Equation Euler

3 Variational Problem and Elastic Torques 8.3.1 Euler Equation 

Consider a nematic liquid crystal layer confined between two glass plates. This structure is of great technical importance. The most of liquid crystalline displays are based on it. The directors at opposite walls (z = 0 and z = d) are rigidly fixed at 

For simplicity we ignore the influence of external fields. The problem is to find that distribution of the director angle tp(z) over cell thickness, which satisfies the minimum of the elastic free energy F for fixed boundary conditions. This is a typical variational problem although very simple in our particular case. The idea of a variational calculation is not to find a value of the integral of a function g z, tp, tp ) over the interval 0 z d for known (p(z), but to find such an unknown function tp(z) that provides the miiumum of the integral. Due to the great importance of this mathematical problem for liquid crystals consider it in more detail. 

Consider a functional F (scalar number, e.g. it might be free energy of the liquid crystal sample)  

Here g is a function of all the three arguments z, rp(z) and dt ldz. The equation is valid for any crmtinuous function g(z) with continuous derivatives g , g defined within interval a, b. For instance, g might be density of free energy of a liquid crystal per unit volume, rp(z) be an angle the director forms with a selected reference axis and d the thickness of the sample. The values of function g are fixed at both ends of the interval (p(a) = tp and tp(h) = tp. In our simplest example, infinitely strong anchoring of the director is assumed at the boundaries. 

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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... 

A combination of open- and closed-type formulas is referred to as the predictor-corrector method. First the open equation (the predictor) is used to estimate a value for y,, this value is then inserted into the right side of the corrector equation (the closed formula) and iterated to improve the accuracy of y. The predictor-corrector sets may be the low-order modified (open) and improved (closed) Euler equations, the Adams open and closed formulas, or the Milne method, which gives the following system... 

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

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) ... 

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. 

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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... 

This formulation results very insightful according to Equation 8.30, particles move under the action of an effective force — We , i.e., the nonlocal action of the quantum potential here is seen as the effect of a (nonlocal) quantum force. From a computational viewpoint, these formulation results are very interesting in connection to quantum hydrodynamics [21,27]. Thus, Equations 8.27 can be reexpressed in terms of a continuity equation and a generalized Euler equation. As happens with classical fluids, here also two important concepts that come into play the quantum pressure and the quantum vortices [28], which occur at nodal regions where the velocity field is rotational. 

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... 

Here it is now understood that Vh = fen + fes and are no longer to be calculated from the approximate TFD Euler equation (4) but rather using a density n(r) given by the ground-state sum... 

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. 

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