Euler ideal-fluid equations

The equations obtained by keeping terms up to order fi are called the Euler ideal-fluid equations. They can be obtained from Eqs. (116)-<118) by setting 17 and A equal to zero. 

The Froude number described above is frequently used for the description of radial and axial flotvs in liquid media when the pressure difference along a mixing device is important. When cavitation problems are present, the dimensionless group (Pj — p,) /pw - called the Euler number - is commonly used. Here p is the liquid vapour saturation pressure and p is a reference pressure. This number is named after the Swiss mathematician Leonhard Euler (1707-1783) who performed the pioneering work showing the relationship between pressure and flow (basic static fluid equations and ideal fluid flow equations, which are recognized as Euler equations). 

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 ... 

Euler s Equation of Motion for an Ideal Fluid. Using the Euler equations... 

The Euler equations are the equations of motion for an ideal fluid, that is, an inviscid fluid. They are easily deduced from the Navier-Stokes equations, simply by omitting the viscous terms. 

Euler s Equation for an ideal, dissipationless fluid is obtained by applying the chain-... 

The ideal gas free energy functional is defined exactly from statistical mechanics, dropping the temperature-dependent terms that do not affect the fluid structure. Free energy functional contribution due to the excluded volume of the segments is calculated from Rosenfeld s (1989) DFT for a mixture of hard spheres. The functional derivatives of these free energy functional contributions, which are actually required to solve the set of Euler-Lagrange equations, are straightforward. 

Special equations for ideal or inviscid fluids can be obtained for a fluid having a constant density and zero viscosity. These are called the Euler equations. Equations (3.7-36)-(3.7-39) for the x, y, and z components of momentum become... 

Here is the scalar value of the y-direction velocity, P is the pressure, and Q is a viscous pressure, which may be real or artificial. In the context of the equation (P + Q) is the component (P + Q)yy of the pressure tensor defined by Eq. (3.28). This component represents the stress normal to the x-z plane. The right-hand side of Eq. (2.7) represents the body force acting on unit volume of the fluid, and is equal by Newton s second law to a rate of formation of momentum. In the absence of the viscosity term, Eq. (2.7) is a version of Euler s equation for the motion of an ideal (nonviscous) fluid. By differentiating out the products in Eq. (2.7) and then substituting from Eq. (2.5) the momentum equation may be reduced to... 

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