Newtonian fluids defined

The shear viscosity is an important property of a Newtonian fluid, defined in terms of the force required to shear or produce relative motion between parallel planes [97]. An analogous two-dimensional surface shear viscosity ij is defined as follows. If two line elements in a surface (corresponding to two area elements in three dimensions) are to be moved relative to each other with a velocity gradient dvfdx, the required force is... 

Fig. 4.3.3 (a) Shear flow of a Newtonian fluid defined as the ratio of the shear stress and trapped between the two plates (each with a shear rate, (b) A polymeric material is being large area of A). The shear stress (a) is defined stretched at both ends at a speed of v. The as F/A, while the shear rate (y) is the velocity material has an initial length of L0 and an gradient, dvx/dy. The shear viscosity (r s) is (instantaneous) cross-sectional area of A. 

The generalized approach of Metzner and Reed AIChE /., 1, 434 [1955]) for time-independent non-Newtonian fluids defines a modified Reynolds number as... 

Reynolds number, dimensionless A1rc. , modified Reynolds number for non-newtonian fluids, defined by Eq, (5.50)... 

For a solid particle settling in a non-Newtonian fluid, defining... 

The absolute or dynamic viscosity is defined as the ratio of shear resistance to the shear velocity gradient. This ratio is constant for Newtonian fluids. 

The dynamic viscosity, or coefficient of viscosity, 77 of a Newtonian fluid is defined as the force per unit area necessary to maintain a unit velocity gradient at right angles to the direction of flow between two parallel planes a unit distance apart. The SI unit is pascal-second or newton-second per meter squared [N s m ]. The c.g.s. unit of viscosity is the poise [P] 1 cP = 1 mN s m . The dynamic viscosity decreases with the temperature approximately according to the equation log rj = A + BIT. Values of A and B for a large number of liquids are given by Barrer, Trans. Faraday Soc. 39 48 (1943). 

The apparent viscosity, defined as du/dj) drops with increased rate of strain. Dilatant fluids foUow a constitutive relation similar to that for pseudoplastics except that the viscosities increase with increased rate of strain, ie, n > 1 in equation 22. Dilatancy is observed in highly concentrated suspensions of very small particles such as titanium oxide in a sucrose solution. Bingham fluids display a linear stress—strain curve similar to Newtonian fluids, but have a nonzero intercept termed the yield stress (eq. 23) ... 

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. 

All fluids for which the viscosity varies with shear rate are non-Newtonian fluids. For uou-Newtouiau fluids the viscosity, defined as the ratio of shear stress to shear rate, is often called the apparent viscosity to emphasize the distiuc tiou from Newtonian behavior. Purely viscous, time-independent fluids, for which the apparent viscosity may be expressed as a function of shear rate, are called generalized Newtonian fluids. 

Laminar and Turbulent Flow, Reynolds Number These terms refer to two distinct types of flow. In laminar flow, there are smooth streamlines and the fuiid velocity components vary smoothly with position, and with time if the flow is unsteady. The flow described in reference to Fig. 6-1 is laminar. In turbulent flow, there are no smooth streamlines, and the velocity shows chaotic fluctuations in time and space. Velocities in turbulent flow may be reported as the sum of a time-averaged velocity and a velocity fluctuation from the average. For any given flow geometry, a dimensionless Reynolds number may be defined for a Newtonian fluid as Re = LU p/ I where L is a characteristic length. Below a critical value of Re the flow is laminar, while above the critical value a transition to turbulent flow occurs. The geometry-dependent critical Reynolds number is determined experimentally. 

The relation between shear stress and shear rate for the Newtonian fluid is defined by a single parameter /z, the viscosity of the fluid. No single parameter model will describe non-Newtonian behaviour and models involving two or even more parameters only approximate to the characteristics of real fluids, and can be used only over a limited range of shear rates. 

In order to predict Lhe transition point from stable streamline to stable turbulent flow, it is necessary to define a modified Reynolds number, though it is not clear that the same sharp transition in flow regime always occurs. Particular attention will be paid to flow in pipes of circular cross-section, but the methods are applicable to other geometries (annuli, between flat plates, and so on) as in the case of Newtonian fluids, and the methods described earlier for flow between plates, through an annulus or down a surface can be adapted to take account of non-Newtonian characteristics of the fluid. 

