## Abstract

This study focuses on two-dimensional fluid flows in a straight duct with free-slip boundary conditions applied on the channel walls *y*=0 and *y*=2*πN* with *N*>1. In this extended wall-bounded fluid motion problem, secondary fluid flow patterns resulting from steady-state and Hopf bifurcations are examined and shown to be dependent on the choice of longitudinal wave numbers. Some secondary steady-state flows appear at specific wave numbers, whereas at other wave numbers, both secondary steady-state and self-oscillation flows coexist. These results, derived through analytical arguments and truncation series approximation, are confirmed by simple numerical experiments supporting the findings observed from laboratory experiments.

## 1. Introduction

The laboratory experiments of Bondarenko *et al*. (1979), Batchaev & Dowzhenko (1983), Obukhov (1983), Kolesnikov (1985*a*,*b*) and Batchaev (1988) into plane fluid flows subject to electromagnetic fields in a channel of finite dimensions revealed primary and secondary flow patterns dependent on the dimensions of the channel and Reynolds number. Differences were found between experimental observations and theoretical predictions based on developed mathematical models. For example, experimentally Bondarenko *et al*. (1979) observed invariance of the longitudinal wave number of secondary flows under variation of the channel length which contradicted the findings of a linear stability analysis of the primary flow.

Thess (1992*a*,*b*, 1993) conducted numerical investigations into linear stability occurring in two-dimensional spatially periodic flows and confirmed that the stability parameters depended non-uniformly on confinement. However, many of his predicted flow patterns were not reported in the laboratory experiments, although theoretical predictions of parameters were in reasonable agreement with experimental results. These laboratory experiments and numerical investigations form part of extensive ranging studies associated with the general problem of plane periodic flows of an incompressible fluid under the action of a spatially periodic force as discussed by Kolmogorov (see Arnold & Meshalkin 1960).

Bondarenko *et al*. (1979) and Thess (1992*a*,*b*, 1993) comprehensively describe the background to the electromagnetically driven fluid motion problem and therefore detailed descriptions of the laboratory experiments and computational procedures are omitted herein. In the proposed mathematical model, the flow velocity field ** v** is defined by a stream function

*ψ*in the dimensionless vorticity equation(1.1)where

*R*denotes the dimensionless Reynolds number, ∇=(

*∂*) and . This equation differs from the conventional Navier–Stokes equation (Meshalkin & Sinai 1961) by the appearance of the term (

_{x},∂_{y}*σ*/

*R*)Δ

*ψ*which describes a friction force proportional to the velocity and by the term (1+

*σ*)

*R*

^{−1}sin

*y*modelling the Lorentz force creating the Kolmogorov flow. That is, the forcing function is chosen such that the stream functiondescribes the Kolmogorov velocity profile and the fluid flow is confined to the region {−∞<

*x*<∞, 0<

*y*<2

*πN*} by non-porous walls at

*y*=0 and

*y*=2

*πN*, imposing the free-slip boundary conditions(1.2)

This boundary assumption implies the fluid motion is restricted to the average velocity field conditionon the fluid domain, which is normally applied to spatial periodic flows as discussed, for example, by Iudovich (1965) to exclude solutions *v*+*c* for any constant *c*≠0.

The parameter *σ*/*R* denotes the strength of the inertial force in relation to the linear friction force. The latter models energy dissipation is inside the bottom Hartman boundary layer. As is discussed by Bondarenko *et al*. (1979) and Dolzhanskiy (1987), this frictional force is crucial to the description of the onset of instability in the experimental results based on the proposed plane fluid motion model.

In contrast to the numerical experiments of Thess (1992*a*,*b*) and Chen & Price (2002) proved that, in a duct bounded by walls at *y*=0 and *y*=2*π* (i.e. *N*=1), all possible secondary flows transitional from the basic flow *ψ*_{0}=sin *y* are self-oscillations. That is, the instabilities arising were analytically proved to be Hopf bifurcations which were subsequently verified by simple numerical predictions. The study herein is an extension of the original investigation with channel walls now positioned at *y*=0 and *y*=2*πN* (*N*>1). To minimize repetition, the findings of the initial investigation by Chen & Price (2002) are quoted herein without proof, and therefore the interested reader may wish to consult with this reference for additional background information.

