AA COORDINATE FREE FORMULATION OF EFFECTIVEDIFFUSION ON CHANNELS
CARLOS VALERO VALDÉSDEPARTAMENTO DE MATEMÁTICASUNIVERSIDAD DE GUANAJUATOGUANAJUATO, MÉXICO
Abstract.
We study diffusion processes in regions generated by “sliding” across section by the phase flow of vector filed on curved spaces of arbitrary di-mension. We do this by studying the effective diffusion coefficient D that ariseswhen trying to reduce the n -dimensional diffusion equation to a 1-dimensionaldiffusion equation by means of a projection method. We use the mathematicallanguage of exterior calculus to derive a coordinate free formula for this coeffi-cient in both infinite and finite transversal diffusion rate cases. The use of thesetechniques leads to a formula for D which provides a deeper understanding ofeffective diffusion than when using a coordinate dependent approach. Introduction
The purpose of this paper is to present a coordinate free formulation of the theoryof effective diffusion on channels. This problem has been studied extensively in theliterature with the use of specific coordinate systems (e.g [7, 1, 11, 5, 4, 6, 8, 9,3, 2, 10]). We tackle the coordinate free formulation by using modern tools fromdifferential geometry; more specifically: vector field flows and exterior calculus.The advantages of taking this point of view is that it provides a unified theory withthe following properties.(1) The formulas obtained hold for channels of any dimension in arbitrary flatand curved spaces.(2) By using this geometric approach one can gain a deeper and more intuitiveunderstanding of the formulas for the effective diffusion coefficient: both inthe finite and infinite transversal diffusion rate case.(3) Our approach has lead us to identify the Fick-Jacobs equation as a standarddiffusion equation. To do this we need to change the metric in the variableparametrizing the cross sections of the channel, which has also lead us tomodify the the definition of effective density function and effective diffusioncoefficient used in most of the literature.(4) Our formulas hold for an arbitrary selection of cross sections of the channel.This generality has lead us to identify the concepts of natural and imposedprojection maps.1.1.
Plan of the paper.
In section 2 we show how to generate channels usingvector fields on arbitrary spaces, and how these provide us with cross sectionswhich allows us to reduce a general diffusion equation to a diffusion equation withonly one spatial variable. In section 3 we present formulas for the effective diffusioncoefficient (both in the infinite and finite transversal diffusion rate cases) avoiding a r X i v : . [ m a t h - ph ] A ug OORDINATE FREE EFFECTIVE DIFFUSION IN CHANNELS 2
Figure 2.1.
A channel generated by “sliding” a cross section S along the integral curves of a vector field. The wall W consists ofthe surface that encapsulates the channel, but not including thecross sections at its extreme sides.the use of the language exterior calculus, so that the main results can be understoodin a non-technical manner. In fact, the main concept needed in our formulas issimply that of the flux of a vector field across a hyper-surface. We show that thereis a special choice of cross section of a channel in which the formulas for the effectivediffusion coefficient for the finite and infinite transversal rate cases coincide. Section4 contains the coordinate free derivation of our formulas using exterior calculus, andwe also how we can recover the coordinate dependent formulas from our generalresult. 2. Channel geometry through vector field flows
We are interested in studying channel-like objects, which we will denote by C ,in an n − dimensional space M . We will construct C using the following procedure(see Figure 2.1). Let S be a ( n − -dimensional hyper-surface with boundary in M and let U a vector field in C . For a given real number u , let S u be the hyper-surfaceobtained by “sliding” S along the integral curves of U for a duration of u . We willrefer to S u as the cross section of C at u . If C is the union of the cross sections S u ,we will say that the vector field U generates C . If ∂ S u is the boundary of S u , thewall W of C is the union of the sets ∂ S u for all u ’s. For x in C we will let u ( x ) beequal to the time it takes for a point in S to reach x (by following an integral curveof U ). In this context, we will refer to u as a projection function for the channel.Notice that S s can be characterized as the set of points in C at which u ( x ) = s . Usually M stands for flat space of dimension either two or three, but our results hold for thegeneral case where M is an arbitrary n -dimensional oriented Riemannian manifold. An integral curve of U is a curve x = x ( t ) in M that satisfies dxdt ( t ) = U ( x ( t )) . Depending on the context will think u as a scalar or as a function. OORDINATE FREE EFFECTIVE DIFFUSION IN CHANNELS 3
Figure 2.2.
Parametric channel with central function c = c ( u ) and width function w = w ( u ) .As a particular case of the above construction consider a parametric channel, obtained by using a parametrization function x = x ( u, v ) where u is a scalar and v belongs to some region in ( n − -dimensional space. In this case the generatingvector field of the channel is U ( x ) = ∂x∂u ( u ( x ) , v ( x )) , where u ( x ) and v ( x ) are the u and v coordinates of the point x in M . Example 1.
