Circumferential velocities

In the Couette flow inside a cone-and-plate viscometer the circumferential velocity at any given radial position is approximately a linear function of the vertical coordinate. Therefore the shear rate corresponding to this component is almost constant. The heat generation term in Equation (5.25) is hence nearly constant. Furthermore, in uniform Couette regime the convection term is also zero and all of the heat transfer is due to conduction. For very large conductivity coefficients the heat conduction will be very fast and the temperature profile will... 

Solid-Body Rotation When a body of fluid rotates in a sohd-body mode, the tangential or circumferential velocity is linearly proportional to radius ... 

After detachment of the flame from the walls, the narrow ever-diminishing hot product zone behind the flame moves owing to the free convection in the centrifugal acceleration field toward the axis of rotation, with a speed scaling with circumferential velocity at the flame location, which reduces the observed flame speed to very low values, and in some cases negative ones. 

The independent variables in these equations are the dimensionless spatial coordinates, x and r. The dependent variables are the dimensionless velocity components (u the axial velocity, v the radial velocity, and w circumferential velocity), temperature , and pressure pm- The viscosity and thermal conductivity are given by p and A, and the mass density by p. Density is determined from the temperature and pressure via an ideal-gas equation of state. The dimen-... 

In these equations the independent variable x is the distance normal to the disk surface. The dependent variables are the velocities, the temperature T, and the species mass fractions Tit. The axial velocity is u, and the radial and circumferential velocities are scaled by the radius as F = vjr and W = wjr. The viscosity and thermal conductivity are given by /x and A. The chemical production rate cOjt is presumed to result from a system of elementary chemical reactions that proceed according to the law of mass action, and Kg is the number of gas-phase species. Equation (10) is not solved for the carrier gas mass fraction, which is determined by ensuring that the mass fractions sum to one. An Arrhenius rate expression is presumed for each of the elementary reaction steps. 

The other boundary conditions are relatively simple. The temperature and species composition far from the disk (the reactor inlet) are specified. The radial and circumferential velocities are zero far from the disk a boundary condition is not required for the axial velocity at large x. The radial velocity on the disk is zero, the circumferential velocity is determined from the spinning rate W = Q, and the disk temperature is specified. 

For comparison. Fig. 21 also shows the dependence of the results on the circumferential velocity of the impeller. Contrary to many assumptions in the literature, this diagram indicates that velocity is not a significant process parameter as far as the disintegration process is concerned. 

At the same time, the results in Fig. 3 - because of the validity here of the relationships represented by Eqs. (25) and (26) - rule out any possibility that the number of circulations Zp made by the particles or the power per unit circulation flow P/qp is important. In the first case there would be a dependence on scale according to Eq. (25), and in the second case there would also be a dependence on the circumferential velocity u nd according to Eq. (26). 

In an earlier phase of this work [9] the intensities of axial and circumferential components of velocity fluctuation were measured in the TC annulus, using Laser Doppler Velocimetry (LDV), for a wide range of cylinder rotation speeds. On average, the intensities of axial velocity fluctuations were found to be within 25% of the intensities of circumferential velocity fluctuations [9]. As in Ronney et al. [5], turbulence intensities were found to be nearly homogeneous along the axial direction and over most of the annulus width, and to be linearly proportional... 

At the end of the Fifties, it became possible - through a turbine-like design and by modification of the ideas of Gaede - to produce a technically viable pump the socalled Turbomolecular pump . The spaces between the stator and the rotor disks were made in the order of millimeters, so that essentially larger tolerances could be obtained. Thereby, greater security in operation vras achieved. However, a pumping effect of any significance is only attained when the circumferential velocity (at the outside rim) of the... 

The extra terms appear because in noncartesian coordinate systems the unit-vector derivatives do not all vanish. Only in cartesian coordinates are the components of the substantial derivative of a vector equal to the substantial derivative of the scalar components of the vector. The acceleration in the r direction is seen to involve w2, the circumferential velocity. This term represents the centrifugal acceleration associated with a fluid packet as it moves in an arc defined by the 9 coordinate. There is also a G acceleration caused by a radial velocity. In qualitative terms, one can visualize this term as being related to the circumferential acceleration (spinning rate) that a dancer or skater experiences as she brings her arms closer to her body. 