The motivation for the present investigation is to show that qualitative differences occur between elementary wall-bounded flows for *N*=1 and extended wall-bounded flows for *N*>1. For example, for *N*=1 only secondary self-oscillations may occur, whereas for *N*>1 all secondary flows for *N*=1 are present and other possible secondary steady-state flows are exhibited, thus supporting the laboratory evidence presented by Bondarenko *et al*. (1979). That is, besides the possible Hopf bifurcation phenomenon with respect to a single critical Reynolds number *R*_{c} ( in the notation of the present paper), the extended wall-bounded flow problem admits *N*−1 extra critical Reynolds numbers , each of which gives rise to a two-dimensional real unstable mode space. On the basis of a truncation series approximation, secondary flows arise at the points (sin *y*, ) (*j*=1,…,*N*−1) in the direction of these real unstable modes supporting the laboratory evidence presented by Bondarenko *et al*. (1979). Furthermore, with the help of simple numerical experiments, we find that the critical Reynolds numbers derived by Thess (1992*a*) for the extended system *N*>1 are the special values in the case *j*=1. Although secondary flows are observed in laboratory experiments, the development of such an analysis enables understanding of the interactive mechanisms underlying the behaviour of supercritical flow regimes inherent in plane parallel flows, especially the onset and type of secondary flows arising from a primary flow. This is achieved through investigating the distribution of secondary flow regimes in the context of a magnetohydrodynamics problem.

The model assumed in this study is described by Thess (1992*a*), who, based on a spatial periodic perturbation approach to the basic Kolmogorov flow, developed a linear stability analysis and through numerical calculations confirmed the experimental observations and findings of Kolesnikov (1985*a*,*b*).

Let us commence our analysis by following the perturbation approach described by Lin (1955). It is noted that the basic steady-state solution is unidirectional and therefore the linear stability analysis of the mathematical model (1.1) and (1.2) reduces to an *x*-periodic linear perturbation problem around *ψ*_{0}. This allows the substitution of the decomposed stream functioninto (1.1) which, after linearization and omitting the superscript ‘hat’, gives the stability eigenvalue equation(1.3)

This equation is subject to the generalized free-slip boundary condition

Therefore, all possible unknown eigenfunctions may be expressed formally in the formor(1.4)where *ϕ*_{n} denotes an unknown complex value and *k*_{x} a wave number. Here the term i^{n} appearing explicitly in the expansion is for convenience only as it aids development of the forthcoming derivation.

For integer 1≤*j*≤*N*, if the eigenvalue *ρ* lies on the imaginary line Re *ρ*=0 of the complex plane, then the associated value of the Reynolds number *R* with respect to the spectral problem (1.3) and (1.4) is represented by . If *j*=*N*, then can be demonstrated numerically by Thess (1992*a*) and is a Hopf bifurcation value due to Hopf bifurcation analysis discussed by Chen & Price (2002). However, as examined in §2, for *j*=1,…,*N*−1, the critical Reynolds number gives rise to the real eigenvalue *ρ*=0 and a pair of real eigenfunctions(1.5)with the wave number subject to the condition

Thess (1992*a*) showed that the flow is linear stable for perturbation disturbances with wave number and Chen & Price (2002) proved that non-real eigenvalues may arise only when with *N*=1. We now find that the value with *N*=1 is also a critical value for the extended system *N*>1 in the following sense: a pair of real eigenfunctions and a pair of non-real eigenfunction may *coexist* with respect to the eigenvalue *ρ* with Re *ρ*=0 when

On the other hand, there exists a pair of real eigenfunctions when and only when

Iudovich (1965) proved that some special steady-state secondary flows exist for a periodic fluid motion problem. However, the postulated secondary flows do not appear in the present wall-bounded fluid motion problem because of the influence of different boundary conditions.

The main purpose of this paper is to provide evidence for the existence of steady-state bifurcation solutions of (1.1) and (1.2) around the point (sin *y*, ) with *j*=1,…,*N*−1. That is, solutions *ψ*_{R}(*x*,*y*) other than the basic flow sin *y* arise when *R* increases across each of the critical values . Unfortunately, owing to the coexistence of the pair of real eigenfunctions is the form of (1.5) in this fluid motion problem, there seems to be no suitable mathematical theorems available to show the existence of bifurcating steady-state solutions. Furthermore there are no flow invariant subspaces containing only a single eigenfunction of (1.5) allowing the application of the steady-state bifurcation theory developed by Krasnoselskii (1965), Nirenberg (1974) and Rabinowitz (1971). To overcome this deficiency, we truncate the Navier–Stokes equation into an infinite dimensional ordinary differential system, which gives exact steady-state solutions undergoing supercritical steady-state bifurcations. The flow patterns defined by these approximate steady-state solutions match well with those observed in the laboratory experiments presented by Bondarenko *et al*. (1979).

