Donations keep this resource free! Give here:
Copyright Brian J. Kirby. With questions, contact Prof. Kirby here.
This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.
This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby
here. Click here for the most recent version of the errata for the print version.
[Return to Table of Contents]
Jump To:
[Kinematics]
[Couette/Poiseuille Flow]
[Fluid Circuits]
[Mixing]
[Electrodynamics]
[Electroosmosis]
[Potential Flow]
[Stokes Flow]
[Debye Layer]
[Zeta Potential]
[Species Transport]
[Separations]
[Particle Electrophoresis]
[DNA]
[Nanofluidics]
[InducedCharge Effects]
[DEP]
[Solution Chemistry]
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 NavierStokes equations.
E.2.1 Nondimensionalization of NavierStokes: Reynolds number
TheReynolds number Re plays several roles, and stems from fluidmechanical considerations in several ways. In
this section, we discuss the nondimensionalization of theNavierStokes equations and the relation of this
nondimensionalization to Re.
Consider the incompressible NavierStokes equations for uniformviscosity, 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 timedependent, 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 NavierStokes problem and four parameters define a steady NavierStokes problem. These lead
to two (for unsteady) or one (for steady) nondimensional parameter(s). In nondimensionalizing the
equations, the structure of the NavierStokes 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 NavierStokes 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_{c}U∕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 NavierStokes equations. For example, Equation E.12 is used
in Chapter 8 to illustrate why the terms on the left hand side of the NavierStokes 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 NavierStokes
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 NavierStokes 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 Πtheorem 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 NavierStokes 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_{c}U∕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 NavierStokes 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 PoissonBoltzmann equation: Debye length and thermal voltage
The PoissonBoltzmann equation can be nondimensionalized using techniques similar to above. The
PoissonBoltzmann equationis a bit different in three key ways. First, since the PoissonBoltzmann equation has
many more parameters, there is more flexibility with regards to how the nondimensional groups are formed. Second,
because of this flexibility, the PoissonBoltzmann equation can be nondimensionalized without using all of the
parameters from the boundary conditions. Thus, the nondimensionalization of the PoissonBoltzmann 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
NavierStokes equations is more common, we make a number of comparisons between the process for
nondimensionalizing the PoissonBoltzmann equation with that for nondimensionalizing the NavierStokes
equation.
We begin with the nonlinear PoissonBoltzmann 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 φ_{0}, 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 PoissonBoltzmann equation is an equilibrium equation and thus has no characteristic
time.
We proceed in an order different from that used for the NavierStokes equations. For the NavierStokes
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 PoissonBoltzmann
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 NavierStokes equation. There, we normalized the
key parameter we were solving for () by a characteristic value from the boundary conditions. Here, the
PoissonBoltzmann 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 righthand 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 NavierStokes 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 PoissonBoltzmann
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
φ_{0}^{*} = Fφ^{*}∕RT . The characteristic length (ℓ) and the characteristic voltage drop across the double layer (φ_{0}) 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
PoissonBoltzmann 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 φ_{0}^{*} is small, the perturbations are
small, the conductivity of the medium remains uniform, and the PoissonBoltzmann equation can be linearized with
minimal error. If φ_{0}^{*} is large, the perturbations are large, ion distributions are drastically changed, and the system is
significantly more complicated.
[Return to Table of Contents]
Jump To:
[Kinematics]
[Couette/Poiseuille Flow]
[Fluid Circuits]
[Mixing]
[Electrodynamics]
[Electroosmosis]
[Potential Flow]
[Stokes Flow]
[Debye Layer]
[Zeta Potential]
[Species Transport]
[Separations]
[Particle Electrophoresis]
[DNA]
[Nanofluidics]
[InducedCharge Effects]
[DEP]
[Solution Chemistry]
Copyright Brian J. Kirby. Please contact Prof. Kirby here with questions or corrections.
This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.
This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby
here. Click here for the most recent version of the errata for the print version.
Ad revenue from these pages is used to support student research. The presence of an advertisement on these pages does not
constitute an endorsement by the Kirby Research Group or Cornell University.
Donations keep this resource free! Give here:
