Vorticity components

Common geometries used to make viscosity measurements over a range of shear rates are Couette, concentric cylinder, or cup and bob systems. The gap between the two cylinders is usually small so that a constant shear rate can be assumed at all points in the gap. When the liquid is in laminar flow, any small element of the liquid moves along lines of constant velocity known as streamlines. The translational velocity of the element is the same as that of the streamline at its centre. There is of course a velocity difference across the element equal to the shear rate and this shearing action means that there is a rotational or vorticity component to the flow field which is numerically equal to the shear rate/2. The geometry is shown in Figure 1.7. 

Figure 11.4 Radial distributions of the RMS of the fluctuating vorticity components at different times for the unheated (left column) and heated (right column) jet. The times are t = 25 (dotted curve), f = 30 (dashed curve), and f = 35 (solid curve)... 

For a two-dimensional axisymmetric problem, all other vorticity components vanish exactly. For the stagnation-flow problem, it has been established that du/dr = 0 thus cog = dv/dz =rdV/dz. It is apparent that 2 = cog/r is a function of z alone. Thus, like the radial velocity, the scaled vorticity also exhibits a radially independent boundary layer. 

A word of caution is needed in going from Eq. 6.47 to Eq. 6.48. Even though there is only one vorticity component for this situation, cog, Eq. 6.47 is still a vector equation. As such, care must be exercised with the substantial-derivative and Laplacian operators, since they involve nonvanishing unit-vector detivatives. The Laplacian of the vector oj produces... 

In the general case transverse surface waves have a vortical component and Eq. (8) is assumed to be an approximation only. This means that Eq. (13) does not allow to calculate the damping coefficient. However, the real part of the complex wave number K for slightly damped capillary waves (ReK > > ImK) can be estimated. 

The jet-like or elongational flow field in these features can be understood as a local separation from the wall. An analysis of the vorticity field however poses a fundamental problem related to the non-slip condition at the wall since the Helmholtz laws are incompatible with this condition. One of Helmholtz s law states that a vortex line has to be closed or that it has to end at a boundary. Since at the wall (xy-plane) the velocities u and v parallel to the wall are identical zero, the same must hold for the vorticity component J normal to the wall. No vortex line can hence be attached to the wall. The main aspects of this problem are described by LIGHTHILL (1963). 

Figure 1.8 Effect of the friction Reynolds number on the vorticity components. (Adapted from...

Stable allowing considerably larger time steps of integration to be used. On the other hand, it requires both more computational memory and more arithmetic operations per time step. A comparison between these two schemes is offered in Figure 1.9, in which the average vorticity components are given as a function of the distance from the wall. It is clearly seen there that the fully implicit scheme with At = 10 and mesh size 96 x 97 x 96 yields the same statistics with the semi-implidt/explidt scheme with At = 2 x 10 and mesh size 144 x 145 x 144. 

Vorticity The relative motion between two points in a fluid can be decomposed into three components rotation, dilatation, and deformation. The rate of deformation tensor has been defined. Dilatation refers to the volumetric expansion or compression of the fluid, and vanishes for incompressible flow. Rotation is described bv a tensor (Oy = dvj/dxj — dvj/dxi. The vector of vorticity given by one-half the... 

In simple shear flow where vorticity and extensional rate are equal in magnitude (cf. Eq. (79), Sect. 4), the molecular coil rotates in the transverse velocity gradient and interacts successively for a limited time with the elongational and the compressional flow component during each turn. Because of the finite relaxation time (xz) of the chain, it is believed that the macromolecule can no more follow these alternative deformations and remains in a steady deformed state above some critical shear rate (y ) given by [193] (Fig. 65) ... 

For a semibatch reaction between A already present in the reactor and B being fed into the reactor, each portion of B introduced is a source of vortices that grow by engulfement of the A-rich environment. The mass balance of component i in this growing zone is ... 

The lack of hydrodynamic definition was recognized by Eucken (E7), who considered convective diffusion transverse to a parallel flow, and obtained an expression analogous to the Leveque equation of heat transfer (L5b, B4c, p. 404). Experiments with Couette flow between a rotating inner cylinder and a stationary outer cylinder did not confirm his predictions (see also Section VI,D). At very low rotation rates laminar flow is stable, and does not contribute to the diffusion process since there is no velocity component in the radial direction. At higher rotation rates, secondary flow patterns form (Taylor vortices), and finally the flow becomes turbulent. Neither of the two flow regimes satisfies the conditions of the Leveque equation. 

Various arrangements at the bottom of the inner cylinder are available in Figure 3.2 an indentation is provided so that an air gap is formed and shearing in the sample below the inner cylinder is negligible. Another arrangement is to make the bottom of the inner cylinder a cone. When one of the cylinders is rotated, a Couette flow is generated with fluid particles describing circular paths. The only non-zero velocity component is ve and it varies in the r-direction. In order to minimize secondary flow (Taylor vortices) it is preferable that the outer cylinder be rotated however, in most commercial instruments it is the inner cylinder that rotates. In this case, the fluid s velocity is equal to IXR, at the surface of the inner cylinder and falls to zero at the surface of the outer cylinder. The shear stress is uniform over the curved surface of the inner cylinder and over the outer cylinder (to the bottom of the annular gap). 

Consider the two-dimensional flow shown in Figure A.l. If the velocity gradient dvjdy is positive it tends to cause the element to rotate in the clockwise direction. Similarly, if dvjdx is positive it tends to cause rotation in the anti-clockwise direction. Thus, the quantity dVy/dx — dvjdy gives the net rate of rotation in the anti-clockwise direction as viewed. It is the clockwise direction about a line parallel to the 2-coordinate as viewed in the positive 2-direction. This quantity is the 2-component of the fluid s vorticity a> ... 

When all three components of the vorticity are zero the flow is said to be irrotational. In irrotational flow the effects of viscosity disappear as will be... 

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] ... 

We can find the magnetic field in the hadronic matter phase from the solution (51) by taking into account that proton vortices in this phase generate a homogeneous mean magnetic field with amplitude B and direction parallel to the axis of rotation of the star [22], For the components of the magnetic field Bp in the hadronic phase (for a [Pg.273]

The effect of heating on the azimuthal and streamwise components of the vorticity field is shown in Fig. 11.1 the effect on the radial component is comparable to that on the azimuthal component, and is therefore not separately shown. The vorticity distributions at the same nondimensional time t = 35 are plotted side by side for the unheated and heated case for each component. The positive and negative values of vorticity are shown by solid and dotted lines, respectively. 

To quantify the vorticity increases due to heating, the total enstrophy and also the enstrophies corresponding to the azimuthal, streamwise, and radial components of vorticity are examined. Computed values are shown using a linear-log scale in Fig. 11.2. In the absence of heating, the total as well as the component enstrophies all fall beyond time t = 25, as would be expected in a fully developed turbulent jet. When heat is applied, there is a virtually exponential rise of the enstrophies after some time. At t = 35, the enstrophies are one order of magnitude higher with heating than without. 

Figure 11.1 Streamwise sections (in the j/z-plane passing through the axis of the jet) of different components of vorticity in the unheated (a), (c) and heated (6), (d) jets at time t = 35 (a), (b) — azimuthal vorticity, (c), (d) — streamwise vorticity. Negative contours are shown using dotted lines in steps of —0.5 starting from —0.5, while positive contours are in solid lines in steps of 0.5 starting from 0.5...

Figure 11.3 Comparison of computed spectra for the azimuthal component of vorticity for the heated (dashed curve) and unheated (solid curve) jets. The spectra are plotted on log-log scales...

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