For n = 2 let the variable v be in re region − / ≤ v ≤ / anddefine (see Figure 2.2) x ( u, v ) = ( u, c ( u ) + vw ( u )) , for scalar valued functions c = c ( u ) and w = w ( u ) . We have that ∂x∂u ( u, v ) = (1 , c (cid:48) ( u ) + vw (cid:48) ( u )) . If we write x = ( x , x ) , then from the formulas x = u and x = c ( u ) + vw ( u ) we obtain v = x − c ( x ) w ( x ) . Hence, the generating vector field of the channel is U ( x , x ) = (cid:18) , c (cid:48) ( x ) + (cid:18) x − c ( x ) w ( x ) (cid:19) w (cid:48) ( x ) (cid:19) . A projection function for this field is u ( x , x ) = x , and the cross section S u is a line parallel to the x -axis intersecting the x -axis at ( u, . OORDINATE FREE EFFECTIVE DIFFUSION IN CHANNELS 4 (a) (b)
Figure 2.3.
Two sets of cross sections associated with two differ-ent generating vector fields of the same channel.
Remark . We constructed the projection function u in terms the vector field U ,by letting u ( x ) be the time it takes for an integral curve of U starting at S toreach x . Alternatively, we could first select a scalar valued function u in C and thenconstruct a generating vector field U in M that satisfies ∇ u ( x ) · U ( x ) = 1 for all x in C . This condition implies that u is a projection function of for U . The initial crosssection S is then chosen so that u ( x ) = 0 for all x in S .2.1. Natural projection functions and fields.
A channel C can have manygenerating fields, which in general produce different sets of cross sections (see Figure2.3 ). This observation leads to the following problem. Problem 3.
Consider a channel C with two fixed cross sections A and B . Is there away to chose a generating vector field U for C such that A and B are cross sectionsof C generated by U , and generated cross in between them “fit” the geometry of C in a “natural way”?We will argue that a “natural way” to choose U is as follows. For two differentscalars a and b let h be a harmonic function (i.e ∆ h = 0) on C such that ∇ h hasno flux across W , and satisfies the boundary conditions(2.1) h ( x ) = (cid:40) a if x is in A ,b if x is in B . We will let (see Figure 2.4) U = H , where(2.2) H ( x ) = ∇ h ( x ) / ||∇ h ( x ) || . This field generates the channel C and has h as projection function. We will referto h and H as a natural projection function and a natural generating field for thechannel C with lateral cross sections A and B . OORDINATE FREE EFFECTIVE DIFFUSION IN CHANNELS 5 (a)
Cross sections (b)
Generating vector field
Figure 2.4.
Cross sections and generating field associated to aharmonic function h on a channel. This function takes the values and on A and B (respectively), and its gradient has no fluxacross the other walls of the channel (i.e in W ). Figure 2.5.
Channel region between two cross section
Remark . If we write h = h a,b to specify that h takes the values a and b in A and B , respectively, then we have that h a,b = a + ( b − a ) h , . Flux functions.
The flux function of a vector field V in channel C is definedas F V ( u ) = flux of V across S u . In the above definition we have assumed that we have fixed a generating vector field U for C (and hence its cross sections). The importance of the generating vectorfield U of a channel is that we will be able to express many quantities of interest in Mathematically, this is the integral over S u of the component of V normal to S u OORDINATE FREE EFFECTIVE DIFFUSION IN CHANNELS 6 terms U . In particular we will consider the flux functions of vector fields V of theform V = λ U , where λ is a scalar valued function in C . In this case we have that F V ( u ) = dc λ du ( u ) , where for (see Figure 2.5 ) C [ u ,u ] = union of the sets S u for u ≤ u ≤ u we defined c λ ( u ) = total concentration of λ in C [0 ,u ] . Example 5.