This equation simply states that the circumferential velocity must be a linear function of r to preserve the orthogonal shape of the element in the absence of shearing. In general, when there is shearing, the lower right-hand corner travels a distance... 

The first term is represents the motion of the lower left-hand comer and the second term, the linear expansion, indicates how the circumferential velocity changes over the length of the element, dr. Since the circumferential velocity w(r) is generally not a linear function of r, there is a relative speedup or slowdown of the right-hand comer relative to the linear rate, meaning that there is shearing as evidenced by da 0 and ds 0. The distance ds... 

The AO = (w/r)dt terms represent the solid-body rotation due to the circumferential velocity w. So, even if there were no shearing (i.e., da = df8 = 0) there would still be a... 

Consider two limiting cases of the fluid flow between a long rod and a fixed concentric cylindrical housing (Fig. 2.10). In the case that the rod is simply translating, the axial velocity u(r) may be taken as a function or r alone. In the case where the rod is simply rotating, the circumferential velocity w(r) may be taken as a function or r alone. 

For long rod-guide systems it is reasonable to assume that the only nonzero velocity component is u, the axial velocity. Inasmuch as the rod and guide may have different axial velocities, it is clear that the fluid velocity must be permitted to vary radially. Given that the radial and circumferential velocities v and w are zero, the mass-continuity equation, Eq. 6.3, requires that... 

The circumferential-momentum equation is a parabolic partial differential that requires solution for the radial dependence of the circumferential velocity. With the velocity profiles in hand, the radial-momentum equation can be used to determine the resulting radial pressure dependence. 

For a specified shaft rotation and a fixed outer shell, the circumferential velocity is bounded as 1 < w < 0. Notice that as the gap becomes very small (i.e., Ar —> 0), r approaches a constant. In this case, r itself tends toward a constant and the differential equations tends toward a planar-coordinate representation. 

Consider a long cylindrical shell whose interior is filled with an incompressible fluid. If the fluid is initially at rest when the cylinder begins to rotate, a boundary layer develops as the momentum diffuses inward toward the center of the cylinder. The fluid s circumferential velocity vu comes to the cylinder-wall velocity immediately, owing to the no-slip condition. At very early time, however, the interior fluid will be only weakly affected by the rotation, with the influence increasing as the boundary layer diffuses inward. If the shell continues to rotate at a constant angular velocity, the fluid inside will eventually come to rotate as a solid body. 

The cylinder-wall circumferential velocity can be an arbitrary function of time, with the fluid velocity still subject to parallel-flow assumptions. The cylindrical analog of Stokes Second problem is to let the cylinder-wall velocity oscillate in a periodic manner. The wall velocity is specified as... 

For a nondimensional oscillation period of tp = 0.1, Fig. 4.15 shows the circumferential velocity profiles at four instants in the period. The wall velocity follows the specified rotation rate exactly, which it must by boundary-condition specification. The center velocity r — 0 is constrained by boundary condition to be exactly zero, incenter = 0. The interior velocities are seen to lag the wall velocity, owing to fluid inertia and the time required for the wall s influence to be diffused inward by fluid shearing action. 

For the steady flow of an incompressible fluid, state the appropriate mass-continuity equation in spherical coordinates. What can be inferred from the reduced continuity equation about the functional form of of the circumferential velocity v 2... 

After substituting the relationship between the friction factor and the nondimensional pressure gradient, solve the nondimensional differential equation to develop an expression for the circumferential velocity profile w(r). The product Re/ should appear as a parameter in the differential equation. Assume no-slip boundary conditions at the channel walls. 

Beginning with a mass-conservation law, the Reynolds transport theorem, and a differential control volume (Fig. 4.30), derive a steady-state mass-continuity equation for the mean circumferential velocity W in the annular shroud. Remember that the pressure p 6) (and hence the density p(6) and velocity V(6)) are functions of 6 in the annulus. 

