Newtonian fluids viscosity

In order to select the pipe size, the pressure loss is calculated and velocity limitations are estabHshed. The most important equations for calculation of pressure drop for single-phase (Hquid or vapor) Newtonian fluids (viscosity independent of the rate of shear) are those for the deterrnination of the Reynolds number, and the head loss, (16—18). 

X cross-channel direction for an unwound channel system y depth direction for an unwound channel system z downstream direction for an unwound channel system p Newtonian fluid viscosity... 

This unit describes a method for measuring the viscosity (r ) of Newtonian fluids. For a Newtonian fluid, viscosity is a constant at a given temperature and pressure, as defined in unit hi. i common liquids under ordinary circumstances behave in this way. Examples include pure fluids and solutions. Liquids which have suspended matter of sufficient size and concentration may deviate from Newtonian behavior. Examples of liquids exhibiting non-Newtonian behavior (unit hi. i) include polymer suspensions, emulsions, and fruit juices. Glass capillary viscometers are useful for the measurement of fluids, with the appropriate choice of capillary dimensions, for Newtonian fluids of viscosity up to 10 Pascals (Newtons m/sec 2) or 100 Poise (dynes cm/sec 2). Traditionally, these viscometers have been used in the oil industry. However, they have been adapted for use in the food industry and are commonly used for molecular weight prediction of food polymers in very dilute solutions (Daubert and Foegeding, 1998). There are three common types of capillary viscometers including Ubelohde, Ostwald, and Cannon-Fenske. These viscometers are often referred to as U-tube viscometers because they resemble the letter U (see Fig. HI.3.1). 

McCarthy, K. L. and Seymour, J. D. 1993. A AmdamenCal approach for the relationship between the Bostwick measurement and Newtonian fluid viscosity. J. Texture Stud. 24(1) 1-10. 

For non-Newtonian fluids, viscosity data are very important. Every impeller has an average fluid shear rate related to speed. For example, for a flat blade turbine impeller, the average impeller zone fluid shear rate is 11 times the operating speed. The most exact method to obtain the viscosity is by using a standard mixing tank and impeller as a viscosimeter. By measuring the power response on a small scale mixer, the viscosity at shear rates similar to that in the full scale unit is obtained. 

For Newtonian fluids, viscosity depends only on pressure and temperature, whereas for non-Newtonian fluids, viscosity also depends on shear rate. 

According to the model, flux is directly proportional to the applied pressure and inversely proportional to the viscosity. For Newtonian fluids, viscosity is primarily controlled by two factors solids concentration (or feed composition) and temperature. Thus, increasing the temperature or increasing the pressure should increase the flux. This happens only under low pressure, low feed concentration, and high feed velocity. When the process deviates from any... 

Newtonian fluids (viscosity independent of shearing stress) under conditions of laminar flow and constant temperature obey the relationship,... 

Liu, T. J., Wu, P. Y., Wen, S. H., and Yu, T. A., Two novel approaches for reducing flow non-uniformities caused by non-Newtonian fluid viscosities in a coat-hanger die, AIChE National Meeting, Atlanta, Georgia, 1994. 

By studying the variations of the moment with the angular rotational velocity, Q, of the inner cylinder, we can thus plot the rheological curve Tre = Tre Dre), using [7.27] and [7.29]. For a Newtonian fluid, viscosity is simply determined by ... 

Newtonian fluid viscosity cP (centipoise) P (poise) Pas (Pascal second) ... 

Rubber and plastic melts can be considered, to a first approximation, as extremely high-viscosity fluids. This is only an approximation and it must be remembered that polymers generally show viscoelastic properties—a combination of viscous flow and elastic recovery. Viscosity, in turn, is the quantitative measure of resistance to flow under a given set of circumstances. The Greek letter that usually designates viscosity is Tj. For an ideal, Newtonian fluid, viscosity is simply the ratio between Shear Stress (t), the pressure placed on the fluid to create flow, and the Shear Rate (y), the rate of flow over time as seen in Equation 16C.1 ... 

