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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.