4.2 Physics of mixing [flow patterning/microfluidic mixing top]

We find it useful to discuss the mixing process in terms of each of the two secondary processes that are governed by the passive scalar convection-diffusion equation. While these phenomena are inseparable, we find we gain considerable insight by isolating these processes and examining their functional dependences. We thus decompose moxing into two processes: (1) diffusion across scalar gradients and (2) shortening of diffusion length scales by motion of the fluid. These phenomena are the diffusive and convective actions of the governing equation.

Diffusion across scalar gradients. Consider an infinite 1D domain, where the scalar for all x < 0 at t = 0 is given by c = c_∞, and the scalar for all x > 0 at t = 0 is given by c = 0. Consider quiescent fluid (Figure 4.2).

The governing equation for this passive scalar diffusion problem is given by

(4.6)

and the solution, achieved by similarity transform (see Exercise 4.5), is

(4.7)

This solution illustrates the action of diffusion in eliminating scalar gradients. For t →∞, the solution approaches c = c_∞∕2 everywhere. An illustration of this solution is shown in Figure 4.3. The distance ℓ = is the distance at which the solution has diffused to (1∕2)c_∞erfc((1∕2)) ≃ 1 4, which is approximately halfway toward to the equilibrium solution. The expression ℓ = is commonly used as the diffusion length scale of this system. The diffusion length scale characterizes how far into the domain the species has diffused as a function of time.

For a given time t, ℓ_diff = denotes the characteristic length over which diffusion has occurred. Similarly, for a given reservoir size R, the time required for diffusion to mix two components in that reservoir is proportional to t_diff = R²∕D. In microscale systems, this time can be long—for example, the time required for a solution of bovine serum albumin (a protein found in cow blood) to diffuse across a quiescent 100 μm channel is about two minutes, and if we were somehow able to measure the time required for a dilute suspension of neutrally-buoyant 10 μm cells to diffuse itself across a quiescent 100 μm channel, we would measure approximately 30 years.

This 1D diffusion problem relates directly to microscale flows of interest. Consider the fluid flow in Figure 4.4, in which two fluids are brought into contact and diffusion transverse to flow occurs while the fluids are convected downstream. In the limit where the channels are shallow as compared to their width, the transverse distribution of depth-averaged species in this case is identical to the 1-D diffusion problem specified above, with only one difference—this is a steady flow with a steady species distribution, and the distribution varies with y∕U (the time since the fluid entered the channel) rather than the time since the experiment started.

Shortening diffusion length scales. Solutions to the one-dimensional diffusion equation show that species mix over a distance ℓ_diff = , which requires that the time for diffusion to mix scalars over a length R is proportional to R²∕D, and this time for many practical microfluidic systems is quite long. The system shown in Figure 4.4 has convection, but this convection is normal to the scalar gradients and does not affect the transverse diffusion process. Thus the geometry in Figure 4.4 is ideal if the mixing is to be minimized so as to keep the two fluid components separate. If mixing is to be maximized, the convection must somehow enhance the process by shortening of diffusion length scales. While convection and diffusion are inseparable, we nonetheless find it useful to describe mixing as a two-step process consisting of (1) convective stirring of fluid over a time t until the spatial separation between components 1 and 2 is reduced to a characteristic length ℓ, and (2) diffusive spreading of the scalar field over a length scale . The characteristic time to mix is then given by the time t such that the length ℓ created by the stirring is on the same order as the diffusive length scale . In the absence of convection, the time is given by R²∕D. If convection actively stirs the fluid and quickly reduces the length scales over which diffusion must occur, the mixing time is much less than R²∕D. This is why we stir things to mix them up, for example, stirring cream added to coffee or chocolate syrup added to milk. The fluid flow generated by the spoon tends to shorten the diffusion length scales, and makes the mixing occur much more quickly. Two types of kinematic structures that shorten diffusion length scales include the baker’stransformation (stretch and fold), denoted in Figure 4.5, and atwist map or vortex, denoted in Figure 4.6. In both cases, the motion of fluid reduces the characteristic size of the scalar domains. These flow structures occur naturally in high-Re flows but are absent in low-Re flows. Thus mixing is often a slow process in microfluidic devices unless system geometries are designed specifically to generate flow structures that shorten diffusion length scales.

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