DIMENSIONAL ANALYSIS OF A SINGLE REACTION-DIFFUSION EQUATION 1 by nak14542

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									               DIMENSIONAL ANALYSIS OF A SINGLE
                 REACTION-DIFFUSION EQUATION

        Abstract. This note illustrates how the dimensional analysis applies to the
        case of a single reaction-diffusion equation for which homogeneous boundary
        conditions are considered. File: Nicolai-scaling.tex




                                1. Model equations
   We consider the following scenario: Let Ω :=]0, [ (0 < < ∞) be the space
domain of interest and Sτ :=]0, τ [ (0 < τ ≤ ∞) be the time interval needed by
a chemical species of initial mass concentration f (x)(x ∈ Ω) to be transported by
molecular diffusion within Ω. If we assume that complete isolation conditions act at
the boundary of Ω and denote by u = u(x, t) the mass concentration of our chemical
species at the point (x, t) ∈ Ω × Sτ , then we get the following reaction-diffusion
model:
                     ∂u     ∂2u
(1.1)                   + D 2 = η(u) for (x, t) ∈ Ω × Sτ
                     ∂t     ∂x
                         ∂u          ∂u
(1.2)                       (0, t) =    ( , t) = 0 for t ∈ Sτ
                         ∂x          ∂x
(1.3)                                              ¯
                         u(x, 0) = f (x), for x ∈ Ω = [0, ],
                                                                     ¯
where D > 0 is the corresponding effective diffusion coefficient, f : Ω → [0, ∞[ is
a positive function that not vanishes on Ω, and η(u) represents the reaction rate.
In order to keep the presentation simple, we assume that the reaction rate is linear
in u, namely η(u) = ku where k > 0 is a given reaction constant. We explain in
section what happens if the reaction rate η is non-linear.

                               2. Scaling procedure
   The procedure we employ in order to perform the scaling follows the lines of
the book by Lin and Segel [1]. The first step we need to do is to summarize the
unknowns, variables and parameters of the model together with the corresponding
units.
        kg
   u    m3    Mass concentration (unknown)
        kg
   f    m3    Mass concentration (parameter)
   t    s     Time (variable)
   x    m     Space (variable)
        m2
   D     s    Diffusion coefficient (parameter)
        1
   k    s     Reaction constant (parameter)
   Independent units in (1.1)-(1.3) are T (time), M (mass) and L (length). As main
scaling quantities we may use a reference mass concentration and length scale, e.g.
A first choice is to employ the length of the domain and the maximum of the
initial mass concentration
                                  u0 := max f (x).
                                             ¯
                                           x∈Ω


  Date: 07/01/2006, Adrian Muntean (Bremen).
                                            1
2        DIMENSIONAL ANALYSIS OF A SINGLE REACTION-DIFFUSION EQUATION


Note that u0 is a strictly positive constant. We choose u0 as reference (intrinsic)
concentration and as typical length scale. Hence, we can write
(2.1)                                       u = U u0
(2.2)                                       x = X ,
where U and X are the dimensionless concentration and space variable. An inter-
esting question1 is: How should one scale the time variable? Since we are dealing
with a reaction-diffusion process, it is an obvious matter to note the two natural
alternatives: The time variable can be scaled either using a characteristic diffusion
time scale or a characteristic reaction-time scale.
2.1. Characteristic reaction-time scale. The equations can be scaled with re-
spect to the maximum of the reaction rate evaluated at the initial time t = 0. More
precisely, we set
(2.3)                                        η0 = ku0 .
Using η0 as scaling quantity, the characteristic time scale of the reaction term is
then given by
                                           u0
(2.4)                                  t=     θ.
                                           η0
In (2.4), θ is the dimensionless time variable and u0 is the (characteristic) reaction
                                                   η
                                                     0


time.
2.2. Characteristic diffusion-time scale. Another option, is to make use of the
characteristic time scale of the molecular diffusion process. In other words, we may
select
                                                    2
(2.5)                                         t=        ρ.
                                                   D
                                                                        2
In (2.5), ρ denotes the dimensionless time variable and                D    is the (characteristic)
diffusion time.

                            3. Forming dimensionless terms
   In this section, we make use of (2.1),(2.2), (2.5) and (2.4) in order to non-
dimensionalize each term arising the model equations (1.1)-(1.3). In the sequel we
show all elementary calculations.
3.1. Scaling of non-transient terms. The scaling of the terms involving spatial
derivatives essentially relies on the use of the reference mass concentration and
length. The diffusion term becomes
                                   ∂2u    ∂ 2 (U u0 )   u0 ∂ 2 U
(3.1)                          D       =D         2   =D 2
                                   ∂x2    ∂(X )            ∂X 2
while the reaction term transforms into
(3.2)                                  ku = ku0 U = η0 U.
Additionally, the boundary conditions transform into
               ∂U           ∂U                 ∂U           ∂U
                   (0, θ) =     ( , θ) = 0 or      (0, ρ) =    ( , ρ) = 0
               ∂X           ∂X                 ∂X           ∂X
following the case, while the initial condition U (X, 0) := U0 becomes
                                              f
(3.3)                                  U0 =      .
                                              u0
    1The scaling procedure is far from being trivial especially if we need to tackle complex reaction-
diffusion scenarios in which several active species may intervene.
        DIMENSIONAL ANALYSIS OF A SINGLE REACTION-DIFFUSION EQUATION                3


Note that if we have f = u0 =const, then (3.3) simplifies to U0 = 1.

