Plane Poiseuille Flow Research Articles

Can isothermal plane Couette flow in fluid overlying porous layer be linearly unstable?

The present study aims to examine the temporal linear stability analysis of isothermal plane Couette flow over a porous layer using the two-domain approach. The flow in the porous layer is described by the unsteady Darcy–Brinkman equations, whereas it is characterised by the Navier–Stokes equations in the fluid layer. In contrast to the Darcy model, it is observed that the isothermal plane Couette flow becomes unstable for such a superposed system on the inclusion of the Brinkman term. From the stability analysis, the two-dimensional mode is found to be least stable, and two modes of instability, namely porous mode and mixed mode are obtained under the consideration of the Darcy–Brinkman model along with advection term (DBA model). For Darcy number $(\delta )=0.01$ , depending on the value of the stress-jump coefficient, mixed mode controls the instability of the system at small values of depth ratio $(\hat {d})$ , and it disappears for relatively high values of $\hat {d}$ , where the porous mode dominates. In addition, it has been observed that when $\hat {d}=0.1$ , the critical mode of instability is found to be mixed for $\delta >0.02$ and porous for $\delta \le 0.02$ . The stress-jump coefficient destabilises the flow in terms of energy production through perturbed stresses at the interface. As observed in the case of isothermal plane Poiseuille flow studied by Chang, Chen & Straughan (J. Fluid Mech., vol. 564, 2006, pp. 287–303), here also depth ratio (Darcy number) stabilises (destabilises) the flow. However, this characteristic does not remain valid when the advection term is eliminated from the considered momentum equation. For a certain range of $\hat {d} (\delta )$ , the destabilising (stabilising) characteristic of the respective parameters are encountered when the fluid mode of instability prevails.

Electrohydrodynamic stability of a two-layer plane Poiseuille flow in the presence of interfacial surfactant: Energy budget analysis

Electrohydrodynamic stability of a two-layer plane Poiseuille flow in the presence of interfacial surfactant: Energy budget analysis

Scale dependence and non-isochoric effects on the thermohydrodynamics of rarefied Poiseuille flow

The plane Poiseuille flow of a rarefied gas in a finite length channel, driven by an axial pressure gradient, is analysed numerically to probe (i) the role of ‘dilatation’ ( $\varDelta ={\boldsymbol \nabla }\boldsymbol {\cdot }{\boldsymbol u}\neq 0$ ) on its thermohydrodynamics as well as to clarify (ii) the possible equivalence with its well-studied ‘dilatation-free’ or ‘isochoric’ ( ${\rm D}\rho /{\rm D}t=0$ ) counterpart driven by a constant acceleration. Focussing on the mass flow rate ${\mathcal {M}}({Kn})$ , which is an invariant quantity for both pressure-driven and acceleration-driven Poiseuille flows, it is shown that while ${\mathcal {M}}\sim \log {{Kn}}$ at ${Kn}\gg 1$ in the acceleration-driven case, the mass flow saturates to a constant value ${\mathcal {M}}\sim {{Kn}}^0$ at ${Kn}\gg 1$ in the pressure-driven case due to the finite length ( $L_x<\infty$ ) of the channel. The latter result agrees with prior theory and recent experiments, and holds irrespective of the magnitude of the axial pressure gradient ( $G_p$ ). The pressure-dilatation cooling ( $\varPhi _p=-p\varDelta <0$ ) is shown to be responsible for the absence of the bimodal shape of the temperature profile in the pressure-driven Poiseuille flow. The dilatation-driven reduction of the shear viscosity and the odd signs of two normal stress differences ( ${\mathcal {N}}_1$ and ${\mathcal {N}}_2$ ) in the pressure-driven flow in comparison with those in its acceleration-driven counterpart are explained from the Burnett-order constitutive relations for the stress tensor. While both ${\mathcal {N}}_1$ and ${\mathcal {N}}_2$ appear at the Burnett order $O({{Kn}}^2)$ in the acceleration-driven flow, the leading term in ${\mathcal N}_1$ scales as $(\mu/p)\varDelta$ due to the non-zero dilatation in the pressure-driven Poiseuille flow which confirms that the two flows are not equivalent even at the Navier–Stokes–Fourier order $O({{Kn}})$ . The heat-flow rate ( ${{\mathcal {Q}}_q}_x=\int q_x(x,y) \,{{\rm d} y}$ ) of the tangential heat flux is found to be negative (i.e. directed against the axial pressure gradient), in contrast to its positive asymptotic value (at ${Kn}\gg 1$ ) in the acceleration-driven flow. Similar to the scale-dependence of the mass flow rate, ${{\mathcal {Q}}_q}_x({{Kn}}, L_x)$ is found to saturate to a constant value at ${Kn}\gg 1$ in finite length channels. The double-well shape of the $q_x(y)$ -profile in the near-continuum limit agrees well with predictions from a generalized Fourier law. On the whole, the dilatation-driven signatures (such as the pressure-dilatation work and the ‘normal’ shear-rate differences) are shown to be the progenitor for the observed differences between the two flows with regard to (i) the hydrodynamic fields, (ii) the rheology and (iii) the flow-induced heat transfer.

