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 [m^{2}∕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} [m^{2}∕s] and the
momentum diffusivity η∕ρ [m^{2}∕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

VIDEO: passive 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:

Figure 4.1: Species fluxes for a Cartesian control volume.

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.