At molecular level, the simplest description of transport phenomena is based on the kinetic theory of diluted monatomic gases, which is also called the Chapman–Enskog theory. The interaction between nonpolar molecules is represented by the Lennard–Jones potential, which has an empirical form of:

where σ is the characteristic diameter of the molecule, r is the distance between the two molecules, and ɛ is the characteristic energy, which is the maximum energy of attraction between the molecules. In (2.1), the term (σ/r)¹² represents the repulsion potential, while the term (σ/r)⁶ represents the attraction potential between the pair of molecules. The coefficients c_ij and d_ij are determined by molecule types and often assumed to be 1. Table 2.1 lists the parameters of some common gases. With the Lennard–Jones potential, the force between the molecules can be derived as:

The Lennard–Jones model results in the characteristic time:

where M is the molecular mass. This characteristic time corresponds to the oscillation period between repulsion and attraction. Furthermore, the model allows the determination of the dynamic viscosity of a pure monatomic gas [2]:

where the collision integral Ω is a function of the dimensionless temperature k_BT/ɛ describing the deviation from rigid sphere behavior, with k_B being the Boltzmann constant. Fig. 2.1 depicts the function of Ω. The value of the collision integral Ω is on the order of 1. The above equation allows the determination of Lennard–Jones parameters σ and ɛ from the measurement of viscosity μ, a macroscopic continuum property.

Example 2.1 (Estimation of gas viscosity using kinetic theory). Estimate the viscosity of pure nitrogen at 25 °C.

Solution. Using the Lennard–Jones parameters of nitrogen listed in Table 2.1, the dimensionless temperature is:

According to the diagram in Fig. 2.1, the collision integral of N₂ at 25°C is 0.95. The estimated viscosity is then:

The most important characteristic length scale in gas dynamics is the mean free path, which is the average distance traveled by a molecule between successive collisions:

where n is the number density (number of molecules per unit volume):

The ratio between the mean free path and the characteristic length of the device, for instance, the channel diameter, is called the Knudsen number:

Because the Knudsen number represents a link between the length scale of a device and the interaction between fluid molecules, it can be used to estimate the right model for describing the transport phenomena. For Kn<10⁻³, the fluid is a continuum. For 10⁻³<Kn<10⁻¹, a continuum model with modified boundary conditions is appropriate. For Kn>10, the fluid can only be described by a free molecular flow model.

Kinetic theory can be applied to liquids as well. In this model, the motion of liquid molecules is confined within a space limited by its neighboring molecules. Based on this theory, the viscosity of a liquid can be estimated as:

where N_A=6.023×10²³ is the Avogadro number or the number of molecules per mole; ħ=6.626068×10⁻³⁴m²kg/s is the Planck constant; is the molar volume; T_b is the boiling temperature; and T is the temperature of the liquid.

The models of viscosity for gas (2.4) and for liquid (2.8) show opposite temperature dependency. While the viscosity of gases increases with higher temperature, the viscosity of liquids decreases.

Example 2.2 (Dynamic viscosity of water). If the molar volume of water at 25°C is 18×10⁻⁶m³/mol, determine the viscosity of water at this temperature.

Solution. The boiling temperature of water under atmospheric pressure is assumed to be 100°C. According to (2.8), the viscosity of water can be estimated as:

The equation overestimates the viscosity of water.

In the previous discussion, continuum properties are derived from the molecular model using statistic methods. If there are not enough molecules for good statistics, numerical tools are used for modeling transport phenomena at the molecular level. There are two numerical methods: molecular dynamics (MD) and direct simulation Monte Carlo (DSMC). While MD is a deterministic method, DSMC is a statistical method.

Molecular dynamics is a numerical method for modeling the motion of single molecules. The interactions between the molecules can be described by the classical second law of Newton. The simplest model of a molecule is a hard sphere of a mass m. The binary interaction between two molecules is determined by the Lennard–Jones force (2.2):

where r=|r_i−r_j| is the distance between the molecules. The bold letter indicates a vector variable. The dynamics of molecule i can be described by Newton’s second law:

where N is the total number of molecules in the modeled system. The basic steps of an MD-simulation are:

Because of its deterministic nature, MD is extremely expensive in terms of computational resources. Less resources would be needed if the system is modeled with a statistic method.

