Viscoplastic fluid

Despite the fact that equation (3.37) is applicable to all kinds of time-independent fluids, numerous workers have presented expressions for turbulent flow friction factors for specific fluid models. For instance, Tomita [1959] applied the concept of the Prandtl mixing length and put forward modified definitions of the friction factor and Reynolds number for the turbulent flow of Bingham Plastic fluids in smooth pipes so that the Nikuradse equation, i.e. equation (3.37) with n = 1, could be used. Though he tested the applicability of his method using his own data in the range 2000 Reg(l — 4 f 3 — / ) 10, the validity of this approach has not been established using independent experimental data. 

In contrast, the semi-empirical equations due to Darby et al. (1992) obviate these difficulties due to their explicit form which is as follows  

This method has been shown to yield satisfactory values of pressure drop under turbulent conditions for D 335 mm, Reg 3.4 x 10 and 1000 He 6.6 X 10. Example 3.7 illustrates the application of this method. 

A 18% iron oxide slurry (density 1170kg/m ) behaves as a Bingham plastic fluid with Tq = 0.78 Pa and yu-s = 4.5 mPa s. Estimate the wall shear stress as a function of the nominal wall shear rate (8F/D) in the range 0.4 V 1.75 m/s for flow in a 79 mm diameter pipeline. 

Over the range of the wall shear stress encountered, the slurry can also be modelled as a power-law fluid with m = 0.16Pa-s and n = 0.48. Contrast the predictions of the two rheological models. 

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Numerical calculations of helical flow in non-linear viscoplastic fluid have been... 

The results of the latest research into helical flow of viscoplastic fluids (media characterized by ultimate stress or yield point ) have been systematized and reported most comprehensively in a recent preprint by Z. P. Schulman, V. N. Zad-vornyh, A. I. Litvinov 15). The authors have obtained a closed system of equations independent of a specific type of rheological model of the viscoplastic medium. The equations are represented in a criterion form and permit the calculation of the required characteristics of the helical flow of a specific fluid. For example, calculations have been performed with respect to generalized Schulman s model16) which represents adequately the behavior of various paint compoditions, drilling fluids, pulps, food masses, cement and clay suspensions and a number of other non-Newtonian media characterized by both pseudoplastic and dilatant properties. 

Schulman ZP, Zadvornyh VN, Litvinov AI (1987) Rheodynamics of nonlinearly viscoplastic fluids in circular channels with movable walls. Acad, of Sc. Bel. SSR, Minsk, Preprint 45 51 c... 

In the case of fluids with yield stress, viscoplastic fluids differ from elastoplastic fluids. With the application of a shear stress o above the yield strength o0. Bingham fluids show a linear dependence of shear stress on shear rate, whereas Casson and Herschel-Bulkley fluids show a nonlinear dependence on these parameters. 

The best activity was observed for PCMEDDAC dissolved in n-hexane. Initial waxy crude oil behaves hke a viscoplastic fluid. Doped by PCMEDDAC, waxy oil approaches a Newtonian liquid, and the shear stress decreases considerably due to the modification of the paraffin crystals by the hydropho-bized macromolecules, hi oily environments, PCMEDDAC forms micelles consisting of a hydrophihc core (made of the betaine groups) and a hydrophobic corona (made of the dodecyl groups). The PPD mechanism of PCMEDDAC with respect to waxy crude oil suggests the adsorption of definite fractions of paraffin molecules on the surface of micelles and further retardation of agglomeration. 

The motion of plastic fluids with finite yield stress to has some qualitative specific features not possessed by nonlinearly viscous fluids. Let us consider a layer of a viscoplastic fluid on an inclined plane whose slope is gradually varied. It follows from (6.2.5) that, irrespective of the rheological properties of the medium, the tangential stress decreases across the film from its maximum value Tjnax = pgh sin a on the solid wall to zero on the free surface. Therefore, a flow in a film of a viscoplastic fluid can be initiated only when the tangential stress on the wall becomes equal to or larger than the yield stress to ... 

