Newtonian fluid, general linear

To be more precise, the general tensor equation of Newton s law of viscosity should be obeyed by a Newtonian fluid (2) however, for onedimensional flow, the applicability of eq 1 is sufficient. For a Newtonian fluid, a linear plot of t versus 7 gives a straight line whose slope gives the fluid viscosity. Also, a log-log plot of t versus 7 is linear with a slope of unity. Both types of plots are useful in characterizing a Newtonian fluid. For a Newtonian fluid, the viscosity is independent of both t and 7, and it may be a function of temperature, pressure, and composition. Moreover, the viscosity of a Newtonian fluid is not a function of the duration of shear nor of the time lapse between consecutive applications of shear stress (3). 

The term rheology dates back to 1929 (Tanner and Walters 1998) and is used to describe the mechanical response of materials. Polymeric materials generally show a more complex response than classical Newtonian fluids or linear viscoelastic bodies. Nevertheless, the kinematics and the conservation laws are the same for all bodies. The presentation here is condensed one may consult other books for amplification (Bird et al. 1987a Huilgol and Phan-Thien 1997 Tanner 2000). We begin with kinematics. 

Non-Newtonian fluids are generally those for which the viscosity is not constant even at constant temperature and pressure. The viscosity depends on the shear rate or, more accurately, on the previous kinematic history of the fluid. The linear relationship between the shear stress and the shear rate, noted in Equation (1), is no longer sufficient. Strictly speaking, the coefficient of viscosity is meaningful only for Newtonian fluids, in which case it is the slope of a plot of stress versus rate of shear, as shown in Figure 4.2. For non-Newtonian fluids, such a plot is generally nonlinear, so the slope varies from point to point. In actual practice, the data... 

Fluids that show viscosity variations with shear rates are called non-Newtonian fluids. Depending on how the shear stress varies with the shear rate, they are categorized into pseudoplastic, dilatant, and Bingham plastic fluids (Figure 2.2). The viscosity of pseudoplastic fluids decreases with increasing shear rate, whereas dilatant fluids show an increase in viscosity with shear rate. Bingham plastic fluids do not flow until a threshold stress called the yield stress is applied, after which the shear stress increases linearly with the shear rate. In general, the shear stress r can be represented by Equation 2.6 ... 

In the previous section we discussed the nature and some properties of the stress tensor t and the rate of strain tensor y. They are related to each other via a constitutive equation, namely, a generally empirical relationship between the two entities, which depends on the nature and constitution of the fluid being deformed. Clearly, imposing a given stress field on a body of water, on the one hand, and a body of molasses, on the other hand, will yield different rates of strain. The simplest form that these equations assume, as pointed out earlier, is a linear relationship representing a very important class of fluids called Newtonian fluids. 

There are two general types of constitutive equations for fluids Newtonian and non-Newtonian. For Newtonian fluids, the relation between the stress tensor, t, and the rate of deformation tensor or the shear stress is linear. For non-Newtonian fluids the relation between the stress tensor and the rate of deformation tensor is nonlinear. The various Newtonian and non-Newtonian rheologies of fluids are shown in Figure 12.2. There are four types of behavior (1) Newtonian, (2) pseudo-plastic, (3) Bingham plastic, and (4) dilatent. The reasons for these different rheological behaviors will also be discussed in subsequent sections of this chapter. But first it is necessary to relate the stress tensor to the rate of deformation tensor. 

In order to use these general momentum conservation equations to calculate the velocity field, it is necessary to express viscous stress terms in terms of the velocity field. The equations which relate the stress tensor to the motion of the continuous fluid are called constitutive equations or rheological equations of state. Although the governing momentum conservation equations are valid for all fluids, the constitutive equations, in general, vary from one fluid material to another and possibly also from one type of flow to another. Fluids, which follow Newton s law of viscosity (although it is referred to as a law, it is just an empirical proposition) are called Newtonian fluids. For such fluids, the viscous stress at a point is linearly dependent on the rates of strain (deformation) of the fluid. With this assumption, a general deformation law which relates stress tensor and velocity components can be written ... 

Fluids for which this constitutive equation is an adequate model are known as Newtonian fluids. We have shown that the Newtonian fluid model is the most general form that is linear and instantaneous in E and isotropic. If the fluid is also incompressible,... 

