Yield-power law fluid

Generalized Bingham or yield-power law fluids are represented by the equation... 

T. Lin and V. L. Shah, Numerical Solution of Heat Transfer to Yield-Power-Law Fluids Flowing in the Entrance Region, 6th Int. Heat Transfer Conf, Toronto, vol. 5, p. 317,1978. 

For Newtonian fluids the dynamic viscosity is constant (Equation 2-57), for power-law fluids the dynamic viscosity varies with shear rate (Equation 2-58), and for Bingham plastic fluids flow occurs only after some minimum shear stress, called the yield stress, is imposed (Equation 2-59). 

Only a very limited amount of data is available on the motion of particles in non-Newtonian fluids and the following discussion is restricted to their behaviour in shear-thinning power-law fluids and in fluids exhibiting a yield-stress, both of which are discussed in Volume 1, Chapter 3. 

Several expressions of varying forms and complexity have been proposed(35,36) for the prediction of the drag on a sphere moving through a power-law fluid. These are based on a combination of numerical solutions of the equations of motion and extensive experimental results. In the absence of wall effects, dimensional analysis yields the following functional relationship between the variables for the interaction between a single isolated particle and a fluid ... 

The same statement can be made about inelastic non-Newtonian fluids, such as the Power Law fluid, from a mathematical solution point of view. In reality, most non-Newtonian fluids are viscoelastic and exhibit normal stresses. For fluids such as those (i.e., fluids described by constitutive equations that predict normal stresses for viscometric flows), theoretical analyses have shown that secondary flows are created inside channels of nonuniform cross section (78,79). Specifically it can be shown that a zero second normal stress difference is a necessary (but not sufficient) condition to ensure the absence of secondary flow (79). Of course, the analyses of flows in noncircular channels in terms of constitutive equations—which, strictly speaking, hold only for viscometric flows—are expected to yield qualitative results only. Experimentally low Reynolds number flows in noncircular channels have not been investigated extensively. In particular, only a few studies have been conducted with fluids exhibiting normal stresses (80,81). Secondary flows, such as vortices in rectangular channels, have been observed using dyes in dilute aqueous solutions of polyacrylamide. Interestingly, these secondary flow vortices (if they exist) seem to have very little effect on the flow rate. 

In this case, the system does not show a yield value rather, it shows a limiting viscosity ri 6) at low shear rates (that is referred to as residual or zero shear viscosity). The flow curve can be fitted to a power law fluid model (Ostwald de Waele)... 

From viscometer measurements, the fluid was found to have a so-called yield-power law behavior with an apparent viscosity of the form... 

The values of the constants were measured to be Tq = 10 Pa, n = 0.630, and m = 0.167 Pa s . This relation is seen to be a combination of the Bingham plastic and power law behavior and is found to fit the measurements to within an accuracy of 1—2%. In Fig. 9.1.3 we have drawn in the velocity distribution for a Bingham plastic fluid using the measured value of the yield stress and a measured value of rja = 0.57 from which we calculate G = 1.4x10 Pam", Mp = 2.28 X 10 Pa s, and 3, = 1.84 m s The agreement between the theory and measurement, although not as excellent as for the yield-power law behavior, is nevertheless seen to be quite good and shows clearly the nature of the non-Newtonian behavior associated with the flow of a colloidal suspension. 

The values calculated from equations (5.13) to (5.15) represent about 400 data points in visco-inelastic fluids (0.4 < n < 1 1 < Re < 1000 10 < Ar < 10 ) with an average error of 14% and a maximiun error of 21%. Finally, in view of the fact that non-Newtonian characteristics exert little influence on the drag, the use of predictive correlations for terminal falling velocities in Newtonian media yields only marginally larger errors for power-law fluids. Finally, attention is drawn to the fact that the estimation of terminal velocity in viscoplastic liquids requires an iterative solution, as illustrated in example 5.4. 

Consider the differential control volume shown in Figure 6.3. The velocity profile is assumed to be fully developed in the direction of flow, i.e. V r). Furthermore, all physical properties including m and n for a power-law fluid and plastic viscosity and yield stress for a Bingham plastic fluid, are assumed to be independent of temperature. 

This expression is for a power law fluid and is valid if the effect of the screw rotation on the melt viscosity can be neglected. In reality, however, the effective viscosity in the clearance will be reduced as a result of the rotation of the screw. The shear stress is composed of a shear stress in the tangential direction and a shear stress in the axial direction. The tangential shear stress can be determined by evaluating the stress at the center of the channel where the axial shear stress is zero. This yields the following expression for the shear stress in the tangential direction ... 

The power law fluid yield stress is zero, and the fluid is deformed as long as the effect of a small force on the fluid. Particle density is greater than that of the fluid. In addition there is a vertical downward force formed by particle gravity and buoyancy force of the particle fluid. Therefore, particles settle. When the particle diameter is small to a certain extent, it will not overcome the yield stress and get a suspension in the fluid. Then sedimentation does not occur, which is known as natural suspended state. When the fluid stops circulating, it can make the solid phase suspension in the annulus to prevent the deposition of the solid phase at the bottom of the borehole. In this case, accidents can be avoided. Conditions for particles sedimentation is shown as follows ... 

The fluid momentum balance applied to the case of laminar, fully developed flow of a power-law fluid in a horizontal pipe of diameter D yields the following expression for the relationship between the pressure drop, AP/L, and the average flow velocity, vav-... 

The behaviour of slurries which exhibit a yield stress can be represented by a model in which the relationship between the effective stress t — ty and the shear rate is either linear, as in Newtonian fluids (Bingham plastic model), or follows a power-law, as in pseudoplastic or dilatant fluids (Herschel-Bulkley model or yield power-law model). The shear stress-shear rate relationship for these models is shown in Figure 4.4. 

The axial flow of a power-law fluid through an annulus under only an imposed pressure gradient with both cylinders stationary was first studied by Fredrickson and Bird [7] and is useful in the analysis of pipe-extrusion dies. In this case, V=0 so, Eq. (10) with A->oo yields... 

The flow of a power-law fluid through an annulus for the case of the inner cylinder in axial motion and no imposed pressure gradient is easily solved independently (rather than by reduction of the generalized Couette flow results because the limit as A oo and A=0 is difficult to evaluate). In this case, AP = 0 so, Eqs. (1) and (2) yield rr =constant. The volumetric flow rate is given by Middleman [3] as... 

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