## Abstract

An asymptotic analysis is presented for the advection–diffusion transport of a chemical species in flow through a small-diameter tube, where the flow consists of steady and oscillatory components, and the species may undergo linear reversible (phase exchange or wall retention) and irreversible (decay or absorption) reactions at the tube wall. Both developed and transient concentrations are considered in the analysis; the former is governed by the Taylor dispersion model, while the latter is required in order to formulate proper initial data for the developed mean concentration. The various components of the effective dispersion coefficient, valid when the developed state is attained, are derived as functions of the Schmidt number, flow oscillation frequency, phase partitioning and kinetics of the two reactions. Being more general than those available in the literature, this effective dispersion coefficient incorporates the combined effects of wall retention and absorption on the otherwise classical Taylor dispersion mechanism. It is found that if the phase exchange reaction kinetics is strong enough, the dispersion coefficient is probably to be increased by orders of magnitude by changing the tube wall from being non-retentive to being just weakly retentive.

## 1. Introduction

Since the classical work of Taylor (1953), the dispersion of a substance that is miscible/mixed with a fluid flowing through a narrow tube has been extensively studied, theoretically and experimentally. The dispersion, or now referred to as Taylor dispersion, is a mechanism that enhances the rate of broadening of a solute cloud in flow through a tube or channel, and can therefore be utilized as an effective means to accomplish dilution or mixing. Taylor dispersion plays a central role in applications as diverse as chromatographic separations in chemical engineering, pollutant transport in the environment, the mixing and transport of drugs or toxins in physiological systems, and so on. As a phenomenon, Taylor dispersion resembles molecular diffusion, but it actually results from an interaction between transverse diffusion and velocity shear, and is therefore a function of the flow type, the geometry of the channel, and also properties of the substance. In whatever contexts, the basic conditions leading to Taylor dispersion are a non-uniform transverse velocity profile and a non-zero axial concentration gradient. Other factors, like flow pulsation, chemical reactions, boundary irregularities, and so on may contribute additional effects to the dispersion. There has been developed a sizable literature presenting alternative approaches to Taylor's analysis, and generalizing his results to applications of all sorts.

This paper concerns the problem of mass transport in oscillatory flow through a small-diameter tube, which is lined with a very thin layer made up of a retentive and reactive material. We consider a piston-driven type of oscillatory flow, which can be induced by a pressure gradient consisting of steady and periodic fluctuating components. The substance that undergoes Taylor dispersion in the fluid is subject to reversible phase exchange with, and irreversible absorption into the wall layer. The primary objective here is to examine the effects of such heterogeneous reversible and irreversible chemical reactions on the dispersion in steady and oscillatory tube flows. To this end, an asymptotic analysis is performed to obtain an overall effective transport equation containing advection and dispersion coefficients that are functions of the chemical reactions and flow oscillation.

There exist a large number of studies on Taylor dispersion under the sole influence of either flow oscillation or chemical reaction. Dispersion of inert matter in pulsating or oscillatory flow has been studied in the contexts of blood flow, airway gas mixing, extraction columns, estuary tidal flow and wave boundary layer. Some classical works in this area may include, among others, Aris (1960), Harris & Goren (1967), Holley *et al*. (1970), Chatwin (1975), Smith (1982), Watson (1983) and Yasuda (1984), while some more recent developments may include Mukherjee & Mazumder (1988), Elad *et al*. (1992), Hazra *et al*. (1996), Bandyopadhyay & Mazumder (1999*a*,*b*) and Ng (2004). The artificial ventilation of the human lung has also motivated studies on dispersion in oscillatory flow, essentially due to waves propagating through the wall of a flexible tube (e.g. Dragon & Grotberg 1991; Hydon & Pedley 1993, 1996). Meanwhile, the effects of phase exchange, partitioning, boundary uptake or chemical reaction on axial dispersion in steady flow were analysed as far back as by Westhaver (1942), Aris (1959), Gupta & Gupta (1972) and Sankarasubramanian & Gill (1973). This type of problem has then been widely extended to applications in chromatography (e.g. Shankar & Lenhoff 1991; Blackburn 2001), in biological or physiological transport (e.g. Davidson & Schroter 1983; Grotberg *et al*. 1990; Phillips *et al*. 1995; Phillips & Kaye 1998; Sarkar & Jayaraman 2002), in environmental contaminant transport (e.g. Smith 1983; Purnama 1988*a*,*b*; Ng & Yip 2001), and so on.

There are, however, much fewer studies that take into account both flow oscillation and chemical reactions. The effect of wall absorption on axial dispersion in oscillatory tube flow has been studied by Mazumder & Das (1992) and Jiang & Grotberg (1993). These authors considered a wall boundary condition that is analogous to a heterogeneous first-order reaction absorbing the substance irreversibly. Also, these authors considered a transport time-scale comparable to the radial diffusion time and paid attention to the fluctuating behaviour of the dispersion process during relatively early times approaching to a developed state. Mazumder & Das (1992) concluded that the wall absorption causes a negatively skewed deviation from Gaussianity, and, therefore, always decreases the dispersion coefficient. Jiang & Grotberg (1993), however, found that, for weak wall conductance, the axial dispersion may be enhanced or diminished by the presence of wall absorption, depending on the oscillation frequency.

In contrast, we consider in the present work a boundary condition for a wall that not only depletes a substance irreversibly, but also allows a reversible exchange of phase with the fluid. A similar reactive boundary condition has been examined by Purnama (1988*b*, 1995) for the effects of absorption and retention on dispersion in steady flow. Purnama (1995) has found that the boundary retention, which increases the dispersion rate, is the dominant effect suppressing the boundary absorption, which decreases the dispersion rate, provided that the absorption rate is not too large. Here, we consider a relatively fast rate of phase exchange and a relatively slow rate of absorption, and the objective is to determine the combined effects of these two kinds of wall reactions on dispersion in oscillatory flow. Phillips & Kaye (1998) have looked into the developing stage of dispersion with phase exchange in steady flow through a pipe which is lined with a thin stationary wall layer of high diffusive resistance. In particular, they presented approximate solutions applicable to the early Taylor (i.e. diffusion extended across the fluid only), intermediate (i.e. non-Gaussian and transient) and the fully developed regimes. In contrast, our attention is more on the long-time asymptotic state of the dispersion, for which analytical expressions can be developed for the transport (advection and dispersion) coefficients as functions of the flow oscillation, phase partitioning, phase exchange kinetics and the wall absorption.

This paper is organized as follows. The flow and transport problems are described in detail in §2, where arguments on the scalings of the various variables and parameters are also discussed. The ratio of the radial to the axial length-scales is a small parameter, based on which we may perform an asymptotic analysis to deduce the effective transport equations valid at different time-scales. In this work, we will follow the homogenization technique (e.g. Mei *et al*. 1996), which is a multiple-scale method of averaging that can be used to derive directly the transport equations for the developed mean concentration containing expressions for the advection velocity and dispersion coefficient as functions of the hydrodynamics, mixing and chemical effects. These equations are, however, valid only at a sufficiently long time when the developed state is attained, but they require an initial condition that depends on the early transient development. We thus adopt the perturbation expansions introduced by Fife & Nicholes (1975), by which the concentrations are separated into developed and transient terms. Perturbation equations for the developed concentration are analysed in §3, all the way to the third order, in order to bring forth the dependence of the dispersion coefficient on the two wall reactions. Analytical expressions, wherever possible, are derived in §4 for the various dispersion coefficient components, due separately to the flow being steady or oscillatory, and to the wall condition being with retention only or with both retention and absorption. The slow-oscillation and fast-oscillation limits for the dispersion coefficient components under oscillatory flow are deduced, respectively, in §§5 and 6. These limits can be compared with those published in the literature for the particular case when the wall is absorptive only. In §7, we offer some numerical discussions on the dispersion coefficient components as functions of the controlling parameters. Some original findings will be presented there.

## 2. The problem under consideration

For fully developed viscous flow along a long uniform tube of circular cross-section, the flow is strictly one-dimensional and the momentum equation is(2.1)where *u*(*r*, *t*) is the axial velocity, ∂*p*/∂*x* is the axial pressure gradient, *ρ* is the fluid density, *ν* is the kinematic viscosity, *x* is the axial distance along the tube, *r* is the radial coordinate and *t* is time. The flow is forced purely by a pressure gradient that comprises a steady and a harmonically fluctuating component,(2.2)where *K*>0 is the steady part of the pressure gradient, *ϕ* is a factor such that *ϕK* is the amplitude of the oscillatory part of the pressure gradient, Re stands for the real part, i is the complex unit and *ω* is the angular frequency of the oscillation. With the no-slip condition at the wall, *u*(*a*, *t*)=0, where *a* is the radius of the tube, the pressure gradient given above will produce the following velocity profile:(2.3)where(2.4)is the steady component of the velocity (where the overbar denotes time averaging over a period of oscillation), in which(2.5)is the sectional average of (where the angle brackets denote averaging across the tube section), and(2.6)is the complex amplitude of the oscillatory component of the fluid velocity. In the equation above, *J*_{0} is the zeroth-order Bessel function of the first kind, and(2.7)where(2.8)is the thickness of the Stokes boundary layer resulting from the oscillation of the flow.

Let us now consider the transport along the tube of a chemical species which is completely miscible with the fluid, but may undergo heterogeneous reactions at the wall. We suppose that on the wall is a thin lining or substratum in which the chemical may be subject to two kinds of first-order reactions, one irreversible and the other reversible. The irreversible reaction can be a reaction by which the chemical is transformed and absorbed irreversibly into the wall material. Meanwhile, a reversible exchange between different phases of the chemical may take place across the fluid-lining interface. It is possible that in the lining the chemical exists in a form different from that in the flowing fluid. To distinguish between the forms of the chemical that exist in the fluid and in the lining, we shall refer to them as the mobile and the immobile phases, respectively. When at equilibrium, these two phases will have their concentrations in a fixed ratio(2.9)where *C* is the concentration (mass of chemical per bulk volume of fluid) of the mobile phase, *C*_{s} is the concentration (mass of chemical per surface area of wall) of the immobile phase and *α* is a partition coefficient which can be taken as a chemical specific constant. When, in general, equilibrium is not attained, the phase exchange will take place in either forward or backward direction with a rate that is supposed to be described by first-order kinetics(2.10)where *k* is the reversible reaction rate constant. Here an assumption is made that the wall lining is so thin that we may need to consider only the integrated effects across the layer. Therefore, the concentration of the immobile phase, after integration over the thickness of the lining, is expressible by the total mass of the phase per unit surface area of the wall. Also, on disregarding the lining thickness, we may specify the reaction conditions on *C* right at the core radius *r*=*a*.

With the heterogeneous reactions specified above, the problem for the transport of the chemical can be formulated as follows:(2.11)together with the boundary conditions(2.12)(2.13)where *D* is the molecular diffusivity of the chemical in the fluid and *Γ* is the irreversible absorption rate constant. One may note that the wall condition (2.13) encompasses both an irreversible absorption process similar to the one considered by Mazumder & Das (1992) and Jiang & Grotberg (1993), and a reversible phase exchange reaction as studied by Phillips & Kaye (1998, although they used a radial diffusion model instead of a first-order kinetic model for the phase exchange).

In order to prepare grounds for the perturbation analysis, further assumptions regarding the scalings of the various physical quantities are made as follows. First, we consider that a sufficiently long time has passed since the discharge of the chemical into the flow so that the length-scale for the axial spreading of the chemical cloud is much greater than the tube radius. By this, we mean that *x*=*O*(*L*) and *r*=*O*(*a*), where *L* is a characteristic longitudinal distance for the chemical transport, and the following is a small ratio(2.14)to be used as the ordering parameter. Second, there are distinct time-scales for the key processes involved in the transport problem. The basic time-scale is of course one oscillation period of the flow, which in this work is supposed to be so short a period of time to result in any appreciable transport effects down the tube. We, however, assume that the tube is so fine in bore that diffusion across the entire section may be accomplished within this short time-scale. Also, the reversible phase exchange is fast enough so that local equilibrium may be largely achieved over a finite number of oscillations. Therefore, we may express(2.15)The irreversible absorption, however, takes place at a much slower rate, which is assumed to be comparable with the advection speed down the tube. These two processes are supposed to be effective on a time-scale that is one order of magnitude longer than *T*_{0}:(2.16)In this work, the Péclet number is supposed to be equal to or greater than order of unity:(2.17)Because of (2.14), axial diffusion/dispersion requires an even longer time to be effective:(2.18)Based on these time-scales, we may introduce accordingly(2.19)which are, respectively, the fast, medium and slow time variables. Also, the relative significance of the terms in the transport equation (2.11) and the boundary conditions (2.12) and (2.13) may now be identified as indicated by the power of *ϵ* below:(2.20)(2.21)(2.22)

Following Fife & Nicholes (1975), the terms involving a time-scale *T*_{1} or longer can be labelled as ‘developed’, while those terms that vanish exponentially as *t*→∞ with relaxation time proportional to *T*_{0} are called ‘transient’ terms. Therefore, the transient terms can be neglected for a time that is a few times longer than *T*_{0}. The leading order transient term may arise as a result of the non-uniform initial distribution of the solute over the tube's cross-section, and vanishes when this distribution is uniform.

The objective here is to deduce a section-averaged diffusion-type transport equation for the developed mean concentration, which represents the long-time behaviour of the solution to the problem (2.20)–(2.22). To solve this equation, an appropriate initial condition is required. It has been shown by Fife & Nicholes (1975) that, rather surprisingly, in the case of a non-uniform cross-sectional distribution of the initial concentration, the initial values of the developed mean is not exactly equal to the mean of the initial concentration.

In order to develop a proper initial concentration, the transience in the early stage must be taken into account. Adopting the asymptotic expansion introduced by Fife & Nicholes (1975), we assume that the concentrations may be expanded in the following manner:(2.23)(2.24)In these expansions, *C*^{(n)} and are the developed terms, which can only be a purely oscillatory function of the short time variable *t*_{0}. It is anticipated that the oscillatory effect does not show up on the zeroth order, and therefore the leading order terms are taken to be independent of this time variable. The *W*^{(n)} and are the transient or short time-scale terms, which depend only on the fast time variable and are required to vanish as *t*_{0}→∞.

A multiple-scale asymptotic analysis then begins upon substitution of the series (2.23) and (2.24) and the following expansion of the time derivative into the equation and boundary conditions (2.20)–(2.22):(2.25)Differential perturbation equations obtained by collecting terms of equal orders are split into pairs of equations for the developed and transient terms. In the following sections, we shall focus only on the derivations involving the developed terms. We have also worked out for this problem the transient terms and the initial concentration, but in the interest of space the details are omitted in this paper.

## 3. Developed concentrations

### (a) Leading order

At *O*(1), (2.20)–(2.22) give(3.1)(3.2)and(3.3)Equations (3.1) and (3.2) obviously imply that the leading order concentration is independent of *r*:(3.4)The boundary condition (3.3) then gives(3.5)As expected, the mobile phase of the chemical is at local equilibrium with the immobile phase at the leading order.

### (b) First order

At *O*(*ϵ*), (2.20)–(2.22) give(3.6)(3.7)and(3.8)Taking time average of (3.6)–(3.8), we get(3.9)(3.10)and(3.11)where the overbar denotes time averaging (with respect to the fast time variable *t*_{0}) over one period of oscillation. We further take section average of (3.9) and may get after using (3.5), (3.10) and the first equality in (3.11):(3.12)where the angle brackets denote averaging across the section, and(3.13)is the retardation factor resulting from the reversible partitioning of the chemical into two phases. To be more specific, the retardation factor can be defined to be the ratio, when at equilibrium, of the total mass of the two chemical phases to the mass of the mobile phase per unit length of tube. Equation (3.12) is the leading order effective transport equation describing the rate of change on the time-scale *T*_{1}, which is controlled by advection at a retarded speed , and a first-order reaction owing to the irreversible wall absorption at a rate of 2*Γ*/*aR*. It is remarkable that for mass transport over this time-scale *T*_{1}, which is much longer than an oscillation period, the wall absorption plays simply the role of a sink that depletes mass continuously from the flow.

We next find expressions for the *O*(*ϵ*) concentrations. Substituting (3.12) back into (3.6) and (3.8) gives(3.14)and(3.15)where (3.5) has been used. These equations suggest that we may let the following relationships for the *O*(*ϵ*) concentrations:(3.16)and(3.17)where *N*(*r*), *N*_{s}, *M*(*r*), *M*_{s}, *B*(*r*) and *B*_{s}, as governed by the boundary-value problems below, are functions of the cross-section and *C*^{(10)}(*x*, *t*) is an undetermined function which is independent of *r* and *t*_{0}. This undetermined function *C*^{(10)} may be absorbed into *C*^{(0)}, and hence can be ignored as far as the transport of the developed mean concentration is concerned.

On matching with the steady terms associated with ∂*C*^{(0)}/∂*x*, we find the function *N*(*r*) to be governed by(3.18)and *N*_{s} to be determined from the boundary conditions(3.19)On matching with the steady terms associated with *C*^{(0)}, we also get the following equations for *M*(*r*) and *M*_{s}:(3.20)and(3.21)

Similarly, on matching with the oscillatory terms, we obtain the following problem for the complex functions *B*(*r*) and *B*_{s}:(3.22)with the boundary conditions(3.23)Solutions to these boundary-value problems are presented in §4, where the dispersion coefficients are to be expressed explicitly.

