Fluids Newtonian

We now have the tools to extend Newton s law to three dimensions. As we have seen, particularly in Example 2.2.3, D is the proper three-dimensional measure of the rate by which the parts of the fluid are being separated. Thus using t and D we can turn eq. 2.1.1 into a tensor relation 

The factor of 2 arises naturally because B = 2D, as we saw in Example 2.2.4, and also because viscosity is normally defined by the shear relation given in eq. 2.1.1. We can see this by applying the Newtonian constitutive relation to the steady simple shear flow of Example 2.2.2. Substituting eq. 2.2.21 for 2D into 2.3.2 gives 

Recall that y is the shear rate, dv /dx2, eq. 2.2.10. The only normal stress in steady shear of an incompressible Newtonian fluid 

The SI unit of viscosity is the pascal-second (Pa-s). The cgs unit, poised. 1 Pa-s, is also often used. One centipoise, cp=10 Pa-s or ImPa-s is approximately the viscosity of water. Table 2.3.1 shows the tremendous viscosity range for common materials. Very different instruments are required to measure over this range. 

Shear rates of cortunon processes can also cover a very wide range, as indicated in Table 2.3.2. Typically, however, very high shear rate processes are plied to low viscosity fluids. Thus shear stresses do not range as widely. 

If the shear stress versus shear rate plot is a straight line through the origin (or a straight line with a slope of unity on a log-log plot), the fluid is Newtonian  

For simplicity, we describe the equations without mass density p. Since the speciflc stress a (x,t) is given in terms of an Eulerian description, we treat here the simplest response for that description. To satisfy the principle of determinism and the principle of local action mentioned in the previous section, the stress ai(x,t) can be written in terms of v and Vv  

The frame indifference of the stress rt(jc, i) is a natural conclusion of Newtonian mechanics in that the force vector is frame indifferent. Since the stretch tensor D is frame invariant by (2.154), we use D instead of Vv. From (2.27) we have 

The invariants iP are calculated by the following characteristic equation for specifying the eigenvalue 

Let us resolve the stress a and stretch tensor D into direct sums of volumetric and deviatoric components, respectively  

Recalling the orthogonality of the volumetric and deviatoric components, each component will give an independent response  

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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... 

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... 

Numerous examples of polymer flow models based on generalized Newtonian behaviour are found in non-Newtonian fluid mechanics literature. Using experimental evidence the time-independent generalized Newtonian fluids are divided into three groups. These are Bingham plastics, pseudoplastic fluids and dilatant fluids. 

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). 

Figure 1.2 Comparison of the rheological behaviour of Newtonian and typical generalized Newtonian fluids...

Typical rheograms representing the behaviour of various types of generalized Newtonian fluids are shown in Figure 1.2. 

Herschel, W.H. and Bulkley, R., 1927. See Rudraiah, N, and Kaloni, P.N. 1990. Flow of non-Newtonian fluids. In Encyclopaedia of Fluid Mechanics, Vol. 9, Chapter 1, Gulf Publishers, Houston. 

Johnson, M. W. and Segalman, D., 1977. A model for viscoelastic fluid behaviour which allows non-affine deformation. J. Non-Newtonian Fluid Mech. 2, 255-270. 

Pearson,. I.R.A., 1994. Report on University of Wales Institute of Non-Newtonian Fluid Mechanics Mini Symposium on Continuum and Microstructural Modelling in Computational Rheology. /. Non-Newtonian Fluid Mech. 55, 203 -205. 

Townsend, P. and Webster, M. I- ., 1987. An algorithm for the three dimensional transient simulation of non-Newtonian fluid flow. In Pande, G. N. and Middleton, J. (eds). Transient Dynamic Analysis and Constitutive Laws for Engineering Materials Vul. 2, T12, Nijhoff-Holland, Swansea, pp. 1-11. 

Luo, X. L, and Tanner, R. L, 1989. A decoupled finite element streamline-upwind scheme for viscoelastic flow problems. J. Non-Newtonian Fluid Mech. 31, 143-162. 

Papaiiastasiou, T. C., Scriven, L. E. and Macoski, C. W., 1987. A finite element method for liquid with memory. J. Non-Newtonian Fluid Mech 22, 271-288. 

MODELLING OF STEADY STATE STOKES FLOW OF A GENERALIZED NEWTONIAN FLUID... 

Similarly in the absence of body forces the Stokes flow equations for a generalized Newtonian fluid in a two-dimensional (r, 8) coordinate system are written as... 

Similarly the components of the equation of motion for an axisymmetric Stokes flow of a generalized Newtonian fluid are written as... 

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