Subcritical transitional flow in two-dimensional plane Poiseuille flow

Recently, subcritical transition to turbulence in the quasi-two-dimensional (quasi-2-D) shear flow with strong linear friction (Camobreco et al., J. Fluid Mech., vol. 963, 2023, R2) has been demonstrated by the 2-D mechanism at $Re = 71\,211$ , and the nonlinear Tollmien–Schlichting (TS) waves related to the edge state were approached independently of initial optimal disturbances. For 2-D plane Poiseuille flow, transition to the fully developed turbulence requires that the Reynolds number is several times larger than the critical Reynolds number $Re_c$ (Markeviciute & Kerswell, J. Fluid Mech., vol. 917, 2021, A57). In this paper, we observed the subcritical transitional flow in 2-D plane Poiseuille flow driven by the nonlinear TS waves by both linear and nonlinear optimal disturbances ( $Re < Re_c$ ) with different quantitative edge states. The nonlinear optimal disturbances could trigger the sustained subcritical transitional flow for $Re \geqslant 2400$ . The initial energy for nonlinear optimal disturbance is more efficient than the linear optimal disturbance in reaching the subcritical transitional flow for $2400 \leqslant Re \leqslant 5000$ . Moreover, the initial energy of linear optimal disturbance is larger than the energy of its edge state. The nonlinear TS waves along the edge state are formed by the nonlinear optimal disturbances to trigger transitional flow, which agrees well with the main conclusions of Camobreco et al. (J. Fluid Mech., vol. 963, 2023, R2), while the required $Re$ of 2-D plane Poiseuille flow is much smaller.

Brownian particle diffusion in generalized polynomial shear flows.

This study presents a mathematical framework for calculating the diffusion of Brownian particles in generalized shear flows. By solving the Langevin equationsusing stochastic instead of classical calculus, we propose a mathematical formulation that resolves the particle mean-square displacement (MSD) at all timescales for any two-dimensional parallel shear flow described by a polynomial velocity profile. We show that at long timescales, the polynomial order of time of the particle MSD is n+2, where n is the polynomial order of the transverse coordinate of the velocity profile. We generalize the method to resolve particle diffusion in any polynomial shear flow at all timescales, including the order of particle relaxation timescale, which is unresolved in current theories. Particle diffusion at all timescales is then studied for the cases of Couette and plane Poiseuille flows and a polynomial approximation of a hyperbolic tangent flow while neglecting the boundary effects. We observe three main stages of particle diffusion along the timeline for Couette and plane Poiseuille flows and four main stages for hyperbolic tangent flow. The particle MSD is distinctly different across these stages due to different dominant physical mechanisms. Thus, higher temporal and spatial resolution for diffusion processes in shear flows may be realized, suggesting a more accurate analytical approach for the diffusion of Brownian particles.

Linear stability analysis of plane Poiseuille flow of a De-Kée–Turcotte fluid

Linear stability analysis of plane Poiseuille flow of a De-Kée–Turcotte fluid

On the stability and accuracy of TRT Lattice-Boltzmann method for non-Newtonian Ostwald-de Waele fluid flows

