## Abstract

The steady-state transition from and to the uniform entry and exit flow profiles is well described, at large aspect ratio, in terms of the stream function by the pipe eigenfunctions. But these latter are unsuited to oscillatory motion or the time evolution of the symmetric piston-driven pipe flow, for which an appropriate solution has a combination of a Fourier series along the finite pipe and a Fourier–Bessel series in the transverse direction. A non-uniqueness requires the identification of a solvability condition and care is needed in demonstrating its satisfaction. An additional result is that the solution must be constructed to satisfy the normal flow conditions identically. Application is made to thermal transpiration, recently explained by the revised Navier–Stokes equations and boundary conditions.

## 1. Introduction

Steady Stokes flow in a pipe exhibits features similar to those seen in a channel. Studies of the latter are more numerous in the literature and comprehensively reviewed by Meleshko (2003). The steady-state flow envisioned here is evidently the superposition of the ‘midpipe’ pressure-driven flow and the transition from and to the uniform entry and exit flow profile, chosen with a particular application in mind. The transition requires by far the more complicated calculation and is well described physically, at large aspect ratio, in terms of the stream function by the Papkovich–Fadle strip or pipe eigenfunctions, given by(sin 2*λ*=±2*λ*), in the channel |*y*|<1, and by(1.1)in the pipe *r*<1, with the two sets of complex eigenvalues having similar distributions. This popular method was applied to similar strip (Katopodes *et al*. 2000) and axisymmetric pipe (Davis 1990) problems and relies on the biorthogonal relations satisfied by the eigenfunctions to determine their individual amplitudes. The successful truncation of the ensuing infinite system of equations was achieved by means of the optimal weighting technique described by Spence (1983). The theory is easily adapted to axisymmetric pipe flow (Davis 1990) without altering the underlying structure. The pointwise convergence of such biorthogonal series has been established by Joseph & Sturges (1978). The smallest eigenvalue indicates that the steady transition occurs within a channel width or pipe diameter. Thus, the efficient use of these eigenfunctions in steady flow is compromised by their biorthogonal relations and their use in describing the time evolution of the flow, or even periodic flow, is evidently not straightforward.

An alternative method (Meleshko & Krasnopolskaya 2001) uses a combination of a Fourier transform along the semi-infinite strip and a Fourier series in the transverse direction and can be readily adapted, as shown below, not only to a combination of a Fourier series along the finite pipe and a Fourier–Bessel series in the transverse direction, but also to the time evolution. One-end and one-side condition are satisfied and the remaining conditions yield an integro-algebraic linear system. Of crucial importance, but not emphasized by the authors, is that the identically satisfied conditions are those of no normal flow (prescribed stream function). The remaining no-slip conditions are intrinsically less demanding. This method was fully described for the rectangle by Meleshko (1996) and by Meleshko & Gomilko (1997), who used Mellin transform techniques to investigate the asymptotic behaviour of the infinite linear systems of algebraic equations. The computational aspects are thus eased, but the solution structure does not mimic the physics of the flow. Convergence of the truncated system can be enhanced, but Davis (2002, 2003) observed that the homogeneous system has a non-trivial solution. While this can be easily eliminated from a numerical solution, there remains the need to satisfy a solvability condition, which requirement is shown in §5 to force the solution construction to be such that the prescribed normal flow conditions are satisfied identically, as noted above. In turn, the solvability condition introduces some delicate convergence considerations which are shown to require precise redefinitions of the unknown function and coefficients. To ensure that the flow evolution is governed by the unsteady Stokes equations, the uniform end profile is imposed continuously from rest in a finite time-interval. A convenient choice for the time dependence is the quarter period 0<*t*<*π*/2*ω* of sin *ωt*. The resulting solution in §3 not only shows how a periodic flow would evolve but also allows the steady flow and viscous decay modes to be deduced in §4 from the oscillatory flow component in the quarter period.

As an example of the modelling capabilities of the revised Navier–Stokes–Fourier equations and associated boundary conditions (Brenner 2004, 2005*a*,*b*), Bielenberg & Brenner (2006) discuss the steady state achieved in Reynolds' (1879) notable experiment, which demonstrated the thermomolecular pressure phenomenon, now termed thermal transpiration. They show how the modified continuum equations render the Maxwell (1879) slip condition unnecessary by allowing the experimentally observed midpipe pressure-driven flow, but do not address the transitions between uniform and parabolic profiles considered below. It is shown in §7 that the theory can describe the evolution of thermal transpiration, provided that the end heating and cooling is introduced sufficiently smoothly.

## 2. Piston flow basic theory

Suppose that the incompressible fluid is bounded rigidly by a cylinder at *r*=*a* and plane ends at *z*=±*L*, 0≤*r*<*a*, where (*r*, *ϕ*, *z*) are cylindrical polar coordinates. The fluid is at rest until, at time *t*=0, the ends begin an axial sine wave oscillation that, after a quarter period, is smoothly replaced by a uniform axial speed *U* such that the Reynolds number *Ua*/*ν*≪1, where *ν* is the kinematic viscosity of the fluid. This low-speed restriction allows the ends to be instantaneously at *z*=±*L*. The unsteady creeping flow is governed by the continuity equation(2.1)and the momentum equation(2.2)in which ** v** is the fluid velocity vector;

*ρ*

_{0}is the ambient fluid density; and

*p*is the fluid pressure. Equation (2.2) implies that the vorticity

**satisfies(2.3)which is the condition for the middle term of (2.2) to be a gradient field. Thus, when forced by prescribed velocities, the motion is governed by (2.1) and (2.3), with the pressure playing a passive role, i.e.**

*Ω**∇*

_{p}is determined from equation (2.2) after

**is known. No slip is applied at**

*v**r*=

*a*, |

*z*|<

*L*and the complete flow evolution can be solved by imposing the end speeds(2.4)which exhibit a jump in acceleration at

*t*=0 but, in contrast to the linear profile, not at

*t*=

*π*/2

*ω*. and given below denote unit vectors. The alternative form of (2.4),where

*H*denotes the Heaviside unit function, shows that the velocity field is the difference of two flows, with the delayed motion a retarded time-integral of the other.

The solution strategy is to construct the time-dependent unidirectional flow that takes account of the imposed flux and then add the time-dependent flux-free flow that achieves the adjustment from the imposed end speed to that of the unidirectional flow. Thus, it is convenient to write(2.5)where *W* is the axial flow that accounts for the prescribed flux and ** V** is the flux-free adjustment dictated by the mismatch of the end flow profiles of

**and . The Laplace transform of**

*v**W*, given by(2.6)is governed by the transform of equation (2.3) in this unidirectional case, namely(2.7)The bounded solution of (2.7) that satisfies no slip at

*r*=

*a*is(2.8)where

*I*

_{0}and

*I*

_{2}are modified Bessel functions. Aided by using a recurrence relation (Watson 1958), the scale factor is determined by ensuring that the mean value in the pipe is the transform of the profile (2.4).

The expressions for *W*(0<*t*<*π*/2*ω*) and *W*(*t*>*π*/2*ω*) obtained from (2.8) by inversion of the Laplace transform consist of residue contributions arising from *s*=±i*ω* or *s*=0 (oscillatory or steady state) and *s*=−*ν*(*λ*_{2m}/*a*)^{2}; *m*≥1 (viscous decay modes), where {*λ*_{2m}; *m*≥1} denotes the positive zeros of *J*_{2}, arranged in ascending order. Since *I*_{n}(i*x*)=i^{n}*J*_{n}(*x*) (*n*≥0) (Watson 1958), the eventual result is(2.9)in which the two series (*t*< or >*π*/2*ω*) consist of eigenfunctions added to ensure that *W* is initially zero and continuous at *t*=*π*/2*ω* throughout the closed pipe. On substitution in (2.2), the evolution of the associated pressure gradient is given by(2.10)

The oscillatory and steady-state terms in (2.9) and (2.10) indicate that the wall-generated rearrangements of the ‘end’ profiles create vorticity in the flow and hence the associated pressure gradient. Note that the zeros of *J*_{2} arise from the imposition of flux, in contrast to the zeros of *J*_{0} invoked by the sudden imposition of a pressure gradient (Batchelor 1967, pp. 193–194). Since *λ*_{21}≃5.136 and *λ*_{01}≃2.405, the limit states here are approached about four times faster. Moreover, *λ*_{2m}≃*λ*_{0,m+1}(*m*≥1) suggests the approximate observation that the slowest decay mode is eliminated when flux is imposed.

The use of (2.2) assumes a time-scale of order *a*^{2}/*ν*, determined by the viscous dissipation and achieved when *ω*=*O*(*ν*/*a*^{2}). The instantaneous imposition of the end flow yields the *ω*→∞ limit of (2.10) (all coefficients in the series equal to unity), which reveals an impulse contribution to the pressure gradient at *t*=0. Here, the continuity of *W* at *t*=0 ensures only an initial jump in the pressure gradient while the continuity of ∂*W*/∂*t* at *t*=*π*/2*ω* ensures a continuous pressure gradient. This latter may be verified by using the *r*-derivative at *r*=*a* of the identity obtained by equating to zero the jump in *W* at *t*=*π*/2*ω*.

Inspection of (2.9) shows that consists of an oscillatory flow that is replaced by the Poiseuille flow at time *t*=*π*/2*ω* and evanescent modes that are generated at both *t*=0 and *t*=*π*/2*ω*. Contributions to the flux-free adjustment flow ** V**(

*r*,

*z*,

*t*) arise from all three component flows and can be efficiently constructed by deducing the decay modes and the steady state from the oscillatory term. In the pipe problem discussed here, there is a symmetric pair of transitions, described by

**, to and from . If**

*V**L*≫

*a*, then

**is expected to be negligible over most of the pipe during the time evolution.**

*V*## 3. Oscillatory flow transition between uniform and Stokes layer profiles

Substitution of equations (2.4) and (2.9) into (2.5) shows that the entry and exit profile of the adjustment velocity field ** V** is given by(3.1)Equation (2.1) allows the introduction of the dimensionless stream function

*ψ*(

*r*,

*z*:

*t*) such that(3.2)Then (3.1) yields(3.3)after imposing(3.4)which ensures no flow through the pipe wall. Equation (3.3) shows that the flux-free adjustment flow consists of the oscillatory mode that is replaced at time

*π*/2

*ω*by steady pressure-driven flow and an infinite set of evanescent modes whose amplitudes change at the same time, i.e.(3.5)

In terms of the stream function, equation (2.3) reduces to(3.6)and hence the modal functions are governed by(3.7)The task of constructing *ψ*_{0}, *ψ*_{ω} and *ψ*_{m} can be reduced to a single calculation either by reverting to Laplace transform methods or, as preferred below, constructing *ψ*_{ω}(*r*, *z*) and subsequently deducing *ψ*_{0} by letting *ω*→0 and *ψ*_{ω}(*r*, *z*) by successively multiplying by and replacing i*ω* by −(*λ*_{2m}/*a*)^{2}*ν*.

Guided by the form of the separated solutions of (3.7), the axial decay and growth of the end values in (3.1) is identified by means of the Fourier–Bessel series (Watson 1958)(3.8)where {*λ*_{1k}; *k*≥1} is the set of positive zeros of *J*_{1}, arranged in ascending order. Hence, a solution of (3.7) that satisfies (3.3) and (3.4) is given by(3.9)The two sets of coefficients are determined by applying the no-slip conditions, ∂*ψ*/∂*r*=0 at *r*=*a* and ∂*ψ*/∂*z*=0 at *z*=±*L*, which respectively yield, after exploiting the orthogonal functions,(3.10)and(3.11)These equations cannot be solved by truncation because the homogeneous system can be shown to have a non-trivial solution. The Fourier–Bessel series (Watson 1958)(3.12)implies that(3.13)Also(3.14)and use of (3.13) and (3.14) shows that the linear system, (3.10) and (3.11), has the homogeneous solution(3.15)Although this can be easily eliminated from a numerical solution, its significance is felt in the need to satisfy an associated solvability condition, as discussed by Davis (2002, 2003).

Since the coefficients of *Y*_{ωk} in (3.10) and *X*_{ωl} in (3.11) are identical, it is evident that the homogeneous adjoint system has the solution *u*_{l}(*l*≥0), *v*_{k}(*k*≥1), given by (3.15). The solvability condition therefore takes the form(3.16)where ** h** and

**denote right-hand side vectors in the linear system. Each series in (3.16) must be convergent, but this is not achieved by the forcing terms in (3.10) and (3.11). Following Meleshko & Krasnopolskaya (2001), convergence is first enhanced by subtracting out the solutions and obtained in the limits**

*c**l*→∞ and

*k*→∞ of (3.10) and (3.11), respectively. In each equation, the series is identified as a Riemann sum associated with , since

*λ*

_{1k}∼

*kπ*(Gradshteyn & Ryzhik 1980), and it follows that(3.17)Thus, and balance part of the forcing term in (3.10), as in Meleshko & Krasnopolskaya (2001) whose method, modified by Davis (2002), is further mimicked by writing(3.18)(3.19)and the new coefficients defined by (3.18) and (3.19) satisfy(3.20)(3.21)where, after using (3.17),(3.22)

