Velocity curl

The vorticity of a fluid corresponds to a vector field given by the velocity curl. Using Eq. (4.47) with timescale yields the Kolmogorov vorticity scale... 

Auton results from the deformation in the velocity curl field produced by the presence of the spherieal particle. The calculation is performed under the sole condition that the shear is small, that is, GatW 1. Auton s result applies mainly to large-size particles and not to small particles in the sense of [16.4], Viscosity cannot be neglected for small particles, and it is found that the lift force given by relation [16.26] or [16.27] is 1/VReo larger than what Auton s calculation predicts, unless J(e) is small in the conditions considered. 

A fluid flow is usually termed a rotating flow when it is possible to identify an Oz axis about which the fluid particles have a movement of rotatioa This notion is simple to visualize when the flow is steady and the trajectory followed by a fluid particle is closed cmve. When this is not the case, the concepts of circulation and velocity curl (or vorticity) need to be introduced in order to define the notion of a rotating flow. 

We are therefore led to classify rotating flows based on whether they are rotational or irrotational. The flow is irrotational at a given point (or in a given domain) if the velocity curl vector is zero at that point (or at all points in that domain). It is rotational at a point if the velocity curl vector is not zero at that point. This classification is relevant, since the property whereby a flow is irrotational or rotational is conserved when a particle is followed in its movement... 

T able 17.1. Navier-Stokes equations, incompressibility, and velocity curl expressed in a cylindrical coordinate system with axis Oz... 

Rotorvaned leaf is usually further cut in a CTC (crush, tear, curl) machine, which is made up of two closely spaced grooved rollers rotating at different velocities. After leaf passes through this equipment, it is more finely divided than that processed by orthodox rolling. 

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... 

The velocity distribution ahead of the flame front in the room coordinate system is shown in Fig. 13. As seen from the figure, the gas motion is accompanied by the rotation and displacement of fluid elements, since the velocities near the axis and walls are directed in opposite directions. The rotation centers of each of the fluid elements are behind the flame front. But the flow is potential and u = curl u = 0. In the coordinate system of the flame front, the velocities near the axis and walls have the same direction (see Fig. 2). 

The load applied to the surface of the polyurethane by the impinging particle is considered to be an important factor in the wear process (Ephithite, 1985). The force is a product of the mass and velocity. The mass of the particle increases greatly as it becomes larger (the volume increases by the cube of its dimensions). When a certain force is reached, rolling wear will commence. Depending on the force, either the elastomer will deform or, if the force is large enough, a particle of polyurethane will be removed in a scallop-like piece or a curled-up leaf. 

Apart from the drag force, there are three other important forces acting on a dispersed phase particle, namely lift force, virtual mass force and Basset history force. When the dispersed phase particle is rising through the non-uniform flow field of the continuous phase, it will experience a lift force due to vorticity or shear in the continuous phase flow field. Auton (1983) showed that the lift force is proportional to the vector product of the slip velocity and the curl of the liquid velocity. This suggests that lift force acts in a direction perpendicular to both, the direction of slip velocity... 

The vorticity vector, w, is a measure of rotational effects, being equal to twice the local rate of rotation (angular velocity) of a fluid particle (i.e., uj = curl(v) = rot(v) = Vx V = 2Q) [168]. Many flows have negligible vorticity, uj curl(v) 0, and are appropriately called irrotational flows. 

In the absence of interactions, electrons are described by the Dirac equation (1928), which rules out the quantum relativistic motion of an electron in static electric and magnetic fields E= yU and B = curl A (where U and A are the scalar and vectorial potentials, respectively) [43-45]. As the electrons involved in a solid structure are characterized by a small velocity with respect to the light celerity c (v/c 10 ) a 1/c-expansion of the Dirac equation may be achieved. More details are given in a paper published by one of us [46]. At the zeroth order, the Pauli equation (1927), in which the electronic spin contribution appears, is retrieved then conferring to this last one a relativistic origin. At first order the spin-orbit interaction arises and is described by the following Hamiltonian... 

The mean vorticity of the fluid phase is the curl of the mean fluid velocity field, coc = "X Uf, and is a measure of the mean rotation of the continuous phase. 

Common examples of pseudo-vectors that will be relevant later include the angular velocity vector f2, the torque T, the vorticity vector co (or the curl of any true vector), and the cross product of two vectors. The inner scaler product of a vector and a pseudo-tensor or a pseudo-vector and a regular tensor will both produce a pseudo-vector. It will also be useful to extend the notion of a pseudo-vector to scalers that are formed as the product of a vector and a pseudo-vector. The third-order, alternating tensor e is a pseudo-tensor of third order as may be verified by reviewing its definition... 

A convenient way to discuss some aspects of flow at high Reynolds number is in terms of the transport of vorticity rather than directly in terms of velocity and pressure. We recall that the vorticity is defined as the curl of the velocity,... 

One way to eliminate the pressure from the Navier-Stokes equation is by transforming it into an equation for the vorticity field defined as the curl of the velocity field, u> = V x v, that characterizes the distribution and direction of the rotational motion in the flow. By taking the curl of both sides of (1.8) and using the vector identity (a V)a = (1/2) V a 2 + (V x a) x a and that for any scalar field ip, V x Vip = 0 one obtains... 

The curl of a vector function is more difficult to visualize than is the divergence. In fluid flow, the curl of the velocity gives the vorticity of the flow, or the rate of turning of the velocity vector. Because of this, the symbol rotF is also sometimes used for the curl of F. 

Equation (8-62) generates three coupled linear second-order partial differential equations (PDEs). Eor complicated two-dimensional flow problems, this force balance and the equation of continuity yield three coupled linear PDEs for two nonzero velocity components and dynamic pressure. In some situations, this complexity is circumvented by taking the curl of the equation of motion ... 

Evaluation of the Curl of the Divergence of the Velocity Gradient. Begin by expressing the velocity gradient tensor using summation notation ... 

Hence, one linear fourth-order PDE must be solved for the stream function from which and Vy can be determined. This approach is the method of choice instead of tackling coupled linear first- and second-order PDEs for three unknowns via the equations of continuity and motion. The PDE of interest, given by (8-91), progressed from a second-order equation to a fourth-order equation by taking the curl of the equation of motion to eliminate dynamic pressure, and relating both velocity components to the stream function. 

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