Torque flow/viscosity

Mechanical for measurement of position, velocity, acceleration, force, pressure, stress, strain, torque, flow velocity, viscosity, etc. 

So far we have introduced four Miesowicz viscosities. Two other viscosities can be proposed by considering the following. The director n in Fig. 4.1(a), if free to move, will rotate due to a viscous torque the viscosity coefficient 71 is introduced to describe this situation and characterises the torque associated with a rotation of n. For this reason 71 is often called the rotational viscosity or twist viscosity. The coefficient 71 generally determines the rate of relaxation of the director. Also, a rotation of n due to body forces will induce a flow. The viscosity coefficient 72 characterises the contribution to the torque due to a shear velocity gradient in the nematic and is sometimes referred to as the torsion coefficient in the velocity gradient it leads to a coupling between the orientation of the director and shear flow. The two viscosities 71 and 72 have no counterpart in isotropic fluids. We therefore have a total of six viscosities four Miesowicz viscosities plus 71 and 72. It turns out, as will be seen in the problems to be discussed in later Sections, that 7i and 72 are precisely the viscosities introduced in the constitutive theory at equations (4.78) and (4.79), namely. 

A fluid with a finite yield. stress is sheared between two concentric cylinders, 50 mm long. The inner cylinder is 30 mm diameter and the gap is 20 mm. The outer cylinder is held stationary while a torque is applied to the inner. The moment required just to produce motion was 0.01 N m. Calculate the force needed to ensure all the fluid is flowing under shear if the plastic viscosity is 0.1 Ns/ni2. 

Resistance functions have been evaluated in numerical compu-tations15831 for low Reynolds number flows past spherical particles, droplets and bubbles in cylindrical tubes. The undisturbed fluid may be at rest or subject to a pressure-driven flow. A spectral boundary element method was employed to calculate the resistance force for torque-free bodies in three cases (a) rigid solids, (b) fluid droplets with viscosity ratio of unity, and (c) bubbles with viscosity ratio of zero. A lubrication theory was developed to predict the limiting resistance of bodies near contact with the cylinder walls. Compact algebraic expressions were derived to accurately represent the numerical data over the entire range of particle positions in a tube for all particle diameters ranging from nearly zero up to almost the tube diameter. The resistance functions formulated are consistent with known analytical results and are presented in a form suitable for further studies of particle migration in cylindrical vessels. 

Figure 4.22 illustrates a tool that can be used to measure viscosity. By measureing the torque required to spin the shallow-angle cone at a constant angular velocity, the viscosity of the fluid in the space between the cone and a stationary plate can be inferred. In the analysis of the flow between the cone and the plate, it can be assumed that vr — 0 and ve = 0. By symmetry it can be assumed that there are no variations. 

For the impeller ribbon viscometer technique, the power number of an impeller is inversely proportional to the impeller Reynolds number (Eq. 1). As the impeller rotational speed increases, the flow will gradually change from laminar to turbulent, passing through a transition region. Parameter c can be obtained from the calibration fluids. If the same value for c is assumed to apply to a non-Newtonian fluid, then Eq. 4 can be used to calculate the apparent viscosity of that fluid. The range of the impeller method is determined by the minimum and maximum torques that can be measured (5). 

Non-Newtonian Viscosity In the cone-and-plate and parallel-disk torsional flow rheometer shown in Fig. 3.1, parts la and 2a, the experimentally obtained torque, and thus the % 2 component of the shear stress, are related to the shear rate y = y12 as follows for Newtonian fluids T12 oc y, implying a constant viscosity, and in fact we know from Newton s law that T12 = —/ . For polymer melts, however, T12 oc yn, where n < 1, which implies a decreasing shear viscosity with increasing shear rate. Such materials are called pseudoplastic, or more descriptively, shear thinning Defining a non-Newtonian viscosity,2 t],... 

Some of the test methods being used to measure the processing stability of polypropylene include melt flow drift measurements at elevated temperatures using an extrusion plastometer (melt indexer), melt viscosity retention measurements using a torque rheometer, retention of melt flow after repeated extrusions, and injection molded spiral test measured by the flow in inches at various temperatures and the retention of melt flow of the injected spirals. The nine commercial resins were evaluated by these methods. 

Accretion. Viscous stresses and gravitational torques within the disk transport angular momentum to the outer regions allowing disk matter to flow inward and accrete onto the star. Because the source of viscosity is still not well understood (see also Chapter 4), it is common to describe the viscosity via a dimensionless parameter a (Shakura Syunyaev 1973). Using this simplification, the viscous dispersal timescale, i.e. the time for the disk to disperse via accretion, becomes inversely proportional to a and increases linearly with the radial distance from the star (Hartmann et al. 1998). While the inner-disk material accretes onto the star, material further out moves in and replenishes the inner disk. Thus, the disk-dispersal timescale from accretion alone is set by the timescale to disperse the mass at the outer disk. 

The experimental methods for the determination of liquid viscosity are similar to those used for gases ( 8.VII F) (i) transpiration, through capillaries, (ii) torque on rotating cylinders, or the damping of oscillating solid discs or spheres, in the liquid, (iii) fall of solid spheres through the liquid, (iv) flow of liquid through an aperture in a plate, (v) capillary waves. Methods (i) and (ii) are mostly used for absolute, the others for comparative, measurements. 

The magnitudes of the viscosities (the a ,- s) for a single small-molecule nematic can differ from one another by an order of magnitude or more. As a result, the fluid s resistance to flow depends strongly on the directions of the flow and the flow gradient relative to the nematic director. In a shearing flow, the viscosities o 2 and o 3 determine director torques in the orientations shown in Fig. 10-7b and 10-7c. If the director is oriented in the flow direction... 

Dense-phase flow properties can be predicted by measuring the viscosity and deaeration rate of a fluidized bed in a laboratory test (117). This method appears to be especially useful for evaluating the flow characteristics in standpipes. The viscosity is measured by means of a Brookfield viscometer, which consists of a cylindrical wire-screen spindle rotated about its axis by an electric motor through a torsion spring. The torque required to rotate the spindle is measured by displacement of the... 

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... 

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