Let ν ( u ) = volume of C [0 ,u ] . For λ = 1 we have that ν ( u ) = c λ ( u ) , and hence dνdu ( u ) = F U ( u ) . An important property of the flux function, that we will use frequently, is thatif we can write λ = λ ( u ) , then F λV ( u ) = λ ( u ) F V ( u ) Effective diffusion on channels
For a given channel C , we are interested in studying the evolution of a densityfunction P = P ( x, t ) that obeys the diffusion equation ∂P∂t ( x, t ) = D ∆ P ( x, t ) . We will assume reflective boundary conditions on the wall W of C , i.e the gradientof P has no flux across W . Using a projection function in C , we will try to reducethe above equation to a diffusion equation in a -dimensional spatial variable. Todo this, we define the total concentration function as c ( u, t ) = total concentration of density P in C [0 ,u ] at time t. and the volume function ν by(3.1) ν ( u ) = volume of C [0 ,u ] . We can now define the effective concentration as p ( u, t ) = ∂c∂u ( u, t ) / dνdu ( u ) . If we let u = u ( ν ) be the value of u that corresponds to volume ν and p ( ν, t ) = p ( u ( ν ) , t ) , then p ( ν, t ) = ∂c∂ν ( ν, t ) . The total concentration is obtained by integrating the function λ in the region C [0 ,u ] . If we think of λ as a function of x (i.e λ = λ ( x )) the expression λ = λ ( u ) means that λ ( x ) = ρ ( u ( x )) for a scalar valued function ρ = ρ ( u ) . Our notation has the advantage of avoidingthe use the extra function ρ . When we speak of volume in n -dimensional space we are referring to n -dimensional volume,i.e length for n = 1 , area for n = 2 , etc. OORDINATE FREE EFFECTIVE DIFFUSION IN CHANNELS 7
Hence, the total concentration of P in the region C [ u ,u ] at time t is given by (cid:90) ν ν p ( ν, t ) dν for ν i = ν ( u i ) . Remark . In most of the literature the effective concentration is defined as(3.2) p ( u, t ) = dcdu ( u, t ) , so that the total concentration of P in C [ u ,u ] is (cid:90) u u p ( u, t ) du. From a mathematical point of view the definition of the effective concentration 3.2is not convenient for the following reason. If we introduce a new variable v = v ( u ) ,with our definition of effective concentration we have that p ( u, t ) = ∂c∂u ( u, t ) / (cid:18) dνdu ( u ) (cid:19) = (cid:18) ∂c∂v ( v ( u ) , t ) dvdu ( u ) (cid:19) / (cid:18) dνdv ( v ( u )) dvdu ( u ) (cid:19) = ∂c∂v ( v ( u ) , t ) dνdv ( v ( u ))= p ( v ( u ) , t ) . This is the proper formula for the change of variable of a function. On the otherhand, if we define p as in 3.2 we have p ( u, t ) = dcdu ( u, t ) = dνdu ( u ) dcdν ( ν ( u ) , t ) = dνdu ( u ) p ( ν ( u ) , t ) , which is the way vectors (not functions) transform under a change of variable.3.1. Infinite transversal diffusion rate.
If we assume that the density function P stabilizes infinitely fast along the cross sections S u of C , which is equivalent to P being constant along them, we arrive at the effective diffusion equation (see Section4.4)(3.3) ∂p∂t ( u, t ) = ∇ · ( D ( u ) ∇ p ( u, t )) , where the effective diffusion coefficient is given by (3.4) D ( u ) = D F ∇ u ( u ) dνdu ( u ) . The divergence and gradient operators ∇· and ∇ in formula 3.3 are the ones asso-ciated to the metric (3.5) g ( u ) = (cid:18) dνdu ( u ) (cid:19) , In the expression for flux function in this formula u is being interpreted as a projectionfunction on the channel C , and its gradient computed with respect to the metric in M . This metric defines a distance function d between values u and u , given by d ( u , u ) = (cid:90) u u (cid:112) g ( u ) du = ν ( u ) − ν ( u ) ,which is the volume of the region C [ u ,u ] . OORDINATE FREE EFFECTIVE DIFFUSION IN CHANNELS 8 i.e ∇ p = 1 g ∂p∂u and ∇ · j = 1 √ g ∂∂u ( √ gj ) . Observe that if we let u = ν then g = 1 , and hence ∇ p = ∂p∂ν , ∇ · j = ∂j∂ν and D ( ν ) = D F ∇ ν . Remark . The effective diffusion coefficient D connects the effective flux j ( u, t ) = F J t ( u ) / dνdu ( u ) where J t = D ∇ P t with the gradient of the effective density function. More concretely, Fick’s first law establishes that (see sections 4.3 and 4.4) j ( u, t ) = −D ( u ) ∇ p ( u, t ) . Remark . The condition of P being constant along the cross sections of the channelimplies that we can write (see Section 4.4) P ( x, t ) = p ( u ( x ) , t ) , where p is the effective density function and u is the projection function. If we hadused the definition of effective concentration found in most of the literature, thisidentity would not hold. Comparison with the generalized Fick-Jacobs equation.
If we let σ ( u ) = dνdu ( u ) and p f ( u, t ) = dcdu ( u, t ) , we can write equation 3.3 as a generalized Fick-Jacobs equation dp f dt ( u, t ) = ∂∂u (cid:18) σ ( u ) D f ( u ) ∂∂u (cid:18) p f ( u, t ) σ ( u ) (cid:19)(cid:19) , where the effective diffusion coefficient is given by D f ( u ) = D F ∇ u ( u ) σ ( u ) . If we define the effective flux j f as j f ( u, t ) = F J t ( u, t ) then we have the continuity equation (see 4.3) ∂p f ∂t + ∂j f ∂u = 0 . Using the Fick-Jacobs equation we conclude that j f = − σ D f ∂∂u (cid:16) p f σ (cid:17) . From these equations and the formulas p = p f /σ and j = j f /σ we obtain j ( u, t ) = −D f ( u ) ∂p∂u ( u, t ) . OORDINATE FREE EFFECTIVE DIFFUSION IN CHANNELS 9
We conclude that the difference between the effective diffusion coefficient given bythe generalized Fick-Jacobs equation and ours is that: in the first case the gradientused in Fick’s first law is that associated to the metric g = 1 , and in the secondcase it is that associated to the metric g ( u ) = σ ( u ) . The formula connecting bothcoefficients is D f ( u ) = D ( u ) σ ( u ) . Observe that when the cross sections of C are parametrized by the volume variable ν , we have that σ ( ν ) = 1 and hence D f ( ν ) = D ( ν ) = D F ∇ ν ( ν ) . Furthermore, in this case both the effective diffusion equation and the Fick-Jacobsequation become the diffusion equation ∂p∂t ( ν, t ) = ∂∂ν (cid:18) D ( ν ) ∂p∂ν ( ν, t ) (cid:19) . Cross section density function.
