Dispersion by Pressure-Driven Flow in Serpentine Microfluidic

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Ind. Eng. Chem. Res. 2002, 41, 4652-4662

Dispersion by Pressure-Driven Flow in Serpentine Microfluidic Channels Brian M. Rush, Kevin D. Dorfman, and Howard Brenner* Department of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139

Sangtae Kim Lilly Research Laboratories, Lilly Corporate Center, Indianapolis, Indiana 46285

Microfluidic hydrodynamic chromatography performed in serpentine microchannels etched on chips is analyzed in the limiting case of chips containing a large number of periodically arrayed turns. Comparison is made between these results and those for a straight channel of the same length as the curvilinear channel, all other things being equal. Explicitly, generalized TaylorAris dispersion (macrotransport) theory for spatially periodic systems is adapted to compute the chip-scale solute velocity U h * and dispersivity D h * for effectively point-size, physicochemically inert Brownian particles entrained in a low Reynolds number, pressure-driven solvent flow occurring within the curvilinear interstices of such serpentine devices. Attention is focused upon relatively thin channels of uniform cross section, enabling the various transport fields pertinent to the problem to be expressed as regular perturbation expansions with respect to a small dimensionless parameter , representing the ratio of channel half-width to curvilinear channel length per turn. The generic leading-order results obtained for U h * and D h *, valid for any sufficiently “thin” channel, formally demonstrate that the serpentine geometry results simply reproduce those for a straight channel, when account is taken of the channel’s “tortuosity,” namely the square of the ratio of curvilinear serpentine length to rectilinear straight channel lengthsa conclusion shown to accord with intuition. 1. Introduction During hydrodynamic chromatographic separation processes involving Brownian solute particles entrained in pressure-driven solvent flows through straight channels, the average separation distance between two species of particles of different size increases with channel length because, as a consequence of hydrodynamic wall effects, different size particles move, on average, with different mean velocities through the channel.1,2 Accordingly, the greater the channel length, the greater the ability to effect a separation of such solute species (assumed to be introduced simultaneously into the channel) by collecting each band of species as it exits the channel. Curvilinear channels of serpentine shapes embossed on chips3,4 provide a convenient scheme for increasing the effective channel length (per unit chip length in the direction of net flow). As such, the ability to quantify the enhanced separation effect arising from multiple turns of the channel, over and above the comparable effect in otherwise straight channels,5 is clearly pertinent to the rational design and operation of such on-chip serpentine devices. It is the purpose of this paper to provide just such a quantitative analysis of the chromatographic separation phenomenon. The several factors governing the asymptotic, longtime transport of a passive spherical Brownian solute particle entrained in a Poiseuille solvent flow within a straight circular tube of radius R, or channel of halfwidth H, are well understood for both point-size6-8 and * To whom all correspondence should be addressed. Telephone: (617) 253-6687. E-mail [email protected].

finite-size2,9 spherical solute particles, thereby furnishing proper metrics for the present analysis. Generalized Taylor-Aris dispersion theory5 reveals that the net unidirectional Brownian particle transport through a rectilinear (straight) channel of large aspect ratio is fully quantified by a pair of position-independent scalar transport coefficients, these being the mean axial solute speed U h * and dispersivity D h *. Explicitly, for the case of a passive, point-size, neutrally buoyant, Brownian solute particle with molecular diffusivity D undergoing convective-diffusive transport in a two-dimensional Poiseuille solvent velocity field flowing at a mean speed vj through a rectilinear (“straight”) channel of half-width H,8 these straight-channel macrotransport parameters are

U h /s ) vj D h /s ) D +

2 (vj H) 105 D

(1.1) 2

(1.2)

Comparable curvilinear Taylor-Aris dispersion analyses exist for several pressure-driven flows in (effectively single turn) circular channels of circular cross section (see, for example, refs 10 and 11 and references therein). Geometrical configurations explored in such studies consisted of toroids or, as minor variations thereof, helices possessing a small, but constant, pitch. Analytical results for these shapes, involving perturbation expansions for otherwise straight tubes of circular cross section bent into curvilinear configurations possessing large longitudinal/transverse (axial/cross-sectional) curvature radius ratios, reveal two competing mechanisms

10.1021/ie020149e CCC: $22.00 © 2002 American Chemical Society Published on Web 07/27/2002

Ind. Eng. Chem. Res., Vol. 41, No. 18, 2002 4653

serving to modify the corresponding dispersion expressions existing for comparably straight circular tubes. All other things being equal, namely the channel crosssectional radius R and mean solvent velocity vj : (i) the effect of longitudinal curvature is to increase the dispersivity D h *, owing to the now asymmetric crosssectional velocity profiles; and (ii) the effect of secondary flows, which exist in longitudinally curved tubes such as toroids,12,13 acts to decrease the dispersivity, owing to the enhanced transverse “mixing” processes stemming from this circulation. The relative importance of these two competing effects depends functionally upon the magnitude of the Reynolds number, Re (based upon tube radius R). For curved tubes characterized by small Re, no secondary flows appear, whence the dispersivity in these bent tubes is larger than for comparably straight tubes. As Re is increased, the appearance of secondary flows begins to dominate, and the dispersivity diminishes below that observed for comparably straight tubes. The present paper addresses the two-dimensional, low Reynolds number transport of a physicochemically passive, point-size, Brownian solute particle entrained in a pressure-driven solvent flow within a serpentine channel, the latter modeled as being indefinitely spatially periodic (so as to avoid having to deal with the issue of “end effects”). For simplicity, in this initial communication, attention is confined to the case where the channel width is small compared with all other pertinent length scales characterizing the serpentine geometry, in particular the smallest of the latter’s radii of curvature. These geometrical simplifications allow use of systematic, regular perturbation expansion techniques within the context of generalized Taylor-Aris dispersion theory for spatially periodic media.5 This scheme permits determining the appropriate macrotransport coefficients, namely the mean solute velocity vector U h * and dispersivity dyadic D h * (each of which ultimately proves to be expressible in terms of respective scalar coefficients, U h * and D h *), quantifying the mean solute transport through the serpentine device as a whole. Moreover, the perturbation analysis to be pursued furnishes a systematic method for computing higher-order corrections arising from the curvature, such corrections being functionally dependent upon the explicit serpentine channel geometry. Such higher-order calculations, however, are not pursued here. The following section details the fluid-mechanical equations governing the entraining solvent flow in terms of the spatially periodic serpentine channel geometry. These equations, embodying the small perturbation parameter , are solved to leading-order in the pressure and velocity fields. Using these data, Section 3 furnishes an explicit macrotransport analysis for computing U h * and D h *, complete with leading-order results valid for small channel curvature ratios (and small particle radius/channel width ratios, the latter restriction owing to our assumption of “point size” particles). These formal results, applicable to any thin serpentine channel geometry of uniform cross section, are rationalized in section 4. Also, a connection is made between our serpentine channel results and those for flow through porous media composed of disconnected pores possessing the same shapes as the serpentine channel. Section 5 concludes with observations pertinent to extending the present analysis to broader classes of

