Rotating sphere viscometer

For a rotating sphere viscometer, the tangential velocity on the surface of the sphere is ft sin 0. This reveals the angular dependence of 1)0 at any radial position, because if one moves into the fluid at larger r and constant 9, and a separation of variables solution to the (f)-component of the equation of motion is valid, then the sin 9 dependence shouldn t change. Hence,... 

In the rotating sphere viscometer, a solid sphere of radius R is suspended from a wire and rotates slowly at constant angular velocity about the long axis of the wire in an incompressible Newtonian fluid. The fluid is quiescent far from the sphere. 

If the entire temperature dependence of viscosity is to be measured, it is necessary to use several methods based on different principles. In the viscosity range 10 —10 dPa s, use is mostly made of rotary viscometers. A platinum cylinder rotates around its axis in the glass melt in a crucible, and the force required for revolving the cylinder at a certain speed is measured. In another arrangement, the external crucible is rotated while the internal one is suspended on a torsion wire. Within the same viscosity region, it is possible to measure with a counterbalanced sphere viscometer a plat inum sphere suspended on a thin wire from the balance arm is immersed in the glass melt in a crucible. The other balance arm is loaded and the speed at which the sphere is withdrawn from the melt is measured. 

Numerous methods for measuring fluid viscosity exist, for example, capillary tube flow methods (Ostwald viscometer), Zahn cup method, falling sphere methods, vibrational methods, and rotational methods. Rotational viscometers measure the torque required to turn an object immersed or in contact with a fluid this torque is related to the fluid s viscosity. A well-known example of this type of system is the Couette viscometer. However, it should be noted that as some CMP slurries may be non-Newtonian fluids, the viscosity may be a function of the rotation rate (shear rate). An example of this is the dilatant behavior (increasing viscosity unda increasing shear) of precipitated slurries that have symmetrical particles [33]. Furthermore, the CMP polisher can be thought of as a large rotational plate viscometer where shear rates can exceed 10 s and possibly affect changes to the apparoit viscosity. The reader can refer to the comprehensive review of viscosity measurement techniques in the book by Viswanath et aL [34]. 

Stormer viscometers, concentric cylinder viscometers, rotating spindles, falling spheres, etc. Because these viscometers expend part of their energy in accelerating the particles, this produces change in their orientation, and because voidage in the bed is affected by the immersed objects, the data on apparent viscosity of fluidized beds have to be carefully examined. 

Consequently, the rheological measurements of MPSs should be carried out such that the dimension of the flow channel is significantly larger than the size of the flow element. For example, the relative viscosity, jjr, of diluted spherical suspensions measured in a capillary instrument depends on the (d/D) factor, where 7) is the sphere diameter and d that of the capillary—for d 107), the error is around 1% [Happel and Brenner, 1983]. Thus, if 1% error is acceptable, the size of the dispersion should be at least 10 times smaller than the characteristic dimension of the measuring device (e.g., diameter of a capillary in capillary viscometers, distance between stationary and rotating cylinders or plates). Following this recommendation is not always possible, which lead to the decline and fall of continuum mechanics [Tanner, 2009]. 

The falling ball viscometer consists of a tube that may be rotated about a horizontal axis, as shown in Figure 7.8. The tube is marked with two lines a and b and contains the fluid of interest maintained at a given temperature. A sphere (steel and glass are the most common materials) with a finely calibrated diameter is inserted into the tube. At the beginning of the test, the ball lies at the bottom. The mbe is rotated by 180°, which brings the ball (sphere) to the top and then it drops through the fluid. The time it takes to traverse a prescribed distance L between the lines a and b is measured. The velocity of the ball is the distance between the two lines divided by the time. 

Any of the flow processes described in sections 6.3, 6.4 and 6.5 can form the basis of absolute viscometers, so that there are viscometers using Poiseuille flow, rotating coaxial cylinders and the terminal velocity of falling spheres. 

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