Biphasic Flow in Microchannels

Transformation from laminar to turbulent flow is characterized by a significant value of a dimensionless quantity known as Reynolds number (Re). Reynolds number relates inertial and viscous forces as represented in Equation 4.2 where p is the fluid density, vs is the fluid velocity, I is the characteristic length (cross section flow in a channel) and p is the fluid viscosity. 

At low values of the Reynolds number (Re 2000), viscous force dominates resulting in a laminar flow. At a high Reynolds number (Re 3000), inertial forces are dominant resulting in a turbulent flow. However, within a certain range of Reynolds number (Re 2000-3000), the flow is neither laminar nor turbulent because the transformation occurs gradually. 

Viscous and inertial forces are related to surface tension by the dimensionless Capillary and Weber numbers. Capillary number (Co), as shown in Equation 4.3, describes the relative importance ofviscosity and surface tension, where p represents the viscosity, u is the velocity and a is the surface tension. 

The Weber number (We) is shown in Equation 4.4 and relates the inertial forces to the surface tension, where p is the density of the fluid, l represents the characteristic channel length, u is the velocity of the fluid and a is the surface tension. 

In a macroscale channel, gravitational force has an effect on the flow pattern of a biphasic system consequently, the flow pattern varies between vertical and horizontal channels. However, in a microchannel, the gravity effect is dominated by the viscous forces that are expressed by the ratio of gravity force and the surface tension using the Bond number (Bo) as expressed in Equation 4.5 where Ap is the density difference between two immiscible liquids, g is acceleration due to gravity, dh is the channel dimension and ct the surface tension. 

The velocity profile of the flow in a channel varies across the diameter of the channel regardless of the flow rate. It has a minimum value ( 0) near the channel walls, and a maximum value at the center of the flow. This variation in velocities 

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Figure 3.33 Biphasic flow in microchannel for narrow RTD (a) schematic illustration and (b) monochrome snapshot of slug flow. (Adapted from Ref. [21] with permission from Elsevier.)...

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