E.2 Nondimensionalization of governing equations [nondimensionalization top]

Many of our nondimensional parameters come from nondimensionalization of governing equations. Nondimensionalizing the governing equations makes the equations simpler, and highlights which terms are the most important. Below, we start with the Navier-Stokes equations.

E.2.1 Nondimensionalization of Navier-Stokes: Reynolds number

TheReynolds number Re plays several roles, and stems from fluid-mechanical considerations in several ways. In this section, we discuss the nondimensionalization of theNavier-Stokes equations and the relation of this nondimensionalization to Re.

Consider the incompressible Navier-Stokes equations for uniform-viscosity, Newtonian fluids with no body forces:

(E.1)

This governing equation has two parameters: ρ and η. In addition, the boundary conditions have a size characterized by a length ℓ and velocities characterized by a velocity U. The characteristic velocity U is a representative fluid velocity in the flow domain, specified perhaps by the velocity at an inlet or at infinity (if available) or by some mean measure of the flow in the flowfield (usually employed if the boundary conditions are specified using pressures). The characteristic length ℓ characterizes the lengths over which the velocities change by an amount proportional to U. Furthermore, if the boundary conditions are time-dependent, then a characteristic time t_c could denote the time over which the boundary condition changes, perhaps the inverse of the frequency if the boundary conditions are cyclic. Thus, five parameters define an unsteady Navier-Stokes problem and four parameters define a steady Navier-Stokes problem. These lead to two (for unsteady) or one (for steady) nondimensional parameter(s). In nondimensionalizing the equations, the structure of the Navier-Stokes equations naturally leads to the definition of the Reynolds number.

We can define nondimensional variables denoted by starred properties, namely

(E.2)

(E.3)

and

(E.4)

Spatial derivatives naturally follow from the nondimensional coordinates, so

(E.5)

and

(E.6)

We normalize the velocity by the characteristic velocity U:

(E.7)

For the time, we choose a characteristic time that describes the fastest process in the system. If the flow is steady or if changes in the boundary condition are slow compared to ℓ∕U, we use the characteristic flow time ℓ∕U:

(E.8)

If the boundary conditions change with a characteristic time t_c < ℓ∕U, we define

(E.9)

For the pressure, we can define the nondimensional pressure either as

(E.10)

or as

(E.11)

With these definitions, we can substitute and rearrange the unsteady Navier-Stokes equations into nondimensional form. If we use p^* = , we obtain

and if we use p^* = , we obtain

where theReynolds number is defined as Re = ρUℓ∕η and theStrouhal number is defined as St = t_cU∕ℓ. The difference between Equations E.12 and E.13 is simply the premultiplier on the pressure term, which varies based on how we normalized the pressure. Note that, mathematically speaking, our choices for how we nondimensionalize our parameters are arbitrary. We can nondimensionalize the equation in any of a variety of forms. We are able to confirm that our nondimensionalization is physically meaningful only when we use the nondimensionalized equation to generate physical insight about the system, perhaps by using the nondimensional parameters to correlate experimental data or to neglect certain terms in the equations. We will know that our nondimensionalization is incorrect if it suggests that we neglect terms that are actually important (this will happen if we define t^* or p^* incorrectly), or if we are unable to correlate experimental data (this will happen if we choose U or ℓ incorrectly).

