Streamwisetravelling viscous waves in channel flows
Abstract
The unsteady viscous flow induced by streamwisetravelling waves of spanwise wall velocity in an incompressible laminar channel flow is investigated. Wall waves belonging to this category have found important practical applications, such as microfluidic flow manipulation via electroosmosis and surface acoustic forcing and reduction of wall friction in turbulent wallbounded flows. An analytical solution composed of the classical streamwise Poiseuille flow and a spanwise velocity profile described by the parabolic cylinder function is found. The solution depends on the bulk Reynolds number R, the scaled streamwise wavelength \(\lambda \), and the scaled wave phase speed U. Numerical solutions are discussed for various combinations of these parameters. The flow is studied by the boundarylayer theory, thereby revealing the dominant physical balances and quantifying the thickness of the nearwall spanwise flow. The Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) theory is also employed to obtain an analytical solution, which is valid across the whole channel. For positive wave speeds which are smaller than or equal to the maximum streamwise velocity, a turningpoint behaviour emerges through the WKBJ analysis. Between the wall and the turning point, the wallnormal viscous effects are balanced solely by the convection driven by the wall forcing, while between the turning point and the centreline, the Poiseuille convection balances the wallnormal diffusion. At the turning point, the Poiseuille convection and the convection from the wall forcing cancel each other out, which leads to a constant viscous stress and to the break down of the WKBJ solution. This flow regime is analysed through a WKBJ composite expansion and the Langer method. The Langer solution is simpler and more accurate than the WKBJ composite solution, while the latter quantifies the thickness of the turningpoint region. We also discuss how these waves can be generated via surface acoustic forcing and electroosmosis and propose their use as microfluidic flow mixing devices. For the electroosmosis case, the Helmholtz–Smoluchowski velocity at the edge of the Debye–Hückel layer, which drives the bulk electrically neutral flow, is obtained by matched asymptotic expansion.
Keywords
Biosensors Electroosmosis Electroosmotic waves Love waves Microfluidics Mixing Shearhorizontal surface acoustic waves Turbulent drag reduction1 Introduction
1.1 Travelling waves in microfluidic systems
Smallscale oscillating flows often feature in microfluidic and microelectromechanical systems. A benefit of the oscillations is the promotion of mixing in the flow, which, as a result of the small length scales and the small velocities involved, is essentially laminar. At such Reynolds numbers, \(R \approx 0.1\)–10, oscillating flows such as that induced by the wall motion (1) can be enforced by surface acoustic waves or by electroosmosis waves.
Surface acoustic waves (SAWs) have been utilized extensively for a wide range of microfluidic applications, such as micromixing, micropumping, drop transport, cell handling, and microejectors [1, 2, 3, 4, 5], although shearhorizontal waves have never been employed in microfluidic flow mixing. SAWs are created by piezoelectric transduction within a thin solid substrate below a fluid, so that electric power causes the mechanical deformation of the substrate, which, in turn, leads to the motion of the fluid.
Wallnormal Rayleigh acoustic waves have been used for mixing of microfluidic flows [6, 7, 8, 9]. However, they generate compression waves in a liquid and suffer from energy dissipation (leaky waves) [6, 10]. When instead inplane motion occurs, thanks to the mismatch of the sound speeds and densities of the substrate material and the fluid, the acoustic propagation is confined within the substrate, while the fluid flow is incompressible. This is a relevant simplification for the analysis of SAWs because the incompressible Navier–Stokes equations with an imposed slip velocity describe the dynamics (i.e. an analytical solution for streamwise standing waves is found on p. 133 of the book by Bruus [11]). Tan et al. [4], in their Fig. 2c, show interfacial standing shear waves which are inplane and sinusoidal as an interesting variant of SAWs. Although no bulk flow is present in the system studied by Neumann et al. [12], this example shows that it is possible to generate shearhorizontal acoustic waves in a thin solid substrate to affect an overlying liquid layer.
Shearhorizontal surface waves, also called Love waves when a layer of lower acoustic velocity is used for increased sensitivity, have also been studied extensively as efficient biosensors and chemical sensors for flowing solutions because of their low dissipation when compared with wallnormal Rayleigh waves [13, 14]. However, these studies are mainly experimental and Lange et al. [13] indeed remark that improved design of these biosensors can be achieved by studying the fluid dynamics generated by the interaction of the spanwise waves and the overriding streamwise flow. Figure 1 depicts a schematic of a biosensor based on shearhorizontal travelling waves, an excellent technological application of the waves studied herein. Shearhorizontal SAWs waves have also been employed by Neumann et al. [12] to manipulate proteins attached to supported lipid bilayers. Love waves have also been used more recently as nonintrusive rheometers. In particular, the interaction between the SAWs and the liquid has been studied to extract the relationship between the wave attenuation and the viscosity [15, 16].
