INTRODUCTION
3.6. DRAG FORCES AND SETTLING VELOCITIES FOR PARTICLES IN NON-NEWTONIAN FLUIDSPARTICLES IN NON-NEWTONIAN FLUIDS
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-thinningpower- law fluids and in fluids exhibiting a yield-stress, both of which are discussed in Volume 1, Chapter 3.
3.6.1. Power-law fluids
Because most shear-thinning fluids, particularly polymer solutions and flocculated suspen- sions, have high apparent viscosities, even relatively coarse particles may have velocities in thecreeping-flowof Stokes’ law regime. CHHABRA(35,36)has proposed that both theoreti- cal and experimental results for the drag force F on an isolated spherical particle of diameterd moving at a velocityumay be expressed as a modified form of Stokes’ law:
F =3π µcduY (3.55)
where the apparent viscosity µc is evaluated at a characteristic shear rateu/d, and Y is a correction factor which is a function of the rheological properties of the fluid. The best available theoretical estimates values of Y for power-law fluids are given in Table 3.9.
Table 3.9. Values ofYfor power-law fluids(35)
n 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
Y 1 1.14 1.24 1.32 1.38 1.42 1.44 1.46 1.41 1.35
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:
2CD =CD=f (Ren, n) (3.56) whereCD andCD are drag coefficients defined by equation 3.4,nis the power-law index andRen is the particle Reynolds number given by:
Ren=(u2−ndnρ)/k (3.57) wherekis the consistency coefficient in the power-law relation. Combining equations 3.55 and 3.56:
CD=(24Ren−1)Y (3.58)
CD =(12Ren−1)Y (3.59)
From Table 3.9 it is seen that, depending on the value of n, the drag on a sphere in a power-law fluid may be up to 46 per cent higher than that in a Newtonian fluid at the same particle Reynolds number. Practical measurements lie in the range 1< Y <1.8, with considerable divergences between the results of the various workers.
In view of the general uncertainty concerning the value ofY, it may be noted that the unmodified Stokes’ law expression gives a acceptable first approximation.
The terminal settling velocityu0 of a particle in the gravitational field is then given by equating the buoyant weight of the particle to the drag force to give:
u0 =
gdn+1(ρs−ρ) 18kY
1/n
(3.60) where (ρs−ρ)is the density difference between the particle and the fluid.
From equation 3.59, it is readily seen that in a shear-thinning fluid(n <1)the terminal velocity is more strongly dependent on d, g andρs−ρ than in a Newtonian fluid and a small change in any of these variables produces a larger change inu0.
Outside the creeping flow regime, experimental results for drag on spheres in power- law fluids have been presented by TRIPATHI et al.(37) and GRAHAM(38) for values of Ren
up to 100, and these are reasonably well correlated by the following expressions with an average error of about 10 per cent:
CD=[(35.2Ren−1.03)2n]+n[1−(20.9Ren−1.11)2n] (3.61a) (0.2< (2−nRen) <24)
CD=[(37Ren−1.1)2n]+[0.36n+0.25] (3.61b) (24< (2−nRen) <100)
CD =[(17.6Ren−1.03)2n]+n[0.5−(10.5Ren−1.11)2n] (3.62a) [0.2< (2−nRen) <24]
CD =[(18.5Ren−1.1)2n]+[0.18n+0.125] (3.62b) [24< (2−nRen) <100]
It may be noted that these two equations do not reduce exactly to the relation for a Newtonian fluid (n=1).
Extensive comparisons of predictions and experimental results for drag on spheres suggest that the influence of non-Newtonian characteristics progressively diminishes as the value of the Reynolds number increases, with inertial effects then becoming dominant, and the standard curve for Newtonian fluids may be used with little error. Experimentally deter- mined values of the drag coefficient for power-law fluids(1< Ren<1000;0.4< n <1) are within 30 per cent of those given by the standard drag curve(37,38).