Direct simulation Monte Carlo is a statistic method for modeling at the molecular level. In DSMC, many molecules are modeled as a single particle. The interactions between the molecules of each particle are determined statistically, while the motion of the particle is modeled deterministically. The basic steps of DSMC are [3]:

At continuum level, transport phenomena are described with a set of conservation equation. Because flow in micromixers is laminar, we do not need to deal with turbulent flow, which is impossible to solve analytically. The three basic conservation equations are:

Solving these three equations will result in three basic variables: the velocity field v, the pressure field p, and the temperature field T. Fluid properties, depending on the thermodynamic state (pressure p and temperature T), such as density, viscosity, thermal conductivity, and enthalpy, can be derived from these variables and fed back into the conservation equations. The above three equations are formulated for a single phase of homogenous composition. In micromixers, most fluids carry one or more species other than the carrying fluids. If the mixers are used as microreactors, chemical reactions also need to be considered. Thus, in addition to the above three conservation equations, two further equations are needed to describe the transport of species in micromixers:

A random walk is the path traced by a particle taking successive steps, each in a random direction. The construction of a simple random walk follows the three basic rules:

Following these rules, random walk of a particle can be realized with a simple program. Considering a one-dimensional random walk on a line, the particle has a random choice of two directions for each of its steps. The distance done by each step is assumed to be s. The position x(n) at a step (n) can be described as

where s_i is a random step, which can take a value of either +s or –s. The squared value of the position x(n) is:

The proportionality comes from the assumption that the same amount of time is taken to make a step . The proportionality factor determines how fast the particle moves, which in reality is the diffusion coefficient. Brownian motion is the random walk with very small steps. Figure 2.7 shows the random walk of a particle calculated for 10,000 steps in two and three dimensions.

Jan Ingenhousz, a Dutch doctor, was the first to observe the irregular motion of coal dust particles on the surface of alcohol in 1785 [5]. However, the botanist Robert Brown, who observed, in 1827, pollen particles floating in water under the microscope [6], is credited for the discovery of this motion. The Brownian motion of particles in a liquid is due to the instantaneous imbalance in the force exerted by the small liquid molecules on the particle. Thus, the diffusion coefficient of this particle can be derived from the force balance equation.

The time evolution of the position of the Brownian particle itself can be described approximately by the force balance equation where the random force of the liquid molecules represents one term in the balance. This equation is called the Langevin equation:

where m is the mass of the particle, β is the friction coefficient, and F(t) is the random force of the liquid molecules. On small time scales, inertial effects are dominant in the Langevin equation. The friction coefficient can be calculated using Stoke’s drag on a spherical particle with a radius of σ_p:

where μ is the viscosity of the surrounding liquid. The force F(t) is random in time; thus, its auto-correlation function should represent the delta function:

where δ is the Dirac function. Solving (2.25) for the one-dimensional case leads to the particle velocity:

The variance of the displacement x(t) can subsequently be determined as [8]:

where is the variance of the particle velocity. For time scale much larger than m/β,

Thus, the diffusion coefficient of the particle can be determined as:

The kinetic energy of the particle is related to the temperature as:

Substituting <u²>=k_BT/m in (2.31) results in the Stokes–Einstein equation of the diffusion coefficient [9]:

The dispersion coefficients are derived by a two-dimensional model, involving one axial and one transversal spatial dimension. G. I. Taylor was the first to present a working model for the transverse average that managed to capture the influences of both transverse diffusion and the transverse variations of the fluid velocity field. The analysis of Taylor [10] is based on the model of a long cylindrical capillary with a radius r₀ (Fig. 2.9). The derivation of the dispersion coefficient given below follows Brenner and Edwards [11]. According to Table 2.2, the velocity distribution in the capillary cross section is:

where is the mean velocity as listed in Table 2.2. For this capillary, the general equation for conservation of species (2.22), also called the convective/diffusive equation, can be formulated in the cylindrical coordinate system with no species generation (r_g=0) as:

where

The boundary and initial conditions of (2.42) are:

Equation (2.42) can be solved numerically with the above boundary and initial condition for c(r, θ, x, t)). Taylor derived an analytical asymptotic solution for (2.42) as follows.