This paper brings a numerical analysis of the TRT modeling for the Lattice-Boltzmann method when solving flow for dilatant and pseudoplastic power-law fluids. Firstly, the method was reviewed to describe the required simulation parameters and the numerical methodology. Secondly, a mathematical procedure was performed to identify the characteristic relaxation frequency as a function of both flow and fluid properties and to work as a guide parameter for LBM operation. Then, a simple shearing Poiseuille flow was employed so its characteristic shear rate could be calculated as a function of fluid properties, given the flow was characterized by the Reynolds and Mach numbers. For this test case, convergence was then explored for a broad range of parameters, and its non-monotonic dependency on the Mach number for a given convergence criterion was shown. Then, stability maps were constructed based on the characteristic relaxation frequency, which showed a strong dependency between consistency and flow index so the simulation could converge. This was explored against the results from the converged tests, which pointed out the usefulness of the characteristic relaxation frequency in predicting stable solutions. Finally, quantitatively, it was shown that for this power-law fluid flow, the ||L2|| relative error depends on the Mach number to the power of 2(2−n) being now a function of the flow index, extending the previously reported dependency of the Mach number to the power of 2 for plane-Poiseuille flow.

KPP fronts in shear flows with cutoff reaction rates

AbstractWe consider the effect of a shear flow which has, without loss of generality, a zero mean flow rate, on a Kolmogorov–Petrovskii–Piscounov (KPP)‐type model in the presence of a discontinuous cutoff at concentration . In the long‐time limit, a permanent‐form traveling wave solution is established which, for fixed , is unique. Its structure and speed of propagation depends on (the strength of the flow relative to the propagation speed in the absence of advection) and (the square of the front thickness relative to the channel width). We use matched asymptotic expansions to approximate the propagation speed in the three natural cases , , and , with particular associated orderings on , while remains fixed. In all the cases that we consider, the shear flow enhances the speed of propagation in a manner that is similar to the case without cutoff (). We illustrate the theory by evaluating expressions (either directly or through numerical integration) for the particular cases of the plane Couette and Poiseuille flows.

Stability patterns in plane porous Poiseuille flow with uniform vertical cross-flow: A dual approach

In this study, we examine the effects of a porous parameter and a uniform vertical cross-flow on the onset of instability in a plane Poiseuille flow (PPF) subjected to infinitesimal disturbances. The linearized Navier-Stokes (N–S) equations, which govern the stability of the fluid flow system, are addressed using the Chebyshev collocation method (CCM). Our approach includes both modal and non-modal analyses, each providing distinct insights into the system's behavior. The modal analysis involves a linear stability assessment using both non-normalized and normalized basic velocity approaches. Assuming a constant pressure gradient, the first approach aligns with the traditional Poiseuille flow framework, where a uniform pressure difference drives the flow. This simplifies calculations and aids in predicting flow characteristics. The second approach, which varies the pressure gradient to maintain a constant maximum streamwise velocity, offers a different perspective by focusing on a specific flow velocity profile through the porous medium. This variation in the pressure gradient allows us to explore changes in flow behavior and the system's responses under these conditions. This analysis includes examining the eigenspectrum, neutral stability curves, and determining critical triplets. On the non-modal front, we explore the intricacies of the non-normal Orr-Sommerfeld (O–S) operator. This involves studying the pseudospectrum and transient energy growth rate (G(t)) plots for optimal perturbations. Identification of regions of stability, potential instability, and instability is achieved using contour plots of maximum transient energy growth (Gmax). Interestingly, our results reveal that the porous parameter acts as a stabilizing agent, mitigating instability. In contrast, the uniform vertical cross-flow plays a dual role, contributing to both stabilization and destabilization of the system. Additionally, our study highlights the impact of examining basic velocity through both normalized and non-normalized approaches, leading to different stability predictions.

Stability of fluid flow in a porous medium with uniform cross-flow and velocity slip

This work examines the evolution of small perturbations in a plane Poiseuille flow through a porous medium bounded by porous walls implemented with the Brinkman flow model. The effect of uniform cross-flow across the channel along with slip imposed on the channel walls is examined for the associated fluid flow. The modified Orr–Sommerfeld equation for the flow system under consideration is derived and then solved numerically using Chebyshev spectral collocation method. The acquired sufficient conditions ensure that fluid motion will remain stable even when the flow system is widely perturbed. The linear stability characteristics and the onset of instability is discussed for a wide range of cross-flow Reynolds number, slip length and porosity parameter. The above mentioned flow parameters have a significant stabilizing effect on the fluid flow that is depicted with neutral stability curves, growth rate dispersion curves and eigenspectrums. Asymmetric slip allows the critical Reynolds number to increase rapidly when the upper channel wall experiences more slip than the lower.