Recently, a bifurcation analysis theory was established by Ma & Wang (2003) with respect to any dimensional eigenfunction spaces. This theory has been successfully applied to Rayleigh–Bénard convection equations and subsequently to the Ginzburg–Landau equation by Ma & Wang (2003) and Ma *et al*. (2004).

Frenkel (1991) introduced a quasi-normal mode approach in examining the linear stability of periodic flows. This idea was further developed by Zhang (1995) and Zhang & Frenkel (1998) to investigate linear stability problems. Zhang (1995) showed that intermediate-scale nonlinear instability of multidirectional periodic flows is mathematically modelled by the Landau equation. Since the critical wave number of instability discussed in the present study is nonzero, in slightly supercritical regimes the nonlinear evolution (from arbitrary initial conditions) of the fluid motion problem described by (1.1) and (1.2) might be described by a Ginzburg–Landau type equation in a manner similar to the approach discussed by Zhang (1995) for the nonlinear stability of periodic flows.

This paper is organized as follows. In §2, by developing the approach of Meshalkin & Sinai (1961) and Frenkel (1991), we prove the existence of the critical Reynolds numbers and associated eigenvalues, and show that the spectral problem with respect to the possible critical value for *N*>1 overlaps with that of the elementary case *N*=1. In §3, we follow the truncation scheme of Chen & Price (2002) to reduce (1.1) and (1.2) into a system of infinite dimensional ordinary differential equations, which display circles of the secondary steady-state solutions approximating to the secondary flows of (1.1) and (1.2). Section 4 presents numerical results of the approximate steady-state solutions obtained in §3. These findings support the laboratory experiments of Bondarenko *et al*. (1979) and show that the critical control parameters observed by Thess (1992*a*) give rise to a pair of real unstable modes.

## 2. Spectral analysis

In order to characterize the eigenvalue *ρ*=*ρ*(*R*) and the eigenfunction *ψ* to derive the proof of the existence of critical Reynolds number for every extended system which depends on the positive integer *N*, let us begin with the formulation of the spectral problem in terms of algebraic equations by following the approach of Meshalkin & Sinai (1961) and Frenkel (1991).

Let us classify the eigenfunctions expressed in (1.5) in accordance with the choice of integer *j* in the form(2.1)

Here, the unknown complex numbers *φ*_{n} and *ϕ*_{n} satisfy the condition(2.2)

It follows from Chen & Price (2002, appendix) that *φ*_{n}≡0 for either *j*=0 or , with *j*=*N* and |*φ*_{n}|+|*ϕ*_{n}|≡0 for *k*_{x}=0. The substitution of equation (2.1) into (1.3) gives for *j*=1,…,*N*, *k*_{x}≠0 and ,(2.3)with *j*=*N*, and(2.4)with *j*=1,…,*N*−1, where denotes the integer set.

If (2.3) has a critical Reynolds number , then Chen & Price (2002, Appendix) proved that (2.3) has exactly a pair of conjugate eigenvalues on the imaginary line Re *ρ*=0 associated with a pair of complex conjugate eigenfunctions {*φ*_{n}} and .

However, for the case *j*=1,…,*N*−1, we multiply the *n*th equation of (2.4) by and sum the resultant equations to give(2.5)

This implies that *ϕ*_{n}=0 when with *j*=1,…,*N*−1. This shows the non-existence of unstable modes for , the result given by Thess (1992*a*).

The main purpose of this section is to show the unique existence of the critical Reynolds number when the corresponding eigenfunction is to form of (2.1) for *j*=1,…,*N*−1 and for the wave number *k*_{x} subject to the condition(2.6)

We see that the real spectral problemwhere *ψ* in the form of (2.1) is equivalent to (2.4), which can be rewritten as(2.7)or, for *n*≥1,

Thus the zeroth equation of (2.7) becomes(2.8)where we have used (2.2).

Multiply this equation by *R* and we obtain(2.9)

It follows from (2.6) thatillustrating that the right-hand side of (2.8) is a strictly monotone function of *R* which tends to 0 and ∞ respectively when *R*→0 and *R*→∞. Thus there exists a unique value satisfying (2.9). This proves the existence of the critical value for *j*=1,…,*N*−1.