### (c) Second order

At *O*(*ϵ*^{2}), (2.20)–(2.22) give(3.24)(3.25)and(3.26)

With the substitution for the following terms (by making use of (3.16), (3.17) and (3.12)):(3.27)(3.28)and(3.29)where the asterisk denotes the complex conjugate, the averaging of (3.24) with respect to time over a period followed by that with respect to *r* across the section will give(3.30)where(3.31)is a higher-order correction factor to the advection velocity,(3.32)is a higher-order correction factor to the decay rate,(3.33)is a dispersion coefficient due to the steady part of the fluid motion, and(3.34)is a dispersion coefficient due to the oscillatory part of the fluid motion. Equation (3.30) is an *O*(*ϵ*) effective transport equation describing the slow time rate of change, which is controlled by advection, first-order decay, diffusion, and more important, dispersion. It is remarkable that while *ζ* and *κ* are solely caused by the irreversible wall absorption, the two dispersion coefficients *D*_{Ts} and *D*_{Tw} are affected only by the reversible phase exchange. In order to find the effects of the wall absorption on the dispersion, we need to advance to the next order.

### (d) Third order

The weak wall absorption has so far manifested itself only as a sink term and a correction factor to the advection speed in the effective transport equations. Its effects on the dispersion will now show up as the transport of *C*^{(2)} is considered. To focus just on this part of deduction, we limit ourselves below only to the details sufficient for the presentation of the dispersion coefficients that are functions of the wall absorption.

Following similar steps as in the previous orders, we may express(3.35)and(3.36)where *U* and *V* are forced by the steady flow and *X* and *Y* by the oscillatory flow. The problem for *U* is(3.37)with the boundary conditions(3.38)and that for *V* is(3.39)with the boundary conditions(3.40)The function *X* is governed by(3.41)with the boundary conditions(3.42)while *Y* is governed by(3.43)with the boundary conditions(3.44)

Substitution of (3.35) and (3.36) into the third-order transport equation, which is then averaged with respect to time and across the section, will give us the two desired dispersion coefficients arising from the wall absorption:(3.45)is the dispersion coefficient associated with the steady flow, and(3.46)is the dispersion coefficient associated with the oscillatory flow.

Finally, we may put these two additional dispersion coefficients into (3.30), which is then combined with (3.12) to give an overall effective transport equation (without the need to separate the time variable any more):(3.47)Despite the two higher-order dispersion coefficients, the equation above is accurate to *O*(*ϵ*), and incorporates the processes of advection, first-order reaction and dispersion. Explicit expressions for the effective transport coefficients will be derived and their properties will be discussed in the following sections.

## 4. Effective coefficients

### (a) Correction factors ζ and κ

The problem (3.20) and (3.21) can be readily solved to give(4.1)and(4.2)where *M*(0) is undetermined, but has no effect on the evaluation of *ζ* and *κ*. One can easily show that on substituting (4.1) and (4.2) into (3.31) and (3.32), all the terms with *M*(0) cancel out to zero. After some algebra upon substitution of *M*, *M*_{s}, *N* and *N*_{s} (where *N* and *N*_{s} are given below), these correction factors can be found to be(4.3)and(4.4)where , , and(4.5)are, respectively, the Damköhler number and the dimensionless absorption number, representing the significance of the kinetics of the phase exchange with, and the rate of absorption into the wall. According to (2.15) and (2.16), these numbers are expected to be of the orders: Da=*O*(1), .

### (b) Dispersion coefficients due to steady flow

#### (i) Wall retention: D_{Ts}

On substituting (2.4) for , the problem (3.18) and (3.19) can readily be solved to yield(4.6)(4.7)where *N*(0) is undetermined unless a uniqueness condition is specified. Again, this uniqueness condition is unnecessary as far as the advection speed correction factor *ζ* and the dispersion coefficient *D*_{Ts} are concerned. One can easily show that on substituting (4.6) and (4.7) into (3.31) and (3.33), the terms with *N*(0) in either expression cancel out to zero. It is also of interest to formally prove that *D*_{Ts} is positive definite. With (3.18) and (3.19), (3.33) can be manipulated as follows (where integration by parts is used in the second step):(4.8)which is always positive and does not depend on the value of *N*(0). Substituting (4.6), we may finally obtain an explicit expression for the dispersion coefficient due to steady flow through a tube with kinetic phase exchange with the tube wall:(4.9)which consists of a component arising from the classical Taylor dispersion mechanism (i.e. interaction between radial velocity gradient and diffusion) but affected by the equilibrium partitioning of the chemical between the mobile and immobile phases, and a component due to the finite rate of mass transfer between the two phases. This dispersion coefficient can be compared with that obtained by Aris (1959), who presented a formal deduction, using the method of moments, for the dispersion of a gaseous solute in a tube under kinetic exchange with a liquid phase flowing in an annular layer held on the wall. Aris' (1959) equation (23) in the case of a very thin and stationary liquid layer can be shown to be identical to (4.9), which has also been obtained by Ng (2000) using the method of homogenization. In the limiting case when the wall is inert (*α*=0 and *R*=1), the well-known dispersion coefficient derived by Taylor (1953) for steady laminar pipe flow is also recovered(4.10)

Introducing the following normalization, the dispersion coefficient in dimensionless form (distinguished by a caret) can be written as(4.11)

#### (ii) Wall absorption:

This coefficient, given by (3.45), can be evaluated upon solving (3.37)–(3.40) for *U*, *V*, *U*_{s} and *V*_{s}. Again, one may show that the undetermined constants in these and other functions will have no effects on . Ignoring the arbitrary constants, the functions *U* and *V* are expressible by(4.12)(4.13)Substituting these and other quantities into (3.45) and after some algebra, the dispersion coefficient can be expressed in dimensionless form as follows:(4.14)which is linearly proportional to the absorption number representing the effect of the mass depletion by wall absorption. In the case of a non-retentive wall ( or *R*=1), the coefficient above reduces to(4.15)

### (c) Dispersion coefficients due to oscillatory flow

#### (i) Wall retention: *D*_{Tw}

The boundary conditions in (3.23) for the problem of *B*(*r*) can first be rewritten as(4.16)where(4.17)

With *u*_{w} given in (2.6) and the boundary conditions given above, the equation (3.22) has the following solution:(4.18)where(4.19)(4.20)in which , Sc=*ν*/*D* is the Schmidt number and *J*_{1} is the first-order Bessel function of the first kind. The solution (4.18) is based on the assumption that *ν*≠*D* or Sc≠1, but as is shown below this condition can be relaxed for the dispersion coefficient that is to be deduced from this solution of *B*(*r*).