3.2. Scaling of transient term for reaction-time scale. By (2.4), we acquire
for the term ∂u a dimensionless version of the time derivative of concentration,
               ∂t
which is relevant at the characteristic reaction time:
                              ∂u     ∂(u0 U )      ∂U
(3.4)                             =           = η0     .
                              ∂t     ∂( u0 θ)
                                        η
                                          0
                                                   ∂θ

3.3. Scaling of transient term for diffusion-time scale. Inserting (2.5) into
the transient term ∂u , we obtain
                   ∂t

                               ∂u   ∂(u0 U )  u0 D ∂U
(3.5)                             =    2     = 2      .
                               ∂t   ∂( D ρ)        ∂ρ

 4. Dimensionless PDE in terms of the characteristic reaction time
   Owing to (2.4), (2.3) and to our normalization of non-transient terms, we gain
the following dimensionless equations:
                          ∂U     u0 ∂ 2 U
                          η0  +D 2         = ku0 U
                           ∂θ       ∂X 2
                            ∂U    1 ∂2U
(4.1)                          + 2         = U,
                            ∂θ   Φ ∂X 2
where Φ2 denotes the Thiele modulus (or the second Damk¨hler number) and is
                                                       o
defined by
                                              η0 2
(4.2)                                  Φ2 =        .
                                              Du0
Notice that the ratios of the coefficients in (4.1) are
                                    Du0             1
                               1:      2
                                         : 1 or 1 : 2 : 1.
                                    η0             Φ

 5. Dimensionless PDE in terms of the characteristic diffusion time
  Similarly the scaled differential equation for the diffusion time scale is obtained:
                         u0 D ∂U     u0 ∂ 2 U
                           2 ∂ρ
                                 +D 2                  = ku0 U
                                        ∂X 2
                                 ∂U     ∂2U
(5.1)                                +                 =   Φ2 U.
                                  ∂ρ    ∂X 2
The final ratios of the coefficients in (5.1) are:
                                       η0 2
                               1:1:         or 1 : 1 : Φ2 .
                                       u0 D

                                      6. Remarks
    (i) We have used two different ways of scaling the time variable, and hence,
        we have obtained two different dimensionless equations. Compare (4.1)
        with (5.1). Nevertheless, with the help of the transformations θ = Φ2 ρ and
        θ = Φ1 ρ we can re-obtain one equation from the other.
              2

   (ii) To write down the dimensionless model we used a relevant dimensionless
        number Φ2 that is also called Thiele modulus. Its role is to compare the
        strength of diffusion with that of reaction, or more precisely, Φ2 com-
        pares the characteristic diffusion time with the characteristic reaction time.
4           DIMENSIONAL ANALYSIS OF A SINGLE REACTION-DIFFUSION EQUATION


            Therefore, if Φ2       1, then the reaction dominates the transport part (dif-
            fusion), and if on the other hand, 0 < Φ2      1 then the transport dominates
            the reaction part.
    (iii)   If u0 = f =const., then a single dimensionless parameter is needed to nor-
            malize the model, namely the Thiele modulus Φ2 . If u0 = f = 0, then the
            solution is trivial, i.e. u = 0 in Ω × Sτ , and no scaling is needed anymore.
    (iv)    There is one major drawback produced by our choice of scaling parameters.
                                                   u
            More precisely, we have that U := u0 < 1!!! This is mainly due to the fact
            that the reaction term η(u) is a positive production term (that is a source),
            which continuously produces concentration u during the time interval of
            interest. It consequently means that u(x, t) > u0 (x) in all points (x, t) ∈
            Ω × Sτ . Note that generally if one speaks about scaling, one always wants
            all parameters to be of order of O(1). This issue can be solved yet again
            for linear reaction rates by a suitable transformation of the problem or a
            maximum principle for parabolic equations.
    (vi)    The scaling procedure described in section 2 allows to replace the linear
            reaction rate by a non-linear one. Surprisingly, the scaled equations stay
            invariant2. Note that if the reaction rate is taken to be non-linear, then,
            generally, we cannot expect that u is of order of O(1).
    (vii)   Such a dimensional investigation can be performed when more reaction-
            diffusion equations or more complex boundary conditions are included into
            the model. Note that in such cases, a drastic reduction in the number of
            model parameters is not to be expected.

                                          References
[1] C. C. Lin and L. A. Segel, Mathematics applied to deterministic problems in the natural
    sciences, Classics in Applied Mathematics, SIAM, Philadelphia, 1988.




   2If the reaction rate is non-linear, then the scaled equations are the same. Clearly, modifications
need to be performed in the definition of η and η0 and in the units of the reaction constant k.

								
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