Newtonian fluids terminal velocity

The viscosity of a Newtonian fluid can be determined by measuring the terminal velocity of a sphere of known diameter and density if the fluid density is known. If the Reynolds number is low enough for Stokes flow to apply (fVRe < 0.1), then the viscosity can be determined directly by rearrangement of Eq. (11-10) ... 

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 and ps — p than in a Newtonian fluid and a small change in any of these variables produces a larger change in no. 

Boussinesq (B4) proposed that the lack of internal circulation in bubbles and drops is due to an interfacial monolayer which acts as a viscous membrane. A constitutive equation involving two parameters, surface shear viscosity and surface dilational viscosity, in addition to surface tension, was proposed for the interface. This model, commonly called the Newtonian surface fluid model (W2), has been extended by Scriven (S3). Boussinesq obtained an exact solution to the creeping flow equations, analogous to the Hadamard-Rybczinski result but with surface viscosity included. The resulting terminal velocity is... 

This method can be easily used to show the logic behind the scale-up from original R D batches to production-scale batches. Although scale-of agitation analysis has its limitations, especially in mixing of suspension, non-Newtonian fluids, and gas dispersions, similar analysis could be applied to these systems, provided that pertinent system variables were used. These variables may include superficial gas velocity, dimensionless aeration numbers for gas systems, and terminal settling velocity for suspensions. 

The dynamic response of a particle in gas-solid flows may be characterized by the settling or terminal velocity at which the drag force balances the gravitational force. The dynamic diameter is thus defined as the diameter of a sphere having the same density and the same terminal velocity as the particle in a fluid of the same density and viscosity. This definition leads to a mathematical expression of the dynamic diameter of a particle in a Newtonian fluid as... 

A solid sphere of radius sphere and density Psphere falls throngh an incompressible Newtonian fluid which is quiescent far from the sphere. The viscosity and density of the flnid are p-auid and pauid, respectively. The Reynolds number is 50, based on the physical properties of the fluid, the diameter of the sphere, and its terminal velocity. The following scaling law characterizes the terminal velocity of the sphere in terms of geometric parameters and physical properties of the fluid and solid ... 

In shear-thinning power-law fluids, therefore, the terminal falling velocity shows a stronger dependence on sphere diameter and density difference than in a Newtonian fluid. 

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. 

In concentrated suspensions, the settling velocity of a sphere is less than the terminal falling velocity of a single particle. For coarse (non-colloidal) particles in mildly shear-thinning liquids (1 > n > 0.8) [Chhabra et al., 1992], the expression proposed by Richardson and Zaki [1954] for Newtonian fluids applies at values of Re(= up to about 2 ... 

As for settling of single particles in Newtonian liquids, the fundamental hydrodynamic characteristic for particle motion in non-Newtonian fluids is again the drag coefficient. Its prediction allows calculations of terminal settling velocities. Note that equation 18.10, which applies to low particle concentrations (below 0.5% by volume) in Newtonian liquids at low Reynolds numbers, can, in principle, also be used non-Newtonian fluids where viscosity // then becomes the apparent viscosity but, depending on the type of the non-Newtonian behaviour (= model), its determination may require an iterative procedure. Each model redefines the particle Reynolds number so that, for example, for a power law fluid characterized by constants n and K... 

A spherical particle is unique in that it presents the same projected area to the oncoming fluid irrespective of its orientation. For nonspherical particles, on the other hand, the orientation must be known before their terminal velocity or the drag force can be calculated. Conversely, nonspherical particles tend to attain a preferred or most stable orientation, irrespective of their initial state, in the free-settling process. Vast literature, although not as extensive as that for spherical particles, is also available on the hydrodynamic behavior of nonspherical—regular as well as irregular type— particles in incompressible Newtonian fluids and these studies have been summarized in the aforementioned references, whereas the corresponding aerodynamic literature has been comprehensively reviewed by Hoerner (1965). 

Show that the terminal velocity for sedimentation of colloidal spherical particles in a Newtonian fluid can be given by the equation... 

A.3. The terminal velocity Ut of a falling sphere in a Newtonian fluid is... 

For the sedimeutatiou of a sphere in a power-law fluid in the Stokes law regime, what error in sphere diameter will lead to an error of 1% in the terminal fafling velocity Does the permissible error in diameter depend upon the value of the power-law index If yes, calculate its value over the range 1 > n >0.1. What is the corresponding value for a Newtonian liquid ... 

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