The mean flow rate velocity of the film flow of a viscoplastic fluid is given by the formula... 

Following [47, 443, 444], let us consider absorption of weakly soluble gases on the surface of a fluid film flowing down an inclined plane. The steady-state velocity distribution inside the film is given by (6.2.8) for nonlinearly viscous fluids and by (6.2.17) for viscoplastic fluids. 

For nonlinearly viscous and viscoplastic fluids, the maximum velocity (7max in formula (6.3.3) can be calculated in the general case by using (6.2.9) and... 

General formulas. The dependence of the shear rate on the stress in flows of viscoplastic fluids in circular tubes can be represented as follows ... 

Up to the different notation (APjL -> pg sin a), formula (6.4.15) coincides with the expression (6.2.5) for shear stresses, which was obtained earlier for film flows. Therefore, we can calculate the velocity profile V in a plane channel (in the region 0 < < h), the maximum velocity f/max, and the mean flow rate velocity (V) for nonlinear viscous fluids by formulas (6.2.8)—(6.2.11) and for viscoplastic fluids by formulas (6.2.17)-(6.2.19) if we formally replace pg sin a by AP/L in these formulas. 

Let us consider the case of an arbitrary viscoplastic fluid with yield stress To (similar results for nonlinear viscous fluids correspond to to = 0). To obtain the temperature profile, we proceed as follows. First, in the near-wall shear region 0 < < h- ho, where ho = toL/AP, we solve Eq. (6.5.2) with the boundary conditions (6.5.1). Then in the quasisolid region h - ho 2 < h, we solve Eq. (6.5.2) with - 0 under the boundary condition (6.5.3). Finally, we match the two solutions on the common boundary = ho. This procedure results in the following temperature distribution in the channel ... 

Viscoplastic fluids. In the case of a spherical bubble in a translational Stokes flow of a viscoplastic Shvedov-Bingham fluid with a small yield stress, the following two-term asymptotic expansion is valid for the drag coefficient [37] ... 

The yield stress of viscoplastic fluids may be estimated by observing the motion/no motion of a sphere. For example, the yield stress for carbopol solutions was evaluated in [182]. 

The key point in the rheological classification of substances is the question as to whether the substance has a preferred shape or a natural state or not [19]. If the answer is yes, then this substance is said to be solid-shaped otherwise it is referred to as fluid-shaped [508]. The simplest model of a viscoelastic solid-shaped substance is the Kelvin body [396] or the Voigt body [508], which consists of a Hooke and a Newton body connected in parallel. This model describes deformations with time-lag and elastic aftereffects. A classical model of viscoplastic fluid-shaped substance is the Maxwell body [396], which consists of a Hooke and a Newton body connected in series and describes stress relaxation. 

Naidenov, V. I., Non-isothermic instability of flow of viscoplastic fluids in tubes, High Temperature, Vol. 28, No. 3,1990. 

In Chapter 6 we consider problems of hydrodynamics and mass and heat transfer in non-Newtonian fluids and describe the basic models for Theologically complicated fluids, which are used in chemical technology. Namely, we consider the motion and mass exchange of power-law and viscoplastic fluids through tubes, channels, and films. The flow past particles, drops, and bubbles in non-Newtonian fluid are also analyzed. 

The Bingham model for viscoplastic fluids (Eq. (101, see next section) is obtained when q T), (intermediate to high shear rates), the term (Xif) I and p is equal to 1/2. As a result. 

The flow behavior of a vtscoplaslir fluid is identified by the. appearance of a yield Stress, i.e., the fluid flows in a viscous manner only after a threshold ha.s been exceeded. Below this threshold, or yield stress, the behavior of the fluid is similar to an elastic solid and should obey Eq. [4) when subjected to a strain or stress sweep. The simplest type of viscoplastic fluid is the so-called Bingham plastic, and its behavior can be expressed by means of the following mathematical model ... 

An additional well-known mathematical expression for viscoplastic fluids is the Casson model ... 