Figure 1. Schematic of a two-dimensional spatially periodic serpentine channel of constant cross-sectional width 2H. The unit cell, denoted by the dashed box, consists of a single complete turn of the serpentine device, possessing an arc length ls (measured along the centerline of the channel, shown by the dashed curve) and macroscopic length l in the direction of mean flow (the latter direction characterized by the unit vector X ˆ ). Both the global coordinate system R(X,Y) and the local intrinsic coordinate system r(s,n) are indicated, with R ≡ X ˆX + Y ˆ Y and r ≡ sˆ s + n ˆ n (the carets denoting unit vectors), respectively, global and local position vectorssthe former defined throughout all fluid points (-∞ < X < ∞, 0 < Y < l) in the serpentine channel, and the latter defined only at the fluid points (0 < s < ls, - H < n < H) within the unit cell. The channel walls define the solid surface sp, while the fluid “interfaces” situated at s ) 0 and s ) ls, formed by the intersection of the boundaries of the unit cell with the fluid within the cell, define the surfaces ∂τ0, collectively representing the entrance and exit domains of the unit cell. In the terminology of macrotransport theory for spatially periodic systems,5 the discrete position vector Xn characterizing the serpentine lattice is given explicitly by the expression Xn ) X ˆ ln, where n is a positive or negative integer, including zero (not to be confused with the normal coordinate n).

serpentine devices as, for example, the case of nonuniform channel widths. 2. Serpentine Geometry and Entraining Solvent Flow 2.1. Serpentine Geometry. Figure 1 depicts a portion of a representative two-dimensional, spatially periodic serpentine channel, with a period of length l in the X-direction. The periodicity of the serpentine channel is manifested in the identification of a repetitive unit cell, consisting in present circumstances of a single complete turn of the channel.18 The unidirectional periodicity of the channel is captured by the single lattice vector, X ˆ l. The device is envisioned as infinitely extended in the X-direction, whereby translations of the unit cell through this lattice vector reproduce the composite device. Of course, real serpentine channels are bounded, with, e.g., a total chip length L in the direction of mean flow (-L/2 e X e L/2). Consequently, the present analysis is strictly valid only in the limit l/L , 1, so that with L ) Nl the channel is assumed to consist of numerous turns, N . 1. The “unrolled” length of a single turn of the serpentine channel is characterized by the arc length ls. With constant cross-sectional width 2H, the unit cell fluid volume is given by τ0 ) 2Hls. (The latter may be regarded as the definition of ls.) Other relevant parameters characterizing the system geometry are described in Figure 1. For the two-dimensional example considered herein, it proves convenient to work in an intrinsic system of locally defined orthogonal curvilinear coordinates r ≡ (s,n),14,19 composed, respectively, of the undisturbed fluid streamlines and their orthogonal trajectories, as indicated in Figures 1 and 2. Both ds and dn, which are generally inexact differentials, represent elements of arc

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Ind. Eng. Chem. Res., Vol. 41, No. 18, 2002

acterizing the steady pressure-driven laminar flow, is assumed to be sufficiently small to allow neglect of (i) fluid inertial effects arising from the curvilinear nature of the streamlines and (ii) particle inertia. The steady-state undisturbed solvent velocity field v ) v(R) and pressure field p ) p(R) therefore satisfy the continuity and Stokes equations

Figure 2. Intrinsic local orthogonal coordinate system (s,n) composed of the curved fluid streamlines s and their orthogonal trajectories n for a two-dimensional flow. The local coordinate system, with tangential unit vector sˆ and normal unit vector n ˆ, as well as differential arc length elements ds and dn, is depicted at the point P.

length, such that the distance |dR| ≡ |dr| between neighboring points is given by |dr|2 ) ds2 + dn2. The local coordinate system (s,n) is defined only in the interior of the fluid domain contained within a single turn,so that with the coordinate s chosen to lie along the central streamline the local coordinate system lies within the domain (0 < s < ls, - H < n < H). With R ≡ (X,Y) the global position vector, defined throughout the infinite extent of fluid in the serpentine channel, the unit tangent vector sˆ , given by

sˆ )

∂R , ∂s

(2.1)

points in the direction of motion of a material fluid element traversing the streamline at point P in Figure 2. Similarly, the unit normal vector n ˆ at point P lies perpendicular to the streamline passing through that point. With k ˆ a unit vector perpendicular to both sˆ and n ˆ , and pointing out of the page, the direction of n ˆ is chosen for definiteness such as to make (sˆ , n ˆ, k ˆ ) a righthanded system of orthonormal vectors. In what follows, the global position vector R ≡ X ˆX + Y ˆ Y will be employed to denote functions which are defined for all X (-∞ < X < ∞). Conversely, the bounded local position vector

r ) sˆ s + n ˆn

(2.2)

will be used for functions which are only defined at fluid points contained within the unit cell (0 < X < l). Use of the local coordinate system, together with the periodicity of the device, enables the particle’s global position vector R to be uniquely specified by giving its discrete unit cell location (“turn number”) Xn ) nl (n ) 0, (1, (2, ..., (∞) (with, e.g., X0 ) 0, the global location of the cell into which the particle was initially introduced into the system at time t ) 0) together with the particle’s local position vector r(s,n) within turn n; that is, functionally, R ≡ (Xn,s,n). These global and local position vectors are explicitly related by the expression

R)X ˆ Xn + r

(2.3)

2.2. Fluid-Mechanical Analysis. The entraining solvent is assumed to be an incompressible Newtonian fluid of constant viscosity µ and density F, flowing steadily at a volumetric flow rate (per unit chip depth in the third dimension) of Q ) 2Hvj through the annular space, with vj the mean velocity of the fluid (in the s-direction). The Reynolds number, Re ) Fvj H/µ, char-

∇‚v ) 0

(2.4)

µ∇2v ) ∇p

(2.5)

subject to the usual no-slip boundary condition prevailing on the solid surfaces of the serpentine channel:

v ) 0 on sp

(2.6)

Owing to the periodic nature of the serpentine channel and the properties of the fluid, the fluid velocity and pressure gradient fields are themselves spatially periodic, thereby satisfying the following periodicity conditions when defined throughout the entire fluid domain:15

v(R + X ˆ l) ) v(R),

(2.7)