Similar to *D*_{Ts}, one may also prove that *D*_{Tw} is positive definite. With (3.22), (4.16), (3.34) can be manipulated to become as follows (where again integration by parts is used):(4.21)which obviously is always positive since(4.22)

Now, we may find after some algebra upon substituting (4.18) into (3.34),(4.23)This is an explicit expression by which one can evaluate for the dispersion coefficient due to the oscillatory component of the flow subject to kinetic phase exchange between the fluid and the wall. Using L'Hospital's rule, one may formally show that the expression above has the following finite limit for the case Sc=1:(4.24)where ‘Im’ stands for the imaginary part.

In the absence of wall retention, the expression (4.23) reduces to the classical result first obtained by Aris (1960) and later re-deduced by Watson (1983) on the dispersion of a passive solute in pulsating flow through a tube. The Bessel functions can be changed into Kelvin's ber and bei functions if we re-define the frequency *ω*=−*ω*′ so that , where *Λ*=*a*(*ω*′/*ν*)^{1/2} is the Womersley number. It follows that(4.25)(4.26)(4.27)(4.28)On substituting these relations into (4.23), we may recover Watson's (1983) expression for a non-retentive wall, i.e. *α*=*β*=0:(4.29)where *P*=*ρKϕ* is the pressure fluctuation amplitude used by Watson (1983), and the functions *T*_{1}, *T*_{2} and *T*_{3}, which were introduced by Joshi *et al*. (1983) to re-express Watson's result, are given by(4.30)(4.31)(4.32)Joshi *et al*. (1983) carried out experiments to determine the dispersion of a gas under oscillatory flow through a circular tube, and found their results in excellent agreement with the theoretical predictions of Watson (1983). The experiments were conducted in the parameter ranges: Pe=*O*(10–10^{3}) and *Λ*=*O*(1–10), with the flow being limited to the laminar regime. The present extended expression (4.23) that incorporates the effects of reversible phase exchange with the wall is expected to be valid for similar ranges of the parameters.

Aris (1960) and Watson (1983) also examined the asymptotic limits of the dispersion coefficient when the oscillation is very fast or very slow. It can be shown that, for a fixed pressure forcing amplitude, the dispersion coefficient will tend to zero very rapidly as the oscillation frequency increases, and will tend to the following limit when the frequency is small and the wall is non-reactive:(4.33)Therefore, for *ϕ*=1 and slow oscillation, the dispersion coefficient due to the oscillatory flow can be as much as half that due to the corresponding steady flow: *D*_{Tw}/*D*_{Ts}=1/2. Although Aris (1960) arrived at basically the same limiting relationship, he mistakenly calculated the ratio to be 1 to 128 and concluded that the periodic flow was probably to contribute rather less than 1% to the total dispersion. In fact, on drawing the conclusion, Aris had neglected a factor of 1/8 that he had earlier introduced in the amplitude of the fluctuations in the pressure gradient. When (1/8)^{2} is pulled out from 1/128, the ratio will then be rectified to one-half. Contrary to Aris' conclusion, the fluctuating flow can indeed contribute considerably to the total dispersion, if the oscillation period is not too short. We will further look into the dispersion under a slower oscillation in §5, where it will be shown that the limiting ratio of *D*_{Tw}/*D*_{Ts}=1/2 is true even when the wall is reactive.

Let us first introduce the following normalization (distinguished by an overhead caret):(4.34)It can be noted that is inversely proportional to the Womersley number. There are two desirable normalization scales for the dispersion coefficient due to oscillatory flow. The first one, which enables a direct comparison between *D*_{Ts} and *D*_{Tw}, is good when the amplitude of the oscillatory pressure gradient is kept constant if the frequency is to be varied. Using the same scale as that for the steady flow, the normalized dispersion coefficient (distinguished by a caret) are expressible by(4.35)where(4.36)in which(4.37)Alternatively, the flow can be under a volume-cycled oscillation such that the tidal or stroke volume is maintained constant when the frequency is varied. In this case, it is desirable to normalize *D*_{Tw} with respect to , where *V*_{T} is the tidal volume that can be defined to be equal to twice the amplitude of the volume fluctuation (Watson 1983):(4.38)Accordingly, an alternative normalized *D*_{Tw} (distinguished by a tilde) may be obtained as below:(4.39)

#### (ii) Wall absorption:

This coefficient, given by (3.46), can be evaluated upon solving (3.41)–(3.44) for *X*, *Y*, *X*_{s} and *Y*_{s}. It may also be shown that the undetermined constants in these and other functions will have no effects on . Since no closed-form analytical solutions are available to the *X* and *Y* problems in general, we have solved these problems numerically by a second-order centred finite-difference scheme. In §§5 and 6, we will deduce and discuss the analytical limits for *D*_{Tw} and when the oscillation becomes very slow or very fast.

## 5. Slow oscillation

### (a) Wall retention: D_{Tw}

In this section, the problem is re-examined for the case when the flow oscillation is so low in frequency that its period becomes comparable to the advection time-scale. The first two time-scales may now be re-defined as follows:(5.1)(5.2)In this case, the oscillatory behaviour occurs on the longer time-scale *T*_{1}, and therefore the fast time variable *t*_{0} may now be dropped. Also, time averaging of the transport equation is not to be performed, and the focus here is to examine how the dispersion coefficient *D*_{Tw} will fluctuate periodically over the time-scale *T*_{1}. Let us temporarily omit also the steady component of the flow and the irreversible wall absorption. The flow velocity is now simplified to(5.3)where *u*_{w}(*r*) is given by (2.6).

The effective transport equations can be deduced following the same procedure of analysis as before, but without taking the time averaging. We present below only the key results that are different from the previous ones. At the leading order, the effective transport equation is(5.4)where(5.5)

The *O*(*ϵ*) concentrations are expressible by(5.6)and(5.7)where *B*(*r*) and *B*_{s} are governed by(5.8)and(5.9)On substituting (2.6) and (5.5), the equations above can be solved to yield(5.10)(5.11)where(5.12)For simplicity, we have dropped all the arbitrary additive constants in (5.10) and (5.11), as they will have no effect on *D*_{Tw} to be deduced.