If we define the area function as A ( u ) = area of S u and let G ( u ) = F ∇ u ( u ) A ( u ) , then we can write D ( u ) = D A ( u ) G ( u ) dνdu ( u ) . Since the cross sections S u are the level sets of u , the vector field ∇ u is orthogonalto them. Hence, if we orient the cross sections S u so that their normal fields havethe same direction as ∇ u , we have that G ( u ) = average value of |∇ u | on S u . This number measures the average density of cross sections near S u , and we willrefer to G as the cross section density function . If the cross sections of the channelare parametrized by the volume variable ν , we have that D ( ν ) = D A ( ν ) G ( ν ) . Finite transversal diffusion rate.
Consider a channel C whose cross sec-tions S u are generated by a vector field U , and let us drop the assumption that thedensity function P = P ( x, t ) stabilizes infinitely fast along these cross sections. Togive a formula for the effective diffusion coefficient D = D ( u ) , we will make use ofthe natural projection function h and the natural generating field H of the chan-nel C with lateral cross sections S u and S u for u < u (see section 2.1). In thiscontext, we will refer to the projection function u and the field U as the imposedprojection function and field (to distinguish them from the natural ones: h and H ).Let ρ be the effective density function of h under the projection map u , i.e ρ ( u ) = F hU ( u ) / dνdu ( u ) . We refer to area as ( n − -dimensional area. For n = 2 this means length, for n = 3 thismeans area in the usual sense, etc. By convention we speak of volume when we want to measurethe “extent” of n -dimensional objects in n -dimensional space, and area when we want to measurethe “extent” of ( n − -dimensional objects in n -dimensional space. OORDINATE FREE EFFECTIVE DIFFUSION IN CHANNELS 10
In section 4.5 we proved that (for u with u ≤ u ≤ u ) the effective diffusioncoefficient D appearing in formula 3.3 can be computed as(3.6) D ( u ) = J (cid:18) dνdu ( u ) (cid:19) / F λU ( u ) , where λ = λ ( x ) is a scalar valued function in C defined by (3.7) λ = ∇ h · U + ( h − ρ ◦ u ) ∇ · U and the constant J is given by J = D F ∇ h ( u ) . Channels with natural projection map.
If for a given channel C we choose the im-posed projection map an generating vector field to be the natural ones, i.e U = H and u = h, then we have that ∇ h · U = ∇ h · ∇ h ||∇ h || = 1 . and ρ ( u ( x )) = F hU ( u ( x )) / dνdu ( u ( x ))= h ( x ) F U ( u ( x )) / dνdu ( u ( x ))= h ( x ) , where we have made use of the formula dνdu ( u ) = F U ( u ) . Hence λ = 1 in formula 3.7, which implies D ( u ) = J (cid:18) dνdu ( u ) (cid:19) ( F U ( u )) − = D F ∇ u ( u ) dνdu ( u ) . We conclude that when using the natural projection function and field of a channel,the formulas for the effective diffusion coefficient in the finite and infinite diffusiontransversal rate cases coincide. Derivation of the effective diffusion coefficient formula
Let M be an oriented Riemannian manifold of dimension n . We are interestedin the diffusion equation ∂P∂t ( x, t ) = ∇ · ( D ( x ) ∇ P ( x, t )) , The gradient and divergence operator appearing in this formula are computed with respectto the metric in M . By using the fact that h is harmonic we showed in 4.5 that the function J ( u ) = F ∇ h ( u ) isa constant function, i.e independent of u. Given the variety of mathematical objects that we will use, throughout this section we won’tfollow the convention of using bold face to denote non-scalar quantities.
OORDINATE FREE EFFECTIVE DIFFUSION IN CHANNELS 11 where P : M × R → R is a time dependent function in M and D ( x ) : T x X → T x X is a linear map for every x in M . The divergence and gradient operators in theabove formula can defined in terms of exterior algebra operations as ∇ · J = ∗ d ∗ J (cid:91) and ∇ P = ( dP ) , where d : (cid:86) k M → (cid:86) k +1 M is the exterior derivative, ∗ : (cid:86) k M → (cid:86) n − k M theHodge star operator, and the musical isomorphisms (cid:93) and (cid:91) allow us to identify1-forms and vector fields. If we let g stand for the metric tensor in M and use localcoordinates x , . . . x n , we can write ∇ · J = 1 | g | / n (cid:88) i =1 ∂∂x i (cid:16) | g | / J i (cid:17) where | g | = det( g ) and ( ∇ P ) i = n (cid:88) j =1 g ij ∂P∂x j where ( g ij ) = ( g ij ) − .. In a homogeneous and isotropic medium the diffusion has the form(4.1) ∂P∂t ( x, t ) = D ∆ P ( x, t ) where ∆ = ∗ d ∗ d and D ∈ R . Channels and projection functions.