Analytical solution of the Poiseuille flow of a De Kee viscoplastic fluid

We provide an explicit analytical solution of the planar Poiseuille flow of a viscoplastic fluid governed by the constitutive equation proposed by De Kee and Turcotte (1980). Formulae for the velocity and the flow rate are derived, making use of the Lambert W function. It is shown that a solution does not always exist because the flow curve is bounded from above and hence the rheological model can accommodate stresses only up to a certain limit. In fact, the flow curve reaches a peak at a critical shear rate, beyond which it exhibits a negative slope, giving rise to unstable solutions.

Modal stability analysis of the density-stratified plane Couette–Poiseuille flow

Shear and density stratification strongly affect the flow mechanism of the different atmospheric and ocean flows. In this paper, we investigate the stability characteristics of plane Couette–Poiseuille (CP) flow with stable density stratification in the vertical direction. A modal stability analysis is carried out to examine the exponentially growing instability of stratified plane CP flow under different controlling parameters. The domain of the flow is periodic in streamwise and vertical directions. The stability problem is solved numerically using the spectral collocation method. The present analysis is carried out for Reynolds number Re=104 with different speeds of moving wall and different strengths of stratification. The results show that the mass diffusivity impact on the flow instability mechanism is almost negligible beyond the Schmidt number Sc≥20. The three-dimensional mode is generally the most unstable mode for density-stratified CP flow. However, the most unstable mode in unstratified CP flow is always two-dimensional. In contrast to unstratified CP flow, the density-stratified CP flow is unstable even moving wall velocity exceeds 70% of the center velocity of the plane Poiseuille flow component. It is also observed that the moving wall velocity and density stratification simultaneously affect the stability of the flow, which shows the importance of the shear and stratification in the flow.

Derivation of extended-OBurnett and super-OBurnett equations and their analytical solution to plane Poiseuille flow at non-zero Knudsen number

In this work, we analyse wall-bounded flows in the continuum to transition regime with the help of higher-order transport equations (super-set of the Navier–Stokes equation). Towards this, we incorporate second-order in Knudsen number accurate terms in the single-particle distribution function, and with this complete representation, we first derive the second-order accurate extended-OBurnett (EOBurnett) and third-order accurate super-OBurnett (SOBurnett) equations. We then demonstrate that these newly derived equations exhibit unconditional linear stability. We finally validate the equations by solving for plane Poiseuille flow and derive closed-form analytical solutions for the pressure and velocity fields. The pressure and velocity results thus obtained have been compared with direct simulation Monte Carlo (DSMC) data in the transition regime. The results from both the EOBurnett and SOBurnett equations are found to yield better agreement with DSMC data than that obtained from the Navier–Stokes equations. This improved agreement is attributed to the presence of additional terms in the proposed equations, which effectively capture the effect of the Knudsen layer near the wall. The obtained higher-order transport equations and the closed-form solution presented in this work are novel. The ability of the equations to describe the flow in the transition regime should form the basis for conducting further realistic analytical studies of wall-bounded flows in the future.

Streaming potential of viscoelastic fluids with the pressure-dependent viscosity in nanochannel

The plane Poiseuille flow of viscoelastic fluids with pressure-dependent viscosity is analyzed through a narrow nanochannel, combining with the electrokinetic effect. When the fluid viscosity depends on pressure, the common assumption of unidirectional flow is unsuitable since the secondary flow may exist. In this case, we must solve the continuity equation and two-dimensional (2D) momentum equation simultaneously. It is difficult to obtain the analytical electrokinetic flow characteristics due to the nonlinearity of governing equations. Based on the real applications, we use the regular perturbation expansion method and give the second-order asymptotic solutions of electrokinetic velocity field, streaming potential, pressure field, and electrokinetic energy conversion (EKEC) efficiency. The result reveals a threshold value of Weissenberg number (Wi) exists. The strength of streaming potential increases with the pressure-viscosity coefficient when Wi is smaller than the threshold value. An opposite trend appears when Wi exceeds this threshold value. Besides, the Weissenberg number has no effect on the zero-order flow velocity, but a significant effect on the velocity deviation. A classical parabolic velocity profile transforms into a wavelike velocity profile with the further increase in Wi. Finally, the EKEC efficiency reduces when pressure-dependent viscosity is considered. Present results are helpful to understand the streaming potential and electrokinetic flow in the case of the fluid viscosity depending on pressure.