The *O*(*ϵ*) effective transport equation is(5.13)where(5.14)is the steady component of the dispersion coefficient resulting from the oscillatory flow, and(5.15)is the time varying component of the dispersion coefficient resulting from the oscillatory flow. Invoking the normalization introduced in (4.34), the two dispersion coefficients in dimensionless form (distinguished by a caret),(5.16)can be found to be as follows:(5.17)(5.18)where .

Several mathematical properties of the dispersion coefficients derived here are noteworthy. First, unlike that has been deduced earlier, the Schmidt number Sc does not show up as an explicit parameter in these dispersion coefficients. This independence of Sc under a slow oscillation is consistent with our numerical observations to be noted later. The oscillation period is now so much longer than the radial diffusion time-scale that the radial diffusion of mass is no longer to match its transport rate with that of momentum across the tube section. See a comparison between (3.22) and (5.8).

Second, as the frequency further decreases to become so small that or , the coefficients in (5.17) and (5.18) will tend to the following limits:(5.19)(5.20)where , given by (4.11), is the dispersion coefficient due to the steady component of flow. Therefore, when *ϕ*=1,(5.21)Such a relationship has been noted previously by Chatwin (1975). When the oscillation period is very long, the flow can be regarded as quasi-steady at a particular instant, and it appears to vary with time only parametrically. Therefore, the dispersion coefficient under a slowly oscillating flow can simply be obtained on replacing by in the expression for the dispersion coefficient due to steady flow of velocity .

Third, one may formally show that is non-negative at any instant of time. With (5.8) and (5.9), and using integration by parts, the dispersion coefficients (5.14) and (5.15) can alternatively be expressed as follows:(5.22)(5.23)where(5.24)Clearly,(5.25)where the triangle inequality has been used in the second step. On the other hand, fluctuates harmonically with time with a minimum value(5.26)One can immediately infer from (5.25) and (5.26) that or must always be non-negative. This property, which is true for the transport of the developed mean concentration, is in sharp contrast to the possibility of a negative dispersion coefficient which may occur at small times during the contraction phase of a solute cloud in an oscillatory flow (e.g. Smith 1982).

### (b) Wall absorption:

Similar to , the slow-oscillation or quasi-steady limit of for *ϕ*=1 is equal to half the value of the coefficient due to the corresponding steady flow, which is given by (4.14):(5.27)which, in the absence of wall retention ( or *R*=1), reduces to(5.28)This slow-oscillation limit of for *R*=1 can be checked to agree with the one reported by Jiang & Grotberg (1993). They numerically calculated the factor to be approximately 9×10^{−5}, where in their normalization a velocity scale equal to twice the corresponding steady flow mean velocity was used. If their velocity scale were adopted, the numerical factor to appear in our expression would be(5.29)which agrees very closely with the limiting value shown in fig. 6 of Jiang & Grotberg (1993). Here, (5.27) is a more general expression for the slow-oscillation limit of incorporating the effects due to the reversible kinetic phase exchange with the wall.

## 6. Fast oscillation

### (a) Wall retention: D_{Tw}

Dispersion in fast oscillatory flow is of practical importance to the artificial ventilation of the lung (Grotberg 1994). When the oscillation frequency is sufficiently high such that , the Stokes boundary layer, where viscous momentum transport is significant becomes a very thin layer adjacent to the tube wall. If further , a concentration boundary layer will also exist near the tube wall.

By ignoring the wall curvature across the thin boundary layers, the momentum equation (2.1) can be simplified to yield the following velocity profile for the oscillatory flow:(6.1)and also the problem (3.22) for *B* can be simplified to produce the solution(6.2)where(6.3)in which is given in (4.37). Based on these limiting expressions, the dispersion coefficient (3.34) can be evaluated again to give(6.4)where *A*_{r} and *A*_{i} are, respectively, the real and imaginary parts of *A*. In the case when the wall is non-retentive (*R*=1 or ), it can be shown that *A*=Sc^{1/2} and the dispersion coefficient above can be written as (recalling is the Womersley number):(6.5)This is an expression that has been given before by Elad *et al*. (1992) and Jiang & Grotberg (1993) for the fast-oscillation limit of *D*_{Tw} in the absence of wall retention.

Normalized with respect to the tidal volume, the dispersion coefficient (6.4) can be written in the following dimensionless form:(6.6)which grows linearly with , as noted by Elad *et al*. (1992).

### (b) Wall absorption:

By virtue of a small , the problems (3.41) and (3.43) can now be much simplified upon discarding the sub-dominant terms, and give the following asymptotic solutions:(6.7)and(6.8)which are valid when the respective conditions for the smallness of are satisfied. Substitution into (3.46) then provides the following relationship for the fast-oscillation limit of :(6.9)which is always negative. In the absence of wall retention , the relationship above reduces to(6.10)where the numerical factor of −1/2 is somehow different from the factor of −1/4 that has been reported by Jiang & Grotberg (1993). We have confirmed our analytical limit by comparing with results obtained by solving numerically the full problems (3.41) and (3.43) with a finite-difference method as mentioned earlier. Indeed, Jiang & Grotberg also reported that, for one example case of moderately fast oscillation, the ratio was calculated to be around −4% when (using the present notation). This implies that the factor is around −40%, which is closer to our factor of −1/2 than theirs of −1/4.

Now, it is clear from the two asymptotic limits (5.28) and (6.10) that, at least for the non-retentive case, the wall absorption may increase or reduce the axial dispersion for sufficiently slow or fast oscillation of the flow, respectively. This has already been discussed by Jiang & Grotberg (1993). We will discuss in §7, as given by the more general asymptotic relations (5.27) and (6.9), the effect of wall absorption on dispersion in the presence of kinetic phase exchange with the wall.