If the observer moves along the flow with the mean velocity , we can consider a new axial coordinate x^∗:

Substituting into (2.42) results in:

For (2.46), the previous boundary and initial conditions with the new spatial variable x^∗ apply. In order to solve (2.46) analytically, the following asymptotic assumptions are made:

Next, an average specie concentration is introduced:

With the assumption of the long-time behavior, the axial concentration gradient is independent of . Thus, (2.46) can be reduced to:

The above equation can be solved for c by integration with respect to r:

where C(x^∗, t) is the function of integration. Substituting (2.49) into (2.47) results in the r-independent function of the integration constant:

Now, the axial concentration (2.49) can be expressed in terms of the average concentration:

In order to introduce the dispersion coefficient or the so-called effective diffusion coefficient D^∗, the conservation of species (2.46) is written in the flux form as:

where J^∗ is the area-averaged axial flux, which consists of both diffusive and convective components:

The convective flux J_conv can be evaluated as:

Substituting (2.51) in (2.54) results in:

The area-averaged axial flux now can be expressed as:

According to Fick’s law, the term in the square bracket of the above equation can be called the effective diffusion coefficient or, more accurately, the dispersion coefficient:

The specie conservation equation can be now formulated as:

The above partial differential equation can be solved analytically. For instance, if the initial condition of the specie concentration is a pulse , where δ(x) is the Dirac function, the transient one-dimensional solution of the average concentration is:

Figure 2.10 shows the typical concentration distribution of the one-dimensional dispersion at different time instances.

A few years after Taylor’s publication, Aris provided a firmer theoretical framework for this theory by using a moment analysis [12]. He also generalized the problem to handle time-periodic flows [13]. Following the works of Taylor and Aris, there are several other contributions to the theory of deriving effective transport models for the transverse averages of solutes flowing through channels with more general cross-sectional geometries and flow properties. Common techniques are the use of asymptotic analysis [14], [15], [16] and [17], the theory of projection operators [18], the center manifold theory [19]. All the above works only dealt with nonreactive problems. Johns and DeGance considered the influences of a system of linear reactions upon Taylor dispersion [20]. Yamanaka and Inui used their projection operator theory to solve problems involving a single irreversible reaction [21]. The following example by Bloechle [22] demonstrates an intuitive approach similar to that of Taylor [10]. The approach is called the mean-fluctuation method commonly used in turbulent flow.

Example 2.8 (Taylor dispersion in Poiseuille flow between two parallel plates[22]). Determine the dispersion coefficient of a Poiseuille flow between two parallel plates with a gap of 2h as depicted in Fig. 2.11.

Solution. Because of the symmetric geometry, only half of the model is considered (−∞<x<∞, 0<y<h). The governing Eqn (2.22) reduces to the two-dimensional form of the parallel plates model (t>0):

where u(y) is the velocity distribution [2]:

The boundary and initial conditions for (2.60) are:

The concentration and velocity can be formulated as the sum of an average component and a fluctuating component:

where

Substituting (2.62) into (2.60) results in:

with

Averaging (2.63) from 0 to h leads to:

The initial condition of the above equation is:

Similar to Taylor’s original approach, the dispersion coefficient can be derived from the above equation if the fluctuation concentration c′ is known. Subtracting (2.64) from (2.63) leads to the partial differential equation for c′:

with

Assuming that

Eqn (2.66) reduces to:

with

Next, (2.69) can be integrated with respect to y:

The symmetry and wall conditions imply that C₀(x, t)=0 and

respectively. Integrating (2.71) with respect to y results in:

C₁(x, t) can be determined by solving the condition:

The final expression for the fluctuating component of the concentration is:

Substituting (2.75) into (2.64) leads to:

Thus, the dispersion coefficient in a Poiseuille flow between two parallel plates is:

where 2h is the gap between the two parallel plates.

For axisymmetric channel geometry, such as a cylindrical capillary, the two-dimensional analysis described in the above subsection is appropriate. For real channel geometry, the cross-sectional velocity profile and, consequently, the dispersion coefficient also depend on the channel shape. In other words, the second transversal spatial dimension needs to be considered in the analysis.

Dutta et al. [23] introduce a factor f into (2.77) to consider the three-dimensional effect of Taylor dispersion:

where d=2h and f=1 for the case of the parallel plate model. The factor f is a function of the aspect ratio d/W, where d is the characteristic length of the shallow channel height and W is the channel width. Based on this definition, the longer cross-sectional dimension is considered as channel width; thus, d/W≤1. Using the Aris approach [12] and numerical simulation, the factor f can be determined for different geometries. Figure 2.12 shows the dispersion factors of typical channel geometries as a function of the aspect ratio d/W.

Because of the velocity gradient at the sharp corners of a rectangular channel, factor f increases from f=1.76 in the case of a square channel cross section (d/W=1) to f=7.95 in the case of a shallow channel (W≫d). The factor of an elliptical channel cross section can be calculated explicitly as:

where is the eccentricity of the geometry.

Ajdari et al. [24] argued that for a shallow channel W≫d with a continually varying height, the dispersion coefficient is not determined by the channel height and aspect ratio. The channel width W is the only geometric parameter that determines the dispersion coefficient. For instance, the dispersion coefficients for channels with triangle, parabolic, and elliptical cross sections are:

respectively.

In the case of an elliptical cross section, a low aspect ratio W≫d refers to an eccentricity of unity ɛ=1. Substituting ɛ=1 in (2.79) results in a factor:

Substituting (2.83) into (2.78) leads to the equation:

Because 5/(192×12)≈0.0022, the approaches of Dutta [23] and Ajdari [24] agree in the case of a shallow elliptical cross section.

The term chaotic advection refers to the phenomenon where a simple Eulerian velocity field leads to a chaotic response in the distribution of a Lagrangian marker, such as a tracing particle [25]. Advection refers to species transport by the flow. A flow field can be chaotic even in the laminar flow regime. Chaotic advection can be created in a simple two-dimensional flow with time-dependent disturbance or in a three-dimensional flow even without time-dependent disturbance. It is to be noted that chaotic advection is not turbulent. For a flow system without disturbance, the velocity components of chaotic advection at any point in space remain constant over time, while the velocity components of turbulence are random. The streamlines of the steady chaotic advection flow across each other, causing the particles to change their paths. Under chaotic advection, the particles diverge exponentially and enhance the mixing between the solvent and solute flows. In a time-periodic system, the condition for chaos is that streamlines cross at two consecutive time instants.

There are few terminologies related to visualization of an Eulerian velocity field. The first and most common terminology is the pathline, also called trajectory of a fluid particle in the flow field. In experiments, pathlines, orbits, or trajectories can be obtained by an image with a long-time exposure of a fluorescent fluid particle.

If the particles are idealized so that they are small enough not to disturb the flow and large enough so that molecular diffusion is neglected, they can move passively with the flow. The particle transport mechanism can simply be described by the advection equations:

Mathematically, pathlines or trajectories can be obtained by solving (2.85) with the initial condition at t=0 (x=x₀, y=y₀, z=z₀). Numerical integration methods, such as the Runge–Kutta method, can be used for determining the positions of the particles.

In a two-dimensional flow, streamlines are given by the solution of:

where t is treated as a constant and s is the independent variable. Streamlines build the image of the flow field at a time instant. In experiments, streamlines can be constructed from the two images recorded with particle image velocimetry (PIV). The flow is traced with fluorescent particles. Particle images are recorded at two successive time instances t and t+Δt. The particle velocities are determined by the recorded particle displacement and the time delay Δt. The streamlines are tangential to the velocity at each point. The streamlines can be depicted as the level sets of the stream function , which is defined as:

A streakline through a point (x, y, z) at a time instance τ is the curve formed by all particles, which (t<τ) passed through this point previously. In experiments, the streakline is the tracing curve of a nondiffusive tracer injected into the flow at the given point.

Example 2.9 (Trajectories and streamlines). An Eulerian velocity field is given as:

All variables and constants are dimensionless. Determine the trajectories and streamlines of particles initially at (x₀=1, y₀=1), (x₀=1, y₀=2), and (x₀=1, y₀=3).

Solution. Solving the differential equations with the initial condition (x₀, y₀) results in the position of the fluid particle as a function of time:

Figure 2.13 shows graphically the above pathlines, with the three initial particle positions, a=1, b=2, and c=π/2.

The stream function can be solved for the explicit form using the equation system (2.87). Taking the time t as constant, u(x, y, t)=x sin(at), and v(x, y, t)=y sin(bt+c), we have:

It is apparent that streamlines are time dependent. Figure 2.14 shows the streamlines as the level sets of stream function .

Equation (2.85) represents a system of coupled ordinary differential equations (ODEs). Similar dynamical systems in engineering and physics have shown a strong chaotic behavior. Poiseuille flow in a straight microchannel is considered as a one-dimensional incompressible flow at low Reynolds number. The dynamics of this flow is simple and nonchaotic. In the case of a two-dimensional flow, the dynamic behavior of the flow is more interesting. The two-dimensional continuity equation:

is fulfilled by the stream function (2.87). Equation system (2.87) has the same form of the Hamilton equation of motion, where the stream function plays the role of the Hamiltonian. Thus, steady two-dimensional incompressible flow and time-independent Hamiltonians with one degree of freedom are integrable and deterministic dynamics. Adding one more dimension to the systems, such as unsteady two-dimensional incompressible flow and independent Hamiltonians, makes the equations nonintegrable and causes chaotic dynamics.