The convergence of the continued fraction is due to Wall (1948, theorem 30.1) and Khinchin (1964, theorem 10). Hence, under the condition of (2.6), the real spectral problem expressed in (2.4) has a unique value *R* and is equivalent to the continued fraction equation (2.8) and the eigenfunctions create a one-dimensional space in the following form:(2.10)

Here ‘const.’ is an arbitrary real constant and(2.11)for with *j*=1,…,*N*−1.

Thus, in summary, we have obtained the following basic results.

*Let σ≥0, and the integers N≥2 and j=1,…,N−1, with wave number k*_{x} *be subject to the condition*

*Under this condition, the spectral problem* (1.3) and (1.5) *with ρ=0 and σ≥0 has a unique critical Reynolds number*

*and exactly two orthogonal real eigenfunctions,*

*where the real coefficients ϕ*

_{n}

*are subject to*(2.8), (2.10) and (2.11).

*Let σ≥*0*, N≥2,* *, and* *be a critical Reynolds number satisfying the spectral problem* (1.3) and (1.5) *with* Re* ρ=*0 *and the eigenfunction ψ in the form*

*Then ρ≠*0 *and the complex coefficients ϕ*_{n} *are defined uniquely up to a complex constant, as proved rigorously by Chen & Price (**2002**, lemma A1).*

Finally, we prove the absence of the non-real eigenfunctions when the wave number .

*Let σ≥0, N≥2 and* *; then the spectral problem* (1.3) and (1.5) *with* Re* ρ=0 and ρ≠0 has no non-real eigenfunctions.*

We see that any eigenfunction of the spectral problem under the condition can be expressed in the formwhere . If *j*=*N*, it follows from lemma 2.2 that eigenfunctions exist only when . Thus we need only consider the case *j*=1,…,*N*−1. It follows from the proof of lemma 2.1 that the spectral problem for this case is equivalent to the following continued fraction equation:and for every *ρ*>0 there exists a unique *R* satisfying this equation. Moreover, this equation implies that *ρ*→−*σ*−*β*_{0} as *R*→0 and *ρ*→∞ as *R*→∞. Thus it suffices to show that this continued fraction equation has a unique eigenvalue *ρ* in the complex space for all *R*.

Indeed, let us suppose that for some *R*>0 in addition to the real eigenvalue *ρ*_{1} derived in the proof of lemma 2.1 there exists a non-real eigenvalue *ρ*_{2}. We rewrite the previous continued fraction equation in the form(2.12)with(2.13)defined by equation (2.11). We see thatand by inductionwhere is given by (2.10) such that *ϕ*_{0}=1/(*β*_{0}−1). Thus subtracting (2.12) with *j*=2 from (2.12) with *j*=1 yieldswhenever *ρ*_{1}≠*ρ*_{2}. By (2.5), this quantity equalsfor the choice of *ϕ*_{0}=1/(*β*_{0}−1). This implies that *ρ*_{1}=*ρ*_{2}. We thus obtain the non-existence of non-real eigenvalues. The proof is complete. ▪

*Thus by* *lemma 2.2* *and the proof of Chen & Price* (*2002**, theorem 1.1), we have actually proved that for any N>1 and* *, the problem defined by* (1.1) and (1.2) *undergoes Hopf bifurcation around the point (*sin* y,* *), provided that* *satisfies the spectral problem* *(1.3)* *with* Re* ρ=*0 *and the associated eigenfunction is in the form*

*The existence of* *, or R*_{c} *in Chen & Price (2002), is based on the numerical computations of* *Thess (1992a)*.

## 3. Approximate steady-state solutions

A truncation scheme based on a Fourier expansion allows demonstration of the fact that the secondary flows are supercritical. Furthermore, the supercritical regimes predicted support the corresponding computational results of Thess (1992*a*) and the laboratory experiments of Bondarenko *et al*. (1979). The evidence to substantiate these findings is developed by first noting that the basic fluid flow and the first unstable mode dominate the existence of secondary flows, and that the unstable modes are of the formsfor *j*=1,…,*N*−1. Let us assume an infinite dimensional truncation scheme given by(3.1)withto approximate the secondary flows around the basic flow, *ψ*_{0}, arising from the real unstable modes. This truncation scheme is referred to as the best approximation of the steady-state solutions on the postulated spaces respectively, as discussed by Chen & Price (2002), expanded by the basic flow and the corresponding unstable mode. Note that for *ϕ*=*ξ _{n}*,

*η*

_{n}and, by applying an integration by parts, we find that

Thus the projection of (1.1) and (1.2) with respect to the infinite dimensional space spanned by {*ξ*_{n}, *η*_{n}, sin *y*} gives(3.2)for *ϕ*=*ξ*_{n}, *η*_{n} and sin *y*. That is, for −∞<*n*<∞,(3.3)

We shall now develop an exact expression of the steady-state solutions of this set of ordinary differential equations.