∇p(R + X ˆ l) ) ∇p(R)

(2.8)

The latter condition is equivalent to the requirement that15

p(R) ) p˜ (R) - X

(∆Pl) ,

(2.9)

where p˜ (R) is a periodic function with period X ˆ l, and

∆P ) p(X, Y) - p(X + l, Y) > 0

(2.10)

is a constant, representing the pressure drop per cell, independent of the values of X and Y appearing in the argument. As the macrotransport analysis to be pursued is intracellular in nature, it proves convenient to introduce an intracellular fluid velocity field v(r) and pressure field p(r), defined only within the unit cell, which fields continue to be governed by eqs 2.4-2.6. By definition, within the unit cell the entraining fluid velocity v(r) is locally of the form

v(r) ≡ sˆ v(s,n),

(2.11)

where v ) |v(r)| is the local speed of the fluid. In terms of the local coordinate system, the prescribed mean velocity vj requires satisfaction of H v(s,n) dn ) vj ∫-H

1 2H

(2.12)

for all s. In the two-dimensional local coordinate system, the gradient operator is given by16

∂ ∂ ∇ ) sˆ + n ˆ , ∂s ∂n

(2.13)

which may be used to recast eqs 2.4 and 2.5 in terms of


local coordinates. For this purpose, the quartet of Serret-Frenet formulas14

∂sˆ ∂n ˆ ∂n ˆ ∂sˆ )n ˆ κ s, )n ˆ κ n, ) - sˆ κs, ) - sˆ κn, ∂s ∂n ∂n ∂s (2.14) prove useful. In the latter, κs and κn denote the respective (algebraically signed) curvatures of the streamlines and their orthogonal trajectories at the point (s,n). With the local intracellular fluid velocity given by 2.11, the incompressibility condition 2.4 assumes the form

cellular, the dimensionless boundary conditions 2.62.8 adopt the respective forms

v(s,n)(1) ) 0,

(2.23)

v(s)0,n) ) v(s)1,n)

(2.24)

∂p ∂p (s)0,n) ) (s)1,n) ∂n ∂n

(2.25)

while the mean velocity condition 2.12 becomes

∫-11v(s,n) dn ) 2

(2.26)

(2.15)

Subject to a posteriori verification, assume the following regular perturbation expansions for the velocity and pressure fields:

whereas the s and n components of the momentum equation, (2.5), become, respectively,

v(r;) ≡ v0 + v1 + ... ) sˆ [v0(n) + v1(s,n) + ...], (2.27)

∂v + κnv ) 0, ∂s

µ

[

)]

∂p ∂2v ∂2v ∂v ∂v + 2 - κs + κsv + κn - κnv ) , 2 ∂n ∂s ∂s ∂s ∂n (2.16)

(

)

[(

(

)]

) (

∂κs ∂κn ∂p ∂v ∂v + v ) + µ 2 κs + κn ∂s ∂n ∂s ∂n ∂n

(2.17)

Crucial to the subsequent analysis is the assumption that the serpentine channel is “thin,” i.e., the channel half-width H is small with the other length scales in the process, such as the unit cell length l and the arc length ls.20 In particular, H is supposed small relative to the magnitudes of the minimum radii of curvature, |κs-1|min and |κn-1|min. Formally, we invoke the following scales to render eqs 2.15-2.17 dimensionless:

s ) lss*, n ) Hn*, v ) vj v* κs )

ls-1κ/s ,

(2.18)

Upon suppressing the asterisks (/) in the above nondimensional entities, eqs. 2.15-2.17 adopt the respective dimensionless forms

(

(2.19)

) (

)

∂v ∂2v ∂v ∂2v ∂p - κs + 2 2 - κs2v + 3 - κnv ) 0, 2 ∂s ∂n ∂s ∂n ∂s (2.20)

[(

) (

)]

∂κs ∂κn ∂v ∂v ∂p - 2 κs + κn + + v ) 0, ∂n ∂s ∂n ∂s ∂n

(2.21)

where is the small parameter

)

H ,1 ls

(2.28)

Upon substituting these expansions into 2.19-2.21 and 2.23-2.26, and retaining only leading-order terms, we obtain

p0(s) ) - s

(2.29)

3 v0(n) ) (1 - n2) 2

(2.30)

In the case of a straight channel, namely ) 0, eqs 2.29 and 2.30 reduce to the classical two-dimensional, flatplate Poiseuille fields. As such, curvature effects do not manifest themselves until the O() terms in eqs 2.27 and 2.28. Satisfaction of eqs 2.19-2.21 and 2.23-2.26 by our leading-order solution, eqs 2.29 and 2.30, furnishes a posteriori verification of the validity of the perturbation expansions 2.27 and 2.28, at least to leading order. 3. Macrotransport Analysis

µvj ls H κn ) 2κ/n, p ) 2 p* ls H

∂v + κnv ) 0 ∂s

p(r;) ≡ p0(s) + p1(s,n) + ...

(2.22)

Because the velocity and pressure fields are now intra-

In what follows, we temporarily abandon the dimensionless notation of the preceding section. Moreover, except where necessary for clarity, we do not explicitly indicate the dependence of the various r-dependent fields. Armed with knowledge of the serpentine channel geometry and the background flow field, macrotransport theory for spatially periodic media5 is employed in this section to compute the mean particle velocity vector U h* and dispersivity dyadic D h *. The ensuing analysis is asymptotic in nature, being valid only for circumstances in which the time scale, l2/D, for solute diffusion within a single unit cell is much less than the nominal solute holdup time, Nls/vj , in the device as a whole, with N (N . 1) being the number of turns. Anticipating the obvious fact that net solute motion through the composite serpentine device will be unidirectional, occurring in the X-direction, we assume subject to a posteriori verification that U h * and D h* possess the respective representations,

U h* ) X ˆU h * and D* ) X ˆX ˆD h*

(3.1)

in terms of their comparable scalar coefficients. These relations furnish a formal link between the macrotransport paradigm5 for hydrodynamic chromatography in

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rectilinear systems and that in curvilinear, spatially periodic systems. Calculation of U h * requires the solution of the following intracellular steady-state equation governing the probability flux density J∞0 (r)5:

∇ ‚ J∞0 (r) ) 0

(3.2)

J∞0 (r) ) v(r)P∞0 (r) - D∇P∞0 (r)

(3.3)

in which

3.1. Mean Solute Velocity. The differential equation governing P ∞0 , obtained by substituting eq 3.3 into eq 3.2, can be expressed in terms of the local (s,n) coordinate system, with use of eq 2.11, as