Let M be an n -dimensional orientedRiemannian manifold. We will say that C ⊂ M is generated by a vector field U ,if C is the union of phase curves of U that have transversal intersection with an ( n − -dimensional sub-manifold with boundary S . We will then say that C is achannel generated by U having S as an initial cross section . A smooth function u : C → R is a projection function for the field U if du ( U ) = 1 . We will usuallychoose S so that S = u − (0) . The cross section S s of C at s is defined by theformula S s = u − ( s ) . Recall that the phase flow { ϕ s : C → C} s ∈ R of U is defined by dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 ( ϕ s ( x )) = U ( x ) , and satisfies(4.2) ϕ s + s = ϕ s ◦ ϕ s . The condition du ( U ) = 1 is then equivalent to u ( ϕ s ( x )) = u ( x ) + s, and hence S s + h = ϕ h ( S s ) . If we let W = ∂ C then W is the union of phase curves of U that intersect ∂ S . Wewill refer to W as the reflective wall of C . We define C [ s ,s ] = u − ([ s , s ]) and W [ s ,s ] = W ∩ C [ s ,s ] . OORDINATE FREE EFFECTIVE DIFFUSION IN CHANNELS 12
Flux functions.
We will let µ ∈ (cid:86) n M stand for global volume form associated withthe metric in M . The orientation in C will the the one induced by the orientationof M , i.e we will let the orientation form be the one obtained by restricting µ to C .Observe that(4.3) du ∧ ι U ( µ ) = du ∧ ( ∗ U (cid:91) ) = < du, U (cid:91) > µ = du ( U ) µ = µ. If we let i u : S u → C be the inclusion map then i ∗ u ( ι U ( µ )) is an ( n − -form in S u which vanishes no-where in S u . We will use this form as an orientation form for S u . For a vector field V in C we define the flux function F V : R → R as F V ( u ) = (cid:90) S u ι V ( µ ) = (cid:90) S u ∗ ( V (cid:91) ) . In particular F ∇ P ( u ) = (cid:90) S u ∗ ( dP ) . Change of variable formulas.
Let u : C → R be a projection function for U and f : R → R a function with positive derivative. If we let v = f ◦ u then dv ( x ) = f (cid:48) ( u ( x )) du ( x ) , which implies that v is a projection function for the field V defined by V ( x ) = U ( x ) /f (cid:48) ( u ( x )) . To simplify notation, we will write the conditions v = f ◦ u and u = f − ◦ v as v = v ( u ) and u = u ( v ) , where in the first equation u is seen as a scalar value and v as a function, and onthe second formula v is seen as a scalar value and u as a function. If we denotea cross sections at u as S u and a cross section at v as S v , then the formulas u − ( s ) = v − ( f ( s )) and v − ( s ) = u − ( f − ( s )) can be simply written as S u = S v ( u ) and S v = S u ( v ) . Furthermore, we have that(4.4) dv = (cid:18) dvdu (cid:19) du and V = (cid:18) dvdu (cid:19) − U, where dvdu ( x ) = f (cid:48) ( u ( x )) . Remark . If for a positive function λ : R → R we let V = ( λ ◦ u ) U , then we canrecover the projection function v = v ( u ) for V as v ( u ) = v + (cid:90) uu (cid:18) λ ( s ) (cid:19) ds for v ∈ R . Some useful identities.
We will now derive some identities that will beuseful in our study of diffusion processes on channels. Let C be a channel generatedby a field U and with a projection function u . In what follows we will make use ofCartan’s magic formula L U = ι U ◦ d + d ◦ ι U , where L U is the Lie derivative with respect to U . OORDINATE FREE EFFECTIVE DIFFUSION IN CHANNELS 13
Lemma 10. If α is an ( n − -form in C and we define f ( u ) = (cid:90) S u α, then f (cid:48) ( u ) = (cid:90) S u L U α. If ω is an n -form in C and for any u ∈ R we define g ( u ) = (cid:90) C [ u ,u ] ω then g (cid:48) ( u ) = (cid:90) S u ι U ( ω ) and g (cid:48)(cid:48) ( u ) = (cid:90) S u ( dλ ( U ) + λ ∇ · U ) ι U ( µ ) where λ = ∗ ω. Proof.