Thus, the pipe friction chart for a Newtonian fluid (Figure 3.3) may be used for shearthinning power-law fluids if Remit is used in place of Re. In the turbulent region, the ordinate is equal to (R/pu2)n 0 fn5. For the streamline region the ordinate remains simply R/pu2, because Reme has been defined so that it shall be so (see equation 3.140). More recently, Irvine(25j has proposed an improved form of the modified Blasius equation which predicts the friction factor for inelastic shear-thinning polymer-solutions to within 7 per cent. 

For many materials, the application of a stress creates a strain rate in a linear fashion, i.e., the rate of strain is proportional to the applied stress. This linear relationship, which defines a Newtonian fluid, does not hold true for polymers. Most molten polymers respond to stresses in a non-linear fashion, such that the greater the applied stress the more effective the stress is at inducing a strain rate. This non-Newtonian behavior is referred to as shear thinning ... 

It should be noted that a dimensional analysis of this problem results in one more dimensionless group than for the Newtonian fluid, because there is one more fluid rheological property (e.g., m and n for the power law fluid, versus fi for the Newtonian fluid). However, the parameter n is itself dimensionless and thus constitutes the additional dimensionless group, even though it is integrated into the Reynolds number as it has been defined. Note also that because n is an empirical parameter and can take on any value, the units in expressions for power law fluids can be complex. Thus, the calculations are simplified if a scientific system of dimensional units is used (e.g., SI or cgs), which avoids the necessity of introducing the conversion factor gc. In fact, the evaluation of most dimensionless groups is usually simplified by the use of such units. 

The model for turbulent drag reduction developed by Darby and Chang (1984) and later modified by Darby and Pivsa-Art (1991) shows that for smooth tubes the friction factor versus Reynolds number relationship for Newtonian fluids (e.g., the Colebrook or Churchill equation) may also be used for drag-reducing flows, provided (1) the Reynolds number is defined with respect to the properties (e.g., viscosity) of the Newtonian solvent and (3) the Fanning friction factor is modified as follows ... 

The Hagen-Poiseuille equation [Eq. (6-11)] describes the laminar flow of a Newtonian fluid in a tube. Since a Newtonian fluid is defined by the relation r = fiy, rearrange the Hagen-Poiseuille equation to show that the shear rate at the tube wall for a Newtonian fluid is given by yw = 4Q/nR3 = 8 V/D. 

There are essentially three different approaches to describing hindered settling. One approach is to define a correction factor to the Stokes free settling velocity in an infinite Newtonian fluid (which we will designate F0), as a function of the solids loading. A second approach is to consider the suspending fluid properties (e.g. viscosity and density) to be modified by the... 

The term viscosity has no meaning for a non-Newtonian fluid unless it is related to a particular shear rate y. An apparent viscosity fia can be defined as follows (using the negative sign convention for stress) ... 

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... 

Define rheology, shear force, shear stress, shear rate, Newtonian fluid, dynamic viscosity, centi-poise, kinematic viscosity, centistokes, viscometry, and viscometer. 

Often times concentrated polymeric solutions cannot be treated as Newtonian fluids, however, and this tends to offset the simplifications which result from the creeping flow approximation and the fact that the boundaries are well defined. The complex rheological behavior of polymeric solutions and melts requires that nonlinear constitutive equations, such as Eqs. (l)-(5), be used (White and Metzner, 1963) ... 

The membrane viscometer must use a membrane with a sufficiently well-defined pore so that the flow of isolated polymer molecules in solution can be analyzed as Poiseuille flow in a long capillary, whose length/diameter is j 10. As such the viscosity, T, of a Newtonian fluid can be determined by measuring the pressure drop across a single pore of the membrane, knowing in advance the thickness, L, and cross section. A, of the membrane, the radius of the pore, Rj., the flow rate per pore, Q,, and the number of pores per unit area. N. The viscosity, the maximum shear stress, cr. and the velocity gradient, y, can be calculated from laboratory measurements of the above instrumental parameters where Qj =... 

The viscosity of Newtonian fluids was defined as the ratio of shear stress to rate of shear r/du/dr (the conversion factor gc being temporarily... 

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