To do so, let us examine the non-zero solution of the steady-state system of (3.3) given byor(3.4)

It follows from (2.4) and (2.10) and the unique existence of the critical Reynolds number obtained in lemmas 2.1 and 2.3 that(3.5)for unknown constants *c*_{1} and *c*_{2}, where *γ*_{±n} are defined by (2.11) with and *ρ*=0.

To define these constants, the substitution of these expressions in (3.5) into the last equation of (3.4) leads tofor *γ*_{0}=1. Defining constants *c*>0 and −*π*/*k*_{x}<*θ*≤*π*/*k*_{x} to bewe obtain

It thus follows from (3.5) that all steady-state solutions branching off the basic solutionof (3.3), at the critical Reynolds number , expressible asfor , −*π*/*k*_{x}<*θ*≤*π*/*k*_{x} and *j*=1,…,*N*−1. These expressions demonstrate a supercritical bifurcation or the bifurcating solutions arises only when . The substitution of these expressions into (3.1) allows an infinite number of steady-state solutions branching off (sin *y*, ), in the range (−*π*/*k*_{x}<*θ*≤*π*/*k*_{x}) and for , to be expressed as(3.6)with

This gives rise to a circle of steady-state solutions bifurcating from *ψ*_{0} at a supercritical Reynolds number value for every *j*=1, 2,…,*N*−1, thus approximating the secondary flows arising from the mathematical model defined by (1.1) and (1.2).

## 4. Numerical experiments

The previous section discusses explicit approximate steady-state stream function solutions, as expressed in (3.6), bifurcating from the basic flow *ψ*_{0}. These bifurcating solutions are defined by the infinite-dimensional truncation system (3.3). There remains uncertainty, however, whether *all* of these bifurcating solutions are linearly stable and thus observable in experiments, since a periodic or even chaotic regime may exist. It is now demonstrated through numerical experiments that the derived bifurcating solutions correspond to stable secondary steady-state flows observed in the liquid metal flow laboratory experiments of Bondarenko *et al*. (1979).

It is readily seen from the previous section that the original system (1.1) and (1.2) and the truncated system (3.3), with and *j*=1,…,*N*−1, have the same critical Reynolds number . Thus these values can be obtained from (3.3) by numerical computation. This section is devoted to the presentation of the approximate steady-state flows (3.6) relating to the secondary flows of (1.1), in conjunction with the control parameters computed by Thess (1992*a*). The numerical computations display the approximate steady-state flows (3.6) and this predicts secondary flows comparable with experimental evidence and measurement as presented by Bondarenko *et al*. (1979). These simple numerical experiments focus on the extended system defined by the parameter *N*=2, 3, 4, since they reflect the generic tendencies of the secondary flows of the overall extended system for *N*≥2. Let us begin this investigation with the mathematical model expressed by Thess (1992*a*) in the form(4.1)where a comparison of notation shows that

This equation is subject to the boundary conditions defined in (1.2). To provide comparative information, we list the critical control parameters, *ν*_{c}, *μ*_{c} and *k*_{x}, together with the corresponding critical control parameters, *R*_{c}(*k*_{x}, *σ*) and *σ*, given by Thess (1992*a*) in tables 1–3 for *N*=2, 3, 4, respectively.

Let us recall that lemma 2.1 ensures the existence of a critical Reynolds number giving rise to the approximate steady-state solutions (3.6) of which the coefficients {*X*_{n}} and the values can be computed from (3.3), where *σ*≥0 and the wave number *k*_{x} satisfy(4.2)for *j*=1,…,*N*−1. The numerical results presented are derived by using the Adams–Bashforth method described by Lambert (1991) to the truncated equations (3.3).

It is readily seen that the wave number *k*_{x} in tables 1–3 satisfies (4.2) for *j*=1, and our numerical computation of (3.3) shows that for *N*=2, 3 and 4 in tables 1–3, respectively. Thus *R*_{c} leads to secondary flows approximated by the steady-state flows (3.6) with *j*=1.

