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Chapter 4 Canned Rolls In this chapter the Taylor vortices in rotating coaxial cylinders are described which I call as “canned rolls” [1]. In initial sections the apparatus and experimental results are described. Then I give the argument for instability in case of inviscid liquids, which is Rayleigh’s criterion. We then consider the plane and circular Couette flows for viscous fluids. The trajectories of particles in Taylor vortices are modeled by a spring twisted in the form of a circle. Such trajectories are generated using computer. Then we consider the reasons for instability in case of viscous fluids. Some other interesting things observed during the project work are then described. I conclude this chapter with recent work on Taylor vortices and future prospects. 4.1 Introduction The hydrodynamic instabilities have puzzled the physicists for centuries. Simplest example is the transition from one flow pattern to another in case of tap water, as the flow rate is increased. Similarly the flow past a circular cylinder has been studied in detail by many researchers. The flow past cylinder goes through many transitions as the flow rate is increased. Initially the flow is streamlined along the sides of cylinder. Then the vortices develop at the back of the cylinder and they are periodically shed out, to form what is called as Karman vortex street. At even higher flow rates, the turbulence sets in. This cascade of transitions in flow patterns is universal that is, it is independent of the system under consideration. There may be differences in actual flow patterns, but the overall transition is from laminar flow to onset of turbulence. These transitions are difficult to describe mathematically, and turbulence remains an unsolved problem till today. The first attempt to account for these transitions was done by Sir G. I. Taylor, who used the linear stability theory to explain the structure of Taylor vortices and the transition from circular Couette flow to Taylor vortices in the fluid between the rotating coaxial cylinders. His approach as well as the physical reasons behind the instability will be explained in the following sections. But the instability was first predicted by Rayleigh for inviscid fluids. Mallock, who used the apparatus to measure viscosity, first experimentally inferred it [2]. 4.2 While I was Canning the Rolls The journey towards obtaining the Taylor vortices wasn't a smooth and simple one. It was not at all clear to start with, how I would manage to construct the apparatus with all its requirements like the outer cylinder should be transparent, the cylinders should rotate independent of each other and so on. So to start with we decided to modify the available viscometer apparatus in our college. The fact that a drinking water glass suited as an outer cylinder was a happy coincidence. To keep the inner cylinder in it and allow it to rotate freely, I put a rubber sheet at the base of it and made a small hole in it, so that the cone of the cylinder easily fits in. Then it was time to look for the ways to make the inner cylinder rotate. After lot of search I got a dc motor of 12 volts for 55 Rs, from Pioneer Electronics, which had sufficient power to rotate the inner cylinder. But now the problem was that of a suitable pulley. Unfortunately the motor dealers don't provide a pulley. After some more searches I found a suitable pulley, but its bore size was less. So I had to get it drilled from Haribhau, our elderly lab assistant. After the pulley the natural question was that of the belt. But we decided to work with all available rubber bands and they worked pretty well. Later I replaced them with tape recorder belts. Finally my initial apparatus took the form as shown in Fig. 4.1. Figure 4.1: My earlier apparatus. Having constructed the apparatus was not all. I was not sure which fluid to take as working fluid, how to visualize the streamlines and see patterns. First I tried dropping ink with a syringe, but it tends to get diffused. Then I tried several combinations of working fluids and suspensions, which are described in Table 4.1. Table 4.1: Description of flow visualization methods. No. Fluids Suspension / Method Observation Comments 1 Water Aluminum Powder Sticks to walls of Though these combinations cylinders. Blurred out form a good suspension, they white solutions. don't provide any good 2 Water Ferrous Oxide Powder Sticks to walls of visualization. cylinders. Blurred out reddish solutions 3 Water Local pH change dye The solution turns yellow Suitable for a closed system. method: add Bromo near one of the We didn't use it much because Thymol Blue indicator and electrodes. This yellow the change wasn't prominent place the electrodes in is carried away by the and large concentrated solution. Apply 3-4 volts. flow, producing a quantities would be required. Bubbles should be avoided streakline. Advantage of this method is [3]. that the dye doesn't spoil the whole solution, it changes back to its original color after some time. 4 Water Fluorescent dye method a In UV light the dye turns The glow is uniform, so it was due called Ruthenium Tris orange from yellow. not clear how it could help in Bipyridine. visualizing the vortices. 5 Water Aluminum Oxide Powder The powder aggregates, This method was initially forming clusters. discarded. 6 Water Aluminum Oxide Powder Good suspension, low Best combination. The addition + little soap solution settling rate. of soap solution stops aggregation, an ingenious idea indeed. There is a drastic difference after adding a small amount of soap solution. 