While equations 3.62a and 3.62b are convenient for estimating the value of the drag coefficient, they need to be re-arranged in order to enable the settling velocity u0 of a sphere of given diameter and density to be calculated, since both CD(CD ) and Ren are functions of the unknown settling velocity. By analogy with the procedure used for Newtonian fluids (equation 3.36), the dimensionless Galileo number Gan which is independent ofu0 may be defined by:
2
3Gan=CD Re[2/(2−n)]=gd[(n+2)/(2−n)](sr−1)(ρ/k)[2/(2−n)] (3.63) where sr is the ratio of the densities of the particle and of the fluid(ρs/ρ).
Equation 3.56 may be written as:
Ren=f (Gan, n) (3.64)
Experimental results comprising about 1000 data points from a large number of sources cover the following range of variables:
1< d <20 (mm); 1190< ρs <16,600(kg/m3); 990< ρ <1190(kg/m3); 0.4< n <1; and 1< Ren <104.
These are satisfactorily correlated by:
Ren=r1(23Gan)r2 (3.65) where r1=0.1{exp[(0.5/n)−0.73n] and r2=(0.954/n)−0.16
Thus, in equation 3.65 onlyRenincludes the terminal falling velocity which may then be calculated for a spherical particle in a power-law fluid.
3.6.2. Fluids with a yield stress
Much less is known about the settling of particles in fluids exhibiting a yield stress.
BARNES(39) suggests that this is partly due to the fact that considerable confusion exists in the literature as to whether or not the fluids used in the experiments do have a true yield stress(39). Irrespective of this uncertainty, which usually arises from the inappro- priateness of the rheological techniques used for their characterisation, many industrially important materials, notably particulate suspensions, have rheological properties closely approximating to viscoelastic behaviour.
By virtue of its yield stress, an unsheared viscoelastic material is capable of supporting the immersed weight of a particle for an indefinite period of time, provided that the immersed weight of the particle does not exceed the maximum upward force which can be exerted by virtue of the yield stress of the fluid. The conditions for the static equilibrium of a sphere are now discussed.
Static equilibrium
Many investigators(35)have reported experimental results on the necessary conditions for the static equilibrium of a sphere. The results of all such studies may be represented by a factorZ which is proportional to the ratio of the forces due to the yield stress τY and those due to gravity.
Thus: Z= τY
dg(ρs−ρ) (3.66)
The critical value ofZwhich indicates the point at which the particle starts to settle from rest appears to lie in the range 0.04< Z <0.2.
Drag force
Under conditions where a spherical particle is not completely supported by the forces attributable to the yield stress, it will settle at a velocity such that the total force exerted by the fluid on the particle balances its weight.
For a fluid whose rheological properties may be represented by the Herschel-Bulkley model discussed in Volume 1, Chapter 3, the shear stress τ is a function of the shear rateγ˙ or:
τ =τY+kHB γ˙nHB (3.67) From dimensional considerations, the drag coefficient is a function of the Reynolds number for the flow relative to the particle, the exponent,nHB, and the so-called Bingham number Bi which is proportional to the ratio of the yield stress to the viscous stress attributable to the settling of the sphere. Thus:
CD =R/pu2f(RemHB, nHB, Bi) (3.68)
where: Bi =τY/kHB/mHB(u/d) (3.69)
Using the scant data in the literature and their own experimental results, ATAPATTUet al.(40) suggest the following expression for the drag on a sphere moving through a Herschel- Bulkley fluid in the creeping flow regime:
CD =12ReHB(1+Bi)−1 (3.70) It may be noted that an iterative solution to equation 3.70 is required for the calculation of the unknown settling velocity u0, since this term appears in all three dimensionless groups, ReHB,Bi, andCD , for a given combination of properties of sphere and fluid.
The effect of particle shape on the forces acting when the particle is moving in a shear-thinning fluid has been investigated by TRIPATHIet al.(37), and by VENUMADHAV AND
CHHABRA(41). In addition,, some information is available on the effects of viscoelasticity of the fluid(35).
It is seen therefore that, in the absence of any entirely satisfactory theoretical approach or reliable experimental data, it is necessary to adopt a highly pragmatic approach to the estimation of the drag force on a particle in a non-Newtonian fluid.
3.7. ACCELERATING MOTION OF A PARTICLE IN THE