Inertial enhancement of the polymer diffusive instability

Beneitez et al. (Phys. Rev. Fluids, vol. 8, 2023, L101901) have recently discovered a new linear ‘polymer diffusive instability’ (PDI) in inertialess rectilinear viscoelastic shear flow using the finitely extensible nonlinear elastic constitutive model of Peterlin (FENE-P) when polymer stress diffusion is present. Here, we examine the impact of inertia on the PDI for both plane Couette and plane Poiseuille flows under varying Weissenberg number ${W}$ , polymer stress diffusivity $\varepsilon$ , solvent-to-total viscosity ratio $\beta$ and Reynolds number ${Re}$ , considering the FENE-P and simpler Oldroyd-B constitutive relations. Both the prevalence of the instability in parameter space and the associated growth rates are found to significantly increase with ${Re}$ . For instance, as $Re$ increases with $\beta$ fixed, the instability emerges at progressively lower values of $W$ and $\varepsilon$ than in the inertialess limit, and the associated growth rates increase linearly with $Re$ when all other parameters are fixed. For finite $Re$ , it is also demonstrated that the Schmidt number $Sc=1/(\varepsilon Re)$ collapses curves of neutral stability obtained across various $Re$ and $\varepsilon$ . The observed strengthening of PDI with inertia and the fact that stress diffusion is always present in time-stepping algorithms, either implicitly as part of the scheme or explicitly as a stabilizer, implies that the instability is likely operative in computational work using the popular Oldroyd-B and FENE-P constitutive models. The fundamental question now is whether PDI is physical and observable in experiments, or is instead an artifact of the constitutive models that must be suppressed.

Hydrodynamics of pitching hydrofoil in a plane Poiseuille flow

Several advanced medical and engineering tasks, such as microsurgery, drug delivery through arteries, pipe inspection, and sewage cleaning, can be more efficiently handled using micro- and nano-robots. Pressure-driven flows are commonly encountered in these practical scenarios. In our current research, we delve into the hydrodynamics of pitching hydrofoils within narrow channels, which may find their potential applications in designing bio-inspired robots capable of navigating through pressure-driven flows in confined channels. In this paper, we have conducted a numerical investigation into the flow characteristics of a National Advisory Committee for Aeronautics (NACA) 0012 hydrofoil pitching around its leading edge within a plane Poiseuille flow using a graphical processing unit accelerated sharp interface immersed boundary method solver. Our study considers variations of the wall clearance from 20% to 50% of the channel width. We have explored the hydrodynamic features such as instantaneous and time-averaged values of lift, drag, input power, and torque for different wall clearance ratios and oscillation frequencies in the range of Reynolds number 100–200 based on the mean velocity and channel width. We have tried to explain the force, torque, and power variations by examining the flow features in the near wake. While the hydrodynamic coefficients showed significant variations with changes in wall clearance and the Strouhal number (St), we did not observe significant variations with alterations in the Reynolds number (Re).

Enhancing large-scale motions and turbulent transport in rotating plane Poiseuille flow

Based on the characteristics of large-scale plume currents in rotating plane Poiseuille flows (RPPF), an injection/suction control strategy is introduced to augment the intensity of plume currents and improve the turbulent transport of passive scalar. The control strategy maintains the conventional non-penetrative condition on the stable side, and applies a wall-normal velocity that varies in the spanwise direction on the unstable side. For comparison, the control with non-penetrative condition on the unstable side and injection/suction on the stable side is also examined. The RPPF at $Re_\tau$ ranging from $180$ to $300$ and $Ro_\tau$ ranging from $0$ to $30$ are studied. Injection/suction with fixed root-mean-squared wall-normal velocity (below $1\,\%$ of the bulk mean velocity) at the wall is considered. Direct numerical simulations reveal that the injection/suction on the stable side has minor effect for all cases studied, whereas the injection/suction with properly distributed slots on the unstable side can significantly enhance the large-scale plume currents and the turbulent transport efficiency at small or moderate rotation numbers. This is attributed to the fact that plumes are generated on the unstable side, and controlling the origin of plumes is more effective. A proper strategy for maximum enhancement of turbulent transport is limiting the distance between injection slots equal to or slightly larger than the intrinsic distance between plume currents. This strategy is valid for all physical and computational parameters currently considered, and is potentially applicable over a much wider parameter range.

