Definition of a Newtonian fluid

Consider a thin layer of a fluid contained between two parallel planes a distance dy apart, as shown in Figine 1.1. Now, if under steady state conditions, the fluid is subjected to a shear by the application of a force F as shown, this will be balanced by an equal and opposite internal frictional force in the fluid. For an incompressible Newtonian fluid in laminar flow, the resulting shear stress is equal to the product of the shear rate and the viseosity of the fluid medium. In this simple case, the shear rate may be expressed as the velocity gradient in the direction perpendicular to that of the shear force, i.e. 

Note that the first subseript on both r and y indieates the direction normal to that of shearing force, while the second subscript refers to the direction of the force and the flow. By eonsidering the equilibrium of a fluid layer, it ean 

The quantity fVf is the momentum in the jc-direction per unit volume of the fluid and hence tyx represents the momentum flu.x in the y-direction and the negative sign indicates that the momentum transfer occms in the direction of decreasing velocity which is also in line with the Fomier s law of heat transfer and Pick s law of diffusive mass transfer. 

Figiue 1.1 and equation (1.1) represent the simplest case wherein the velocity vector which has only one component, in the jc-direction varies only in the y-direction. Such a flow configuration is known as simple shear flow. For the more complex case of three dimensional flow, it is necessary to set up the appropriate partial differential equations. For instance, the more general case of an incompressible Newtonian fluid may be expressed - for the jc-plane - as 

Similar sets of equations can be drawn up for the forces acting on the y- and z-planes in each case, there are two (in-plane) shearing components and a 

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As per the definition of a Newtonian fluid, the shear stress, ct, and the shear rate, y, are proportional to each other, and a single parameter, t], the viscosity. 

Thus, the complete definition of a Newtonian fluid is that it not only possesses a constant viscosity but it also satisfies the condition of equation (1.9), or simply that it satisfies the complete Navier-Stokes equations. Thus, for instance, the so-called constant viscosity Boger fluids [Boger, 1976 Prilutski et al., 1983] which display constant shear viscosity but do not conform to equation (1.9) must be classed as non-Newtonian fluids. 

The emphasis in this chapter is on the viscous behavior of polymeric fluids and in particular their pseudoplastic behavior. The chapter is arranged in the following manner. First, in Section 2.1 we review the definition of a Newtonian fluid, and then we present empiricisms for describing the viscosity of polymeric fluids. In Section 2.2... 

The number of components in the matrix in Eq. 3.5 is reduced for an incompressible Newtonian fluid in shear flow. Referring to Table 2.8, the definition of a Newtonian fluid, and Eq. 3.1 (the kinematics for shear flow) one can show that the stress components are... 

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. 

Rheology is not just about viscosity, but also about another important property, namely the elasticity. Complex fluids also exhibit elastic behaviour. Similar to the viscosity defined above being similar to the definition of a Newtonian viscosity, the elasticity of a complex material can be defined similar to its idealised counterpart, the Hookean solid. The modulus of elasticity is defined as... 

The method of definition of boundary extent of circulation of a liquid in impact-sluggish apparatuses is based on laboratory definition of concentration of slurry above which it loses properties of a Newtonian fluid. This concentration will answer concentration of operating fluid, which cannot be exceeded if it is required to secure with a constant of efficiency of a dust separation. 

As indicated in Section 3.7.9, this definition of ReMR may be used to determine the limit of stable streamline flow. The transition value (R ur)c is approximately the same as for a Newtonian fluid, but there is some evidence that, for moderately shear-thinning fluids, streamline flow may persist to somewhat higher values. Putting n = 1 in equation 3,140 leads to the standard definition of the Reynolds number. 

In the case of non-Newtonian flow, it is necessary to use an appropriate apparent viscosity. Although the apparent viscosity (ia is defined by equation 1.71 in the same way as for a Newtonian fluid, it no longer has the same fundamental significance and other, equally valid, definitions of apparent viscosities may be made. In flow in a pipe, where the shear stress varies with radial location, the value of fxa varies. As pointed out in Example 3.1, it is the conditions near the pipe wall that are most important. The value of /j.a evaluated at the wall is given by... 

Figure H1.1.5 Empirical models that are used to predict the complete flow curve of non-Newtonian fluids or portions of the complete curve. In the full-curve models, K is a constant with time as its dimension and m is a dimensionless constant. See text for definition of other variables in equations.

The dynamic response of a particle in gas-solid flows may be characterized by the settling or terminal velocity at which the drag force balances the gravitational force. The dynamic diameter is thus defined as the diameter of a sphere having the same density and the same terminal velocity as the particle in a fluid of the same density and viscosity. This definition leads to a mathematical expression of the dynamic diameter of a particle in a Newtonian fluid as... 

