Divergence free

The symbols in this equation are defined below). It was shown by Gordon [323], and further discussed by Pauli [104] that, by a handsome tr ick on the four current, this can be broken up into two parts J" = djgj (each divergence-free),... 

Figure 6. (a) Diverging free jet (b) shroud too short to contain the jet (c) minimum shroud length required to contain jet. 

Note that the divergence-free character of Eq. (458) implies that f °d does not contribute to Eq. (459). 

The initial conditions for the velocity components are set up so that there is a tubular shear layer aligned along the 2 -direction at time t = 0. The tv-velocity has a top-hat profile with a tan-hyperbolic shear layer. Stream wise and azimuthal perturbations are introduced to expedite roll-up and the development of the Widnall instability. The details can be found in [7]. The initial velocity field is made divergence-free using the Helmholtz decomposition. The size of the computational domain (one periodic cubical box) is 4do on each side. 

If e = = 0, the electric field is divergence free this is the general condition for a transverse field. Except possibly at frequencies where e = 0, therefore, the medium cannot support longitudinal fields. 

Therefore, M and N have all the required properties of an electromagnetic field they satisfy the vector wave equation, they are divergence-free, the curl of M is proportional to N, and the curl of N is proportional to M. Thus, the problem of finding solutions to the field equations reduces to the comparatively simpler problem of finding solutions to the scalar wave equation. We shall call the scalar function ip a generating function for the vector harmonics M and N the vector c is sometimes called the guiding or pilot vector. 

The first equation is scalar, and has a wave solution with velocity Vi = -J c /p). This is the longitudinal wave of eqn (6.7). It is sometimes called an irrotational wave, because V x u = 0 and there is no rotation of the medium. The second equation is vector, and has two degenerate orthogonal solutions with velocity v = s/(cu/p)- These are the transverse or shear waves of eqn (6.6) the degenerate solutions correspond to perpendicular polarization. They are sometimes called divergence-free waves, because V u = 0 and there is no dilation of the medium. Waves in fluids may be considered as a special case with C44 = 0, so that the transverse solutions vanish, and C = B, the adiabatic bulk modulus. 

We start by constructing an orthonormal basis a, b, c (where c is a unit vector along the wavevector k, with a and b in the two-dimensional vector space orthogonal to k). The significance of this ansatz is that any vector function F(x,y,z) is divergence free if and only if its Fourier coefficients F(k) are orthogonal to k, that is if k -F(k) =0. Thus, F(k) is a linear combination of a(k) and b(k). Lesieur defines the complex helical waves as... 

In order to obtain Green s identities for the flow field (u,p), a vector z is defined as the dot product of the stress tensor a(u, p) and a second solenoidal vector field v (divergence-free). The divergence or Gauss Theorem (10.1.1) is applied to the vector z... 

Equivalently, we can decompose Eij into a divergence-free and trace-free tensor Eij, a divergence-free vector Ei and a scalar E ... 

Thus, with the choice of the Coulomb gauge, the electrical field, according to (8.36), is decomposed into a curl-free part, - Vf7, and a divergence free part, iu>A. 

Using (22), (23) and (24), it is easy to see that Jab x) is a divergence free vector field in 17 (4 U B). Indeed, after some straightforward algebraic manipulations, one arrives at... 

The pressure-based method was introduced by Harlow and Welch [67] and Chorin [30] for the calculation of unsteady incompressible viscous flows (parabolic equations). In Chorines fractional step method, an incomplete form of the momentum equations is integrated at each time step to 3ueld an approximate velocity field, which will in general not be divergence free, then a correction is applied to that velocity field to produce a divergence free velocity field. The correction to the velocity field is an orthogonal projection in the sense that it projects the initial velocity field into the divergence free... 

In application of the fractional step method to the incompressible Navier-Stokes equations, the pressure may be interpreted as a projection operator which projects an arbitrary vector field into a divergence-free vector field. 

Here e1 is the fundamental transverse microscopic electric held operator and b is the corresponding magnetic held operator. The superscript on the electric held operator designate its transverse character with respect to the direction of propagation, redundant in the case of the magnetic held as it is intrinsically transverse, namely, divergence-free, since it arises from the curl of a vector potential held a(r). Since the electric held also derives from a(r), we concentrate first on the second-quantized form of this vector potential, which is cast in terms of a summation over radiation modes as follows ... 

In effect, (7-29) represents a decomposition of the general vector field a into an irrotational part, associated with V, and a solenoidal (or divergence-free) part, represented by V A 0AVX). It should be noted that general proofs exist that show not only that (7-29) can represent any arbitrary vector field a but also that an arbitrary, irrotational vector field can be represented in terms of the gradient of a single scalar function

solenoidal vector field can be represented in the form of the second term of (7-29). Because... 

Helmholtz analysis. So, a vector field with the desired curl is evaluated and projected in the space of divergence-free vectors. This concept leads to the next assertion ... 

Under normal conditions most fluids are not compressed much in a flow. In general, if the typical flow velocity (U) is much smaller than the speed of sound (c) in the medium (cair 340 m/s, cwater 1500 m/s), i.e. the Mach number, Ma = U/c, is small, then the fluid is essentially incompressible. In this case the velocity field is a divergence-free (solenoidal) vector field... 

A key quantity is the vector potential A f,t) = A f), which satisfies a wave equation analogously to the wave equation for the electron field operators ip ). It is chosen divergence free V A r) = 0. Invoking zero scalar potential and this choice is usually referred to as the Coulomb gauge. 

In the event that the vector field is divergence free, we have fu + gv = which implies that the numerator and denominator are identical, and it follows that... 

Euler s method is not symplectic for a general Hamiltonian system. Similarly for a general divergence free vector field, Euler s method is not volume preserving. Find conditions on the vector field that imply that Euler s method is volume preserving. Are there special Hamiltonian systems for which Euler s method is a symplectic method ... 

The volume in phase space (as the vector field is divergence free). 

An important consequence of the skew symmetry of Cf for a divergence-free... 

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