Order–disorder transitions within deformable particle suspensions in planar Poiseuille flow

Three-dimensional simulations of the ordering of elastic capsule suspensions within planar Poiseuille flow channels are reported. The simulations utilize the immersed boundary method coupled with the lattice Boltzmann method to capture the complex flow-induced capsule deformations and hydrodynamic interactions within the suspensions. A parametric study is presented whereby the confinement ratio and the particle deformability are varied independently within a two-dimensional range relevant to this ordering phenomenon. The initial distribution of capsules is random, and the simulations evolve the system from a disordered state to an ordered one, while an order parameter that quantifies the fraction of capsules belonging to one-dimensional train assemblies is computed throughout time. A monotonic increase in ordering is observed with increasing deformability. However, an optimal confinement ratio is identified corresponding to a peak in the order parameter. This peak is attributed to the competition between increasing long-range capsule attractions and decreasing in-plane capsule density (with fixed volume fraction) as the confinement ratio increases. Simulations are also performed to understand how dispersity in capsule size and deformability impact the degree of ordering. It is shown that ordering is quite sensitive to dispersity in capsule size, and much less sensitive to dispersity in deformability. Overall, the results provide important insights for the design of microfluidic devices.

Three-dimensional diffusive-thermal instability of flames propagating in a plane Poiseuille flow

The three-dimensional diffusive-thermal stability of a two-dimensional flame propagating in a Poiseuille flow is examined. The study explores the effect of three non-dimensional parameters, namely the Lewis number Le, the Damköhler number Da, and the flow Peclet number Pe. Wide ranges of the Lewis number and the flow amplitude are covered, as well as conditions corresponding to small-scale narrow (Da≪1) to large-scale wide (Da≫1) channels. The instability experienced by the flame appears as a combination of the traditional diffusive-thermal instability of planar flames and the recently identified instability corresponding to a transition from symmetric to asymmetric flame. The instability regions are identified in the Le-Pe plane for selected values of Da by computing the eigenvalues of a linear stability problem. These are complemented by two- and three-dimensional time-dependent simulations describing the full evolution of unstable flames into the non-linear regime. In narrow channels, flames are found to be always symmetric about the mid-plane of the channel. Additionally, in these situations, shear flow-induced Taylor dispersion enhances the cellular instability in Le<1 mixtures and suppresses the oscillatory instability in Le>1 mixtures. In large-scale channels, however, both the cellular and the oscillatory instabilities are expected to persist. Here, the flame has a stronger propensity to become asymmetric when the mean flow opposes its propagation and when Le<1; if the mean flow facilitates the flame propagation, then the flame is likely to remain symmetric about the channel mid-plane. For Le>1, both symmetric and asymmetric flames are encountered and are accompanied by temporal oscillations.

Forced convection past a circular cylinder in a planar Poiseuille flow

In this work the transfer of a passive scalar (temperature) from a circular cylinder (radius R) in a planar Poiseuille flow (height 2h) is investigated with numerical simulations. The study varies the channel half-height to cylinder radius ϵ=h/R, the inertial parameter Λ=(UcR/ν)(h/R)2 and the Péclet number Pe=UcR/α, where Uc is the midplane velocity, ν is the kinematic viscosity and α is the thermal diffusivity. There has been significant amount of work for ϵ≫1, therefore this work will focus on the range, 0.03≤ϵ≤1, which has been chosen to show the effect of geometry and inertia on the flow features and the subsequent transfer. As ϵ is decreased the Nusselt number increases and this behaviour is explained with velocity and temperature profiles.Particularly for ϵ=0.03, the local Nusselt number has a significant vertical variation (in the z-direction) when Λ and Pe are increased past certain values. For Λ≪1 and Pe>103 this is due to the non-uniform incident flow present in the Hele-Shaw cell while for Λ=O(1), it is the secondary flow that is induced at the sides of the cylinder. For these latter flows the transfer in the midplane is reduced relative to near the side walls, due to the triple turning point radial velocity profile adjacent to the circular cylinder. For Λ≤1, the Nusselt number is only a function of Pe. However, for Λ>1, the secondary flow results in a slightly increased Nusselt number.