(

Assume the following regular perturbation expansions:

with P∞0 (r) the unconditional probability density. On the solid surfaces sp of the serpentine channel, this flux density obeys the following no-flux condition

ν ‚ J∞0 (r) ) 0 on sp

(3.4)

with ν a unit vector normal to the channel walls. The field P∞0 (r) satisfies the following periodicity conditions on the fluid boundaries ∂τ0 of the unit cell

||P∞0 (r)|| ) 0, ||∇P∞0 (r)|| ) 0 on ∂τ0

(3.5)

where τ0 denotes the fluid domain within a unit cell. In the present unidirectional periodicity case, the “jump operator” appearing in eq 3.5 is defined (for an arbitrary tensor valued function f) as

||f|| ) f(s)ls,n) - f(s)0,n)

(3.7)

P∞0 (r)

[and, by eq Eventual knowledge of the solution 3.3, J∞0 (r)] enables the mean solute velocity vector to be computed via the unit cell quadrature5

∫τ J∞0 (r) d2r 0

h* D∇ ‚ [P∞0 (r)∇B(r)] - J∞0 (r)‚∇B(r) ) P∞0 (r)U

(3.15)

∞ ∞ ) τ0-1 ) const and J0(0) (n) ) sˆ τ-1 j P 0(0) 0 v0(n)v (3.16)

||B(r)|| ) -||r||, ||∇B(r)|| ) 0 on ∂τ0

(3.9)

(3.10)

(3.11)

Equations 3.9-3.11 together serve to uniquely define the B-field, albeit only to within an arbitrary, additive constant vector. This ambiguity proves inconsequential as regards the final result obtained for the dispersivity dyadic, which is computed via the following unit cell quadrature, valid for point-size particles:

P∞(r)[∇B(r)]†‚∇B(r) d2r τ 0 0

with † being the transposition operator.

∫τ v0dr2 + O()]

(3.12)

(3.17)

0

In terms of the present dimensional variables, the preceding integral becomes, with use of eqs 2.27, 2.18, and 2.30

{

U h * ) vj

}

(3.18)

∫0l sˆ ds ) R(s)ls,n) - R(s)0,n) ) Xˆ l

(3.19)

3 4Hls

H ∫0l sˆ ds∫-H [1 - (Hn ) ] dn + O() 2

s

With the help of eq 2.1 we have that

whence we obtain

U h* ) X ˆU h*

(3.20)

wherein

[( )

U h * ) vj

as well as the following no-flux condition arising from eq 3.4:

υ‚∇B(r) ) 0 on sp

U h * ) vj [τ0-1

s

subject to the jump conditions

∫

∞ ∞ (n) + J0(1) (s,n) + ... J∞0 (r;) ≡ J0(0)

Substitution of eq 3.14 into eq 3.13, together with use of eqs 2.27 and 2.30, reveals that the leading-order probability density and flux density fields satisfying boundary conditions 3.4 and 3.5 are

(3.8)

Calculation of D h * requires knowledge of the so-called B-field,5 the latter field being governed by the steadystate equation

D h* ) D

(3.14)

(3.6)

0

U h*)

∞ ∞ + P 0(1) (s,n) + ... P ∞0 (r;) ≡ P 0(0)

From eqs 2.27, 3.3, 3.8, and 3.16, the mean velocity vector can now be calculated to leading order via the quadrature

as well as the normalization condition

∫τ P∞0 (r) d2r ) 1

)

∂2P ∞0 ∂2P ∞0 ∂P ∞0 ∂P ∞0 ∂ + + κ )0 (vP ∞0 ) - D κ n s ∂s ∂s ∂n ∂s2 ∂n2 (3.13)

]

l + O() ls

(3.21)

thereby confirming a posteriori the assumed form of the first of eq 3.1. Consequently, for any sufficiently thin channel of uniform cross section throughout its length, the mean solute velocity in the direction X of net flow is smaller than the comparable straight channel TaylorAris result, eq 1.1, by an amount equivalent to the inverse square root of the “tortuosity,” ls/l, of the serpentine channel, a result rationalized in section 4. 3.2. Dispersivity. Assume, subject to a posteriori verification, a trial solution for B of the form

B* (s,n) l

B)X ˆ B(s,n) ) X ˆ

(3.22)

With use of the latter equation, as well as eqs 2.18, 2.27, 3.16 and 3.20, eq 3.9 may be rendered dimensionless


and expressed in the local coordinate system (with the asterisk affix suppressed) as

() (

)

ls ∂ 2B U ∂B ∂2B ∂B h* Pe - κs + Pev + 2 2 + 2 vj l ∂n ∂s ∂n ∂s 3 ∂B κn ) 0 (3.23) ∂s with the velocity field v given by the expansion 2.27. The Peclet number, Pe ) vj H/D, appearing above will be assumed to be of O(1) in order to subsequently compare our serpentine Taylor-Aris parameters with those for a straight channel of width 2H, which themselves are valid to O(1) terms in Pe. Upon substituting eqs 3.21 and 2.27 into eq 3.23, we obtain

[

] { }

ls ∂B ∂2 B - Pe 1 + O() - κs + Pe[v0 + 2 l ∂n ∂n ∂2B ∂B ∂B + 2 2 + 3κn ) 0. (3.24) O()] ∂s ∂s ∂s The dimensionless no-flux boundary condition 3.11 now takes the form

∂B ) 0 at n ) (1 ∂n

(3.25)

while the dimensionless jump conditions 3.10 become (upon suppressing in the arguments appearing below),

B(s ) 1,n) - B(s ) 0,n) ) -1

(3.26)

∂B ∂B (s ) 1,n) (s ) 0,n) ) 0 ∂s ∂s

(3.27)

Assume, subject to a posteriori verification, a trial solution of the form

B(s,n;) ) - s + PeB ˜ (n) + O(2)

(3.28)

where, to leading-order terms in , B ˜ (n) is, from eqs 3.24 and 3.25, the solution of the equation

˜ d2B ) 1 - v0(n) dn2

(3.29)

dB ˜ ) 0 at n ) (1 dn

(3.30)

The jump conditions 3.26 and 3.27 are automatically satisfied by the assumed form eq 3.28 of the trial solution. Substitute eq 2.30 into eq 3.29, and integrate twice, using eq 3.30 to eventually arrive at21

11 4 n - n2 + c 42

(

D h* )

)

(3.31)

with c an arbitrary constant of integration. This satisfaction of the governing equation eq 3.9 and boundary conditions 3.10 and 3.11, at least to leading order in , verifies, a posteriori, our assumed trial solutions eqs 3.22 and 3.28.

