Uniform shear viscosity

However, this expression assumes that the total resistance to flow is due to the shear deformation of the fluid, as in a uniform pipe. In reality the resistance is a result of both shear and stretching (extensional) deformation as the fluid moves through the nonuniform converging-diverging flow cross section within the pores. The stretching resistance is the product of the extension (stretch) rate and the extensional viscosity. The extension rate in porous media is of the same order as the shear rate, and the extensional viscosity for a Newtonian fluid is three times the shear viscosity. Thus, in practice a value of 150-180 instead of 72 is in closer agreement with observations at low Reynolds numbers, i.e.,... 

Here v(r) is the velocity field at any space point r, p is the pressure, po is the shear viscosity of the solvent containing salt ions and counterions, and F(r) is the force arising from any potential field in the solution. As an example, if electrical charges are present in the solution under an externally imposed uniform electrical field E, F(r) is given by... 

Notation, k is the wavenumber, Us the sound velocity, F the damping coefficient (11), r the shear viscosity, po the uniform mass density, k the heat conductivity, and Cp the specific heat capacity at constant pressure [4]. 

Choi, G. R. and Krieger, 1. M. 1986. Rheological studies on sterically stabilized model dispersions of uniform colloidal spheres II. Steady-shear viscosity. J. Colloid Interface Sci. 113 101-113. 

When discussing the morphology it is useful to use the microrheology as a guide. At low stresses in a steady uniform shear flow, the deformation can be expressed by means of three dimensionless parameters the viscosity ratio, the capillarity number, and the reduced time, respectively ... 

Han and Funatsu [1978] studied droplet deformation and breakup for viscoelastic liquid systems in extensional and non-uniform shear flow. The authors found that viscoelastic droplets are more stable than the Newtonian ones in both Newtonian and viscoelastic media they require higher shear stress for breaking. The critical shear rate for droplet breakup was found to depend on the viscosity ratio it was lower for < 1 than for A, > 1. In a steady extensional flow field, the viscoelastic droplets were also found less deformable than the Newtonian ones. In the viscoelastic matrix, elongation led to large deformation of droplets [Chin and Han, 1979]. 

Einstein s result is remarkable, since it says that for uniform shear the relative viscosity does not depend on the size or size distribution of the spheres but only on the volume fraction, provided the solution is very dilute. A physical explanation for this follows from the diluteness criterion, which may be restated as the interparticle distance being large enough that the motion of any particle is unaffected by that of any neighboring particles. As a result, the increased energy dissipation arising from the presence of the particles must be proportional to the particle number density. Therefore the relative viscosity is simply linear in the particle volume fraction. 

Taylor (T2) also treated the other extreme case where e oo, but only for the special case of the very viscous drop —that is, - 0. When e is large the viscous forces greatly exceed the interfacial forces, and Taylor ignores the latter as a first approximation. The shape of the droplet is then only a function of the viscosity ratio. For a simple uniform shear, Taylor finds, on... 

Since Oq is very small (ca. 0.5-3°, or 0.0087-0.0523 radians), sin (jt/2 -So) will close to one, and t wiU be nearly independent of position [see Eq. (59)] that is, the tested material between the gap will experience uniform shear stress. This is the advantage of cone-plate compared to other geometries (i.e., capillary tube and parallel disk). For example, at an angle of 1°, the pereentage difference in shear stress between cone and plate is 0.1218% (Fredrickson, 1964). This is within the precision of measurements that must be made therefore, one can assume that shear stress, and, hence, shear rate and apparent viscosity, are uniform throughout the fluid. 

Uniform fibers were formed at high concentrations, i.e., C/C 10. Dependence of fiber diameter on concentration was determined, i.e., fiber diameter (C/C ]. The fiber diameter was found to vary with the zero shear viscosity of the solutions as, ... 

In contrast, if the surfactant film has a low surface shear viscosity, another type of hydrodynamic instabihty may develop in which the film shape becomes asymmetric and the hquid in the dimple rapidly escapes into the meniscus, leaving a film that is relatively flat. It is imder these circumstances that the uniform film models described above provide a first approximation of the subsequent drainage. Because the dimple disappears quickly, overall drainage time is much faster when asymmetric drainage occurs, as shown by Figure 7.8 for films made with sodiiun dodecyl sulfate (SDS)/dodecanol mixtures. 

Rotational rheometer n. An instrument for measuring the viscosity of molten polymers (any many other fluid types) in which the sample is held at a controlled temperature between a stator and a rotor. From the torque on either element and the relative rotational speed, the viscosity can be inferred. The most satisfactory type for melts is the cone-and-plate geometry, in which the vertex of the cone almost touches the plate and the specimen is situated between the two elements. This provides a uniform shear rate throughout the specimen. It may be operated in steady rotation or in an oscillatory mode. 

It is difficult to measure 7] experimentally. Kiani et al. [17] further assumed that the shear viscosity is uniform within a powder compact so that it is eliminated from Eqn. (7). The virtual power principle can then be written as... 

In steady-state uniform shear flow at low concentrations and stresses, the drop deformation can be expressed using the three dimensionless parameters the capillarity nrunber, k, the viscosity ratio, k, and the reduced deformation time, t [5] ... 

It should be realized that the duration of the experiment is limited to the time taken for the moving end of the sample to traverse the length of the constant-temperature bath so that the entire experiment may be over in as few as four or five seconds. Errors arise if the sample temperature rises or if the sample does not deform uniformly. Also, it is necessary for the end of the sample to go instantly from rest to a finite predetermined velocity at the inception of stretching. In addition, especially for vertical instruments, the density of the supporting liquid has to match that of the polymer to prevent the force of buoyancy from influencing the results. Finally, the method is suitable only for polymers having a zero-shear viscosity in excess of about 1(H Pa sec at the temperature of interest. 

The uniaxial extensiometers described so far are suitable for use with viscous materials only. They cannot, for example, be used to measure the steady extensional viscosity of such commercially important polymers as nylons and polyesters used in the textile industry, and which may have shear viscosities as low as 100 Pa sec at processing temperatures. As a consequence, other techniques are needed but these invariably involve nonuniform stretching. Here one cannot require that the stress or the stretch rate be constant. Also, the material is usually not in a virgin (stress-free) state to begin with. One can therefore not obtain the extensional viscosity directly from these measurements. Nonetheless, data from properly designed non-uniform stretching experiments can be profitably analyzed with the help of rheological constitutive equations. In addition, such data provide a simple measure of resistance that polymeric fluids offer to extensional deformation. 

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