Figure 1 displays the critical control parameters for *N*=2 and *j*=1 with *σ*=8.48 selected from table 1. As already obtained analytically, is bounded by the line in the (*R*, *k*_{x}) plane and this property is revealed in figure 1. When *k*_{x} tends to , the critical Reynolds number .

The numerical profiles of the approximate steady-state solutions (3.6) with *θ*=0 and the control parameters listed in tables 1–3 are deduced from the numerical computations of (3.3) to obtain ({*X*_{n}}, {*Y*_{n}}, *Z*) by taking the initial value ({*X*_{n}(0)}, {*Y*_{n}(0)}, *Z*(0)) close to the basic solution ({*X*_{n}}, {*Y*_{n}}, *Z*)=(0,0,1) of (3.3), such that *X*_{0}(0)=−0.001, *X*_{n}(0)=0 for *n*≠0, *Y*_{n}(0)≡0 and *Z*=1. A selection of typical numerical results are displayed in figures 2–5. Figures 2 and 3 show the approximate steady-state solutions (3.6) with *θ*=0 and *N*=2 for Reynolds number values *R*≈52.5 and 63.1, respectively, just above and well above the critical value . They clearly display the differing secondary flow patterns excited driven electromagnetically and reflect experimental findings. Moreover, to provide further understanding of the secondary flows for general integer value *N*, figure 4 (*σ*=11.98, *R*=52.5, ) and figure 5 (*σ*=11.98, *R*=63.1, ) display the flows associated with (3.6) with *θ*=0 and *N*=4. These provide descriptions of typical secondary flows for a large *N* value.

It should be noted from the unstable modesthat for 1≤2*j*≤*N*, where for example, for *j*=1, . Additionally, for each *j*=1,…,*N*−1 there exist infinite branches of the approximate steady-state solutions (3.6) bifurcating from the first flow *ψ*_{0}=sin *y* when . Each of such branches is labelled by *θ* for −*π*/*k*_{x}<*θ*≤*π*/*k*_{x}. However, from the expression (3.6) we see the flow pattern at a particular value of *θ* overlaps that at the value zero after the parallel shift in the *x* direction (*x*′, *y*′)=(*x*+*θ*, *y*) is made. Thus all such secondary flows for −*π*/*k*_{x}<*θ*≤*π*/*k*_{x} are the same and so we only consider their numerical profile at the value of *θ*=0.

Finally, in order to address the connection with laboratory experiments given by Bondarenko *et al*. (1979), we substitute the transformation(4.3)into equation (1.1), which becomes(4.4)

This leads to the nondimensional formulation of the fluid motion equations examined by Bondarenko *et al*. (1979, eqn. 7), if the additional substitution is made to transform the forcing term sin *y* into .

In their experiments Bondarenko *et al*. (1979) selected *σ*=20. Let us adopt (3.3) and (4.3) with *N*=4 and *j*=1 to obtain the first critical control parameter of (4.4). Namely,with respect to all *k*_{x}, which is displayed in figure 6. This shows that (*k*_{x}, 20) attains approximately its minimal value, 1768, when *k*_{x}≈0.63. Thus, additional numerical experiments were performed based on (3.1) and (3.3) to obtain approximate steady-state solution in the form of (3.6) and (4.3) in order to illustrate predicted flow patterns. Figure 7 shows the secondary steady-state flow of (4.4) for in the range just above the minimal critical valueand figure 8 illustrates the flow for

In both cases, the predicted secondary flows displayed are at the centre of the duct away from the boundaries.

The laboratory experiments conducted by Bondarenko *et al*. (1979, figs 4 and 5) show the secondary flows in the centre of a duct under the dimensionless forcing cos *y* for the critical control parameters *k*_{x}≈0.68±0.05 and the critical Reynolds number≈2000. In comparison with the laboratory experiments, figure 7 with *R* chosen just above critical value shows very favourable agreement with the experimental evidence (Bondarenko *et al*. 1979, fig. 4). For *R* selected well above the critical value, agreement between prediction and experimental data is again very good as displayed in figure 8 here and in fig. 5 produced by Bondarenko *et al*. (1979). This agreement implies that the secondary flows observed by Bondarenko *et al*. (1979) are modified forms of the bifurcating steady-state solutions, which can be approximated well by the flows defined by (3.6).

## Footnotes

- Received December 25, 2002.
- Accepted December 1, 2004.

- © 2005 The Royal Society