M. van den Tempel. Surface Rheology, in Journal of Non-Newtonian Fluid Mechanics 2 (1977) 205-19. (Review with a phenomenological emphasis operational definitions and coupling to bulk rheology.)... 

REYNOLDS NXJMBER FOR NON-NEWTONIAN FLUIDS. Since noti-newtonian fluids do not have a single-valued viscosity independent of shear rate, Eq. (3.8) for the Reynolds number cannot be used. The definition of a Reynolds number for such fluids is somewhat arbitrary a widely used definition for power-law fluids is... 

This is the definition of the Reynolds number Nr, given in Eq. (3,9). This Reynolds number reduces to the Reynolds number for a newtonian fluid when n = 1, and it reproduces the linear portion of the logarithmic plot of / versus Nr, with a slope of — 1, for the laminar flow of newtonian fluids. 

Here the absolute value of the velocity gradient is called the shear rate. For a newtonian fluid it is known that in this simple shear flow only the shear stress zxy is nonzero. However, it is possible that all six independent components of the stress tensor may be nonzero for a non-newtonian fluid according to its definition. For simple shearing flow of an isotropic fluid it can be proven [6] that the total stress tensor can have the general form... 

Fig. 7.2 Illustration of the definition to the Newtonian fluid in the shear flow field. / is the shear force, and A is the shear area...

Finally, the shear rate as deflned by Equation 2.36b is clearly the appropriate argument for the viscosity function only for one-dimensional flows like the one used here. We need a quantity that reduces to dvx/dy for the one-dimensional flow but is properly invariant to the way in which we choose to deflne our coordinate system. The appropriate function, which follows directly from the principles of matrix algebra, is one half the second invariant of the rate of deformation, which is usually denoted Ud- Ud is shown in Table 2.6, where it is identical to the dissipation function O divided by r] for the special case of Newtonian fluids. (It is important to keep in mind that the function /) in Table 2.6 is the proper form for the dissipation only for a Newtonian fluid, whereas IId is a universally valid definition that depends only on the velocity field.) For an arbitrary flow field, then, the power-law and Carreau-Yasuda equations would be written, respectively. 

The method of ultrasound Doppler velocimetry (UDV) [9] was proposed to measure the viscosity of non-Newtonian fluids over a wide range of shear rates and in a short period of time. This is a noninvasive, nondisturbing, quick, and accurate procedure. The distribution of shear-stress can be found by pressure drop. At a radial position, the ratio of shear-stress to shear rate, by definition, yields the viscosity at that point. Thus, for the shear rate range in the flow, viscosity values can be obtained by means of only one online experiment. This is a method known in the literature as pointwise rheological measurement [10,11]. 

The definitions various types of non-Newtonian fluids along with exanqiles of common teal systems falling in each category are given in Ihble 2.1. Detailed discussions relating to non-Newtonian fluids are available in a number of books [18-27] as weU as other review articles [28-33]. 

Development of an understanding of turbulence requires consideration of the details of turbulent motion. Much of our intuitive sense of fluid flow is based on what we can observe with the naked eye, and much of this intuitive sense can be applied to an understanding of turbulence, if we proceed with some care. We begin with the classical definition of simple shear flow, as shown in Figure 2-6. In this figure a Newtonian fluid is placed between two flat plates. The top plate moves with velocity Vx, requiring a force per unit area of plate surface (F/A) to maintain the motion. The force required is in proportion to the fluid viscosity. 

Next, determine the viscosity of the chemical slug. For this example, assume the chemical slug is a Newtonian fluid so that variations of viscosity with frontal-advance rate do not have to be considered. That is, the viscosity is a function of composition but does not change with shear rate. The viscosity of the chemical slug is obtained from the definition of mobility. It is first necessary to determine the relative permeability of the chemical slug in the presence of residual oil. The ROS to chemical flooding is 0.2. From... 

The flow of compressible and non-compressible liquids, gases, vapors, suspensions, slurries and many other fluid systems has received sufficient study to allow definite evaluation of conditions for a variety of process situations for Newtonian fluids. For the non-Newtonian fluids, considerable data is available. However, its correlation is not as broad in application, due to the significant influence of physical and rheological properties. This presentation is limited to Newtonian systems, except where noted. 

The usual approach for non-Newtonian fluids is to start with known results for Newtonian fluids and modify them to account for the non-Newtonian properties. For example, the definition of the Reynolds number for a power law fluid can be obtained by replacing the viscosity in the Newtonian definition by an appropriate shear rate dependent viscosity function. If the characteristic shear rate for flow over a sphere is taken to be V/d, for example, then the power law viscosity function becomes... 

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