## 7. Numerical discussions on the dispersion coefficients

We need to specify and Da in order to compute and , and need to further specify *ϕ*, and Sc in order to compute and . The absorption number , which is only a linear proportionality factor, is also required for the computation of and . While the parameter is the ratio of chemical mass distributed between the phase retained by the wall and that carried by the flow, the Damköhler number Da is the ratio of the phase exchange rate to the diffusion rate. In particular, for a non-retentive or inert boundary. It is clear that in the expression (4.11) for , the terms inside the parentheses are due to the equilibrium partitioning of the chemical between the fluid and wall phases, and the last term is due to the kinetics of the phase exchange. Clearly, stronger kinetics (or a slower exchange rate or smaller Da) will lead to a larger value of .

The other dispersion coefficient depends also on and Sc, which are, respectively, the ratio of the Stokes boundary thickness to the tube radius, and the ratio of kinematic viscosity to molecular diffusivity. The number increases with the oscillation period. Therefore, the higher the frequency, the smaller the value of . The Schmidt number Sc is of order unity when the fluid is a gas, and is much greater than unity when the fluid is a liquid. For dispersion to be significant, one may expect that, if the oscillation is pressure gradient-cycled, or the period of oscillation must be long enough for the Stokes boundary-layer thickness to be at least comparable with the tube radius. The opposite would be true, i.e. or the period of oscillation must be so short that the Stokes boundary layer is just a tiny fraction of the tube radius, if the oscillation is volume-cycled instead. The phase partitioning and exchange kinetics now affect in a more complicated manner through the parameter .

### (a) and

Let us further examine with numerical plots how the dispersion coefficients will vary with the parameters. From figure 1, we note that a higher value of Sc may lead to smaller , but the effect is significant only when is not too large and . The dispersion coefficient turns out to be rather insensitive to Sc when Sc itself is small, or when the oscillation period is long enough (say, when for , and when for ). As noted already in §5, the dispersion coefficient is indeed independent of Sc when the slow-oscillation limit is approached (i.e. the oscillation period becomes much longer than the cross-sectional diffusion time-scale).

Recall that the two dispersion coefficients and are based on different normalization scales: the former is for a pressure gradient-cycled oscillation, while the latter is for a volume-cycled oscillation. As shown in figure 2, the oscillation frequency will have qualitatively different effects on these two normalized dispersion coefficients. While increases monotonically with (i.e. a larger dispersion coefficient with a lower frequency), the opposite is true for . Therefore, under a volume-cycled oscillation, a larger dispersion coefficient will result from a higher frequency, since for which a stronger pressure forcing needs to be applied to maintain a constant tidal volume. For , the rate of increase obviously turns from very steep to almost flat as the oscillation period increases over the range indicated. The value of beyond which becomes almost constant depends on Sc and . In the cases shown here, when is greater than 1.5 or so, becomes virtually independent of both Sc and . This is to signify the asymptotic approaching of the dispersion coefficient to its slow-oscillation limit.

We illustrate the effects of the phase partitioning and kinetics on the steady and oscillatory dispersion components and , as shown in figure 3*a*,*b*. It is clear that the phase exchange effects on are quite similar to those on . The figure confirms that, as already pointed out above, stronger kinetics of the phase exchange (i.e. smaller Da) will give rise to a larger value of either dispersion coefficient. It is remarkable that the dispersion coefficients may be increased sharply, by as much as almost 2 orders of magnitude, as the tube wall just turns from perfectly inert to slightly retentive. Even a value of as small as 0.05–0.1 can enhance the dispersion coefficients appreciably under a slow rate of exchange. It can be checked that, from (4.11), the gradient of the curve at is(7.1)which means a rather sharp increase of the dispersion coefficient in response to a slight departure from the inert wall condition, if the phase exchange is strongly kinetic. The dispersion coefficients reach their peak values as the phase partition ratio is in the range 0.2–0.3. Further increase in will cause the dispersion coefficients to decrease, essentially because they are then much weighted down by the retardation factor. It is possible that, especially for fast kinetics (Da≥1.0), the dispersion coefficients subject to a sufficiently large phase partitioning are smaller than their inert counterparts. Also, to see the significance of relative to , the ratio for the plotted cases is shown in figure 3*c*. This ratio varies between 0.1 and 0.4 for Da=0.1, but has an almost constant value of 0.45 for Da=1.0 and 10.0. Except for some very small values of Da and , the ratio is, in general, a finite fraction of unity, and tends to the limiting ratio of 0.5 as increases.

### (b) and

Unlike *D*_{Tw} and *D*_{Ts}, which are positive definite, the absorption-induced dispersion coefficients and can be positive, zero or negative depending on the parameters. As remarked by Jiang & Grotberg (1993), many previous studies for steady tube flows (e.g. Sankarasubramanian & Gill 1973; Smith 1983) have claimed that the wall absorption is to decrease the dispersion. This is in fact not necessarily true when the absorption is weak, as is considered in the present work. Figure 4 shows how may change in sign as the oscillation frequency varies, for the cases in the absence/presence of wall retention. In particular, figure 4*a* for actually presents the same results as fig. 6 of Jiang & Grotberg (1993), although the two figures were plotted using different normalized variables. These authors also offered some physical arguments to explain the effects of wall absorption when it is weak: it diminishes dispersion when the oscillation is fast, but enhances dispersion when the oscillation is slow (including the quasi-steady limit). The critical at which the dispersion coefficient changes in sign increases with Sc, because as the molecular diffusivity is decreased, the radial diffusion time becomes larger and a longer oscillation period is required for the slow-oscillation argument to be valid. Here, as shown in figure 4*b*,*c*, the results are extended to the cases with kinetic phase exchange with the wall. Note that different ranges of are shown for the three cases plotted in figure 4. This dispersion coefficient can dramatically be increased in magnitude by having the phase partition ratio to be just as small as 0.1, and the increase is even more amplified when the phase exchange kinetic is strong. In case (*c*), the dispersion coefficient for Sc=1.0 and 10.0 will not turn positive (it has to do so before approaching to the quasi-steady limit) until a rather large value of is attained. By and large, the kinetic phase exchange with the wall is to considerably magnify the effect of wall absorption on dispersion.