In general terms simple liquids are described as Newtonian, in that they follow Newton s Postulate in the Principia [1687] in that their viscosity [resistance to flow] is a constant value at a given temperature and pressure. Although other types of viscosity can be defined, it is most common to use shear viscosity, where the force is applied in the same direction as flow occurs and layers of fluid slide past one another like playing cards in a deck. Newtonian fluid viscosity [q] can therefore be described as a point value, and only the temperature of measurement need be stated. Materials that obey this rule are simple low molecular weight liquids such as water, organic solvents, glycerol, sucrose... 

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 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... 

In the simplest case of Newtonian fluids (linear Stokesian fluids) the extra stress tensor is expressed, using a constant fluid viscosity p, as... 

Theoretically the apparent viscosity of generalized Newtonian fluids can be found using a simple shear flow (i.e. steady state, one-dimensional, constant shear stress). The rate of deformation tensor in a simple shear flow is given as... 

Incorporation of viscosity variations in non-elastic generalized Newtonian flow models is based on using empirical rheological relationships such as the power law or Carreau equation, described in Chapter 1. In these relationships fluid viscosity is given as a function of shear rate and material parameters. Therefore in the application of finite element schemes to non-Newtonian flow, shear rate at the elemental level should be calculated and used to update the fluid viscosity. The shear rale is defined as the second invariant of the rate of deformation tensor as (Bird et at.., 1977)... 

In generalized Newtonian fluids, before derivation of the final set of the working equations, the extra stress in the expanded equations should be replaced using the components of the rate of strain tensor (note that the viscosity should also be normalized as fj = rj/p). In contrast, in the modelling of viscoelastic fluids, stress components are found at a separate step through the solution of a constitutive equation. This allows the development of a robust Taylor Galerkin/ U-V-P scheme on the basis of the described procedure in which the stress components are all found at time level n. The final working equation of this scheme can be expressed as... 

Let us consider the flow in a narrow gap between two large flat plates, as shown in Figure 5.19, where L is a characteristic length in the a and y directions and h is the characteristic gap height so that /z < L. It is reasonable to assume that in this flow field il c iq, Vy. Tlierefore for an incompressible Newtonian fluid with a constant viscosity of q, components of the equation of motion are reduced (Middleman, 1977), as... 

We start with the governing equations of the Stokes flow of incompressible Newtonian fluids. Using an axisymraetric (r, z) coordinate system the components of the equation of motion are hence obtained by substituting the shear-dependent viscosity in Equations (4.11) with a constant viscosity p, as... 

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). 

For some materials the linear constitutive relation of Newtonian fluids is not accurate. Either stress depends on strain in a more complex way, or variables other than the instantaneous rate of strain must be taken into account. Such fluids are known collectively as non-Newtonian. Many different types of behavior have been observed, ranging from fluids for which the viscosity in the Navier-Stokes equation is a simple function of the shear rate to the so-called viscoelastic fluids, for which the constitutive equation is so different that the normal stresses can cause the fluid to flow in a manner opposite to that predicted for a Newtonian fluid. 

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) ... 

One simple rheological model that is often used to describe the behavior of foams is that of a Bingham plastic. This appHes for flows over length scales sufficiently large that the foam can be reasonably considered as a continuous medium. The Bingham plastic model combines the properties of a yield stress like that of a soHd with the viscous flow of a Hquid. In simple Newtonian fluids, the shear stress T is proportional to the strain rate y, with the constant of proportionaHty being the fluid viscosity. In Bingham plastics, by contrast, the relation between stress and strain rate is r = where is... 

Heat Exchangers Using Non-Newtonian Fluids. Most fluids used in the chemical, pharmaceutical, food, and biomedical industries can be classified as non-Newtonian, ie, the viscosity varies with shear rate at a given temperature. In contrast, Newtonian fluids such as water, air, and glycerin have constant viscosities at a given temperature. Examples of non-Newtonian fluids include molten polymer, aqueous polymer solutions, slurries, coal—water mixture, tomato ketchup, soup, mayonnaise, purees, suspension of small particles, blood, etc. Because non-Newtonian fluids ate nonlinear in nature, these ate seldom amenable to analysis by classical mathematical techniques. 

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