Nonlinearly Viscous Fluids

Nonlinear viscous fluid (non-Newtonian) Linear viscous fluid (Newtonian) Inviscid fluid (Pascalian)... 

Most common fluids of simple structure are Newtonian (i.e., water, air, glycerine, oils, etc.). However, fluids with complex structures (i.e., high polymer melts or solutions, suspensions, emulsions, foams, etc.) are generally non-Newtonian. Examples of non-Newtonian behavior include mud, paint, ink, mayonnaise, shaving cream, polymer melts and solutions, toothpaste, etc. Many two-phase systems (e.g., suspensions, emulsions, foams, etc.) are purely viscous fluids and do not exhibit significant elastic or memory properties. However, many high polymer fluids (e.g., melts and solutions) are viscoelastic and exhibit both elastic (memory) as well as nonlinear viscous (flow) properties. A classification of material behavior is summarized in Table 5.1 (in which the subscripts have been omitted for simplicity). Only purely viscous Newtonian and non-Newtonian fluids are considered here. The properties and flow behavior of viscoelastic fluids are the subject of numerous books and papers (e.g., Darby, 1976 Bird et al., 1987). 

Rigid Solid (Euclidian) Linear Elastic Solid (Hookean) Nonlinear Elastic Solid (Non-Hookean) Visco-Elastic Fluids and Solids (Non-Linear) Nonlinear Viscous Fluid (Non-Newtonian) Linear Viscous Fluid (Newtonian) Inviscid Fluid (Pascalian)... 

Figure 6.1. Typical flow curves for nonlinearly viscous fluids...

Under one-dimensional shear, many Theologically stable fluids of complex structure (whose rheological characteristics are time-independent) have a flow curve other than Newtonian. If the flow curve is curvilinear but still passes through the origin in the plane 7, r, then the corresponding fluids are said to be nonlinearly viscous (often they are said to be purely viscous, anomalously viscous, or sometimes non-Newtonian). 

Nonlinearly viscous fluids are further classified into pseudoplastic fluids, whose flow curve is convex, and dilatant fluids, whose flow curve is concave (both cases are shown by dashed lines in Figure 6.1). 

At present, there exist several dozens of rheological (mostly empirical) models of nonlinear viscous fluids. This is due to the fact that for the vast variety of fluid media of different physical nature, there is no rigorous general theory, similar to the molecular kinetic theory of gases, which would enable one to calculate the characteristics of molecular transport and the mechanical behavior of a medium on the basis of its interior microscopic structure. 

Table 6.1 gives the most widespread rheological models of nonlinearly viscous fluids. Most of these models do not describe all aspects of the actual behavior of nonlinear viscous fluids in the entire range of the shear rate. Instead, they explain only some specific characteristic features of the flow. Table 6.1 contains quasi-Newtonian relations of two types, namely,... 

Nonlinearly viscous fluids. Generalformulas. In the general case of nonlinearly viscous fluids, it is convenient to define the shear rate as a function of the stress ... 

The motion of plastic fluids with finite yield stress to has some qualitative specific features not possessed by nonlinearly viscous fluids. Let us consider a layer of a viscoplastic fluid on an inclined plane whose slope is gradually varied. It follows from (6.2.5) that, irrespective of the rheological properties of the medium, the tangential stress decreases across the film from its maximum value Tjnax = pgh sin a on the solid wall to zero on the free surface. Therefore, a flow in a film of a viscoplastic fluid can be initiated only when the tangential stress on the wall becomes equal to or larger than the yield stress to ... 

Following [47, 443, 444], let us consider absorption of weakly soluble gases on the surface of a fluid film flowing down an inclined plane. The steady-state velocity distribution inside the film is given by (6.2.8) for nonlinearly viscous fluids and by (6.2.17) for viscoplastic fluids. 

For nonlinearly viscous and viscoplastic fluids, the maximum velocity (7max in formula (6.3.3) can be calculated in the general case by using (6.2.9) and... 

Up to the different notation (APjL -> pg sin a), formula (6.4.15) coincides with the expression (6.2.5) for shear stresses, which was obtained earlier for film flows. Therefore, we can calculate the velocity profile V in a plane channel (in the region 0 < < h), the maximum velocity f/max, and the mean flow rate velocity (V) for nonlinear viscous fluids by formulas (6.2.8)—(6.2.11) and for viscoplastic fluids by formulas (6.2.17)-(6.2.19) if we formally replace pg sin a by AP/L in these formulas. 

Let us consider the case of an arbitrary viscoplastic fluid with yield stress To (similar results for nonlinear viscous fluids correspond to to = 0). To obtain the temperature profile, we proceed as follows. First, in the near-wall shear region 0 < < h- ho, where ho = toL/AP, we solve Eq. (6.5.2) with the boundary conditions (6.5.1). Then in the quasisolid region h - ho 2 < h, we solve Eq. (6.5.2) with - 0 under the boundary condition (6.5.3). Finally, we match the two solutions on the common boundary = ho. This procedure results in the following temperature distribution in the channel ... 

For nonlinear viscous fluids, the basic parameters of heat exchange are given by formulas (6.5.9)-(6.5.11) with ro = 0. 

For weakly nonlinearly viscous fluids that obey the viscous friction law t = r(7), the local friction coefficient can be found by using the approximate formulas... 

Pavlov, K. B., Boundary-layer theory in non-Newtonian nonlinearly viscous media, Fluid Dynamics, Vol. 13, No. 3, pp. 360-366, 1978. 

Experimental testing (Seleemah and Constantinou 1997) has shown that the behaviOT of viscous fluid dampers corresponds to a nonlinear viscous dashpot and can be modeled by the following nonlinear force-velocity relation ... 

A knowledge of the viscous and thermal properties of non-Newtonian fluids is essential before the results of the analyses can be used for practical design purposes. Because of the nonlinear nature, the prediction of these properties from kinetic theories is as of this writing in its infancy. Eor the purpose of design and performance calculations, physical properties of non-Newtonian fluids must be measured. 

The convective terms are the ones most responsible for nonlinearity in the fluid-flow conservation equations. As such they are often troublesome both theoretically and practically. There are a few situations of interest where the convective terms are negligible, but they are rare. As a means of exploring the characteristics of the equations, however, it is interesting to consider how the equations would behave if these terms were eliminated. For the purpose of the exercise, assume further that the flow is incompressible, single species, constant property, and without body forces or viscous dissipation. In this case the governing... 

Mqtqi can be considered to be the matrix element, which represents a process in which a viscous mode of wavevector q is annihilated and two thermal (heat) modes of wavevectors q and q — q are created. Thus in this treatment the transport modes are assumed to be coupled to each other nonlinearly, and the disturbances in the fluid can be transmitted back and forth between the various modes. 

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