D τ0

H ∫-H ∫0l ∇B‚∇B ds dn s

(3.32)

From eqs 2.13, 2.18, 3.28. and 3.31, the gradient of the scalar B field required above is, in dimensional variables

∇B(s,n;) )

( ){

[( ) ( )]} + O()

Q n 3 l n - sˆ + n ˆ ls 2D H H

(3.33)

Using the fact that the vectors (sˆ ,n ˆ ) constitute an orthonormal vector set leads eventually to the expression

{[

D h* ) D 1 +

( ) ](ll )

2 vj H 105 D

2

2

s

}

+ O()

(3.34)

Comparison with eq 1.2 shows that, to terms of dominant order in the curvature, the dispersivity in a serpentine channel is given by its classical rectilinear, straight channel form, augmented by the inverse tortuosity factor (l/ls)2, whose presence is rationalized in the subsequent Discussion. Similar to the mean solute velocity in eq 3.21, this result is valid for any arbitrarily shaped, albeit thin, serpentine channel of sufficiently small local curvature. 4. Discussion While the preceding section furnished formal results for the leading-order effect of curvature upon the mean solute speed U h * and dispersivity D h *, it behooves us to provide an intuitive rationalization of these results. This will be done by employing two very different approaches, one mathematical and the other physical. In addition, in the context of these rationalizations, the present results will be extended to the case of transport through the interstices of a spatially periodic, model porous medium. 4.1. Mathematical Rationale. In the context of Taylor-Aris dispersion theory for spatially periodic systems,5 the asymptotic, unit-cell-averaged solute “concentration” or conditional probability density, P h (R h ,t), is defined as

P h (R h ,t) =

subject to the boundary condition

B ˜ (n) )

With use of eqs 3.16 and 3.22, the expression 3.12 for the dispersivity adopts the form of the second of eqs 3.1, in which the scalar dispersivity is given by

1 τ0

∫τ {n}P(Rn,r,t|R0 ) 0,r′) d2r f

(4.1)

where R h denotes a “smoothed,” coarse-grained, continuous version of the discrete position vector Rn, and the integration is performed over the fluid domain τf{n} h within cell n.22 In solute “concentration” terms, ΓP represents the number of solute molecules per unit of superficial volume at time t in the cell centered at the point R h , for circumstances in which Γ (non-hydrodynamically interacting) particles were originally introduced into the serpentine channel at time t ) 0 at the point r′ located within the cell centered at R h 0 ≡ R0 ) 0. In the long-time, asymptotic Taylor-Aris limit, the preceding integral is independent of the initial local position r′, which is why the symbol r′ has been omitted from the argument (R h , t) of P h. This averaged conditional probability density asymptotically obeys the following unsteady-state, generalized

4658


Taylor-Aris convection-diffusion macrotransport equation5:

∂P h +U h *‚∇P h )D h *:∇∇P h, ∂t

(4.2)

where ∇ h ≡ ∂/∂R h is a coarse-grained gradient operator. In the above equation, U h * and D h * are the directional quantities respectively defined by eqs 3.8 and 3.12. Upon introducing into eq 4.2 the pair of equations in eq 3.1 and noting that X h )X ˆ ‚R h , where X h is a smoothed, continuous version of the discrete coordinate Xn (namely, X ˆ ‚Rn ≡ nl), the macrotransport equation reduces to the scalar convective-diffusive equation

h ∂2P ∂P h ∂P h )D h* 2 +U h* ∂t ∂X h ∂X h

(4.3)

On the other hand, the corresponding macrotransport equation for the hypothetical rectilinear channel created by straightening the serpentine channel (as one would “straighten” a coiled garden hose) is23

∂P h ∂P h ∂2P h )D h /s +U h /s ∂t ∂X h ∂X h 2 s

(4.4)

s

where X h s ≡ nls; that is, X h s is the continuous version of the discrete variable nls. Given that X h ≡ nl and X h s ≡ nls, we have that the respective differential lengths δX h and δX h s are related by the expression

δX h )

()

l δX h δX hs ≡ δX hs l δX hs s

(4.5)

Upon substituting the latter into eq 4.3 and comparing the result with eq 4.4, we find that the respective pairs of macrotransport coefficients are related by the expressions

U h* )

()

()

l / l 2 / U h s and D h* ) D hs ls ls

(4.6)

Substitution of eqs 1.1 and 1.2 into these relations furnishes respective expressions for U h * and D h * which accord with our formal leading-order results for the mean solute speed in eq 3.21 and dispersivity in eq 3.34. The preceding analysis demonstrates, as would be intuitively expected, that no fundamental difference exists between the curvilinear and rectilinear results for circumstances in which curvature effects are negligible. The analysis also clearly suggests that differences are to be expected in circumstances where curvature effects are sensible, inasmuch as the informal arguments embodied in eq 4.5, connecting the pair of macrotransport eqs 4.3 and 4.4, have no formal viability when account is taken of curvature. This is, inter alia, an immediate consequence of the fact that the infinitesimal curvilinear arc length ds underlying the length term appearing in eq 4.4 is, strictly speaking, an inexact differential. 4.2. Physical Rationale. It is instructive to contemplate the significance of the basic eqs 4.6, connecting the respective transport processes in the serpentine and straight channels, by more physically insightful arguments than those given by the above algebraic manipulation of symbols. Consider first the nominal solute

Figure 3. Schematic of a V-shaped serpentine channel (-∞ < X < ∞) with superficial area A h X and cross-sectional width 2H. The channel possesses a serpentine length ls over the macroscopic distance l.

holdup time within the serpentine channel. While traversing one complete turn (period) of the channel, an average solvent molecule, and hence a point-size, passive (i.e., unbiased) Brownian solute “molecule,” travels a physical distance ls at a mean speed vj . Accordingly, the nominal holdup time th of the solute particle, namely the mean time between its entrance into and exit from that one complete turn of the channel, is given by the expression th ) ls/vj . During the course of executing this single turn, the particle has traversed a distance l in the net, axial X-direction. Consequently, the particle’s mean Lagrangian displacement per unit time in that direction (for the specified turn, and hence for the serpentine channel as a whole, consisting of N such turns) may be expressed as the serpentine-scale mean solute velocity

U h* )

()