Given the apparent arbitrariness of the assumptions in a purely continuum-mechanics-based theory and the desire to obtain results that apply to at least some real fluids, there has been a historical tendency to either relax the Newtonian fluid assumptions one at a time (for example, to seek a constitutive equation that allows quadratic as well as linear dependence on strain rate, but to retain the other assumptions) or to make assumptions of such generality that they must apply to some real materials (for example, we might suppose that stress is a functional over past times of the strain rate, but without specifying any particular form). The former approach tends to produce very specific and reasonable-appearing constitutive models that, unfortunately, do not appear to correspond to any real fluids. The best-known example is the so-called Stokesian fluid. If it is assumed that the stress is a nonlinear function of the strain rate E, but otherwise satisfies the Newtonian fluid assumptions of isotropy and dependence on E only at the same point and at the same moment in time, it can be shown (see, e.g., Leigh29) that the most general form allowed for the constitutive model is... 

Here, a, ft, and 5 are material coefficients that can depend on the thermodynamic state, as well as the invariants of E, namely tr E, det E, and (tr E - tr E2). The Stokesian model appears to be a perfectly obvious generalization of the Newtonian fluid model. However, no real fluid has been found for which the model with 5 f 0 is an adequate approximation. We should perhaps, not be surprised by this result as the examples in the all seem to suggest that the assumptions of isotropy, plus instantaneous and linear dependence on E, all seem to break down at the same time in real, complex fluids. 

Let us choose a Bingham plastic (Eq. 9.1.7) for our generalized Newtonian fluid. From Eq. (9.1.11), the shear stress is seen to vary linearly from 0 at the pipe axis to jGa at the pipe wall, where following convention (Section 2.2), we take the shear stress to be positive. In the pipe core the shear stress t is... 

The Flux Expressions. We begin with the relations between the fluxes and gradients, which serve to define the transport properties. For viscosity the earliest definition was that of Newton (I) in 1687 however about a century and a half elapsed before the most general linear expression for the stress tensor of a Newtonian fluid was developed as a result of the researches by Navier (2), Cauchy (3), Poisson (4), de St. Venant (5), and Stokes (6). For the thermal conductivity of a pure, isotropic material, the linear relationship between heat flux and temperature gradient was proposed by Fourier (7) in 1822. For the difiiisivity in a binary mixture at constant temperature and pressure, the linear relationship between mass flux and concentration gradient was suggested by Pick (8) in 1855, by analogy with thermal conduction. Thus by the mid 1800 s the transport properties in simple systems had been defined. 

Newton s law of motion for liquids describes a linear relationship between the deformation of a fluid and the corresponding stress, as indicated in Equation 22.16, where the constant of proportionality is the Newtonian viscosity of the fluid. The generalized Newtonian fluid (GNF) refers to a family of equations having the structure of Equation 22.16 but written in tensorial form, in which the term corresponding to viscosity can be written as a function of scalar invariants of the stress tensor (x) or the strain rate tensor (y). For the GNF, no elastic effects are taken into account [12, 33] ... 

A non-Newtonian fluid is one whose flow curve (shear stress versus shear rate) is non-linear or does not pass through the origin, i.e. where the apparent viscosity, shear stress divided by shear rate, is not constant at a given temperature and pressure but is dependent on flow conditions such as flow geometry, shear rate, etc. and sometimes even on the kinematic history of the fluid element under eonsideration. Such materials may be conveniently grouped into three general elasses ... 

In general, fluid velocity is given by the Navier-Stokes and continuity equations. For fluids that are Newtonian (shear stress linearly related to fluid shear rate) and incompressible, the Navier-Stokes equation is written as... 

There does not appear to be a general, explicit solution for constant A. Thus, in most cases a numerical scheme has to be used to determine the value of A. A simple solution can be found for the special case of a Newtonian fluid with equal wall temperatures. In this case, A depends only on the Nahme number with the following linear... 

When the viscosity is a function of shear rate, then the relationship between shear stress and shear rate is given by equation (2.9). Since its form is similar to equation (2.36) except for Ae shear rate dependent viscosity, the equation is said to represent a Generalized Newtonian fluid. In such a fluid, the presence of normal stresses defined by equations (2.10) and (2.11) is considered to be negligible for a specific flow situation. In effect, equation (2.5b) represents the constitutive equation for a Generalized Newtonian fluid. The hypothesis of a Generalized Newtonian fluid differs from the simple Newtonian case by the assumption that the functional relationship between the stress tensor and the kinematic variable need not be only linear. It holds, however, the suggestion that only the kinematic variable of the first order can influence the state of stress in the fluid and no attempt is to be made to describe the normal stresses in it. 

Newtonian Flow. Newtonian flow behavior is characterized by a linear relation between the shear stress and shear rate. In this case, the viscosity is independent of the applied shear rate or stress and is a material constant. Materials exhibiting shear-independent viscosity are called newtonian fluids. Newtonian flow behavior is generally exhibited by solvents, dilute polymer solutions, and dispersions. Thick-film pastes rarely show newtonian flow behavior, although the concept is sometimes used in the theoretical analysis of screen printing behavior. 

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