Vorticity, equation

Let us consider the rotational dynamics of a two-component neutron star taking into account the pinning and depinning of neutron vortices. Equations of motion of the superfluid and normal components have the following forms [15, 17] ... 

One important use of the stream function is for the visualization of flow fields that have been determined from the solution of Navier-Stokes equations, usually by numerical methods. Plotting stream function contours (i.e., streamlines) provides an easily interpreted visual picture of the flow field. Once the velocity and density fields are known, the stream function field can be determined by solving a stream-function-vorticity equation, which is an elliptic partial differential equation. The formulation of this equation is discussed subsequently in Section 3.13.1. Solution of this equation requires boundary values for l around the entire domain. These can be evaluated by integration of the stream-function definitions, Eqs. 3.14, around the boundaries using known velocities on the boundaries. For example, for a boundary of constant z with a specified inlet velocity u(r),... 

In this expression one term vanishes because V V = 0 for an incompressible flow and Vw = 0 because the divergence of the curl of a vector vanishes (vorticity is the curl of the velocity vector). For the same reason the last term on the right-hand side of the vorticity equation also vanishes. As a result the vorticity-transport equation is further reduced to... 

Recognizing the definition of the substantial derivative, the vorticity equation can be written compactly as... 

In this form one sees an analogy in the vorticity equation to the other transport equations— a substantial-derivative description of advective transport, a Laplacian describing the diffusive transport, and possibly a source term. It is interesting to observe that the vorticity equation does not involve the pressure. Since pressure always exerts a normal force that acts through the center of mass of a fluid packet (control volume), it cannot alter the rotation rate of the fluid. That is, pressure variations cannot cause a change in the vorticity of a flow field. 

A further reduction of the vorticity equation is possible by restricting attention to two-dimensional flows. Here, since the vorticity vector is orthogonal to the velocity vector, the term (u> V) V vanishes. To retain the two-dimensional flow, the body force f must remain two-dimensional. 

The stream-function-vorticity equation, taken together with the vorticity transport equation, completely replaces the continuity and momentum equations. The pressure has been eliminated as a dependent variable. The continuity equation has been satisfied exactly by the stream function, and does not need to be included in the system of equations. The... 

For the steady, constant-viscosity, axisymmetric stagnation flow, assuming no body forces, the vorticity equation emerges as a scalar equation for the circumferential vorticity field,... 

The vorticity equation is obtained by eliminating the pressure between the two momentum equations, i.e., by taking the y-derivative of Eq. (8.123) and subtracting from it the jc-derivative of Eq. (8.124). This gives ... 

In terms of these variables, the vorticity equation becomes ... 

It will be seen from the results given in Fig. E8.8 that Nu does appear to be essentially independent of Pr when Pr > 2. It will also be seen that the change in Nu over the entire Pr range considered is small. This indicates that the inertia term in the vorticity equation, i.e. ... 

RCm) is essentially of the same order of magnitude as the ratio of the induced magnetic field strength to the imposed magnetic field strength. The equation of motion is augmented by the MHD force density term j x B, and Ohm s law by the o(v B) term, where 0 is the bulk electrolyte conductivity. Of particular importance is the vorticity equation... 

By cross-differentiating and combining the primitive equations (3.23a and 3.236) and using the continuity equation (3.25), one obtains the vorticity equation... 

It turns out that there are significant differences in some aspects of turbulence between three- and two-dimensional systems. As the vorticity vector points in the direction perpendicular to the plane of the flow u> T v, fluid motion can be fully described by a scalar field u (x,y). A consequence of two-dimensionality is that the vortex stretching term u> Vv vanishes in the vorticity equation (1.11) that becomes... 

The central tenet of the MHD approach Is the rotational contribution to the overall force field by the magnetic force term In the classical vorticity equation, whose expanded form... 

Vorticity Equation for Creeping Flow of an Incompressible Newtonian Fluid. Since all scalar components of the velocity vector are exact differentials, it is permissible to reverse the order of mixed second partial differentiation without affecting the final result. If this procedure is performed twice, then inspection of summation representations of the following two vector-tensor operations reveals that they are equivalent ... 

At first glance, three coupled linear third-order PDEs must be solved, as illustrated above. However, each term in the x and y components of the vorticity equation is identically zero because =0 and Vj and Vy are not functions of z. Hence, detailed summation representation of the vorticity equation for creeping viscous flow of an incompressible Newtonian fluid reveals that there is a class of two-dimensional flow problems for which it is only necessary to solve one nontrivial component of this vector equation. If flow occurs in two coordinate directions and there is no dependence of these velocity components on the spatial coordinate in the third direction, then one must solve the nontrivial component of the vorticity equation in the third coordinate direction. 

In the low-Reynolds-number limit for incompressible Newtonian fluids, one calculates the stream function from the z-component of the vorticity equation, as described on page 180 ... 

Analytical Solution of the Vorticity Equation for (r,6). Equation (8-120) reveals that... 

For creeping viscous flow in spherical coordinates, the -component of the vorticity equation requires that... 

For low-Reynolds-number flow of an incompressible Newtonian fluid, the r and 6>-components of the equation of motion are useful to calculate dynamic pressure after the vorticity equation allows one to determine ir, Vr, and V0. Hence,... 

To study the behavior of the vorticity in these events under the influence of the molecules one has to solve the vorticity equation in the advected flow field with changing rheology. 

In the outer flow with respect to the wall, the l-vortex model will be dominant. Since in this region the vorticity equation is tractable in the form of eq. (2.3) we approximate the outer flow field of the event by this model. The tornado like wall-near flow then represents a boundary condition for the i1-vortices. 

Since the direction of the elongated particles does not coincide with the axial direction as a consequence of the spiraling of the streamlines as well as of the perpendicular shear, we find that extensional viscosity can act in an anisotropic way in the plain perpendicular to the axial movement. As long as the molecules are stretched the viscosity used in eq. (2.1) therefore becomes time dependent and its value increases with wall distance.. It can easily be seen from the integration (2.9) of the vorticity equation (2.3) that... 

In isobaric coordinates the vorticity equation takes a simpler form. Defining the vertical component of vorticity in pressure coordinates as = dvjdx — dujdy with... 

The vorticity equation describes how vorticity is changed by various properties of the flow. Only in very special circumstances would the vorticity be conserved following the flow. Kelvin s circulation theorem describes how an integral measure of vorticity is conserved but is valid only for barotropic flow and furthermore requires a knowledge of the time evolution of material surfaces. There does exist a quantity, referred to as the Ertel potential vorticity, that is conserved under more general conditions than either the vorticity or the circulation. It may be shown by combining the curl of the momentum equation [Eq. (26a)] with the continuity equation [Eq. (26c)] and the thermodynamic equation [Eq. (26b)] expressed in terms of potential temperature 0 that... 

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