X-distance traveled l Nl ≡ vj (4.7) ) nominal holdup time Nth ls

which is precisely the result obtained in eq 3.21. The parameter ls/l represents the mapping or projection of the solute transport along the serpentine channel onto the X-direction, the direction of net motion. This mapping is expressed concisely via the differential relationship 4.5. To physically rationalize the inverse tortuosity factor appearing in the second part of eq 4.6, governing the dispersivity, consider a simple, nonconvective solute transport process involving steady-state Fickian molecular diffusion through the spatially periodic serpentine channel shown in Figure 3. The unit cell, bounded between the dashed lines in the figure, is chosen to consist of a single V-shaped conduit of constant width 2H and centerline length ls. As in previous problems the channel is assumed to be “thin,” corresponding to the inequality ) H/ls , 1. The physical problem of molecular diffusion in this spatially periodic serpentine channel is easily shown to be equipollent to the problem of molecular diffusion through a single V-shaped channel, shown in Figure 4, between two well-stirred reservoirs, whose respective left- and right-side solute concentrations are cL and cR (cL > cR). As indicated, the two reservoirs are separated by a distance l in the Xdirection. The steady-state solute concentration field, c ≡ c(s,n), at a point (s,n) within the channel satisfies the diffusion equation

∇ 2c ) 0

(4.8)


unit time, M ˙ , between the two reservoirs is M ˙ ) jsAs. Explicitly

M ˙ ) 2HD

Figure 4. Schematic for the two-dimensional, steady-state solute diffusion between two well-mixed reservoirs of differing solute concentrations, cL and cR (cL > cR), in the respective left- and righthand reservoirs connecting a single turn of the channel depicted in Figure 3. The point i corresponds to the intersection of the leftand right-side channel centerlines. The “zero-intersectional volume” model considered herein assumes that one may neglect the local transport processes existing within the overlap region (the boxed region indicated by the horizontal shading), whose area is 4(ls)2, in favor of the strictly global aspects of the transport within that regionsa presumably valid assumption in the context of our leading-order , 1 results.

c ) cL at s ) 0

(4.9)

c ) cR at s ) ls

(4.10)

∂c ) 0 at n ) (H ∂n

(4.11)

To terms of dominant order in , eqs 4.8-4.11 may be solved separately in the left- and right-hand portions, (0 < s < ls/2) and (ls/2 < s < ls), respectively, of the V-shaped channel and the two solutions joined at the intermediate “plane” (s ) ls/2, n) by the dual requirements that both the concentration, c, and normal solute flux, -D(∂c/∂s), be continuous at this intermediate point, i. (The normal flux continuity condition ensures that the rate M ˙ at which solute mass leaves the left side is identical to that entering the right side.) Thus, in the complete V-shaped region, (0 < s < ls), we obtain the linearly-decreasing concentration field

c(s) ) -

∆c s + cL ls

(4.12)

with the constant ∆c ) cL - cR being the reservoir concentration difference. To the degree of approximation implicit in the above solution, the Fick’s law solute mass (or number) flux density vector, j ) -D∇c, is

j(s) ) sˆ js, js ) D

∆c ) const ls

(4.13)

Consequently, with As ) 2H denoting the cross-sectional area of the channel (per unit height out of the plane of the paper), the corresponding solute mass flow rate per

(4.14)

In place of the detailed “microscopic” fields, c and j, leading up to the above calculation of M ˙ , one can define corresponding “macroscopic” solute concentration and flux fields, cj and J h , as equivalently determining this mass transport rate. These are the quantities serving to define the “effective” or macroscopic dispersivity dyadic, D h *, the quantity of interest in the present calculation, via the relation J h ) -D h *‚∇ h cj. Clearly, in present circumstances, the appropriate macroscopic vector solute concentration gradient is ∇ h cj ) - X ˆ ∆c/l, while the dispersivity dyadic is necessarily of the form D h* ) X ˆX ˆD h *. These lead to the fact that J h )X ˆJ h X, where h *∆c/l. However, by definition, J hX ) M ˙ /A h X, where J hX ) D the area A h X is the area (per unit height out of the plane of the paper) shown in Figure 4. Explicitly, the “projected area” A h X is related to the corresponding crosssectional area As ) 2H of the channel by the trigonometric relation A h X ) As /cosθ, where θ is the angle between the directions X ˆ and sˆ . Consequently, cosθ ) |X ˆ ‚sˆ | ≡ l/ls. In terms of these macroscopic variables, this ˙ l/2Hls, or in terms of the effective makes J hX ) M diffusivity

M ˙ ) 2HD h*

subject to the (approximate) boundary conditions

∆c ls

()

ls 2∆c l ls

(4.15)

Comparison of the latter with eq 4.14 yields the relation

D h* )

()

l 2 D ls

(4.16)

which is presumably correct to leading-order terms in . Equation 4.16 represents the pure molecular diffusion (vj ) 0) analogue of our more general Taylor-Aris dispersion result, eq 3.34, valid for all sufficiently thin channels irrespective of their shape (e.g., smoothly curved vs V-shaped). Indeed, the latter scaling eq 4.16 has been reported elsewhere17 for purely molecular diffusion through tortuous porous media, albeit with an additional factor accounting for bed porosity, a result rationalized in section 4.3. The present contribution has rigorously extended the intuitive scaling result in eq 4.16 to the Taylor-Aris dispersivity for pressure driven flow occurring in thin channels, at least up to leading order. The argument of the preceding paragraph, wherein the relation A h X ) Asls/l was derived geometrically for the V-shaped serpentine channel shown in Figure 3, is not sufficiently general to see its universality for all sufficiently thin serpentine channel shapes. The issue is revealed, for example, by Figure 1, where the choice of the required area, A h X, would appear to depend on where within the unit cell, one arbitrarily inserted the (vertical) plane, X ) const. Explicitly, the cosine, |X ˆ ‚sˆ |, of the “angle” θ, which was constant in the V-shaped channel example of Figure 3, would vary with the position s of centerline distance coordinate in the smoothly curvilinear channel depicted in Figure 1. The issue of universality is resolved, however, by recognizing that the unit-cell-defined solute flux J h X appearing in the

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Figure 5. Schematic of a nonconnected, spatially periodic porous medium. The fluid portion of the medium possesses a superficial area A h X(f), while the solid portion possesses a superficial area A h X(s).