(3.23)Not surprisingly, the inverse factor in (3.19) resembles the terms in the eigenvalue equation solved by Shankar *et al*. (2003) in their study of oscillatory eddies in a rectangular container. When (3.15), (3.22) and (3.23) are substituted into (3.16), most of the multiple summations can be evaluated by reversing their orders. It is shown in appendix A that the terms in and exactly cancel each other and, by use of (3.13) and (3.14), it is easily found that the other terms vanish identically. Thus, the solvability condition is satisfied.

## 4. The viscous diffusion modes and the steady state

Having completed the construction of *ψ*_{ω}, an expression for *ψ*_{m}(*r*, *z*) can be deduced from (3.9) by multiplying by , and then replacing i*ω* by −(*λ*_{2m}/*a*)^{2}*ν*. Thus,(4.1)Since (3.13) implies thatit follows from (3.17) that the limit values and are both zero and hence (3.18) and (3.19) are replaced by(4.2)and(4.3)respectively. The new coefficients defined by (4.2) and (4.3) satisfy equations that replace (3.20)–(3.23), namely(4.4)(4.5)Note that *ψ*_{m} needs to be modified if the aspect ratio *L*/*a* is such that for some integer pair (*K*, *N*). Then, in (4.1) must be replaced by *Z*_{m}. With *z*_{m} defined similarly, the *k*=*K* equations in (4.3) and (4.5) yieldrespectively. Also, the *k*=*K* terms in the *l*=*N* equation of (4.4) are *O*(1). So the last series in *ψ*_{m} has one term and its associated equation deleted.

The steady limit of the additional stream function is given by(4.6)which, according to (3.1), must have end values(4.7)This yields the imposed zero net flux. The associated Fourier–Bessel series, deduced from (3.5), features in the steady-state solution deduced from (3.6), by substitution into (4.6), to be given by(4.8)The equations corresponding to (3.10) and (3.11) are deduced similarly and the homogeneous solution (3.12) is unaltered. The limit solutions, , maintain their role, being given by the *ω*→0 limit of (3.17), namelyThe transformation(4.9)deduced from (3.18) and (3.19), yields the following linear system, deduced from (3.20)–(3.23):(4.10)(4.11)A simplified form of appendix A confirms the zero net contribution of and to the limit form of the solvability condition (3.13).

## 5. Failure of alternative constructions

For simplicity, attention here is restricted to the steady state. The solution (4.8) is constructed to satisfy the Dirichlet conditions identically and completed by subsequent imposition of the Neumann conditions. Ostensibly, there appear to be three alternative constructions but the need to achieve both *ψ*_{0}=*O*(*r*^{2}) as *r*→0 and *ψ*_{0}=0 at *r*=*a* forces the use of the Bessel functions in (4.7), leaving only one sensible alternative to consider. A solution satisfying the steady case in (3.7), with *ψ*_{0}=0 at *r*=*a* and ∂*ψ*_{0}/∂*z*=0 at *z*=±*L*, is given by(5.1)Note that the different Fourier cosines allow a term whose significance, in contrast to all others in (4.8) and (5.1), is not confined to a pipe wall or end layer. Though having the same algebraic form as the end values in (4.7), the term is not redundant because it generates tangential slip. The two sets of coefficients are determined by applying the normal flow condition (4.7) and the no-slip condition, ∂*ψ*_{0}/∂*r*=0 at *r*=*a*, which respectively yield, after exploiting the orthogonal functions,(5.2)and(5.3)The homogeneous solutions are *X*_{0l}=1(*l*≥0) and *Y*_{0k}=1(*k*≥1), and the equality of the coefficients of *Y*_{0k} in (5.2) and *X*_{0l} in (5.3) ensures that the homogeneous adjoint equations have the same solution. Moreover, the respective *l*→∞ and *k*→∞ limits of these equations both yield . The arbitrariness of their common value is made possible by the homogeneous solutions above. Thus, *X*_{0l}→0 and *Y*_{0k}→0 may be assured, which allows the interchange of orders of summation when terms in (5.3) are summed from *k*=1 to ∞. The net result, after substituting (5.2), is that, since , the solvability condition is not and *cannot* be satisfied, due to the inability to make helpful use of the limit values, , . Thus, it is imperative to use the solution construction that identically ensures no flow through the tube walls.

