Coaxial cylinder geometry

The coaxial cylinder geometry has the advantage of low heat loss from the ends by having a large length to diameter ratio. However, there is a problem of error due to it being very difficult to make the test piece fit accurately. This is not the case with a molten polymer and such apparatus has allowed measurements on polymers under pressure40. 

The coaxial cylinder geometry is illustrated in Fig. 2. The cylinder is of length L, outside radius and inside radius /-. The temperature difference between the inner and outer surfaces is A. Eq. 1 can be integrated to give... 

For rotating coaxial cylinder geometry, the radial gradient in pressure across the gap between the cylinders, measured, at a point well removed from either end, would in the absence of hole effects be related to the primary normal stress difference ... 

A somewhat less direct method in which there is no specific determination of force involves coaxial cylinder geometry in which the outer cylinder is oscillated through a small angle. The inner cylinder, suspended from a torsion wire of suitable stiffness, responds with an angular oscillation whose amplitude and phase depend on the viscoelastic properties of the material in the gap as well as the moment of inertia of the inner cylinder and the stiffness of its support. From the ratio of amplitudes and the phase difference between the two motions, which are usually de-... 

This velocity profile is commonly called drag flow. It is used to model the flow of lubricant between sliding metal surfaces or the flow of polymer in extruders. A pressure-driven flow—typically in the opposite direction—is sometimes superimposed on the drag flow, but we will avoid this complication. Equation (8.51) also represents a limiting case of Couette flow (which is flow between coaxial cylinders, one of which is rotating) when the gap width is small. Equation (8.38) continues to govern convective diffusion in the flat-plate geometry, but the boundary conditions are different. The zero-flux condition applies at both walls, but there is no line of symmetry. Calculations must be made over the entire channel width and not just the half-width. 

Figure 6-4. Rotation plastimeter geometry, (a), (b) and (c) Coaxial cylinder types (d) and (e) concentric disc types ((d) is the Mooney geometry). A is usually the stator and B the rotor, C is the rotating shaft and r is the cylinder radius (much larger than the clearance between A and B). xy indicates the mid-plane along which the chamber can be opened for filling.

Steady state methods are usually based on parallel plate geometry, although coaxial cylinders are also suitable. The unguarded hot plate apparatus is a development of Lee s disk first described in 1898 and beloved of school physics laboratories for many decades. The general arrangement is shown in Figure 14.1. 

Taylor (1923) first observed the instability of fluid when the inner cylinder exceeds a critical speed between two coaxial cylinders in his established work. The laminar flow confined within the annulus region between two coaxial cylinders with the inner one differentially rotating with respect to the outer suffers centrifugal instability depending on the geometry and rotation rates. Taylor (1923) showed that an inviscid rotating flow to be... 

Axial Motion Between Coaxial Cylinders (Segel-Pochettino Geometry)... 

In the last decade of the nineteenth century, Maurice Couette invented the concentric cylinder viscometer. This instrument was probably the first rotating device used to measure viscosities. Besides the coaxial cylinders (Couette geometry), other rotating viscometers with cone-plate and plate-plate geometries are used. Most of the viscometers used nowadays to determine apparent viscosities and other important rheological functions as a function of the shear rate are rotating devices. 

Normal stress differences can be observed in Couette flow, cone-plate and plate-plate geometries, and capillary flow. The only nonzero components of the stress tensor in coaxial cylinders are a. e(r), cSrr(f), CTee(r), and... 

Both strain- and stress-controlled rotational rheometers are widely employed to study the flow properties of non-Newtonian fluids. Different measuring geometries can be used, but coaxial cylinder, cone-plate and plate-plate are the most common choices. Using rotational rheometers, two experimental modes are mostly used to study the behavior of semi-dilute pectin solutions steady shear measurements and dynamic measurements. In the former, samples are sheared at a constant direction of shear, whereas in the latter, an oscillatory shear is used. 

The traditional way to measure thermal conductivity is with steady-state instruments, in which a measured heat flux is compared to a temperature difference between surfaces. Most often the geometry is coaxial cylinders, a thin wire inside a cylinder, or parallel plates. In such instruments, eliminating convection currents is crucial many old data taken with steady-state instruments are unreliable because of convection. Multiple experiments at different heat fluxes are often performed to verify the absence of convection. With good design and operation, such instruments may achieve accuracy in the 1% to 3% range. 

Figure 10. Rheological measurements performed with a coaxial cylinder viscometer and different geometries. The fact that the curves are not superimposed implies that the measurements are affected by wall slip. (Reproduced with permission from reference 13. Copyright 1991 E.irF.N. Spon.)...

For a smectic-A that is confined in the SEA crossed cylinder geometry, there is a competition between the homeotropic alignment on mica and the tendency to form layers of equal thickness d. As a result, dislocations must arise (Fig. 3.18, inset). As the local geometry aroimd the contact point is equivalent to a sphere-plane geometry, the loops are expected to be circular and centered on the contact point. Consider thus an array of torus-like cells, coaxial to the loops (Fig. 3.18, inset). Each cell is defined by an inner radius r, corresponding to a thickness h ri) = Uid, and an outer radius rj+i, with h ri+i) = (rij - - l)d, and contains a circular dislocation loop of radius pi- The cells are independent, because the strain patterns produced by the dislocations decays exponentially outside a parabola of equation = z, where A ... 

Flow between Two Rotating Coaxial Cylinders. The geometry of two coaxial... 

FIGURE 4.3. The Prederiks transition in different geometries (a) nematic between two coaxial cylinders (b) quasi-homeotropic orientation with opposite director tilt at the boundaries and (c) hybrid aligned (homeoplanar) nematic cell. 

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