The Eulers equation

The net force on a volume T of an inviscid fluid, bounded by a surface S, arising 

This equation means that the force per unit volume of fluid is simply —Vp. 

Applying the Newton s law gives the Euler s equation consider an elementary volume of fluid of density p experiencing a pressure gradient and placed in an external field of intensity g per unit of mass then we have that 

12 is valid for an ideal inviscid liquid. It was established by Euler in 1755. Using eqs 1.8 and 1.12 leads to 

This equation constituted the first main equation of hydrodynamics. 

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S is the path length between the points a and b. The Euler equation to this variation problem yields the condition for the reaction path, equation (B3.5.14). A similar method has been proposed by Stacho and Ban [92]. 

It has been observed by [27, 24] that the equations of motion of a free rigid body are subject to reduction. (For a detailed discussion of this interesting topic, see [23].) This leads to an unconstrained Lie-Poisson system which is directly solvable by splitting, i.e. the Euler equations in the angular momenta ... 

The inequality like (1.59) is called a variational inequality. It was obtained from a minimization problem of the functional J over the set K. In the sequel we will look more attentively at a connection between a minimization problem and a variational inequality. Now we want to underline one essential point. We see that the problem (1.58) is more general in comparison with the minimization problem on the whole space V. It is well known that the necessary condition in the last problem coincides with the Euler equation. The variational inequality (1.59) generalizes the Euler equation. Moreover, ior K = V the Euler equation follows from (1.59). To obtain it we take U = Uq +u and substitute in (1.59) with an arbitrary element u gV. It gives... 

This exactly coincides with the Euler equation. 

The set K in Theorem 1.11 may coincide with the space V. For a differentiable functional J it guarantees the solvability of the Euler equation... 

The Euler equation, assuming simple one-dimensional flow theory, is the theoretieal amount of work imparted to eaeh pound of fluid as it passes through the impeller, and it is given by... 

A positive vane angle produees prewhirl in the direetion of the impeller rotation, and a negative vane angle produees prewhirl in the opposite direetion. The disadvantage of positive prewhirl is that a positive inlet whirl veloeity reduees the energy transfer. Sinee Vg is positive aeeording to the Euler equation defined by... 

With positive prewhirl, the first term of the Euler equation remains H = U Vg — UiVgi Therefore, Euler work is redueed by the use of positive prewhirl. On the other hand, negative prewhirl inereases the energy transfer by the amount U Vg. This results in a larger pressure head being produeed in the ease of the negative prewhirl for the same impeller diameter and speed. 

The power developed by the flow in an impulse turbine is given by the Euler equation... 

Taking a typical centrifugal pump (Figure 32.2) the Euler equation can be written, at best efficiency flow, in... 

Electron Nuclear Dynamics (48) departs from a variational form where the state vector is both explicitly and implicitly time-dependent. A coherent state formulation for electron and nuclear motion is given and the relevant parameters are determined as functions of time from the Euler equations that define the stationary point of the functional. Yngve and his group have currently implemented the method for a determinantal electronic wave function and products of wave packets for the nuclei in the limit of zero width, a "classical" limit. Results are coming forth protons on methane (49), diatoms in laser fields (50), protons on water (51), and charge transfer (52) between oxygen and protons. 

A relationship, known as Euler s formula, exists between a complex number [x + jy] (x is the real part, y is the imaginary part of the complex number (j = P )) and a sine and cosine function. Many authors and textbooks prefer the complex number notation for its compactness and convenience. By substituting the Euler equations cos(r) = d + e -")/2 and sin(r) = (d - e t )l2j in eq. (40.1), a compact complex number notation for the Fourier transform is obtained as follows ... 

Thus, as expected, the Euler equation equivalent to extremizing E is the Schrodinger equation. 

The equation satisfied by the wave function T, the Schrodinger equation, is obtained by minimizing the functional [T] with respect to T, with the energy of the system appearing as a Lagrange multiplier to ensure the normalization of the wave function. Similarly in DFT, the equation for the density is obtained by minimizing the functional E[p with respect to the density p and leads to the Euler equation... 

Assume that to the left of the combustor outlet boundary, rr = 0, there exists a stationary solution of the Euler equations p = po, P = Po, u = uq, where po, po, and Mo are the constant pressure, density, and velocity. Flow velocity has a single nonzero component, mq, along the x axis. The flow is assumed subsonic, i.e., M = uq/cq < 1, where cq is the speed of sound. We consider the solution of the nonstationary Euler equations and linearize the problem in the vicinity of the stationary solution by assuming that... 

The energy and spectral optimization problems are convex programs so when there are multiple solutions the solution sets form a convex set. The following corollary characterizes how these convex sets of solutions relate to solutions of the Euler equation. In the formulation of this corollary we use the notion of optimal gap Ao—the gap achieved by optimal P and S. The optimal gap is a characteristic of the energy problem, depending only on H and S. 

The virtue of this theorem is that it reduces the dual problem to the question of solving the Euler equation PQ = 0, a second-order algebraic equation for the... 

Proof. By property R5 listed at the end of Section II, the elements of the Pauli subspace S are traceless, from which we infer by Theorems 12 and 13 that the energy problem and the spectral optimization problem have optimal solutions. By Theorem 10 these solutions are characterized by the Euler equation PQ = 0. ... 

In Section IV, we obtained the Euler equations (44)-(46), which yield the optimum bound on the ground-state energy. Usually, the simultaneous solution of... 

In an infinite solid this set of critical points obeys a number of theorems, the chief being the Euler equation (eqn (14.1)) ... 

The components of this equation are sometimes referred to as the Navier-Stokes equations when the viscosity is set equal to zero (inviscid fluid), then these equations reduce to the Euler equations. 

There has been very much effort devoted to the solution of the diffusion equation of motion for a reactant particle executing Brownian motion. The Euler equation of diffusion... 

Discuss the conditions under which the viscous terms vanish, leading to the Euler equations for inviscid flow,... 

Form the scalar product of the Euler equations and an arbitrary displacement vector as... 

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