relation M ˙ )J h XA h X is, because of the coarseness of the scale upon which J h X is defined, necessarily based upon average volumetric issues, involving the unit cell as a whole, rather than upon areal issues associated with a particular choice of a locally positioned plane, X ) const, within the unit cell. Consequently, the same must be true of the definition of the “area” A h X, since the physical quantity M ˙ has the same value for all planes, X ) const. Accordingly, we recognize that the quantity A h X is strictly a “volumetric,” unit cell entity, rather than an “areal” entity, and that its proper generic definition is A h X ) τf/l s namely, the fluid volume per unit length of channel. Since, for all sufficiently thin channels, the unit cell fluid hX volume is τf ) Asls, the universality of the relation A ) Asls/l for all sufficiently thin channels immediately follows. 4.3. Applications to Porous Media. Relations similar to eq 4.6, which use tortuosity factors to express Taylor-Aris macrotransport parameters for transport along curvilinear trajectories in terms of comparable parameters for rectilinear trajectories, can be adapted to treat transport problems in porous media (albeit, noninterconnected, spatially periodic models of porous media). To see how this may be accomplished, for simplicity, initially consider the problem of pure molecular diffusion through the model porous medium shown in Figure 5, the latter periodically extending indefinitely in both the X and Y directions. This problem is easily analyzed using techniques similar to those employed in connection with the derivation of eq 4.16. In this context, it will be helpful to refer first to Figure 6, consisting of “nested” noninterconnected, serpentine channels, extending periodically and indefinitely in both the X and Y directions, where the channel “walls” separating the channels are infinitesimally thin (and hence occupy no volume). On the basis of our calculations in section 4.2 for the pure molecular diffusion case, it is apparent that eq 4.16 applies equally to the (zero solids volume fraction) “porous medium” depicted in Figure 6, where the solute present in the two reservoirs of the section 4.2 example (both extending indefinitely in the Y-direction) enters the channels without lateral bias, and where D h * is the h * ∂cj/∂X h , with quantity defined by the relation J hX ) - D cj(X h , t) the instantaneous solute concentration at the ˙ /A hX “point” X h , independent of Y. The quantity J hX ) M

Figure 6. Schematic of “nested,” noninterconnected, serpentine channels with infinitesimally thin walls. With respect to the medium depicted in Figure 5, the present “porous medium” corresponds to the case A h X(s) ) 0.

appearing above is independent of Y. At the macroscopic length scale at which the dispersivity (effective diffusivity) is defined for a porous medium, the latter is regarded as being a continuum (devoid of the concept of separate subcontinuum interstitial fluid and solid “phases,” wherein the various fields are regarded as being defined at each point X h of the space-filling, interpenetrating, continua). Accordingly, since the mass flow rate M ˙ is a physically measurable quantity for a specified (superficial) area in the direction of X of the net solute transport, the area A h X must be understood as referring to the superficial area of the porous medium in Figure 5, rather than to the actual physical (fluid) area A h X(f) available for transport, as in Figure 6. Given the Y dimensions shown in Figure 5, we have h X(f) + A h X(s). As is evident from Figure here that A hX ) A h X ≡ φ, e.g., constitutes the volume fraction of 5, A h X(s)/A solids in the porous medium. Consequently, A h X ) (1 h X(f) represents the superficial area A h X in terms of φ)-1A the available area A h X(f) through which the mass transport M ˙ is actually occurring. As in the last several paragraphs of section 4.2, we have that A h X(f) ) As/cosθ h x ) (1 - φ)-12Hls/l. In ≡ 2Hls/l, whence we find that A hX ≡ turn, this yields for the porous medium M ˙ ) J h XA h by -D h * (∂cj/∂X h )(1 - φ)-12H(ls/l). Upon replacing ∂cj/∂X the Darcy-scale macroscopic gradient, -∆c/l, and comparing the resulting expression for M ˙ with that given by eq 4.14, we obtain

()

D h * ) (1 - φ)

l 2 D ls

(4.17)

for the effective molecular diffusivity in porous media. Of course, when φ ) 0, as in the “porous medium” of Figure 6, the preceding formula reduces to that of eq 4.16. More general macrotransport results, valid for all such “thin,” noninterconnected, spatially periodic, model porous media, and applicable to pressure-driven flows in porous media, can be derived by procedures similar to those leading from eqs 4.2 and 4.4 to eq 4.6. In particular, the issue of the solids concentration immediately arises in such circumstances when going from eq 4.4, with P h (now referred to as P h A) an area-averaged quantity, to the volume-averaged quantity P h appearing in eq 4.3 (now referred to as P h V). In particular, in h /s appearing in eq relation to the quantities U h /s and D 4.4, the mean solute concentration P h A refers to the “true”


solute concentration in the fluid within the channels, whereas the definitions of the quantities U h * and D h * in eq 4.3 are such that P h V refers to the superficial solute concentration, that is the solute mass per unit of superficial volume. Accordingly, since (1 - φ) is the volume of fluid per unit of superficial volume, these two concentrations are related by the expression P hV ) (1 - φ)P h A. Proceeding as was done in section 4.1 to relate eqs 4.3 and 4.4 via relation 4.5, and replacing P h therein h A, respectively, we ultimately obtain the pair by P h V and P of relations

()

U h * ) (1 - φ)

()

l / l 2 / U h and D h * ) (1 - φ) D hs ls s ls

(4.18)

where, as before, the straight-channel Taylor-Aris parameters are given in eqs 1.1 and 1.2. Of course, for the case of pure molecular diffusion, the second of the above relations reduces to eq 4.17. In the trivial case where the porous medium consists of straight channels, the tortuosity factor is unity, whence the first part of eq 4.18 reduces to U h* ) (1 - φ)vj, the latter simply representing the usual Darcyscale porous medium relation between the so-called seepage velocity U h * and the Lagrangian interstitial (or “pore”) velocity vj . 5. Conclusions The preceding fluid mechanical and macrotransport analyses have furnished systematic, rigorous, regular perturbation results for the effects of channel curvature upon the mean velocity and dispersivity of a point-size particle entrained in a two-dimensional, pressuredriven, low Reynolds number fluid flow through a spatially periodic serpentine channel. The leading-order results obtained are valid for any arbitrarily shaped, albeit thin, channel geometry. Given a specific channel shape [thereby rendering the channel curvatures κs and κn functionally dependent upon the local coordinates (s,n)], the procedure developed herein could be employed to calculate higher-order corrections to U h * and D h * in terms of the aspect ratio , 1. We have also outlined several informal and intuitive arguments for rationalizing the formal results obtained. As a consequence of its generality, the present preliminary analysis provides a fundamental basis for subsequent extensions from the case of point size particles to (passive) spherical Brownian particles of finite size (relative to the channel width). In such circumstances, the resulting steric volume-exclusion and conventional hydrodynamic “wall effects”2,9 would permit the hydrodynamic chromatographic separation of different size particles based upon their different mean velocities U h /j (j ) 1, 2, ...) (all of which are greater than the projected entraining solvent velocity, lvj /ls) through the serpentine device. Moreover, the detailed scheme presented herein may be readily modified5 so as to incorporate effects arising during affinity chromatography in serpentine channels, where the solute particless although effectively point-sizesare no longer physicochemically inert but instead tend to be adsorbed onto the channel walls. These subjects are reserved for a future communication, as too are calculations for the nonuniform case where the channel width 2H is no longer independent of axial position s, but is, rather, functionally dependent upon the latter (albeit in a spatially periodic manner).

