4.1 Passive scalar transport equation [flow patterning/microfluidic mixing top]

In the following sections, we first describe the basic sources of scalar fluxes. These fluxes, when applied to a control volume, lead to the basic conservation equations for scalars, the scalar convection-diffusion equation.

4.1.1 Scalar fluxes and constitutive properties

The two mechanisms that lead to flux of scalars into or out of a control volume are diffusion and convection. Diffusion refers to the net migration of a fluid property owing to random thermal fluctuations in the fluid system. While we typically treat the fluid as a continuum and ignore any properties on the atomistic scale, the molecules nonetheless exhibit extensive motion on the molecular scale, and the essentially random nature of this motion leads to random fluctuations in the distribution of apassive scalar. The net effect is a flux of the scalar in the direction opposite to the local scalar gradient.

In the ideal solution limit (which is applicable for temperature and most chemical species in the conditions used in microfluidics; see Section B.3.5), Fick’s lawis a constitutive relation that links the flux density of a scalar to both the gradient of the scalar and the diffusivity of the scalar in the solvent:

where _diff is thediffusive scalar flux density (i.e., the amount of the scalar moving across a surface per unit area due to diffusion), D [m²∕s] is the diffusivityof the scalar in the fluid, and c is the scalar, which in this text is often the concentration c_i of a chemical species. Fick’s law is a macroscopic representation of the summed effect of the random motion of species owing to thermal fluctuations. Fick’s law is analogous to the Fourier lawfor thermal energy flux caused by a temperature gradient as well as the Newtonian model for momentum flux induced by a velocity gradient; the species diffusivity D is analogous to the thermal diffusivity α = k∕ρc_p [m²∕s] and the momentum diffusivity η∕ρ [m²∕s].

In addition to the random fluctuations of a scalar due to thermal motion, the deterministic transport of the scalar due to fluid convection also leads to a convective species flux:

where _conv is theconvective scalar flux density (i.e., the amount of the scalar moving across a surface per unit area due to macroscopic fluid motion) and  is the velocity of the fluid.

4.1.2 Scalar conservation equation

Given the above fluxes, theconservation equation for a scalar c can be written as

(4.3)

where V  is a control volume with differential element dV, S is its control surface with differential element dA,  is a unit outward normal vector, and  is the total scalar flux density owing both to diffusion and convection. Application of the fluxes described above to a differential control volume (such as the Cartesian control volume shown in Figure 4.1) leads to the differential form of the scalar convection-diffusion equation, written for uniform D as:

As compared to the Navier-Stokes equations for momentum transport, the passive scalar transport equation is simpler owing to its linear dependence on  and the absence of the pressure term. Nondimensionalization of this equation for a flow with steady boundary conditions (see Section E.2.2 in the appendix) leads to the following form:

in which starred properties have been nondimensionalized, andthe Peclet number Pe = Uℓ∕D. This nondimensional form highlights that the relative magnitude of the convective fluxes as compared to the diffusive fluxes is proportional to the Peclet number. Thus, a system with high Peclet number has negligible diffusion, and scalars move about primarily by fluid convection, while a system with low Peclet number has a large amount of diffusion, and the scalar distribution is spread out quickly by diffusive processes.

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