Confined fluid shear stress

Viscosity is defined as the shear stress per unit area at any point in a confined fluid divided by the velocity gradient in the direc tiou perpendicular to the direction of flow. If this ratio is constant with time at a given temperature and pressure for any species, the fluid is caUed a Newtonian fluid. This section is limited to Newtonian fluids, which include all gases and most uoupolymeric liquids and their mixtures. Most polymers, pastes, slurries, waxy oils, and some silicate esters are examples of uou-Newtouiau fluids. 

An important issue in the thermodynamics of confined fluids concerns their symmetry which is lower than that of a corresponding homogeneous bulk phase because of the presence of the substrate and its inherent atomic structure [52]. The substrate may also be nonplanar (see Sec. IV C) or may consist of more than one chemical species so that it is heterogeneous on a nanoscopic length scale (see Sec. VB 3). The reduced symmetry of the confined phase led us to replace the usual compressional-work term —Pbuik F in the bulk analogue of Eq. (2) by individual stresses and strains. The appearance of shear contributions also reflects the reduced symmetry of confined phases. 

If a confined fluid is thermodynamically open to a bulk reservoir, its exposure to a shear strain generally gives rise to an apparent multiplicity of microstates all compatible with a unique macrostate of the fluid. To illustrate the associated problem, consider the normal stress which can be computed for various substrate separations in grand canonical ensemble Monte Carlo simulations. A typical curve, plotted in Fig. 16, shows the oscillatory decay discussed in Sec. IV A 2. Suppose that instead... 

Viscoelasticity was introduced in Section 11.5. A polymer example may be useful by way of reeapitulalion. Imagine a polymer melt or solution confined in the aperture between two parallel plates to which it adheres. One plate is rotated at a constant rate, while the other is held stationary. Figure 11-3la shows the time dependence of the shear stress after the rotation has been stopped, r decays immediately to zero for an inelastic fluid but the decrease in stress is much more gradual if the material is viscoelastic. In some cases, the residual stresses may... 

Viscosity is defined as the shear stress per nnit area at any point in a confined fluid, divided by the velocity gradient in the direction perpendicular to the direction of flow. The absolute viscosity T is the shear stress at a point, divided by the velocity gradient at that point. The SI unit of viscosity is Pa s [1 kg/(m s)], but the cgs unit of poise (P) [1 g/(cm- s)] is also commonW used. Because many common fluids have viscosities on the order of 0.01 P, the unit of centipoise (cP) is also frequently used (1 cP =1 mPa s). The kinematic viscosity V is defined as the ratio of the absolute viscosity to density at the same temperature and pressure. The SI unit for V is mVs, but again cgs units are very common and v is often given in stokes (St) (1 cmVs) or centi-stokes (cSt) (0.01 cmVs). 

An equally remarkable feature to whidi we shall turn now is the fact that confined fluids may sustain a certain shear stress without exhibiting structural features normally pertaining to solid-like phases that is, they do not necessarily assume any long-range periodic order. We tacitly assumed this from the very beginning of this book in our development of a thermodynamic description of cuiifiiied fluids, wliich closely resembles that appropriate for solid-like bulk phases (see Section 1) (12). 

As a quantitative measure of the extent to which a confined phase is capable of resisting a shear deformation, we introduce in Section 5.6.2 the shear stress Txz. For a fluid bridge a typical shear-stress curve r (aSxo) is plotted in Fig. 5.18. Regardless of the thermodynamic state and the thickness (i.e., s ) of a bridge phase, a typical stress curve exhibits the following features ... 

The behavior and characteristics of confined fluids is more complex than that of bulk liquids or of simple solvated systems described in the chapters mentioned above. One must consider the complexities of the interface between confining walls and the fluid along with confinement-induced phase transitions, critical points, the stratification of the fluid near the confining walls, the idea that confined fluids may sustain certain shear stress without exhibiting structural features normally associated with solid-like phases, and... 

Problem 3-17. Plane Couette Flow Driven by Oscillating Shear Stress. Consider an initially motionless incompressible Newtonian fluid confined between two infinite plane boundaries separated by a gap width d as depicted in the figure. Suppose the plane y = 0 is oscillated back and forth with an oscillatory shear stress r = to sin cot. 

Darcy Equation. The bulk resistance to flow of an incompressible fluid through a solid matrix, as compared to the resistance at and near the surfaces confining this solid matrix, was first measured by Darcy [23]. Since in his experiment the internal surface area (interstitial area) was many orders of magnitude larger than the area of the confining surfaces, the bulk shear stress resistance was dominant. 

Happel further assumed that each cell remains spherical and that the outside surface of the cell is frictionless that is, the shear stress vanishes at the outer boundary of the cell. The disturbance due to any particle is therefore confined to the fluid cell. 

In a dynamic experiment, a small-amplitude oscillatory shear is imposed to a molten polymer confined in the rheometer. The shear stress response of the polymeric system can be expressed as in Equation 22.14. In this equation, G and G" are dynamic moduli related to the elastic storage energy and dissipated energy of the system, respectively. For a viscoelastic fluid, two independent normal stress differences, namely, first and second normal stress differences can be defined. These quantities are calculated in terms of the differences of the components of the stress tensor, as indicated in Equation 22.15a and 22.15b, and can be obtained, for instance, from the radial pressure distribution in a cone-and-plate rheometer [5]. Some other experiments used in the determination of the normal stress differences can be found elsewhere [9, 22] ... 

For such flows, the range of flow rates is such that the variations in normal stresses are inconsequential relative to the shear stresses. Even when the fluid is highly elastic, the flow geometry and confinement may be such that the fluid elasticity has only a minor influence on the streamline orientations. On the other hand, shear-thinning effects upon the flow characteristics and the pressure distribution may appear to be quite important, even for low values of the shear rate. Such flow problems can be solved by considering constitutive equations that are of analogous (although, not necessarily the same) forms as those adopted for Newtonian flows as follows ... 

We now discuss that in the jetting regime radius selection is dominated by simple hydrodynamics [46]. We assume that the shear stress and the velocity difference across the membrane can be neglected. For the analysis of the hydrodynamic flow profile, we can hence focus on the fluid motion inside and outside of the tube. In the experiments shown in Figure 11.10, the outer silicate solution is confined to a glass cylinder of radius J cyi(l-1 cm)- Along its central axis, buoyant cupric sulfate solution ascends as a cyHndrical jet of radius R. The cylindrically symmetric velocity fields v(r), which solve the Navier-Stokes equations, are... 

The basic notion of solution viscosity is illustrated in Fig. 2.7 of a volume of fluid in a shear field (for instance a film of polymer solution confined between parallel plates where one is stationary and the other is moving in the x-direction at a constant velocity v). Assuming no slippage between the liquid and the plate, the force per unit area applied on the volume, the shear stress t, results in a rate of deformation or a strain rate y where... 

The Couette cell is a classical geometry used to study the rheological properties of the complex fluids (see Figure 8.Id). One typically imposes a shear rate and measures the resulting shear stress and velocity profile [23]. In a 3D Couette cell, the granular material is confined between two coaxial cylinders. The wall friction can be controlled by coating the surface of each cylinder with a layer of randomly... 

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