From the formula S u + h = ϕ h ( S u ) we obtain f ( u + h ) − f ( u ) = (cid:90) S u + h α − (cid:90) S u α = (cid:90) S u ( ϕ ∗ h α − α ) , and hence f (cid:48) ( u ) = lim h (cid:55)→ (cid:90) S u h ( ϕ ∗ h ( α ) − α ) = (cid:90) S u L U α. To prove the second part of the lemma observe that g ( u + h ) − g ( u ) = (cid:90) C [ u,u + h ] ω, and hence g (cid:48) ( u ) = lim h (cid:55)→ g ( u + h ) − g ( u ) h = lim h (cid:55)→ h (cid:90) u + hu (cid:90) S t ( ι U ( ω )) dt = (cid:90) S u ι U ( ω ) . Combining the previous results we obtain g (cid:48)(cid:48) ( u ) = (cid:90) S u L U ( ι U ( ω )) . Using Cartan’s magic formula it is easy to verify that L U ( ι U ( ω )) = ι U ( L U ( ω )) . We can write ω = λµ for λ = ∗ ω , and hence L U ( ω ) = L U ( λµ ) = ι U ( dλ ) µ + λ L U µ. Using this and the fact that L U µ = ( ∇ · U ) µ , we conclude that ι U ( ω ) = ( dλ ( U ) + λ ( ∇ · U )) ι U ( µ ) (cid:3) OORDINATE FREE EFFECTIVE DIFFUSION IN CHANNELS 14
The effective continuity equation.
If we let the metric tensor in the u variable be g ( u ) = (cid:18) dνdu ( u ) (cid:19) , then the divergence and gradient operators are given by the formulas ∇ · j = g − / ∂∂u (cid:16) g / j (cid:17) and ∇ p = g − ∂p∂u . Consider a concentration function P = P ( x, t ) and the flux vector field J = J ( x, t ) .Let us write P t ( x ) = P ( x, t ) and J t ( x ) = J ( x, t ) , and for a channel C define the effective flux function as j ( u, t ) = F J t ( u ) / dνdu ( u ) where F J t = (cid:90) S u ∗ J (cid:91)t . and the effective concentration as p ( u, t ) = ∂c∂u ( u, t ) / dνdu ( u ) where c ( u, t ) = (cid:90) C [0 ,u ] ∗ P t . By Lemma 10 we have that ∂c∂u ( u, t ) = (cid:90) S u ι U ( ∗ P t ) and d F J t du ( u ) = (cid:90) S u L U ( ∗ J (cid:91)t ) = (cid:90) S u ( d ◦ ι U + ι U ◦ d )( ∗ J (cid:91)t ) . If we assume reflective boundary conditions on the wall W of C , we get (cid:90) S u d ( ι U ( ∗ J (cid:91)t )) = (cid:90) ∂S u ι U ( ∗ J (cid:91)t ) = 0 . Using the above formulas and the continuity equation ∗ ∂P∂t ( x, t ) + d ∗ J (cid:91) ( x, t ) = 0 we obtain d F J t du ( u ) = (cid:90) S u ( ι U ◦ d )( ∗ J (cid:91)t ) = − (cid:90) S u ι U (cid:18) ∗ ∂P∂t (cid:19) , and hence (cid:90) S u ι U (cid:18) ∗ ∂P∂t (cid:19) = ∂∂t (cid:90) S u ι U ( ∗ P t ) = ∂∂t (cid:18) ∂c∂u ( u, t ) (cid:19) . We conclude that ∂∂t (cid:18) ∂c∂u ( u, t ) (cid:19) + d F J t du ( u ) = 0 , which implies that ∂∂t (cid:18) g − / ( u ) ∂c∂u ( u, t ) (cid:19) + g − / ( u ) ∂∂u (cid:18) g / ( u ) g − / ( u ) d F J t du ( u ) (cid:19) = 0 . This last equation is known effective continuity equation and can be re-written as(4.5) ∂p∂t ( u, t ) + ∇ · j ( u, t ) = 0 . OORDINATE FREE EFFECTIVE DIFFUSION IN CHANNELS 15
Infinite transversal diffusion rate.
The assumption of an infinite transver-sal diffusion rate is expressed mathematically by letting P ( x, t ) = ρ ( u ( x ) , t ) for a function ρ : R × R → R . The effective density function can then be written as p ( u, t ) = g ( u ) − / (cid:90) S u u ∗ ( ρ t ) ι U µ = ρ ( u, t ) g ( u ) − / (cid:90) S u ι U ( µ ) . From the formula (cid:90) S u ι U ( µ ) = dνdu ( u ) = g ( u ) / , we conclude that p ( u, t ) = ρ ( u, t ) . Using Fick’s law J t = − D ∇ P t , we obtain F J t ( u ) = (cid:90) S u ∗ J (cid:91)t = − D (cid:90) S u ∗ ( dP t ) = − D (cid:90) S u ∗ ( u ∗ ( dρ t )) . Since (for s equal to the identity map in R ) u ∗ ( dρ t ) = u ∗ (cid:18) ∂ρ t ∂s ds (cid:19) = u ∗ (cid:18) ∂ρ t ∂s (cid:19) du, we obtain F J t ( u ) = − D ∂ρ∂u ( u, t ) (cid:90) S u ∗ ( du ) . Using this last formula and the fact that ρ = p , we obtain j ( u, t ) = g ( u ) − / F J t ( u ) = − (cid:18) D g ( u ) / (cid:90) S u ∗ ( du ) (cid:19) g ( u ) − ∂p∂u ( u, t ) . Substitution of this formula for j in the effective continuity equation 4.5 leads tothe effective diffusion formula (4.6) ∂p∂t ( u, t ) = ∇ · ( D ( u ) ∇ p ( u, t )) , where the effective diffusion coefficient is given by D ( u ) = D (cid:18)(cid:90) S u ∗ ( du ) (cid:19) g ( u ) / . = D F ∇ u ( u ) dνdu ( u ) . Finite transversal diffusion rate.