In addition to mechanical wall motions, travelling wave on the walls of a microchannel of the form (1) can also be engendered via electroosmosis [17, 18, 19]. Surface electrodes driven by AC currents below a fluid can generate a uniform plug flow within a very thin charged Debye–Hückel layer, which drags the overlying uncharged fluid by shear stresses. As remarked by Ajdari [18], since these layers are usually much thinner than the radii of curvature of the surface and the channel height, it can be assumed that the uncharged fluid is simply affected by an imposed effective slip velocity, which is linearly related to the electric field, and that the bulk flow is described by the incompressible Navier–Stokes equations. A concise explanation of the physics of wallbased electroosmosis in microchannels is found in Chang and Yang [20]. These shear motions have also been proposed as an electroosmotic pumping device to drive fluid along a channel [21, 22]. Micromixing has also been successfully achieved through electroosmotic wall forcing [23, 24, 25, 26, 27].
In line with these microfluidic mixing applications, we complement our theoretical/numerical results with ideas on the laboratory realization of the waves engendered by (1) through surface acoustic forcing by piezoelectric crystals and through electroosmosis actuation for microfluidic flow mixing (refer to Sect. 8).
1.2 Travelling waves for turbulent drag reduction
The wall wave motion given by (1) has also been studied beneath wallbounded turbulent flows, first via direct numerical simulations in a turbulent channel flow by Quadrio et al. [31] and Quadrio and Ricco [32], and experimentally in a pipe with rotating sections by Auteri et al. [28] and in a windtunnel flow over a deformable Kagome lattice surface by Bird et al. [29, 30]. Drag reduction or drag increase occurs depending on the forcing parameters \(\lambda \) and U. For the pipe with rotating sections [28] and the windtunnel flow over the Kagome surface [29, 30], the bulk Reynolds number is obviously much larger than unity and \(\lambda \) is either comparable or a few times larger than the pipe radius in the case of Auteri et al. [28] or the boundarylayer thickness in the case of Bird et al. [29, 30].
It is obviously very different to investigate the flow engendered by the waves given by (1) in the laminar regime or in the fully developed turbulent regime. Nevertheless, there is ample evidence that the laminar profiles generated by spanwise wall motion are very useful to study various aspects of the corresponding fully turbulent flow. Choi et al. [33] and Quadrio and Ricco [32] have indeed verified that the unsteady spaceaveraged spanwise profile may closely match the corresponding laminar solution. The good agreement occurs when the wall forcing acts on a time scale which is much shorter than the life time of the nearwall turbulent structures. Under these conditions, the drag reduction scales with the thickness of the spanwise boundary layer, which is computed through the laminar solution. Furthermore, the nearwall laminar solutions have been instrumental for the accurate computation of the power spent for moving the wall against the viscous flow resistance, the optimal layer thickness which leads to maximum drag reduction, or the smallest period of wall forcing which guarantees drag reduction [32]. Choiet al. [33] and Ricco et al. [34] have also utilized the laminar Stokes layer solution to define a scaling parameter for drag reduction prediction and Choi [35] has taken advantage of the spanwise laminar flow behaviour to interpret the changes of the nearwall turbulent structures.
1.3 Objectives and structure of the paper
Motivated by the possibility of microfluidic flow manipulation offered by shearhorizontal waves, by their extensive use as bio and chemical sensors, and by the importance of the laminar solutions for the study of turbulent drag reduction by spanwise forcing, a complete study on the laminar spanwise flow engendered by the wall motion given by (1) is presented herein. The investigation is based on numerical calculations and on asymptotic analysis. The spanwise momentum equation is first simplified to a secondorder ordinary differential equation and solved numerically by a secondorder finitedifference scheme. Its solution is also expressed analytically through the parabolic cylinder function (hereinafter referred to as PCF).
The Reynolds number R, the wave speed U, and the wavelength \(\lambda \) are treated as asymptotic parameters, thus deriving asymptotic analytical solutions utilizing the boundarylayer and the Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) theories. Employing the boundarylayer scaling, the thicknesses of the nearwall viscous layers are quantified, while the WKBJ solution gives the correct flow structure across the whole channel. When the streamwise diffusion is negligible, the convection term due to the wave motion may balance the streamwise Poiseuille flow convection term in a specified range of wave phase speeds. For these cases, alternative solutions found using WKBJ turningpoint composite expansions and the method of Langer [36, 37] are derived. All the asymptotic solutions show excellent agreement with the numerical solutions and offer the further advantage over the numerical approach that the dominant physical balances are revealed. The final aim of our work is to discuss how these waves can be generated in a laboratory via electroosmosis and surface acoustic forcing, and to propose them as microfluidic flow mixers.
