Navier-Stokes, Euler, and Bernoulli Equations

The force balance written for a volume element dx dy-dz leads to the Navier-Stokes equation of motion  

Here dv/dt is the substantial derivative of the velocity or the sum of derivatives with respect to time and space. The symbols g and p denote the acceleration due to gravity and the pressure, respectively. Dealing with ideal fluids the viscosity is zero and the Navier-Stokes equation can be simplified to the Euler equation  

The integration of the Euler equations for the special case of a one-dimensional steady-state flow in z -direction leads to the Bernoulli equation  

The derivative of the momentum with respect to the time is equal to the sum of all forces Y.F  

Multiplying the mass flow M = AM/dt with the velocity w yields the momentum flux M-w. The momentum flux density is the momentum flux based on the cross-sectional area /  

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