Shearing between two plates

Figure 3. Examples of the NESS, (a) An electric current / flowing through a resistance R and maintained by a voltage source or control parameter V. (b) A fluid sheared between two plates that move at speed v (the control parameter) relative to each other, (c) A chemical reaction A — B coupled to ATP hydrolysis. The control parameters here are the concentrations of ATP and ADP.

Figure 1. No slip boundary condition steady laminar velocity profile for fluid sheared between two plates.

For example, when a liquid is sheared between two plates parallel to the xy plane, we have y = dvjdz.) A typical plot of y vs. x is shown in Figure 5.52a. For low and high shear rates, we observe Newtonian behavior (r = const.), whereas in the intermediate region a transition from the lower shear rate viscosity, r o, to the higher shear rate viscosity, takes place. This is also visualized in Figure 5.52b, where the viscosity of the colloidal dispersion, q is plotted vs. the shear rate, y note that in the intermediate zone q has a minimum."" ... 

In EME the polymer is sheared between two plates, one stationary and one rotating. The extruder makes use of the viscoelastic properties of polymer melts. When a viscoelastic fluid is sheared, the normal stresses develop in the fluid, trying to push the shearing plates apart. Thus, leaving a hole in the center of the stationary plate makes it possible for the melt to flow continuously from the rim toward the center then out [Maxwell and Scalora, 1959 Blyler, 1966 Eritz, 1968, 1971 ... 

If a disklike specimen is sheared between two end plates by rotation of one over the other to obtain the shear modulus, then at any moderate twist angle the strain (and strain rate) vary along the radius, so only an effective shear modulus is obtained. For better results the upper plate is replaced with a cone of very small angle. Figure 4 shows fche cone-and-plate and two other possible test geometries for making shear measurements. 

There are many examples of nonequilibrium states. A classic example of a NESS is an electrical circuit made out of a battery and a resistance. The current flows through the resistance and the chemical energy stored in the battery is dissipated to the environment in the form of heat the average dissipated power, Pdiss = VI, is identical to the power supplied by the battery. Another example is a sheared fluid between two plates or coverslips and one of them is moved relative to the other at a constant velocity v. To sustain such a state, a mechanical power that is equal toVoc r v has to be exerted on the moving plate, where p is the viscosity of water. The mechanical work produced is then dissipated in the form of... 

In the case of elastic fluids and for simple shear flow, the first normal stress difference is N[ =on — o22- When shearing a fluid between two plates (x, direction), the first normal stress difference N( forces the plates apart (x2 direction). The first normal stress difference N i is shown together with the measured shear stress x as a function of the shear rate in Fig. 3.9. In the range of shear rates investigated, the shear stress in the case of silicone oil is substantially greater than the normal stress difference and we see substantially greater normal stress differences for viscoelastic PEO solution than for viscous silicone oil. 

Colloidal suspensions of beads crystallized between two plates Shear and electrostatic forces nm to (Jim 45,46... 

It is important to realize that this type of behavior is not just a simple addition of linear elastic and viscous responses. An ideal elastic solid would display an instantaneous elastic response to an applied (non-destructive) stress (top of Figure 13-74). The strain would then stay constant until the stress was removed. On the other hand, if we place a Newtonian viscous fluid between two plates and apply a shear stress, then the strain increases continuously and linearly with time (bottom of Figure 13-74). After the stress is removed the plates stay where they are, there is no elastic force to restore them to their original position, as all the energy imparted to the liquid has been dissipated in flow. 

The experimental device constructed to orient uniformly thick samples in simple shear is schematically represented in Fig. 3. It is basically a sliding-plate rheometer, the polymer sample being sheared between two temperature-controlled parallel plates. The upper plate is fixed whereas the lower plate can be displaced both horizontally and vertically with two pneumatic jacks. 

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. 

Shear apparatus. A simular apparatus as discribed by Hadziioannou et al. (10) was used to orientate the block copolymer. The polymer was sheared between two heatable metal plates by backward and forward sliding of the plates at 130°C for 4h under nitrogen atmosphere. 

Carbon tetrachloride at 1 atm and 20°C is sheared between two long horizontal parallel plates, 0.5 mm apart, with the bottom plate fixed and the top plate moving at a constant velocity v. Each plate is 2 m long and 0.8 m wide. The strain rate (or velocity gradient) is 5000 s . Calculate the shear stress if the viscosity and density of the carbon tetrachloride are 0.97 cP and 1590kg/m, respectively. 

It has been shown that when a fluid is sheared with a shear stress t, its strain rate (or deformation rate) is proportional (for most fluids) to the shear stress. The proportionality constant is termed the fluid viscosity n. For a fluid sheared between two long parallel plates, the local velocity (at any height y) varies from zero at the fixed plate to v at the upper moving plate. As described above, the derivative of the local velocity (m) with respect to the height y (i.e., duldy) is termed the velocity gradient, strain rate, or deformation rate. The shear stress is related to duldy by the equation... 

We have already discussed confinement effects in the channel flow of colloidal glasses. Such effects are also seen in hard-sphere colloidal crystals sheared between parallel plates. Cohen et al. [103] found that when the plate separation was smaller than 11 particle diameters, commensurability effects became dominant, with the emergence of new crystalline orderings. In particular, the colloids organise into z-buckled" layers which show up in xy slices as one, two or three particle strips separated by fluid bands see Fig. 15. By comparing osmotic pressure and viscous stresses in the different particle configurations, tlie cross-over from buckled to non-buckled states could be accurately predicted. 