7 Paraffin Aluminum Oxide Powder Good suspension, Good combination, this gave oil appreciable settling rate. us the clearest results in the beginning 8 Paraffin Aluminum Oxide Powder This gave even better oil colored with blue ink and Good suspension, low visualization due to color dried under lamp. settling rate. contrast with the inner cylinder. 9 Castor Aluminum Oxide Powder Good suspension, very For all available rotation rates, oil low settling rate. only circular flow was seen. This combination will require very high rotation rates. We got the best results with the combinations 6, 7, 8 in the above list. Now the reader would have realized the amount of efforts involved in constructing a new apparatus and setting up an experiment. But this work had rewarding outcomes for me. Firstly it gave me confidence to build an apparatus. Secondly I came to know the various resources involved. Thirdly, I experienced the joy of a successful experiment. Having secured some good results, we confidently proceeded to make a professional apparatus, for more rigorous study in the near future. With available infrastructure and some more investment, we could prepare a well-finished apparatus, with innovative ideas and efforts on part of the workshop persons. The newer experimental setup for studying the Taylor vortices is given in Fig. 4.2. The two cylinders are placed coaxially one inside the other. The inner cylinder rests on the outer one with a cup and cone arrangement. The outer cylinder is a glass pipe, which is fixed in a groove at the base by Araldite. The inner cylinder is held at the upper side using a ball bearing arrangement. Both the cylinders are provided with tape recorder belts and can be rotated independent of each other using separate motors. Figure 4.2: Taylor vortices apparatus. 4.3 Observations When the inner cylinder is rotated slowly, the fluid moves in circular lines. But as we go on increasing the rotation rate, something interesting happens: some horizontal lines are seen in the flow and when the inner cylinder is rotated above a certain critical rate, the flow breaks up into horizontal vortices. The particles in the vortex help in identifying the sense of rotation of the vortex. It can be easily seen that the adjacent vortices are counter rotating. Apart from 2-3 vortices near top and near bottom, which are distorted due to end effects, all the other vortices have same width. 4.3.1 Taylor Vortices Using Old Apparatus This apparatus is shown in Fig. 4.1. With the working fluid as paraffin oil and suspension powder as blue colored Al2O3 Powder, we saw nicely separated bands (see Fig. 4.3). The width of bands was observed to be equal to the gap between the cylinders. Figure 4.3: Taylor Vortices in paraffin oil. 4.3.2 Taylor Vortices Using New Apparatus Figure 4.3: Taylor vortices in water. Since for this apparatus the gap was smaller, the vortices seen in this figure are smaller. 4.4 Theoretical Explanations 4.4.1 Argument for instability in an inviscid liquid Consider a fluid with zero viscosity. Practically no such fluid exists, except for the Helium3 superfluid at very low temperatures. And it gives Helium3 some of its astonishing properties. It has this property that it keeps rotating once it is set into motion! It also has this amazing property to creep over the walls of the container, so just keeping it in our apparatus won't do! We will require confining it into the concerned region. But air can be taken as a practical substitute as a very low viscosity fluid. Now we can easily imagine that if only the air is present in the gap between the cylinders, then even the slightest rotation of inner cylinder would cause lot of disturbance in it. In fact the flow will be a turbulent one. Actually even without rotation of the inner cylinder the flow in air is not stable. I hope this gives the reader a sufficient judgement of the behavior of fluids of low viscosity. But water has some significant viscosity. It will be unstable after certain rotation rates, but it does show a stable circular flow for low rotation rates. The reason for instability in an inviscid liquid is the following: the fluid near the inner cylinder continues in its state of motion, by Newton's first law, as no force acts on it. But the inner cylinder is rotating and the liquid has to stay in the confined region. So it hits the outer wall and bounces in a different direction, since the walls are curved. The inner cylinder's motion is required to throw the fluid near it towards the outer cylinder again and again. Then, to replace this outgoing fluid, some fluid must come from beneath it. The fluid coming towards the inner cylinder then occupies the place of this fluid. In this manner the convection rolls are set up, and these canned rolls are nothing but the Taylor vortices. This can be quantitatively explained as follows: Suppose a fluid layer at distance , circulating with angular velocity is displaced to a distance , where < . Because there is no external torque on this fluid particle, the angular momentum of this fluid particle must remain the same. Hence its new angular velocity will be 2 . The centrifugal force of this fluid particle, with angular velocity will 2 exceed the available centrifugal force at , if . Using the angular momentum relation, we can write this condition as: d r 2 0 . ..... (4.1) dr That is, the radial gradient of angular momentum r 2 is negative. This condition is called as Rayleigh’s criterion for instability in inviscid Couette flow [3]. If the two cylinders are rotating in same sense, then the flow is either every where stable or everywhere unstable. Substituting the radii a1 and a2 in Rayleigh’s criterion, we get the condition as 1a12 2 a2 2 .....