## 6. Steady thermal transpiration

The motivating application is to the steady state achieved in Reynolds' (1879) notable experiment, which demonstrated the thermomolecular pressure phenomenon, now termed thermal transpiration. A single component fluid, liquid or gas, is contained in a long (*a*≪*L*) closed capillary tube and obeys the single component isobaric law of adiabatically additive volumes (Brenner 2005*a*), which assumes that pressure effects on density are small compared with those of temperature. Brenner's modified continuum equations (Brenner 2004, 2005*a*,*b*) then include, for steady flow, both (2.1) and the time-independent form of (2.2), with *νρ*_{0} replaced by a constant viscosity *μ*. However, ** v** is identified as the volume velocity and distinguished from the barycentric (mass) velocity

*v*_{m}by the relation(6.1)where

*α*is the thermal diffusivity and

*T*is the temperature. The use of

*α*and

*T*here, instead of the volume diffusivity and the specific volume (inverse of density), is facilitated by the above assumption that pressure effects on density are small compared with those of temperature.

At a rigid boundary, the tangential component(s) of ** v** and the normal component of

*v*_{m}are zero. In particular, according to (6.1),

**=0 at a thermally insulated rigid boundary, which property was possessed by Reynolds' tube wall and is retained here. Thus, the thermal insulation implies no slip for**

*v***, but axial temperature gradients at the tube ends can now generate a flow. Let the differential heating be such that(6.2)where**

*v**T*

_{0}is a constant temperature. The linearized form, ∇

^{2}

*T*=0, of Fourier's steady internal energy equation and the sidewall insulation condition, ∂

*T*/∂

*r*=0 at

*r*=

*a*, show that which, according to (6.1), implies that(6.3)where

*α*

_{0}is a typical value of

*α*. Since this end condition matches the steady part of (2.4) when

*U*=

*ϵα*

_{0}/

*L*, the assumption

*ϵaα*

_{0}≪

*Lν*allows the steady flow to be identified as a multiple of that fully analysed in §4.

As an example of the modelling capabilities of the revised Navier–Stokes–Fourier equations and associated boundary conditions, Bielenberg & Brenner (2006) discuss the steady flow in Reynolds' experiment. They construct only the Poiseuille flow contribution to *W* in (2.9) that suffices to render the Maxwell (1879) slip condition unnecessary and eliminate the factor of 2 discrepancy with the experimental data. The flow details of their ‘end-cap’ cylindrical regions of length *O*(*a*), where the mass fluid motion *v*_{m} must reverse direction, are given by (4.8). The physics of the flow is better described in terms of the pipe eigenfunction (1.1), as outlined by Katopodes *et al*. (2000) for piston forcing and used by Davis (1990) to show that a body falling within a pipe ‘sees’ the bottom wall at *O*(*a*) distances. These calculations show that a large aspect ratio, *L*/*a*, is essentially immaterial in the ‘end-caps’, being supplanted by the real part, 4.463, of the smallest zero of in the first quadrant. A related result is that the net pressure drop in creeping flow through an orifice is well approximated by assuming Poiseuille flow throughout the pore and Sampson flow outside (Dagan *et al*. 1982).

## 7. Evolution of thermal transpiration

Given that the thermal law of adiabatically additive volumes remains valid in time-dependent flows (Brenner 2004), suppose that the end temperatures in (6.2) evolve from *T*_{0} in the same manner as the end velocities in (2.4). Then their Laplace transforms are given by(7.1)The linearized form,(7.2)of Fourier's internal energy equation and the sidewall insulation condition, ∂*T*/∂*r*=0 at *r*=*a*, then yield(7.3)Inversion of this transform gives(7.4)which is continuous at *t*=*π*/2*ω* and shows that the insulated walls not only allow a solution independent of *r* but also force it to be asymmetric with respect to the midplane *z*=0. With pressure effects on temperature assumed negligible, the latter evolves according to (7.4) regardless of the fluid motion generated by thermal transpiration at each end of the tube.