From the format of Equations E.12 and E.13, we can see that the Reynolds and Strouhal numbers are measures of the relative magnitude of the different terms in the Navier-Stokes equations. For example, Equation E.12 is used in Chapter 8 to illustrate why the terms on the left hand side of the Navier-Stokes equations can be ignored at low Reynolds number. While not relevant for microscale flows, Equation E.13 can be used to derive the Euler equations, valid for high Reynolds number. Consider a steady flow (for which the Strouhal number is unity) and consider the Re → 0 and Re →∞ limits. Clearly, the Re → 0 limit will lead to elimination of the convection and unsteady terms, and the Re →∞ limit will lead to elimination of the viscous term. However, the role of the pressure term is currently unclear and seems to depend on which definition we use for p^*. To clarify this, recall that the system to be solved has four equations (one for mass and three for momentum) and four unknowns (one pressure and three velocity components). If the pressure term is eliminated, we have four equations in three unknowns. Thus, the Navier-Stokes equations in general cannot be solved if the pressure term is neglected, and the pressure gradients in a fluid system can be neglected only in rare exceptions (for example, Couette or purely electroosmotic flow). The Reynolds number helps us to identify which velocity terms to keep, but the value of the Reynolds number never motivates us to eliminate the pressure term. A consequence of this is that, when using the Reynolds number to eliminate terms, our physical and mathematical motivations lead to the same choice for the form of p^*. At low Re, we use Equation E.10 because physically we know that the pressure gradients are primarily caused by viscous effects, and further because mathematically we know that Equation E.10 will retain the pressure term in the Re → 0 limit. At high Re, we use Equation E.11 because physically we know that the pressure gradients are caused primarily by inertial effects, and further because mathematically we know that Equation E.11 will retain the pressure term in the Re →∞ limit.

The Reynolds number does more than eliminate terms in certain limits. We refer to two flows as beingdynamically similar if they have the same Reynolds number and if their geometry is similar. The nondimensional solution to the Navier-Stokes equations will be identical for two systems if the geometry and Reynolds number are matched. Thus fluid mechanics results can be meaningfully compared across many different experimental realizations. In microscale flows, the Reynolds number is small compared to unity and small compared to St, and thus we often solve the Stokes flow equations (Chapter 8), for which the unsteady and convective terms are neglected.

E.2.2 Nondimensionalization of the passive scalar transfer equation: Peclet number

Now we discuss the nondimensionalization of the passive scalar transfer equation (applicable to passive transport of mass or temperature) and the relation of this nondimensionalization toPe. We ignore active transport mechanisms and source terms, such as electromigration of charged chemical species or chemical reaction.

Consider the mass transfer equations for dilute solutions of species i in the absence of an electric field:

(E.14)

where c_i is the molar concentration of species i and D_i is the binary diffusivity of species i in the solvent. If n species are being considered, these equations have n parameters in the governing equations—the n species diffusivities D_i. The boundary conditions have two characteristic parameters (U and ℓ) if the boundary conditions are steady, and a third parameter (t_c) if the boundary conditions are unsteady. This equation has two units (length and time), and thus the Buckingham Π-theorem predicts that a system with unsteady boundary conditions will have n+3 parameters minus 2 fundamental physical quantities, leading to n+1 nondimensional parameters that govern the system. Following a similar approach to that used for the Navier-Stokes equations, we find:

Where the mass transfer Peclet numberfor each species i is defined as Pe_i = Uℓ∕D_i and the Strouhal number is given by St = t_cU∕ℓ. For each species, the Peclet number is a measure of the relative magnitude of the diffusion term in the mass transfer equations as compared to the convection term. For all species, the Strouhal number gives the relative magnitude of the unsteady term to the convection term, and for steady boundary conditions, the Strouhal number is unity. As compared to the Navier-Stokes equations, the passive scalar transport equation is different because D_i for the species we study in microdevices varies widely from species to species, and is often orders of magnitude smaller than η∕ρ for water; thus Pe varies more widely then Re and is less often small than Re is.

E.2.3 Nondimensionalization of the Poisson-Boltzmann equation: Debye length and thermal voltage

The Poisson-Boltzmann equation can be nondimensionalized using techniques similar to above. The Poisson-Boltzmann equationis a bit different in three key ways. First, since the Poisson-Boltzmann equation has many more parameters, there is more flexibility with regards to how the nondimensional groups are formed. Second, because of this flexibility, the Poisson-Boltzmann equation can be nondimensionalized without using all of the parameters from the boundary conditions. Thus, the nondimensionalization of the Poisson-Boltzmann equation can be carried out independently of its voltage and length boundary conditions. Third, the standard nomenclature that stems from this nondimensionalization focuses on the characteristic length and voltage that arise from manipulation of the governing equation, rather than the resulting nondimensional groups. The process is therefore largely the same, but the resulting nomenclature focuses on different parameters. Since nondimensionalization of the Navier-Stokes equations is more common, we make a number of comparisons between the process for nondimensionalizing the Poisson-Boltzmann equation with that for nondimensionalizing the Navier-Stokes equation.