(4.2) When the two cylinders are counter rotating, the region near the inner cylinder is unstable while the region near the outer cylinder is stable. So the inner cylinder rotations have a destabilizing effect, while the outer cylinder’s rotations have a stabilizing effect. 4.4.2 Plane and circular Couette flow. Our task is now to find the velocity profile in case of viscid fluid between the rotating cylinders. Naturally, due to cylindrical symmetry, the best coordinate system to choose is the cylindrical polar coordinate system. But the Navier Stokes equations in cylindrical coordinate system are complicated compared to those in Cartesian coordinate system. So, as any good Physicist would do, let’s first try to approximate the situation by a simpler one and try to solve the Navier Stokes equations there. First let’s neglect the end effects due to finite size and consider the infinitely long cylinders. Then the velocities will be independent of the positions on z-axis. Let’s restrict ourselves to low rotation rates for the time being. Then there are no vortices and the fluid just moves in circles as observed. So there is no dependence of velocity on the angle. At a given radius, the velocity is same over the entire vertical cylinder at that radius. If we consider a small vertical section of the cylinder and magnify the picture then we will have the flow in 2 almost parallel plates, just like the earth looks flat in the surroundings we see. This limiting case we are considering is equivalent to an apparatus with large radii of cylinders and small gap between the cylinders. The outer plate is at rest and inner moves at a constant velocity in one direction, let's say x-direction. Due to no slip condition, the velocity of the fluid near the outer plate is 0 and at the inner plate it will be equal to that of the inner plate. We can set up the coordinate system with x-y axes parallel to the two plates and z-axis perpendicular to them. Then we can use Navier Stokes equations to find the velocity distribution as a function of z. [4]. Z u a X Figure 4.4: Geometry of plane Couette flow. Refer to the appendix D for components of Navier Stokes equations in Cartesian coordinates. For steady state, time derivative is zero. Also we may neglect any external forces. Also u u x ( z )i i.e., velocity is in x direction only and its magnitude changes with z alone. Then the left-hand side of Navier Stokes equations is zero. On the right- hand side we may take external forces to be zero, also there is no external pressure applied, hence the pressure gradient is also zero. Then the Navier Stokes equations reduce to just 2u 0. since only the x-component of velocity is nonzero, the solution of this equation is , with the boundary conditions u=0 at z=0 and u= V at z=a, V ux ( z) z. a 10 Vz/a=u height z from lower plate 8 6 4 2 0 0 20 40 60 80 100 velocity in x direction Figure 4.5: Velocity profile in plane Couette flow. Note that this flow profile is different from the flow considered in chapter 2, for an open, wide, shallow channel. There, the velocity profile was quadratic and here it is linear. There, the velocity was pressure driven. Here it is induced by the viscous action alone. There, the velocity of upper layer was selected by the pressure gradient. Here it is imposed as an external condition. So we see the difference between the flow profiles even though the boundaries are same, due to different physical reasons for the flow. Now we are ready to solve for the velocity profile in our cylindrical geometry. We set our coordinates as follows: we choose the cylindrical polar coordinate system such that the axis of coordinate system is that of our cylinders. As discussed before, the flow is only tangential, that is the velocity is tangent to any circle drawn perpendicular to the z-axis. So only u is nonzero. Also u is the function of r alone. We can expect the solution to be of the form r + some function of r as the solution of plane Couette flow was linear. Figure 4. 6: The geometry of circular Couette flow. Because u is the function of r alone, the continuity equation is automatically satisfied. Referring to appendix D, we can easily see that the z component of Navier Stokes equations also gets satisfied automatically. The component Navier Stokes equations reduces to d 2u 1 du u 0 ( ). dr 2 r dr r 2 The velocity at the inner cylinder is 1a1 and that at the outer cylinder is 2a2.This can be u easily solved using the variable , with the above boundary conditions, to give [3], r B u (r ) Ar . r ( 2 a 2 1 a12 ) 2 (1 2 )a12 a 2 2 Where, A , and B a 2 a12 2 (a 2 a12 ) 2 . See how the 1/r term absorbs the curvature of the geometry compared to the plane Couette flow. Also note that this velocity distribution is independent of ! That means, for a given geometry of apparatus and for given rotation rates, the same flow exists for all d (ur ) liquids! We can easily see from these expressions that 2 A . So according to dr rayleigh’s criterion, the flow will be unstable if A < 0 and flow will be stable if A > 0. The r component reduces to u2 dp . r dr The important point to be noted here is that though there is no external pressure, the pressure develops intrinsically, due to the velocity distributions. Here we are having the pressure due to the centrifugal force. Putting the velocity as a function of r, we can easily find out the pressure. Now let's look at the plot of the velocity profile in this circular couette flow. In our case, 2 0 . Taking the the constants to A and B to be 1 and –1 respectively, we get the following velocity distribution. 10 u = -r +1/r derivative(ur)= - 2r tangential velocity u 5 0 -5 -10 0 2 4 6 8 10 a2 distance from axis r Figure 4.7: Unstable velocity profile in circular Couette flow.