Over the years, many empirical expressions have been proposed as a result of straightforward cmve fltting exercises. A model based on sound theory is yet to emerge. Three commonly used models for viscoplastic fluids are briefly described here. 

In a liquid flowing down a surface, a velocity distribution wUl be established with the velocity increasing from zero at the surface itself (y = 0) to a maximum at the free surface (y = H). For viscoplastic fluids, it can be expected that plug-like motion may occur near the free surface. The velocity distribution in the film can be obtained in a manner similar to that used previously for pipe flow, bearing in mind that the driving force here is that due to gravity rather than a pressure gradient which is absent everywhere in the film. 

Because of the no-slip boimdary condition at both solid walls, i.e. air = oR and r = / , the velocity must be maximum at some intermediate point, say at r = XR. Then, for a fluid without a yield stress, the shear stress must be zero at this position and for a viscoplastic fluid, there will be a plug moving en masse. Equation (3.76) can therefore be re-written ... 

The shear stress and pressiue distributions necessary for the evaluation of the integrals implicit in equation (5.1) can, in principle, be obtained by solving the continuity and momentum equations. In practice, however, numerical solutions are often necessary even at low Reynolds numbers. Since detailed discussions of this subject are available elsewhere, [Chhabra, 1993a] oifly the significant results are presented here for power-law and viscoplastic fluid models. 

The flow of viscoplastic fluids through beds of particles has not been studied as extensively as that of power-law fluids. However, since the expressions for the average shear stress and the nominal shear rate at the wall, equations (5.41) and (5.42), are independent of fluid model, they may be used in conjimction with any time-independent behaviour fluid model, as illuslrated here for the streamline flow of Bingham plastic fluids. The mean velocity for a Bingham plastic fluid in a circular tube is given by equation (3.13) ... 

Lawal, A., Railkar, S., Kalyon, D. M., Mathematical Modeling of Three-Dimensional Die Flows of Viscoplastic Fluids with Wall SUp, SPE-ANTEC (1999)... 

Housiadas KD, Tanner RI (2009) On the rheology of a dilute suspension of rigid spheres in a weakly viscoelastic matrix fluid. J Non-Newtonian Huid Mech 2009 162 88-92 Hughes TJR, Liu WK, Zimmermann TK (1981) Lagrangian-Eulerian finite element formulation for incompressible viscous flows. Comput Methods Appl Mech Eng 29 329-349 Huilgol RR (2006) On the derivation of the symmetric and asymmetric Hele-Shaw flow equations for viscous and viscoplastic fluids using the viscometric fluidity function. J Non-Newtonian Fluid Mech 138 209-213... 

Mendoza R, R gnier G, Seiler W, Lebrun JL (2003) Spatial distribution of molecular (mentation in injection molded iPP influence of processing conditions. Polym 44 3363-3373 Mitsoulis E (2010) Fountain flow of pseudoplastic and viscoplastic fluids. J Non-Newtonian Fluid Mech 165 45-55... 

Adachi, K. and N. Yoshioka, On creeping flow of a viscoplastic fluid past a circular cylinder, Chem. Eng. Sci. 25 215 (1973). 

Chhabra, R. P. and P. H. T. Uhlherr, Static equilibrium and motion of spheres in viscoplastic fluids. Encyclopedia of Fluid Mechanics, Vol. 7, (N. P. Cheremisinoff, ed.). Gulf, Houston, 1988, Chap. 21. 

The rheology of yield-stress (or viscoplastic) fluids is complex and often time dependent. Considerable insight can be gained, however, by considering the simplest example, the Bingham material. The classical Bingham material is defined for a shear flow with a positive shear rate as... 

Current practice for laminar flow through various fittings is to present the loss coefficient as a function of an appropriate Reynolds number. Different Reynolds numbers developed for non-Newtonian fluids have been evaluated to determine their ability to establish the necessary requirement of dynamic similarity for flow of viscoplastic fluids in various fittings. 

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