Cylinder translational flow

Figure 3.1 Schematic of the experimental setup, (a) Side view showing a rotating cylinder translating over a flat plate, (b) Same setup as viewed from the top of the tunnel. H= adjustable distance of the cylinder from the plate, Uoo = flow speed, c= translational speed of the cylinder, and Q= angular velocity of the cylinder. x= distance of cylinder from the leading edge of the plate. This arrangement creates a captive vortex (at the center of the cylinder) that can be made to travel at predetermined speed.

Low Reynolds numbers. In [216, 382] the problem on a circular cylinder of radius a in translational flow of viscous incompressible fluid with velocity Ul at low Reynolds numbers was solved by the method of matched asymptotic expansions. The study was carried out on the basis of the Navier-Stokes equations (1.1.4) in the polar coordinates 1Z, 6. Thus, the following expression for the stream function was obtained for IZ/a 1 ... 

Circular cylinder. The mass exchange between a circular cylinder of radius a and a uniform translational flow whose direction is perpendicular to the generatrix of the cylinder was considered in [186,218] for low Peclet and Reynolds numbers Pe = Sc Re and Re = aU-Jv. For the mean Sherwood number (per unit length of the cylinder) determined with respect to the radius, the following two-term expansions were obtained ... 

Cylinder of arbitrary shape. Let us consider mass exchange for cylindrical bodies of arbitrary shape in a uniform translational flow of viscous fluid at small... 

Diffusion to an elliptic cylinder in a translational flow at high Peclet numbers was considered in [166]. 

Figure 4.3. Heat exchange between a cylinder of an arbitrary shape and a translational flow (a) the original system of rectangular coordinates (b) the plane of the new variables ip, ip...

While the lift on a stationary cylinder in an air stream is zero [Fig. 6.8 (a)], that for a rotating cylinder [Fig. 6.8 (6)] is not zero. Air is dragged along with the rotating cylinder. This circulation, when combined with the translational flow, causes the velocity on the top side of the cylinder to be higher than that on the bottom side. As a consequence of the Bernoulli equation [Eq. (5.15)], the pressure on the bottom side of the cylinder will be higher than that on the top side, giving rise to an upward lift (L). 

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. 

The strategy is as follows. We start by rewriting the equations in cylindrical coordinates (r, ,z). The variables we consider are the layer displacement u (now in the radial direction) from the cylindrical state, the director n, and the fluid velocity v. The central part of the cylinder, r < Ri, containing a line defect, is not included. It is not expected to be relevant for the shear-induced instability. We write down linearized equations for layer displacement, director, and velocity perturbations for a multilamellar (smectic) cylinder oriented in the flow direction (z axis). We are interested in perturbations with the wave vector in the z direction as this is the relevant direction for the hypothetical break-up of the cylinder into onions. The unperturbed configuration in the presence of shear flow (the ground state) depends on r and 0 and is determined numerically. The perturbations, of course, depend on all three coordinates. We take into account translational symmetry of the ground state in the z direction and use a plane wave ansatz in that direction. Thus, our ansatze for the perturbed variables are... 

In contrast, when the translation speed of the cylinder was increased to (c = 0.772Uoo) (Case 2), there was no violent breakdown of the flow, as shown in Fig. 3.3. This indicates that the boundary layer is insensitive to the vortex convecting at higher speeds. For the range of translation speed investigated, it was found that slower the translation speed of the vortex, greater was the effect on boundary layer stability, when other parameters were kept the same. 

To explain the above aspect of results, note that a rotating and translating cylinder of diameter d induces a disturbance stream function in the inviscid irrotational part of the flow field that is given in Robertson (1969) by,... 

In Fig. 3.5, visualization sequences are shown for the Case 3. In this case of non-rotating translating cylinder, no violent instability was seen to occur for two reasons. Firstly the imposed disturbance field, as given by Eqn. (3.3.1) has no captive vortex i.e.F = 0) as the cylinder does not rotate while translating. Secondly, if there are shed vortices present, they will be very weak and Benard- Karman vortex street is seen to affect the flow weakly far downstream of the translating cylinder - only at earlier times. 

Problem 7-22. The Viscosity of a Multicomponent Membrane. An interesting generalization of the Einstein calculation of the effective viscosity of a dilute suspension of spheres is to consider the same problem in two dimensions. This is relevant to the effective viscosities of some types of multicomponent membranes. Obtain the Einstein viscosity correction at small Reynolds number for a dilute suspension of cylinders of radii a whose axes are all aligned. Although there is no solution to Stokes equations for a translating cylinder, there is a solution for a force- and torque-free cylinder in a 2D straining flow. The result is... 

Note that in some problems of heat and mass transfer and chemical hydrodynamics, the velocity fields near the body can be determined by the flow laws of ideal nonviscous fluid. This situation is typical of flows in a porous medium [75, 153, 346] and of interaction between bodies and liquid metals (see Section 4.11, where the solution of heat problem for a translational ideal flow past an elliptical cylinder is given). 

Mass and Heat Transfer Between Cylinders and Translational or Shear Flows... 

At high Peclet numbers, for an nth-order surface reaction withn=l/2, 1,2, Eq. (5.1.5) was tested in the entire range of the parameter ks by comparing its root with the results of numerical solution of appropriate integral equations for the surface concentration (derived in the diffusion boundary layer approximation) in the case of a translational Stokes flow past a sphere, a circular cylinder, a drop, or a bubble [166, 171, 364], The comparison results for a second-order surface reaction (n = 2) are shown in Figure 5.1 (for n = 1/2 and n = 1, the accuracy of Eq. (5.1.5) is higher than for n = 2). Curve 1 (solid line) corresponds to a second-order reaction (n = 2). One can see that, the maximum inaccuracy is observed for 0.5 < fcs/Shoo < 5.0 and does not exceed 6% for a solid sphere (curve 2), 8% for a circular cylinder (curve 3), and 12% for a spherical bubble (curve 4). 

In Chapters 1 and 2 we study fluid flows, which underlie numerous processes of chemical engineering science. We present up-to-date results about translational and shear flows past particles, drops, and bubbles of various shapes at a wide range of Reynolds numbers. Single particles and systems of particles are considered. Film and jet flows, fluid flows through tubes and channels of various shapes, and flow past plates, cylinders, and disks are examined. 

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