Euler s equation

It is a property of this family of differential equations that the sum or difference of two solutions is a solution and that a constant (including the constant i = / ) times a solution is also a solution. This accounts for the acceptability of forms like A (t) = Acoscot, where the constant A is an amplitude factor governing the maximum excursion of the mass away from its equilibrium position. The exponential form comes from Euler s equation... 

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

While the general form of the generalized Euler s equation (equation 9.9) allows for dissipation (through the term Hifc) expression for the momentum flux density as yet contains no explicit terms describing dissipation. Viscous stress forces may be added to our system of equations by appending to a (momentarily unspecified) tensor [Pg.467]

The incompressible Navier-Stokes equations are obtained by substituting the above form for into the generalized Euler equation (equation 9.9) and by using the incompressibility condition (5 ) dvijdxi = 0 equation 9.4) and Euler s equation dvijdt = -Y k Vkidvi/dxk) - dp/dxi) equation 9.7) ... 

In this section we show how the fundamental equations of hydrodynamics — namely, the continuity equation (equation 9.3), Euler s equation (equation 9.7) and the Navier-Stokes equation (equation 9.16) - can all be recovered from the Boltzman equation by exploiting the fact that in any microscopic collision there are dynamical quantities that are always conserved namely (for spinless particles), mass, momentum and energy. The derivations in this section follow mostly [huangk63]. 

Euler s equation (equation 9.7) may be recovered from Boltzman s equation as a consequence of the conservation of momentum, but only in the zeroth-order approximation to the full distribution function. Setting k — mvi in equation 9.52 gives, in component form. 

Euler s equation is thus recovered as a direct consequence of momentum conservation, but only via the zeroth-order approximation to the full solution to the Boltzman-equation. 

The LG version of Euler s equation (equation 9.100) thus becomes... 

Recall that in the continuum case we derived the Navier-Stokes equations (9.16) by allowing for the possibility of having nonzero viscous terms in the form for the momentum tensor appearing in Euler s equation (9.9). While their LG analogs may be derived in essentially the same manner, however, the lack of Galilean invariance tend to make the calculations more involved. We will outline the procedure below. 

The theory for this problem is well known. The following necessary condition, which must be satisfied, is Euler s equation for isoperimetric problems (usually used to construct the solution) ... 

Note that one mav consider a similar problem that requires finding a function of sever 1 independent variables and several dependent variables with derivatives of order higher than the first. In that case, one obtains a more complicated form of Euler s equation. 

Recall that the equation Lu = f x) is Euler s equation related to such a functional /[ ]. 

Legendre transformation does not affect the essential nature of a function and all of the different potentials defined above still describe the internal energy, not only in terms of different independent variables, but also on the basis of different zero levels. In terms of Euler s equation (2) the internal energy consists of three components... 

From Euler s equation applied to ay, and assuming a quasi-constant density, the displacement propagation can be expressed as ... 

This is a simple example of the general form of Euler s equation (1744), derived directly from a variational expression. 

In this example, Euler s equation takes the form of the geodesic equation... 

Thus v % = X and uf j = Y. Euler s equation then takes the form... 

Using this formula and = X+Jy, Euler s equation implies... 

Euler s equation of fluidal motion for the velocity field v,... 

Euler s equation will suffice as long as the time step, At, is kept small. Jelinek et al. (61) and Mori et al. (62) developed techniques for calculating the time step and the composition derivative and restated the component balance equations accordingly. 

To simplify Euler s equation we introduce the shorthand notation... 

On inserting (1.25.4b) into (1.25.4c) and canceling out a common factor k11 1 we obtain Euler s equation in the form... 

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