Stress Newtonian fluid

In the case of fluids without yield stress, viscous and viscoelastic fluids can be distinguished. The properties of viscoelastic fluids lie between those of elastic solids and those of Newtonian fluids. There are some viscous fluids whose viscosity does not change in relation to the stress (Newtonian fluids) and some whose shear viscosity T] depends on the shear rate y (non-Newtonian fluids). If the viscosity increases when a deformation is imposed, we define the material as a shear-thickening (dilatant) fluid. If viscosity decreases, we define it as a shear-thinning fluid. 

Non-Newtonian flow processes play a key role in many types of polymer engineering operations. Hence, formulation of mathematical models for these processes can be based on the equations of non-Newtonian fluid mechanics. The general equations of non-Newtonian fluid mechanics provide expressions in terms of velocity, pressure, stress, rate of strain and temperature in a flow domain. These equations are derived on the basis of physical laws and... 

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... 

Bingham plastics are fluids which remain rigid under the application of shear stresses less than a yield stress, Ty, but flow like a. simple Newtonian fluid once the applied shear exceeds this value. Different constitutive models representing this type of fluids were developed by Herschel and Bulkley (1926), Oldroyd (1947) and Casson (1959). 

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... 

For a generalized Newtonian fluid the components of the extra stress and... 

Note that convected derivatives of the stress (and rate of strain) tensors appearing in the rheological relationships derived for non-Newtonian fluids will have different forms depending on whether covariant or contravariant components of these tensors are used. For example, the convected time derivatives of covariant and contravariant stress tensors are expressed as... 

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) ... 

The shear stress is hnear with radius. This result is quite general, applying to any axisymmetric fuUy developed flow, laminar or turbulent. If the relationship between the shear stress and the velocity gradient is known, equation 50 can be used to obtain the relationship between velocity and pressure drop. Thus, for laminar flow of a Newtonian fluid, one obtains ... 

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... 

Capillary viscometers are useful for measuring precise viscosities of a large number of fluids, ranging from dilute polymer solutions to polymer melts. Shear rates vary widely and depend on the instmments and the Hquid being studied. The shear rate at the capillary wall for a Newtonian fluid may be calculated from equation 18, where Q is the volumetric flow rate and r the radius of the capillary the shear stress at the wall is = r Ap/2L. 

Gla.ss Ca.pilla.ry Viscometers. The glass capillary viscometer is widely used to measure the viscosity of Newtonian fluids. The driving force is usually the hydrostatic head of the test Hquid. Kinematic viscosity is measured directly, and most of the viscometers are limited to low viscosity fluids, ca 0.4—16,000 mm /s. However, external pressure can be appHed to many glass viscometers to increase the range of measurement and enable the study of non-Newtonian behavior. Glass capillary viscometers are low shear stress instmments 1—15 Pa or 10—150 dyn/cm if operated by gravity only. The rate of shear can be as high as 20,000 based on a 200—800 s efflux time. 

In most rotational viscometers the rate of shear varies with the distance from a wall or the axis of rotation. However, in a cone—plate viscometer the rate of shear across the conical gap is essentially constant because the linear velocity and the gap between the cone and the plate both increase with increasing distance from the axis. No tedious correction calculations are required for non-Newtonian fluids. The relevant equations for viscosity, shear stress, and shear rate at small angles a of Newtonian fluids are equations 29, 30, and 31, respectively, where M is the torque, R the radius of the cone, v the linear velocity, and rthe distance from the axis. 

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. 

Non-Newtonian fluids include those for which a finite stress 1,. is reqjiired before continuous deformation occurs these are c ailed yield-stress materials. The Bingbam plastic fluid is the simplest yield-stress material its rheogram has a constant slope [L, called the infinite shear viscosity. 

The stress has an isotropic contribution due to fluid pressure and dilatation, and a deviatoric contribution due to viscous deformation effects. The deviatoric contribution for a Newtonian fluid is the three-dimensional generalization of Eq. (6-2) ... 

In a perfectly viscous (Newtonian) fluid the shear stress, t is directly proportional to the rate of strain (dy/dt or y) and the relationship may be written as... 

Now for a Newtonian fluid, the shear stress, Xy, is related to the viscosity, t], and the shear rate, y, by the equation... 

In a fluid under stress, the ratio of the shear stress, r. to the rate of strain, y, is called the shear viscosity, rj, and is analogous to the modulus of a solid. In an ideal (Newtonian) fluid the viscosity is a material constant. However, for plastics the viscosity varies depending on the stress, strain rate, temperature etc. A typical relationship between shear stress and shear rate for a plastic is shown in Fig. 5.1. 

For a Newtonian Fluid the shear stress, r, is also given by... 

With most fluids the shearing stress t is linearly proportional to the change of velocity hence viscosity /t is not a function of dv/dy. A fluid having these characteristics is called a Newtonian fluid. 

Non-Newtonian fluids vary significantly in their properties that control flow and pressure loss during flow from the properties of Newtonian fluids. The key factors influencing non-Newtonian fluids are their shear thinning or thickening characteristics and time dependency of viscosity on the stress in the fluid. 

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