The scaling and the simplification of the spanwise momentum equation are discussed in Sect. 2, while Sect. 3 presents the analytical solution in terms of the PCF. The numerical results are shown in Sect. 4, the boundarylayer theory results are discussed in Sect. 5, while Sect. 6 presents the WKBJ results and Sect. 7 contains the Langer solution. A discussion on future applications of the travelling waves for flow mixing is contained in Sect. 8 and a summary is found in Sect. 9.
2 Governing equation
3 Analytical solution in terms of the parabolic cylinder function
4 Numerical results
Figure 5 presents profiles for steadywave flows at high Reynolds number \(R=1000\) and with long wavelength (left: \(\lambda =100\) and right: \(\lambda =1000\)). The streamwise viscous diffusion effects, represented by the last term in (7), are negligible. Like the flows at orderone Reynolds number shown in Fig. 3, the spanwise flow occupies the entire channel, with the viscous region becoming thinner as \(\lambda /R\) decreases. For the case with \(\lambda =1000\) (Fig. 5, right), the spanwise velocity is finite at the centreline.
Profiles for \(\lambda \ll R\) resemble the classical Stokes solution for high positive U and for negative U. Trends similar to the exponentially decaying profiles shown in Fig. 4 are found when \(\lambda \ll 1\). However, the profiles vary significantly from the classical Stokes solution when \(U \approx 1\) or smaller (and positive) and \(\lambda \) is sufficiently larger than 1 and smaller than R. Figure 6 shows four sets of profiles for \(\lambda =50\), \(R=1000\), and \(0.5 \le U \le 1.25\). The trends are similar to the profiles of the Stokes layer only in the upper portion of the viscous layer, while at lower locations the trends show oscillatory behaviour which is distinctly different from the Stokes layer. For example, profiles for \(U=0.75\) with \(w=0.7\) at \(y=0\) may decay and change their curvature as y increases without crossing the \(w=0\) line for the whole channel extent. The spanwise velocity may therefore be positive along the whole channel, which is never the case for the Stokes layer.
Solving the governing equation (7) represents a simple numerical exercise. However, it is clear from the numerical results presented that restricting the analysis to a computational endeavour severely limits the understanding of the physical problem. Therefore, asymptotic methods, i.e. the boundarylayer theory in Sect. 5, the WKBJ theory in Sect. 6, and the Langer theory in Sect. 7, are employed. These analyses are useful because approximate analytical solutions of (7) are obtained and because insight on the physics is gained, which cannot be revealed either through the full numerical solution or the PCF analytical expression (10). In particular, the asymptotic approach quantifies the thickness of the spanwise viscous layer, highlighting the physical balance very near the wall, and explains the occurrence of the wiggly behaviour for wave speeds comparable with the bulk velocity, shown in Fig. 6 (refer to Sect. 6). The theoretical analysis precisely identifies the parameter range for this turningpoint regime, i.e. \(R^{1} \ll \lambda \ll R\) and \(0 \le U \le 3/2\), and quantifies the thickness of the thin turningpoint layer and of the other two orderone regions which confine this layer. The physical balances in these three layers are revealed, which explains the mathematical forms of their asymptotic solutions. The solid lines in Fig. 6, located at \(y = 1  \sqrt{1  2U/3}\), indicate the turningpoint location. It will be further shown that the asymptotic analysis is also useful for the design of the proposed micromixer based on the travelling waves. The boundarylayer theory indeed identifies the cases where the spanwise flow is confined to a very thin wallbounded layer, which are clearly not candidates as efficient mixers because the bulk flow, where the mixing is required, is largely unaffected by the wall motion.