An additional feature of the revised equations, not previously relevant here, is that the inertia term in the momentum equation is(Brenner 2005*b*) so that the linearized equation (2.2) is replaced by(7.5)Owing to (6.1), this change leaves (2.3) unaltered and hence its sole effect is to modify the axial pressure gradient, because *T*=*T*(*z*, *t*). The entry and exit profile of the volume velocity ** v** is determined by setting to zero the tangential component of

**and the normal component of**

*v*

*v*_{m}at the rigid ends. Thus,

*v*_{z=±L}is obtained from (6.1) by setting

*v*_{m}=0 at

*z*=±

*L*. The steady term appeared in (2.4), as did the oscillatory term, now modified by a complex amplitude factor. The flow generated by the ‘heat diffusion’ series can be deduced from the oscillatory flow, as earlier for the ‘viscous diffusion’ series, by replacing i

*ω*by −(

*nπ*/

*L*)

^{2}

*α*

_{0}(

*n*≥1). Thus, by comparison with (2.9) and (3.5), the flow induced by the applied end temperatures,is given by(7.6)where(7.7)(7.8)

In each case, the second series is the response to the evolving temperature gradient at the ends, while the first series consists of eigenfunctions added to ensure that *W*_{T} and *ψ*_{T} are initially zero throughout the closed pipe. Note that the diffusion of heat introduces a second infinite set of decay rates, with time-scale *L*^{2}/*α*_{0} which, for consistency, must be *O*(*a*^{2}/*ν*) or larger. If (Prandtl number) *ν*/*α*_{0}≫*a *^{2}/*L*^{2}, the flow is seen to adjust relatively quickly to the evolving temperature field. The appearance in (7.8) of and the denominator indicates an assumption that for any integers *m*, *n*≥1. Otherwise, the consequent double pole in the Laplace transform of *W*_{T}(*r*, *t*) generates a resonance between the two disparate modes.

The function *Ψ*_{n}(*r*, *z*) in (7.8) is deduced from (3.9) by replacing i*ω* by −(*nπ*/*L*)^{2}*α*_{0} and in (3.10), (3.11) for determination of its coefficients Note that *Ψ*_{n} needs to be modified, like *ψ*_{m} in (4.1), if the aspect ratio *L*/*a* and Prandtl number are such that for some integer pair (*K*, *M*).

The additional term in ∂*p*/∂*z*, due to having *v*_{m} instead of ** v** in the momentum equation (7.5) is found from (6.1), (7.4) and (7.6) to bewhich is continuous at

*t*=

*π*/2

*ω*but at

*t*=0 reduces to

*ϵρ*

_{0}

*α*

_{0}

*ω*/

*L*times the seriesThis anomalous feature is due to the Fourier series in the additional pressure, an odd function of

*z*, having discontinuous sums at

*z*=∓

*L*, exhibited by . It can be removed, at the expense of more algebra, by replacing sin

*ωt*in (2.4) by, for example, sin

^{2}

*ωt*. Indeed, the pressure gradient can be made to evolve as smoothly as required by suitably introducing the temperature disparity between the ends of the tube.

## 8. The flow structure

The evolution from rest of the unidirectional flow is essentially a standard calculation. Equation (2.9) displays an oscillatory flow that is replaced by the Poiseuille flow at time *t*=*π*/2*ω* and evanescent modes that are generated at both *t*=0 and *t*=*π*/2*ω*. Contributions to the flux-free adjustment flow ** V**(

*r*,

*z*,

*t*) arise from all three component flows whose physical structure can be made apparent in terms of the pipe eigenfunction (1.1) and their time-dependent modifications. Thus,

*ψ*

_{0}(

*r*,

*z*) is a linear combination of functionwhere . Similarly, from (3.7),

*ψ*

_{ω}(

*r*,

*z*) is a linear combination of functionwhere(8.1)while

*ψ*

_{m}(

*r*,

*z*) and, in the transpiration application,

*Ψ*

_{n}(

*r*,

*z*) are likewise constructed by again replacing i

*ω*by −

*ν*(

*λ*

_{2m}/

*a*)

^{2}and −(

*nπ*/

*L*)

^{2}

*α*

_{0}, respectively. Note that there are infinitely many pipe eigenfunctions for each

*m*and

*n*. Nevertheless, the adjustment flow

**(**

*V**r*,

*z*,

*t*) is essentially confined to end-cap cylinders of length

*O*(

*a*). Evidently, the eigenvalue equation (8.1) is the axisymmetric analogue of that solved by Shankar

*et al*. (2003).

## Acknowledgments

The author thanks Professor Howard Brenner for stimulating his interest in the revised continuum equations and boundary conditions.

## Footnotes

- Received April 11, 2006.
- Accepted October 6, 2006.

- © 2006 The Royal Society