The terminologies for chaotic advection can be borrowed from the more established field of dynamical systems theory. The advection equations (2.85) can be solved explicitly for the fluid particle position (x, y, z) at a given time t. The solution of (x, y, z) is then used for describing the motion of fluid particles in a region R, such as a channel cross section, and thus can be called a mapping function S[26]. From an initial condition (t=0), R can be transformed into a new region using S. This transformation or mapping can mathematically be described as S(R). Each transformation is called an advection cycle, which corresponds to a mixing element in different micromixer designs such as sequential lamination, discussed later in this book. Repeating these mixing elements n times refers to repeated application of S to R, or Sⁿ(R). With the discrete number of advection cycles, the transformation S is understood as a discrete, not continuous, operation as in the case of the time function.

If the fluid is assumed to be incompressible, then the volume (in a three-dimensional case) or the area (in a two-dimensional case) are preserved after each transformation. Thus, the above mapping function S is called a volume-preserving transformation or a area-preserving transformation.

A trajectory of a point after applying many discrete transformations is called an orbit. If a point p returns to the same place after N transformations, the orbit is periodic with a period of N. The two typical periodic orbits are the stable elliptic orbit and the unstable hyperbolic orbit. Elliptic orbits lead to a region that does not mix with the surrounding fluid and thus are bad for mixing. Hyperbolic orbits lead to squeezing and stretching of fluid regions and thus are good for mixing. In an unstable orbit, if the fluid particle changes its path at the intersection of the same orbit, the orbit is called homoclinic. If the fluid particle changes its path at the intersection with another orbit, the orbit is called heteroclinic.

The trajectories of a chaotic three-dimensional flow are complicated. Three-dimensional positions of fluid particles can be reduced into a two-dimensional map called the Poincaré section. In a time-periodic system, Poincaré section is a collection of intersections of trajectories with a plane. The continuous trajectories become discrete points of the transformations P_n→P_n+1. The time needed between the two points P_n and P_n+1 does not need to be the period of the system. In three-dimensional space-periodic systems, the plane is taken at the same position of the repeated spatial structure. A trajectory intersects all these periodic planes at several points. The collection of these points forms the Poincaré section. In this case, the transformation P_n→P_n+1 is the advection cycle.

Electrokinetic effects are based on an electric double layer at the interface between a solid and a liquid or between two liquids. This double layer is also called the Debye layer. This section focuses on the interface between a solid and a liquid. In general, there are four basic electrokinetic effects:

Electrophoresis is the motion of a charged particle relative to the surrounding liquid in an electric field (Fig. 2.34). Because of the small size and low Reynolds number involved, the Stokes model can be assumed for the motion of the particle:

where q_surf, , and r_p are the surface charge, the particle velocity and the radius of the particle. The charge density on the particle surface is:

If the radius of the particle is much larger than the Debye length (r_p/λ_D≫1), the surface charge can be calculated as follows:

Combining (2.174) and (2.176) results in the velocity of the particle:

At a constant dynamic viscosity μ and a constant zeta potential ζ, the electrophoretic velocity is proportional to the field strength E_el. The proportional factor is called the electrophoretic mobility:

If the particle radius is much smaller than the Debye length (r_p/λ_D<<1), the electrophoretic mobility approaches the electroosmotic mobility on a flat wall:

The electrophoretic mobility of a particle can generally be formulated as follows:

where the correction factor C is a function of the ratio r_p/λ_D :

Dielectrophoresis (DEP) is the motion of a dielectric particle in a dielectric fluid. Because the particle is charge neutral, dielectric force is caused by the inhomogeneity of the electric field. Assuming a homogenous linear dielectric fluid with a susceptibility of χ, the polarization field P of the fluid is given as:

The displacement field D of the fluid is:

where ɛ_f=ɛ₀(1+χ) is the permittivity of the fluid. The relation between the displacement field and the charge density is:

The dielectric force acting on a dipole moment in an inhomogeneous electric field is:

For a spherical particle with the permittivity ɛ_p, the polarization leads to a dipole moment m:

Although magnetic forces are body forces and therefore do not scale favorably in micromixers, high field gradients can be achieved with integrated microcoils. The use of liquids with magnetic properties and an external actuating magnetic field promises to be a niche for inducing transversal transport and chaotic advection in micromixers. The best candidate for this concept is ferrofluid. Pure substances such as liquid oxygen also behave like a magnetic liquid or ferrofluid. However, the term ferrofluid is commonly referred to as colloidal ferrofluid. The magnetic property of this fluid is credited to ferromagnetic nanoparticles, usually magnetite, hematite, or some other compounds containing iron 2+ or 3+. These nanoparticles are solid, single-domain magnetic particles that are suspended in a carrier fluid. The particles are coated with a monolayer of surfactant molecules to avoid them to stick to each other. Because the size of the particles is on the order of nanometers, Brownian motion, which represents the kinetic or thermal energy of the particles, is able to disperse them homogenously in the carrier fluid. The dispersion is strong enough that the solid particles do not agglomerate or separate even under strong magnetic fields. A typical ferrofluid is opaque to visible light. It is to be noted that the term magnetorheological fluid (MRF) refers to liquids similar in structure to ferrofluids but differing in behavior. MRF particle sizes are on the order of micrometers that are one to three orders of magnitude larger than those of ferrofluids. MRF also has a higher volume fraction on the order of 20–40%. Exposing MRF to a magnetic field can transform it from a light viscous fluid to a thick solid-like material [39].

Ferromagnetic nanoparticles are fabricated based on size reduction through ball milling, chemical precipitation, and thermophilic iron reducing bacteria. In ball milling, magnetic material of several micrometers in size such as magnetite powder is mixed with carrier liquid and surfactant. The ball milling process takes approximately 1000h. Subsequently, the product mixture undergoes centrifuge separation to filter out oversize particles. The purified mixture can be concentrated or diluted in the final ferrofluid.

Synthesis by chemical precipitation is a more common approach in which the particles precipitate out of solution during chemical processes. A typical reaction for magnetite precipitation is:

The reaction product is subsequently coprecipitated with concentrated ammonium hydroxide NH₄OH. Next, a peptization process transfers the particles from water-based phase to an organic phase, such as kerosene with a surfactant, for example, oleic acid. The oil-based ferrofluid can then be separated by a magnetic field.

Another approach for fabrication of ferrofluid is based upon thermophilic bacteria that reduce amorphous iron oxyhydroxides to nanometer-sized iron oxides. The thermophilic bacteria are able to reduce a number of different metal ions. Thus, this approach allows incorporating other compounds, such as Mn(II), Co(II), Ni(III), Cr(III)), into magnetite. Varying the composition of the nanoparticles can adjust magnetic, electrical, and physical properties of the substituted magnetite and consequently of the ferrofluid. Ferromagnetic particle with extremely low Curie temperature can be designed with this method. Most of the particle materials commonly used in ferrofluid have much higher Curie temperatures. The temperature dependence of magnetic properties can be used for micromixing applications.

At the typical channel size of microfluidics (about 100μm), ferrofluid flow in a microchannel can be described as a continuum flow. The governing equations are based on the conservation of mass and conservation of momentum. In the case of temperature-dependent magnetic properties, the conservation of energy may be needed for calculating the temperature field. The Navier–Stokes equation for a ferrofluid has the following form:

Compared to other types of fluid, ferrofluid flow in microchannel has an additional term for magnetic force [39]:

where μ₀=4π×10⁻⁷ Hm⁻¹ is the permeability of space, M is the intensity of magnetization, v is the specific volume, and H is the magnetic field strength in A/m. The magnetic term can be grouped together with static pressure to form an apparent pressure. Thus, the conservation of momentum can be reduced to the conventional Navier–Stokes equation. Because magnetic force is a body force, ferrofluid flow in microchannel should have the same velocity distribution as a pressure-driven flow.

The first term in (2.189) shows that the magnetic force is a body force, which is proportional to the volume. According to the scaling law, or the so-called cube-square law, magnetic force will be dominated by viscous force in microscale. However, the second term in (2.189) may have advantages in microscale due to the high-magnetic-field gradient that is achievable with integrated microcoils. As mentioned previously, ferrofluid with low Curie temperature is readily available. Magnetization can be controlled by adjusting the temperature from room temperature to an acceptably low Curie temperature. The temperature dependence of magnetization can be implemented in the first term of (2.189); the magnetic force then has the form [40]:

It is clear from (2.190) that the high-temperature gradient in microscale can be another advantage for driving ferrofluid in microchannels. Thus besides microcoils, microheaters can be another tool for controlling ferrofluid-based micromixers.