We will now consider the case whendensity function P = P ( x, t ) is not necessarily constant along the cross sections ofthe channel. In general it is not possible to define define D = D ( u ) such that theeffective density function p = p ( u, t ) satisfies the -dimensional diffusion equation4.6 exactly, but for many cases of narrow channels it is possible to find D such that p satisfy 4.6 to a very good approximation. In any case, if such a D existed wecould recover it from of a stable solution ρ = ρ ( u ) to 4.6. In fact, if ρ is such afunction we have that ∇ · ( D∇ ρ ) = 0 , which is equivalent to(4.7) ∂∂u (cid:18) D σ dρdu (cid:19) = 0 where σ = g / = dνdu . OORDINATE FREE EFFECTIVE DIFFUSION IN CHANNELS 16
Hence, we can find a constant
J ∈ R such that(4.8) D ( u ) = J σ ( u ) (cid:18) dρdu ( u ) (cid:19) − . Remark . If we introduce a new variable v = v ( u ) , then we have that D ( v ) = D ( u ( v )) , since D ( v ) = J dνdv ( v ) (cid:18) dρdv ( v ) (cid:19) − = J dνdu ( u ( v )) dudv ( v ) (cid:18) dρdu ( u ( v )) dudv ( v ) (cid:19) − = D ( u ( v )) . We will now assume that ρ is the effective concentration function of a stable solu-tion h = h ( x ) to the full diffusion equation 4.1 (with reflective boundary conditionson W ). We then have that ρ ( u ) = 1 σ ( u ) (cid:90) S u hι U ( µ ) = 1 σ ( u ) dcdu ( u ) for c ( u ) = (cid:90) C [0 ,u ] hµ, and hence dρdu = ddu (cid:18) σ dcdu (cid:19) = 1 σ (cid:18) d cdu − ρ d νdu (cid:19) . Using Lemma 10 we obtain d cdu = (cid:90) S u ( dh ( U ) + h ∇ · U ) ι U ( µ ) , and since ν ( u ) = (cid:90) C [0 ,u ] µ then d νdu = (cid:90) S u ( ∇ · U ) ι U ( µ ) . We conclude that D ( u ) = J σ ( u ) / F λ U ( u ) = J F U ( u ) (cid:18) F U ( u ) F λU ( u ) (cid:19) where λ = dh ( U ) + ( h − u ∗ ( ρ )) ∇ · U. OORDINATE FREE EFFECTIVE DIFFUSION IN CHANNELS 17
Computation of J . By definition, we have J ( u ) = D ( u ) σ ( u ) dρdu ( u ) . Using Fick’s laws j ( u ) = −D ( u ) ∇ ρ ( u ) J ( x ) = − D ∇ h ( x ) and the formulas j ( u ) = 1 σ ( u ) (cid:90) S u ∗ J (cid:91) ∇ ρ ( u ) = 1 σ ( u ) dρdu ( u ) we obtain J ( u ) = D (cid:90) S u ∗ ( dh ) = D F ∇ h ( u ) . The function J = J ( u ) is in fact a constant function (i.e independent of u ), sincefor any two values u and u we have that (by Stokes Theorem and the reflectiveboundary conditions on W ) J ( u ) − J ( u ) = D (cid:90) S u − S u ( ∗ dh ) = (cid:90) C [ u ,u d ∗ dh = (cid:90) C [ u ,u ∗ ∆ h = 0 . Lateral boundary conditions.
It is important to notice that formula 4.8 holds onlyunder the assumption that ρ (cid:48) ( u ) (cid:54) = 0 for all u ∈ R . We can achieve this if for α (cid:54) = β we fix boundary the conditions(4.9) ρ ( a ) = α and ρ ( b ) = β. For fixed values of α and β we will denote the stable solution to 4.6 satisfying theseboundary conditions by ρ α,β . Using the linearity of equation 4.7 we obtain ρ α,β = α + ( β − α ) ρ , . If we denote the constant J associated to ρ α,β by J ( α, β ) then D = σ J ( α, β ) (cid:18) dρ α,β du (cid:19) − . Since D is independent of the choice of α and β we must have J ( α, β ) (cid:18) dρ α,β du (cid:19) − = J (0 , (cid:18) dρ , du (cid:19) − , from which we obtain the formula J ( α, β ) = ( β − α ) J (0 , . Apparently J depends on u , but we will show below that J is actually a constant function(as required for the formula we computed for the effective diffusion coefficient D ) OORDINATE FREE EFFECTIVE DIFFUSION IN CHANNELS 18
The boundary conditions 4.9 can be written in terms of H (using Lemma 10) as σ ( a ) (cid:90) S a hι U ( µ ) = α, σ ( b ) (cid:90) S b hι U ( µ ) = β. If we choose h so that it is has constant value h a in S a and constant value h b in S b ,the above conditions become h a = α and h b = β. Channels defined by harmonic conjugate functions.
