because it expresses conservation of total energy, this is sometimes referred to as the energy balance equation of continuous media. The first law is used to derive the non-conservation form of the Navier–Stokes equations. [3]