Dimensionless groups capillary number

The study of emulsion rheology was pioneered by Geoffr Taylor (1,2), who not only experimentally identified e dimensionless groups (capillary number and viscosity ratio) that control droplet deformation in an emulsion in simple shear and hyperboUc flow fields, but also proposed a linear theory for droplet deformation in flow. The droplet Cs illary number is defined as Ca f -... 

Since the capillary number dominates the dispersion of the fluids, that dimensionless group must also be maintained constant. Since the rotational speed and the diameter have already been dealt with, the only remaining parameter in Ca is the channel depth, which must be maintained constant. Hence... 

If gravitational settling can be neglected and if the droplet Reynolds number Re = payout 9s is small, then the droplet deformation and possible breakup in the flow are controlled by two dimensionless groups, namely the ratio of viscous to capillary forces, or capillary number... 

The main objective of surfactant flooding is to reduce residual oil saturation, which is closely related to capillary number. Therefore, the concept of capillary number is discussed first. Analysis of the pore-doublet model yields the following dimensionless grouping of parameters (Moore and Slobod 1955), which is a ratio of the viscous-to-capillary force ... 

Here, Fv and Fc are viscous and capillary forces, respectively v is the pore flow velocity of the displacing fluid in their derivation p is the displacing fluid viscosity and a is the interfacial tension between the displacing and displaced phases. The dimensionless group is called a capillary number, Nc. A set of consistent units is used so that the dimensionless group is dimensionless. For example, v is in m/s, p in mPa s, and a in mN/m or dyne/cm. 

A second dimensionless quantity is represented by the left side of Eq. (10.2.7). However, this quantity is a variable, not a parameter, since dH/dt is not an imposed velocity. When there is an imposed characteristic velocity, call it U, then the group represented by the form of the left side is called the capillary number. This number measures the ratio of the viscous force to surface tension force and is defined by... 

A number of authors have studied the ratio of viscous to capillary forces and the effect of the ratio or dimensionless group on oil recovery. Table 2 lists several of these groups which have been described by various authors (11,15,21,38-51). 

The capillary forces which cause trapping or resist mobilization can be overcome by viscous pressure gradient or buoyancy forces. The ratios of gravity to capillary forces and viscous to capillary forces are expressed as dimensionless groups, known respectively as the Bond number (Ng), and capillary number (N ). 

Components As most polymers are highly viscous and have closely-matched densities, buoyancy, inertia and Brownian motion can generally be neglected. In that case, the dynamics of isolated Newtonian droplets in a Newtonian matrix is determined by two independent dimensionless groups the capillary number Ca and the viscosity ratio p [29]. The capillary number represents the ratio of the deforming hydrodynamic stresses over the restoring interfacial stresses ... 

Before proceeding with the boundary conditions, it is interesting to examine the relevant dimensionless numbers in polymer processing. The dimensionless groups are calculated at a reference temperature, here taken as the temperature of the process, Tq. As a characteristic length, it is usually assumed the smallest dimension, for example, in a capillary tube its radius, R. As a characteristic speed, it is usually assumed the average velocity defined by... 

A similar technique can be used to grow crystals without wall contact by melting a narrow zone in the solid that can be contained by the surface tension of the melt. A dimensionless group called the Bond number relates the hydrostatic pressure in the floating zone to the capillary or surface tension pressure containing the melt. The Bond number Bo is defined as... 

A few additional comments about when and under what conditions one must use a nonlinear viscoelastic constitutive equation are discussed here. At this time it seems that whenever the flow is unsteady in either a Lagrangian DvIDt 0) or a Eulerian (9v/9r 0) sense, then viscoelastic effects become important. In the former case one finds flows of this nature whenever inhomogeneous shear-free flows arise (e.g., flow through a contraction) and in the latter case in the startup of flows. However, even in simple flows, such as in capillaries or slit dies, viscoelastic effects can be important, especially if the residence time of the fluid in the die is less than the longest relaxation time of the fluid. Then factors such as stress overshoot could lead to an apparent viscosity that is higher than the steady-state viscosity. In line with these ideas one defines a dimensionless group referred to as the Deborah number ... 

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