Relative shear velocity

Various models will be used for the interface between the fiber and the matrix. For bonded interfaces, complete continuity of all components of the velocity will be invoked. The simplest model for a weak interface is that a shear drag equal to r opposes the relative shear velocity jump across the interface. The direction of the shear drag is determined by the direction of the relative velocity. However, the magnitude of r is independent of the velocities. This model is assumed to represent friction occurring mainly because of roughness of the surfaces or due to a superposed large normal pressure on the interface. Creep can, of course, relax the superposed normal stress over time, but on a short time scale the parameter r can be assumed to be relatively invariant. No attempt will be made to account for Coulomb friction associated with local normal pressures on the interface. 

Figure 5.6 The director is strongly anchored in the x-direction on the boundaries, parallel with the bounding plates which are placed at a distance d = 2h apart. The lower plate is fixed with velocity set to zero while the upper plate has a relative shear velocity of V. The orientation angle y) of the director is defined as shown for the director given by equation (5.93)...

As a result of the above scaling properties, the shear flow problem outlined previously in subsection 5.5.4 can be reduced from one involving the gap width d and the relative shear velocity to a problem with unit gap width and relative velocity V where... 

For applications in the field of micro reaction engineering, the conclusion may be drawn that the Navier-Stokes equation and other continuum models are valid in many cases, as Knudsen numbers greater than 10 are rarely obtained. However, it might be necessary to use slip boimdaty conditions. The first theoretical investigations on slip flow of gases were carried out in the 19th century by Maxwell and von Smoluchowski. The basic concept relies on a so-called slip length L, which relates the local shear strain to the relative flow velocity at the wall ... 

A thin layer of a molten polymer of 2 mm thickness is sandwiched between two plates. If a shear stress of 120 kPa is applied to the melt, and the apparent viscosity of the melt is 4 x 10 kg(m s) calculate the relative sliding velocity of the two plates. 

In counter-rotating screws, the roll-off process between the screw crest and screw root and between the screw flanks creates a calendar effect. The shear velocity available to wipe the boundary layers is proportionately lower due to the small relative velocity between the surfaces. 

In ICRR the roll-off process between the screw crest and screw root and between the screw flanks simulates the action of a calender. The necessary shear velocity required to wipe the boundary layers is proportionately lower because of the low relative velocity. Counter-rotating screws require greater clearances between them since their mode of action is rather like a two-roll mill, passing material through the nip between them [Schoengood, 1973]. The material is drawn into the roller gap and is squeezed onto the... 

Integrating over all the spheres of the same layer and then over all the layers, we see that the resultant value vanishes. Therefore, the shear stress at the planes y = yj and y = -y is the same as before the introduction of the particles, and the only effect of the perturbation flow from all the particles is to produce a decrease of the relative horizontal velocity in these planes. 

A layer of molten poly(methyl methacrylate) at 190°C is of uniform thickness 3 mm, and is sandwiched between two flat, parallel plates. A shear stress of 100 kPa is applied to the melL Imd the relative slidiiig velocity of the plates, using data for apparent viscosity from Figure 7.13. 

The shear strain-rate in the melt is identically equal to the transverse velocity gradient. Hence, the relative sliding velocity v of the plates is given by... 

In CORI the material is transferred from one screw to another in a tangential path (Salden 1978). Here, edges of one screw crest wipe the flanks of the other screw with a tangentially oriented, constant relative velocity. There is a high relative velocity and hence sufficiently high shear velocity to wipe the boundary layers. 

As described above, the viscosity of a fluid at relatively low velocities generates laminar flow. When the velocity exceeds a critical level, the inertial force becomes sig-niflcant compared to the viseosity. TTiis is called turbulent flow. A thorough analysis of turbulent flow is very difficult. However, it is recognized that the shearing stress is proportional to the square of the velocity gradient perpendicular to the solid/fluid interface as ... 

Couette flow n. Shear flow in the annulus between two concentric cylinders, one of which is usually stationary while the other turns. By measuring the relative rotational velocity and the torque required to maintain steady flow, one can infer the viscosity of the liquid. Flow in the metering section of a single-screw extruder resembles Couette flow, modified by the presence of the flight and, normally, by the pressure rise along the screw. 

To calculate the intrinsic viscosity, a particular velocity gradient must be selected. Because the relevant quanhty is the difference in the local velocity of the fluid and the subchain, it is convenient to note that, in simple shear, the center of mass should be moving at the macroscopic fluid velocity. The local unperturbed relative fluid velocity is then. Using the same... 

---