Figure 5 shows that the coefficient may respond rather differently to a change in the wall retention condition when the flow oscillation is fast or slow. For corresponding to a relatively fast oscillation (figure 5*a*), the wall absorption is to diminish the dispersion, regardless the values of and Da. This is consistent with the fast-oscillation asymptotic limit given by (6.9). Let us recall a physical explanation by Jiang & Grotberg (1993). When the oscillation is fast, the concentration is nearly uniform across the section except in the boundary layer near the wall. The chemical species at the upstream side of the solute cloud is largely confined to this thin layer near the wall, where the species is most susceptible to loss by wall absorption. The mass depletion at the upstream side is in effect to decrease the variance of the concentration axial distribution, thereby diminishing the dispersion. This diminishing effect is aggravated by the presence of wall retention, because the uptaking of mass from the flow to store as an immobile phase in the wall will only further facilitate the depletion by absorption.

For corresponding to slower oscillation (figure 5*b*), the wall absorption will instead enhance the dispersion, but only for less than a certain critical value if Da is small. This is exemplified by the case of Da=0.1: is increased by some four orders of magnitude when is raised from 0 to 0.1, but then drops with increase in and becomes negative when . Let us follow again the argument by Jiang & Grotberg (1993). When the oscillation is slow, the mass depletion by wall absorption is more intensive around the solute cloud peak than at the upstream side, causing a flatter axial distribution of the concentration. The variance of the distribution is thus increased, or the dispersion is enhanced. The presence of strongly kinetic phase exchange with the wall, as long as the partition ratio remains small, will add to this enhancement effect. This is essentially due to the slow release of the immobile phase back to the flow at the upstream side, resulting in a long tail of the solute cloud. This is to further increase the variance of the axial distribution of the concentration, or the dispersion. However, when the partition ratio becomes sufficiently high so that a larger fraction of the mass is retained for a longer time as an immobile phase at the wall, the mass will be depleted substantially by wall absorption before it may be released back into the flow. This is in some sense similar to the case with strong wall absorption. Not only will the long tail not to appear, but also the concentration at the upstream side, where the mass is nearer the wall, will sharply decay to zero. The variance and therefore the dispersion is therefore decreased.

The effects of wall absorption, with the influence of wall retention as well, on the dispersion in steady flow are shown in figure 6. One can see that the effects are similar to those of slow oscillation described above. The initial sharp increase of with for small Da is extremely remarkable. One may find that, from (4.14), the gradient of the curve at is(7.2)which means a very sharp increase (even sharper than that of ) of the dispersion coefficient in response to an even very small degree of wall retention, if the phase exchange is sufficiently kinetic. It is probably that, although is very small, the boosting-up effect due to wall retention may render the coefficients and to be comparable with their leading order counterparts.

To enable one to determine whether the effect of the wall absorption is to increase or to decrease the dispersion, we show in figure 7 the demarcation of the -space into regions, where (for steady flow) and (for oscillatory flow) are positive or negative. For , the demarcation is also a function of Sc and ; only the case Sc=0.1 is shown here. While the quantitative effects have been seen and discussed in the preceding figures, figure 7 provides a broader view on the sign of these dispersion coefficients as a function of the parameters. In particular, for steady flow, one may infer from (4.14) that is always positive when . When Da is below this value, is positive only for small . For the oscillatory flow case shown in figure 7*b*, one may see that can turn from negative to positive as the oscillation goes slower. Also, when the wall is non-retentive (i.e. ), is zero for , and is negative for smaller . When is, for example, equal to 0.2, a very small increase of from zero will, however, cause the coefficient to turn back positive. This further supports the argument that the presence of wall retention, no matter how weak it is, may lead to results that can be dramatically different from that of the perfectly non-retentive case, both quantitatively and qualitatively.

## 8. Summary and concluding remarks

Applying the multiple-scale method of homogenization, we have derived an effective transport equation (3.47) governing the advection and dispersion of a substance in tube flow with steady and oscillatory components under the influences of kinetic phase exchange with the wall and first-order decay at the wall. Analytical expressions, given by (4.9) and (4.23), have been obtained for the two leading-order dispersion coefficient components *D*_{Ts} and *D*_{Tw}, which are due, respectively, to the steady and the oscillatory components of the fluid motion, and are affected only by the phase partitioning with the wall. The higher-order dispersion coefficient components, and , which arise from the combined effects of wall absorption and retention, are, respectively, given by the analytical expression (4.14) and the formal expression (3.46) in terms of functions to be solved numerically. Analytical expressions have also been obtained for the slow- and fast-oscillation limits of *D*_{Tw} and , as given by (5.21), (5.27), (6.4) and (6.9). In all these expressions, the coupling effects between the phase partitioning, phase exchange kinetics, the Schmidt number and oscillation frequency (for *D*_{Tw} only) are accounted for explicitly.

It has been shown with some numerical calculations that a combination of the following effects may lead to a higher value of the overall dispersion coefficient: (i) smaller Sc (the effect of Sc, however, diminishes as increases); (ii) larger corresponding to a lower frequency of oscillation for , and the opposite for ; (iii) smaller Da for stronger kinetics of phase exchange; (iv) a small but non-zero value of for a relatively small extent of retention and retardation effect. The presence of wall absorption may, however, cause a diminishing or enhancing effect on the dispersion, depending on the oscillation frequency and the phase exchange kinetics and partition ratio. One remarkable finding is that, if the phase exchange is sufficiently kinetic (i.e. phase equilibrium not readily achieved), the dispersion coefficient when subject to even very weak wall retention can be dramatically different, quantitatively or qualitatively, from its zero-retention counterpart. Such a high sensitivity of the dispersion coefficient to the kinetic phase exchange may lead to interesting phenomena especially during the early developing stage of the transport; this deserves further investigations.

## Acknowledgments

The work was supported by the Research grants Council of the Hong Kong Special Administrative Region, China, through project nos. HKU 7199/03E and HKU 7192/04E.

## Footnotes

- Received January 31, 2005.
- Accepted September 13, 2005.

- © 2005 The Royal Society