We begin with the nonlinear Poisson-Boltzmann equation, written as

(E.16)

This equation has 2n+6 parameters, where n is the number of chemical species. The governing equation has n+4 parameters, namely ε, T , F , R, and the species valences z_i, while the boundary conditions have n+2 parameters, namely the surface potential φ₀, a characteristic length scale ℓ, and the bulk species concentrations c_i,∞. The fundamental physical quantities in this equation are fourfold (C, V, m, K), and thus we expect 2n+2 nondimensional groups. The Poisson-Boltzmann equation is an equilibrium equation and thus has no characteristic time.

We proceed in an order different from that used for the Navier-Stokes equations. For the Navier-Stokes equations, we presumed we knew how to nondimensionalize the terms, and substituted the nondimensional forms into the equation, resulting in two nondimensional parameters (Re and St). The process for the Poisson-Boltzmann equation can start by simply rearranging the governing equation with no attention to the boundary condition. First, we notice that the argument of the exponential term must be dimensionless, and it already highlights a nondimensional ratio. We thus define a nondimensional potential as follows:

(E.17)

This effectively normalizes the potential by the thermal voltage RT ∕F , whichis a measure of the voltage (about 25 mV at room temperature) that induces a potential energy on an elementary charge equal to the thermal energy. This leads to

(E.18)

This is philosophically different from the steps we used for the Navier-Stokes equation. There, we normalized the key parameter we were solving for () by a characteristic value from the boundary conditions. Here, the Poisson-Boltzmann equation provides a parameter (RT ∕F ) that can be used for this nondimensionalization–the boundary condition is not necessary.

Next, we normalize concentrations by the ionic strength of the bulk solution:

(E.19)

leading to

(E.20)

This is more closely akin to our previous approach, in that we are normalizing a property by a characteristic value specified at the boundary (in this case, at infinity).

Next, we note that both right and left sides of this equation have units of length^-2. Thus the premultiplier of the sum on the right-hand side can be interpreted as a characteristic length to the -2 power. We thus define theDebye length λ_D as follows:

and normalize all spatial variables x, y, and z by the Debye length:

(E.22)

(E.23)

(E.24)

In so doing, we define a nondimensional del operator:

(E.25)

Again, the key difference here as compared to nondimensionalization of the Navier-Stokes equations is that the governing equation provides a characteristic length, and the boundary conditions need not be employed. Implementing the nondimensional del operator leads to the nondimensional form of the Poisson-Boltzmann equation:

The 2n+2 nondimensional parameters that govern this system are the n valences z_i, the n normalized bulk concentrations c_i,∞^*, the normalized characteristic length (ℓ∕λ_D), and the normalized double layer potential φ₀^* = Fφ^*∕RT . The characteristic length (ℓ) and the characteristic voltage drop across the double layer (φ₀) are not in the nondimensional governing equation, but are found in the boundary conditions. We have given names (thermal voltage and Debye length) to the characteristic length and voltage that evolve from the Poisson-Boltzmann equations, rather than to their nondimensional forms as employed in the boundary conditions.

The Debye length gives an estimate of the length scale over which an electrostatic perturbation (such as a charged surface) is shielded by rearrangement of ions. This process is analogous in some ways to the electrical permittivity—while the electrical permittivity describes how the polarization of a medium cancels out much of the field a charge would have caused if it were a vacuum, the Debye length describes the characteristic length scale over which an electrolyte cancels out the remaining electric field by rearrangement of ions. The normalized length ℓ∕λ_D is a measure of how large an object is as compared to the electrical double layer surrounding it. Chapter 13, for example, uses a^* = a∕λ_D to characterize the electrophoretic mobility of a particle with radius a.

The thermal voltage nondimensionalized the surface potential. The nondimensionalized surface potential indicates how much of a perturbation the surface makes on ion concentrations—if φ₀^* is small, the perturbations are small, the conductivity of the medium remains uniform, and the Poisson-Boltzmann equation can be linearized with minimal error. If φ₀^* is large, the perturbations are large, ion distributions are drastically changed, and the system is significantly more complicated.

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