The flow configurations discussed in this chapter are mainly axisymmetric, namely flow in the z-r plane. The circumferential velocity w can be nonzero, although it can have no circumferential variation. The rotating disk is probably the most useful application of this flow. 

Deriving the axisymmetric stagnation-flow equations begins with the steady-state three-dimensional Navier-Stokes equations (Eqs. 3.58, 3.60, and 3.60), but considering flow only in the z-r plane. In general, there may be a circumferential velocity component ui, but there cannot be variations of any variable in the circumferential direction 0. The derivation depends on two principal conjectures. First, the velocity field is presumed to be described in terms of a streamfunction that has the separable form... 

These equations are written to isolate the pressure-gradient terms on the left-hand side to emphasize the point that the right-hand sides are functions of z alone. If there is no circumferential velocity (i.e., w = 0), then it is apparent that the right-hand sides depend only on z. If there are circumferential velocities, then the further assumption is made that... 

If there is a circumferential velocity component, the circumferential momentum equation follows from substitutions into Eq. 6.185,... 

The governing equations for the rotating disk must include a circumferential momentum equation, and the circumferential velocity becomes a dependent variable. Also the circumferential velocity contributes to the radial-momentum equation. As simplified by the general equations of Section 6.2, the nonreacting, constant-property equations are summarized... 

Inclusion of the circumferential-momentum equation demands two new boundary conditions on the scaled circumferential velocity W = w/r. At the rotating surface z = 0, W = 2, where Q is the rotation rate (rad/s). Usually the outer flow is considered to have no circumferential velocity, W = 0. In general, there can be a swirl component in the outer flow. However, as discussed in Section 6.7.3, an inlet-swirl velocity can destabilize the flow. For the pure rotating disk situation with a semi-infinite outer environment the pressure-curvature eigenvalue will vanish. However, the eigenvalue is retained in the analysis because it will be needed for the analysis of fixed-gap rotating-disk situations. 

There is a natural draw rate for a rotating disk that depends on the rotation rate. Both the radial velocity and the circumferential velocity vanish outside the viscous boundary layer. The only parameter in the equations is the Prandtl number in the energy equation. Clearly, there is a very large effect of Prandtl number on the temperature profile and heat transfer at the surface. For constant properties, however, the energy-equation solution does not affect the velocity distributions. For problems including chemistry and complex transport, there is still a natural draw rate for a given rotation rate. However, the actual inlet velocity depends on the particular flow circumstances—there is no universal correlation. 

The boundary-layer thickness is a function of the rotation rate and can be derived from the nondimensional velocity profiles. Boundary-layer thickness can be defined in different ways, but generally it represents the thickness of the viscous layer. Defining the boundary-layer thickness as the point at which the circumferential velocity is 1% of its surface value gives zi% = 5.45. 

Notice that a choice was made to scale the circumferential velocity using the inlet velocity U. The rotation-rate scale is used in the boundary-condition specification. With these variables the nondimensional equations are... 

The radial velocity profile is linear and the circumferential velocity is zero outside the viscous boundary layer, which indicates that the vorticity is constant in that region. Thus, for substantial ranges of the flow and rotation Reynolds numbers, the flow is inviscid, but rotational, outside the viscous boundary layer. For sufficiently low flow, the boundary-layer can grow to fill the gap, eliminating any region of inviscid flow. 

The disk rotation is specified by a boundary condition for W at z = 0. In principal, a nonzero circumferential velocity could also be specified at the inlet. Physically, however, inlet swirl can lead to difficulties. When the flow swirls and the stagnation surface is stationary, a tomadolike circumstance is created. Fluid tends to be drawn radially inward near the stationary surface, which has deleterious consequences that are similar to starved flow. 

Written in this form, these equations clearly show that the pressure-gradient terms are both functions of r alone as long as the circumferential velocity w is either zero or a function of r alone. The axial momentum equation has been divided by z, so the term... 

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