Let M be a 2-dimensional oriented surface. We will say that u, v : M → R are harmonic conjugateif dv = ∗ du, or equivalently ∇ v = i ∇ u. Observe that in this case ∗ dv = ∗ ∗ du = − du. The existence of a harmonic conjugate v for u implies that u and v are harmonic,since ∆ u = ∗ d ∗ du = ∗ ( d v ) = 0 , ∆ v = ∗ d ∗ dv = − ∗ ( d u ) = 0 . For fixed value v , v ∈ R , consider a channel C defined as C = { x ∈ M | v ≤ v ( x ) ≤ v } , If we use a harmonic conjugate u of v as projection function for this channel, then u is a harmonic function with reflective boundary conditions on W . The channels C has generating field U = ∇ u |∇ u | . The effective diffusion coefficient both in the infinite and finite transversal diffusionrate cases coincide and is given by the formula D ( u ) = J dνdu ( u ) , where J = (cid:90) S u ∗ du = (cid:90) S u dv = v − v and dνdu ( u ) = (cid:90) S u ∗ du |∇ u | = (cid:90) S u dv |∇ v | . Observe that we can parametrize a cross section S u with a curve x : [ t , t ] → C with ˙ x ( t ) = ∇ v ( x ( t )) , so that A ( u ) = (cid:90) t t | ˙ x ( t ) | = (cid:90) t t |∇ v ( x ( t )) | |∇ v ( x ( t ) | . OORDINATE FREE EFFECTIVE DIFFUSION IN CHANNELS 19
Hence A ( u ) = (cid:90) S u dv |∇ v | . Parametric channels.
In this section we will assume that the channel
C ⊂ M can be parametrized by a map ϕ : [ a, b ] × Ω → M, where Ω is a ( n − -dimensional sub-manifold with boundary of R n − . In localcoordinates we will write the elements of [ a, b ] × Ω as ( u, v ) for u ∈ [ a, b ] and v = ( v , . . . , v n − ) ∈ R n − . If denote the of points in C by x then we have that x = ϕ ( u, v ) , which we will simply write as x = x ( u, v ) . We will let the generatingvector field for C be U = ϕ ∗ (cid:18) ∂∂u (cid:19) , which has u as a projection function. To compute the effective diffusion coefficientfor C (in both the finite and infinite transversal diffusion rate cases) we will needto compute dνdu , F ∇ u , ρ, dh ( U ) and ∇ · U, where h is a natural projection function for C and ρ its corresponding effectivedensity function. To compute the above quantities in ( u, v ) -coordinates we willmake use of the metric tensor g = ϕ ∗ ( g M ) , where g M is the metric in M . We havethat g = (cid:32) ∂x∂u · ∂x∂u ∂x∂u · ∂x∂v (cid:0) ∂x∂u · ∂x∂v (cid:1) T g v (cid:33) , where ∂x∂u · ∂x∂v = (cid:18) ∂x∂u · ∂x∂v , . . . , ∂x∂u · ∂x∂v n − (cid:19) and g v is the matrix with entries ( g v ) i,j = ∂x∂v i · ∂x∂v j . The volume form in C is given by µ = det( g ) / du ∧ dv, and hence dνdu ( u ) = (cid:90) Ω ι ∂∂u ( µ ) = (cid:90) Ω det( g ( u, v )) / dv A ( u ) = (cid:90) Ω det( g v ( u, v )) / dv Observe that ∇ u = a ∂∂u + n − (cid:88) i =1 a i ∂∂v i where a a ... a n − = g − ... . OORDINATE FREE EFFECTIVE DIFFUSION IN CHANNELS 20
Since a = det( g v )det( g ) , we conclude that F ∇ u ( u ) = (cid:90) Ω det( g ( u, v )) ι ∇ u ( du ∧ dv )= (cid:90) Ω (cid:18) det( g v ( u, v ))det( g ( u, v )) (cid:19) dv. (4.10)The divergence of U can be computed using the the formula d ( ι U ( µ )) = ( ∇ · U ) µ .In our case we have that d ( ι U ( µ )) = d (det( g ) / dv ) = ∂ det( g ) / ∂u du ∧ dv, and hence ∇ · U = ∂∂u (cid:0) det( g ) / (cid:1) det( g ) / = 12 ∂∂u (log(det( g ))) . If h is the natural projection map on the channel then dh ( U ) = ∂h∂u and ρ = (cid:18)(cid:90) Ω h ( u, v ) det( g ( u, v )) / dv (cid:19) / (cid:18)(cid:90) Ω det( g ( u, v )) / dv (cid:19) References [1] R.M. Bradley. Diffusion in a two-dimensional channel with curved midline and varying width.
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