5 Boundarylayer theory
5.1 The small\(\varepsilon \) regime: \(\lambda \ll R\)
In Eq. (13a), the term involving \(Y^2\) is always negligible with respect to the term involving Y. However, if neither of the other two terms multiplying \(\overline{W}\) (the convective term containing U and the xdiffusion term containing \((\lambda R)^{1}\)) is \(O\,(1)\) and at least one balances the term \(\varepsilon ^{1/2} Y\), there is no term to balance the y viscous diffusion term. This may occur when \(U=O\bigl (\sqrt{\lambda /R}\bigr )\) and \(\lambda =O\,(R^{1/3})\), when \(U=O\bigl (\sqrt{\lambda /R}\bigr )\) and \(\lambda \gg R^{1/3}\), or when \(U \ll \sqrt{\lambda /R}\) and \(\lambda =O\,(R^{1/3})\). When \(U \ll \sqrt{\lambda /R}\) and \(\lambda \gg R^{1/3}\), the term proportional to Y alone balances the y viscous diffusion term, leading to the steadywave case studied by Viotti et al. [42]. The scaling with \(\beta =1/2\) therefore breaks down because the term proportional to \(\overline{W}\) is always smaller than the wallnormal viscous term and the scaling with \(\beta =1/3\) applies.
5.2 The orderone\(\varepsilon \) and large\(\varepsilon \) regimes: \(\lambda = O\,(R)\) and \(\lambda \gg R\)
The full PCF solution (10) applies when \(\lambda = O\,(R) = O\,(1)\) and \(U = O\,(1)\). When the streamwise diffusion is negligible, i.e. for \(\lambda \gg 1\) and \(U = O\,(1)\), the PCF solution (10) remains valid, albeit with \(a = \mathrm{i}^{3/2} \sqrt{\pi R/(3 \lambda )} (U3/2)  1/2\). These two solutions are still valid when \(U \ll 1\), with \(a =  \sqrt{\mathrm{i}\pi /(3 \lambda R)} \left[ 3 \mathrm{i} R/2 + 2 \pi /\lambda \right]  1/2\) when \(\lambda = O\,(1)\), and \(a =  \mathrm{i}^{3/2}\sqrt{3 \pi R/(4 \lambda )}1/2\) when \(\lambda \gg 1\). When the streamwise diffusion effects dominate, i.e. when \(\lambda = O\,(R) \ll 1\) and \(U = O\,(1)\) or smaller, Eq. (7) simplifies as only the last term in parenthesis is retained. Hence the solution is given by Eq. (17).
When \(\lambda \gg R\), Eq. (7) simplifies to Eq. (22). The solutions are (15), (16), and (17) when \(U=O\,((\lambda R)^{1})\), \(U \gg (\lambda R)^{1}\), and \(U \ll (\lambda R)^{1}\), respectively. We close this section by pointing out that the boundarylayer analysis reveals no information on the physics behind the peculiar profiles shown in Fig. 6. This is studied through the WKBJ theory in Sect. 6.
6 WKBJ theory
Figure 9 shows the profiles of the real and imaginary parts of \(W\left( y \right) \) for \(\lambda =1\), \(R=1\) and \(U=0\) (left) and \(\lambda =1\), \(R=1\) and \(U=10\) (right) given by (25). Both the real (dashed) and imaginary (dotted) theoretical profiles are compared to the numerical profiles (thin solid lines), with excellent agreement obtained, although \(\varepsilon = \lambda /R = 1\) in both cases. The corresponding spanwise velocities \(w=\mathfrak {R}\,\left[ W\left( y \right) \exp \;\left( 2 \pi \mathrm{i} \xi \right) \right] \) for these cases are shown in Figs. 4(left) and 3(left), respectively.
6.1 Turningpoint solution by matched asymptotic expansion
As shown in Appendix C, the asymptotic solution (27) shows excellent agreement with the numerical solution. The changing nature of the flow on either side of the turning point can be explained by the changes in the dominant balances in (11). In this parameter range, the dominant behaviour close to the wall is governed by a balance between the unsteady convection and the wallnormal viscous stresses. Moving away from the wall, the streamwise Poiseuilledriven convection increases. Between the wall and the turning point, the streamwise Poiseuilledriven convection remains smaller than the convection resulting from unsteady wall wave forcing. At the turning point, a constantviscousstress balance exists between the streamwise Poiseuilledriven convection and the convection due to the unsteady wave forcing. Above the turning point the convection due to the Poiseuille flow is more significant than the contribution due to the wall wave forcing and the y viscous diffusion is present again.
7 Langer theory
8 Application to flow mixing

Active mixing of laminar flows by surface acoustic waves.
Rayleigh waves, i.e. wallnormal displacement streamwisetravelling acoustic waves, have been utilized for microfluidic mixing. Sritharan et al. [1] and Tseng et al. [45] experimentally verified that Rayleigh waves can mix cofluent streams with very different passive scalar concentrations. However, in liquids these waves suffer from severe energy dissipation due to the compression waves engendered by the wallnormal displacement (leakywave phenomenon). Also, although the induced smallscale secondary recirculatory motion is beneficial for mixing, it may interfere with the smoothness of the streamwise flow and create additional pressure gradients and therefore additional losses. We instead propose to use shearhorizontal waves to mix the cofluent streams studied by Sritharan et al. [1] and Tseng et al. [45], which, to the best of our knowledge, have never been employed in microfluidic mixing. A schematic of the microfluidic SAW mixer is shown in Fig. 11(left).