Hydrated peptide-lipid mixtures can be manually aligned between two parallel, flat surfaces. The procedures are described in references 5 and 6. For OCD and neutron diffraction experiments, we used two silica plates. For X-ray diffraction experiments, we used a polished beryllium (Be) plate and a silica plate. The sample thickness (1-80 fxm) was controlled by a spacer between two plates. A circular hole was made in the spacer to provide a cavity to hold the sample. The thicker the sample, the smaller the area of monodomain region will be. The desirable radius of the cavity is about 8 mm for thick ( 80-pm) samples. A sample was aligned between two plates by hand using the procedure of shearing and compression-dilation first described in reference 5. 

Wc shall limit discussion to the simplest case of shear viscosity, illustrated in Figure 8.1(a). Fluid is contained between two plates parallel to the ry-plane and a distance h apart along the. c-axis. The plates are maintained in relative motion at a constant velocity V in the x-direction by a shearing force F applied to the upper plate and a force — F to the lower plate. 

Consider the flow shown in Fig. 1, where a fluid between two plates is sheared as the top plate moves with velocity E in the x direction. The... 

It is interesting to see what we can learn from the reptation model. The best way to see it is to look at a simple experiment. Take a polymer melt or a concentrated solution and place it between two plates, similar to the geometry shown in Figure 12.1 (in practice it can also be a gap between two co-axial cylinders). At time t = 0, apply a constant shear stress a, and measure the relative deformation, or strain 7, as it develops in time after the stress is switched on at f = 0. If a is small, the deformation will be proportional to the stress ... 

The sliding plate viscometer, also known as sliding plate micro-viscometer (Figure 4.4a), measures absolute or dynamic viscosity. The apparatus comprises a loading system that applies a shear stress and a recording system of flow as a function of time. The bitumen sample is placed between two plates so as to create a very thin film of 5-50 pm. The apparatus can measure the viscosity only in the range of 10 to 10 Pa s thus, it is not suitable for low-viscosity measurements. 

The dynamic viscosity can be illustrated with the Couette flow (Figure 3.17). A fluid is located between two plates at the distance H. If a shearing force F is applied, a velocity gradient in the fluid is built up. The maximum velocity will occur at the point where the stress is applied, whereas the velocity is zero at the opposite side due to the wall adherence. For a Newtonian fluid, the velocity gradient is constant across the distance between the two plates. With A as the surface area of the upper plate, the shear stress r is defined as... 

We have discussed frictional properties within the framework of a model of a sheared monolayer embedded between two plates. The model accounts for the coupled lateral and normal motions in the system, and includes the relevant parameters... 

Now let us consider the mechanical experiment shown schematically in Figure 9.1. A material, huid or solid, is placed between two plates that are separated by a distance H. One plate is oscillated sinusoidally relative to the other, with amplitude L and frequency co. We measure the shear stress r on the upper plate as a function... 

If a liquid crystal is sheared between two infinite plates parallel to the xz-plane, a constant velocity gradient parallel to the y-axis results. Assuming stationary flow the stress component a y can be calculated from... 

The SmC phase is sheared between two parallel plates of infinite dimension. The distance between the plates is so large that the influence of the surface alignment at the plates can be neglected in the bulk. The orientation of the directors, the velocity and the velocity gradient is the same as already discussed for the shear flow experiment (Fig. 18). Furthermore, a rotation of the director c is now allowed. Calculation of the torque according to Eq. (104) and F =0 gives... 

From Table 5.1, the viscosity of mercury is approximately 2.3 x lO" Reyn. If mercury is sheared between two flat plates as in Fig. 5.1 that are 0.005 in. apart and = 100 fpm, estimate the resisting force on the upper... 

Figure 1.1.3 shows the results of a different kind of experiment on a similar rubber sample. Here the sample is sheared between two parallel plates maintained at the same separation X2. We see diat the shear stress is linear with the strain over quite a wide range howevo additional stress components, notmal stresses Ti i and T22, act on the block at large strain. In the introduction to this part of the text, we saw that elastic liquids can also generate normal stresses (Figure 1.3). In rubber, the normal stress difference depends on the shear strain squared... 

When a viscoelastic material is sheared between two parallel surfaces at an appreciable rate of shear, in addition to the viscous shear stress T 2, there are normal stress differences Wi s Tn - 722 and N2 s 722 - T23. Here 1 is the flow direction, 2 is perpendicular to the surfaces between which the fluid is sheared, as defined by eq 1.4.8, and 3 is the neutral direction. The largest of the two normal stress differences is N, and it is responsible for the rod climbing phenomenon mentioned at the beginning of this book. For isotropic materials, Ni has always been found to be positive in sign (unless it is zero). In a cone and plate rheometer this means that the cone and plate surfaces tend to be pushed q>art. N2 is usually found to be negative and smaller in magnitude than Ni typically the ratio —N2/N1 lies between 0.05 and 0.3 (Keentok et al., 1980 Ramachandran et al., 1985). Figure 4.2.1 shows the... 

It is well known from elementary fluid dynamics that an isotropic fluid with constant viscosity rj undergoing a shear between two parallel plates, separated by the distance d = 2h as described above in Fig. 5.6, is characterised by (cf. Landau and Lifshitz [159, p.55j)... 

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