Note Added after ASAP Posting This article was posted ASAP on 7/27/2002 before author revisions were inserted. The revised version was posted on 7/30/2002. Acknowledgment This work was partially supported by a grant from Eli Lilly and Co. to H.B. and a Graduate Research Fellowship from the National Science Foundation to K.D.D. Literature Cited (1) Dimarzio, E. A.; Guttman, C. M. Separation by Flow. Macromolecules 1970, 3, 131. (2) Brenner, H.; Gaydos, L. J. The Constrained Brownian Movement of Spherical Particles in Cylindrical Pores of Comparable Radius: Models of the Diffusive and Convective Transport of Solute Particles in Membranes and Porous Media. J. Colloid Interface Sci. 1977, 58, 312. (3) Jacobson, S. C.; Hergenroder, R.; Koutney, L.; Warmak, R.; Ramsey, J. M. Effects of Injection Schemes and Column Geometry on the Performance of Microchip Electrophoresis Devices. Anal. Chem. 1994, 66, 1107. (4) Gas, B.; Kenndler, E. Dispersive Phenomena in Electromigration Separation Methods. Electrophoresis 2000, 21, 3888. (5) Brenner, H.; Edwards, D. A. Macrotransport Processes; Butterworth-Heinemann: Boston, MA, 1993. (6) Taylor, G. I. Dispersion of Soluble Matter in Solvent Flowing through a Tube. Proc. R. Soc. A 1953, 219, 186. (7) Aris, R. On the Dispersion of a Solute in a Flowing Fluid through a Tube. Proc. R. Soc. A 1956, 235, 67. (8) Woodling, R. A. Instability of a Viscous Fluid of Variable Density in a Vertical Hele-Shaw Cell. J. Fluid Mech. 1960, 7, 501. (9) Mavrovouniotis, G. M.; Brenner, H. Hindered Sedimentation, Diffusion, and Dispersion Coefficients for Brownian Spheres in Circular Cylindrical Pores. J. Colloid Interface Sci. 1988, 124, 269. (10) Berger, S. A.; Talbot, L.; Yao, L. Flow in Curved Pipes. Annu. Rev. Fluid Mech. 1983, 15, 461. (11) Daskopoulos, P. H.; Lenhoff, A. M. Dispersion Coefficient for Laminar Flow in Curved Tubes. AIChE J. 1988, 34, 2052. (12) Dean, W. R. Note on the Motion of Fluid in a Curved Pipe. Philos. Mag. 1927, 4, 208. (13) Dean, W. R. The Sreamline Motion of Fluid in a Curved Pipe. Philos. Mag. 1928, 5, 673. (14) Aris, R. Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Dover: New York, 1962. (15) Nitsche, L. C.; Brenner, H. Eulerian Kinematics of Flow through Spatially Periodic Models of Porous-Media. Arch. Ration. Mech. Anal. 1989, 107, 225. (16) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics. Kluwer: Dordrecht, The Netherlands, 1983. (17) van Brakel, J. Pore Space Models for Transport Phenomena in Porous Media: Review and Evaluation with Special Emphasis on Capillary Liquid Transport. Powder Technol. 1975, 11, 205. (18) For a specified geometry, the mean velocity U h * and dispersivity D h * must prove to be independent of the particular choice adopted for the unit cell configuration.5 Consequently, the location of the point s ) 0 and the number of turns in a unit cell are arbitrary, albeit subject to the caveat that translation of the unit cell through its lattice vector X ˆ l reproduces the composite device. Indeed, the leading-order results obtained herein do not require specifying a particular serpentine geometry, being valid for any thin serpentine channel. However, for the sake of definiteness, it is convenient to envision the unit cell as being composed of the semi-circular and two-quarter-circular annuli shown in Figure 1. Accordingly, Figure 1 suggests that the channel height is the same as the period, although the results to be obtained transcend this assumption. (19) It will be clear from the context of their use whether the symbols s and n refer to the intrinisic coordinate system (s,n), or to the similar symbols respectively denoting a “straight” channel (as in U h /s ) and the turn number, n.

4662


(20) Of course, these two lengths are proportional to one another, so that only one of the two represents an independent length-scale parameter. (21) As an aside, for the case of plug flow, where the nondimensional velocity profile is flat (i.e., v0 ) 1), the RHS of eq 3.29 is zero. This leads to the trivial solution B ˜ (n) ) 0, whence the dispersivity that would eventually be obtained for this case is equivalent to that for pure molecular diffusion, as would be expected in such circumstances. (22) Prior to eq 4.1, we have only considered cases wherein the volume τ0 of the unit cell is equivalent to the volume of fluid in one turn of the serpentine channel. In what follows, the unit cell will be permitted to consist of both a fluid filled region, as before, as well as a solid region. We will retain the symbol τ0 to denote the superficial volume of the unit cell, while employing τf{n} to refer to the volume of fluid contained within cell n. (23) Equation 4.4 derives from the usual rectilinear TaylorAris macrotransport equation, ∂P h A/∂t + U h /s (∂P h A)/∂xs ) D h /s (∂2P h A/ 2 ∂xs ) wherein P h A(xs,t) is the asymptotic, area-averaged (over the

channel width 2H and chip depth) solute concentration or conditional probability density field, and xs (-∞ < xs < +∞) is the (continuous) axial coordinate. [Within a single “unit cell” (0 < xs < ls) of the straight channel, xs is equivalent to the curvilinear arc length variable, s, at least to within lowest-order terms in .] In the preceding equation, U h /s and D h /s are respectively given in terms of the prescribed microscale data for the straight channel problem by expressions 1.1 and 1.2. Equation 4.4 then derives from the preceding equation in this footnote by integrating the latter over the length ls and subsequently dividing by ls to obtain the transport equation 4.4 for the now volumetrically averaged solute )X h s+(ls/2) density P h (X h s,t) ) (1/ls)[∫xxss)X h A(xs,t) dxs], where, as defined h s-ls/2 P above, X h s ≡ nls.

Received for review February 21, 2002 Revised manuscript received June 11, 2002 Accepted June 24, 2002 IE020149E

Dispersion by Pressure-Driven Flow in Serpentine Microfluidic

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