The main advantages over the Rayleigh waves would be (i) less energy dissipation (higher efficiency) because the shearhorizontal waves do not suffer from acoustic streaming energy loss due to the absence of wallnormal motion, as thoroughly discussed by Lange et al. [13] and Rocha et al. [46], which implies (ii) mixing along longer streamwise distances [46]. Furthermore, (iii) as the mixing occurs through the spanwise velocity, the streamwise flow remains smooth and no additional induced pressure gradients must be accounted for. A fourth advantage is that (iv) the spanwise waves would be better mixers than twodimensional Rayleigh waves given the concentration distribution at the inlet, shown in Fig. 11(left), which is uniform along the wallnormal direction, but strongly varying along the spanwise direction. Although the streamlines of the streamwise flow are unchanged when the SAWs are implemented, the mixing is required primarily along the spanwise direction where the concentration variations are most intense.
The passive scalar equation to be solved iswhere \(\theta \) is the passive scalar concentration (mass or temperature, for example), \(\mathrm{Pe}=U_\mathrm{b}^* h^*/\alpha _\mathrm{p}^*\) is the Peclet number, and \(\alpha _\mathrm{p}^*\) is the diffusion coefficient for mass transfer and the thermal diffusivity for heat transfer. The spanwise waves w(x, y, z, t) would act, through the boxed term in (33), on the spanwise gradient of \(\theta \), which would be most intense at the upstream channel location, where the two cofluent streams start interacting. The Reynolds numbers in the experiments of Sritharan et al. [1] and Tseng et al. [45] are in the range \(10^{2}1\) and the forcing wavelengths are comparable or smaller than the channel height, which leads to \(\varepsilon =\lambda /R\) in the range 1–100. This corresponds to the orderone\(\varepsilon \) and large\(\varepsilon \) regimes with U belonging to any columns of Fig. 8 because a wide range of wave speeds can be generated by wave interference, as proved by the standing wave study by Ding et al. [8].$$\begin{aligned} \frac{\partial \theta }{\partial t} + u \frac{\partial \theta }{\partial x} + \boxed {w \frac{\partial \theta }{\partial z}} = \frac{1}{\mathrm{Pe}} \left( \frac{\partial ^2 \theta }{\partial x^2} + \frac{\partial ^2 \theta }{\partial y^2} + \frac{\partial ^2 \theta }{\partial z^2}\right) , \end{aligned}$$(33)We finally note that the boundarylayer analysis of Sect. 5 is also here useful because it allows identifying the regimes where the spanwise flow is confined very near the wall, which are bound not to be efficient mixers as the mixing is required across the whole channel height. The turningpoint regimes, for which the wave speed is comparable with the bulk velocity, could instead qualify as candidates for good mixing because the flow may extend along the channel height (refer to Fig. 6).

Active mixing of laminar flows by electroosmotic waves.
The travelling wave flow produced by (1) could also be generated by electroosmotic waves and used for microfluidic flow mixing, as shown in Fig. 11(right). To the best of our knowledge, these waves have never been created through electroosmosis, but proper design of an unsteady and spatially inhomogeneous electric field could achieve this purpose. They would be an unsteady and streamwisemodulated version of the waves suggested by Ajdari [18], a streamwisemodulated variant of the oscillatory flow studied numerically by Dutta and Beskok [47], or an optimized variant of the micromixer proposed by Oddy et al. [23], where the wall forcing would be spanwise and streamwisemodulated instead of simply oscillatory and spatially uniform. Extending the analyses of Dutta and Beskok [47] and Meisel and Ehrhard [25], the governing equation for the spanwise velocity with electroosmotic effects included issubject to the noslip boundary condition at the wall and to the zerogradient condition at the centreline (we only consider half channel for simplicity). Here \(\ell _\mathrm{d}^*\) is the thickness of the Debye–Hückel layer, \(E_\mathrm{z}^*=E_{\mathrm{z},0}^*\mathfrak {R} \big [\mathrm {e}^{2 \pi \mathrm{i} \left( x^*  U^* t^* \right) /\lambda ^*} \big ]\) is the spanwise electric field, \(\overline{\varepsilon }^*\) is the permittivity, and \(\zeta ^*\) is the zeta potential (we have set the ionic energy parameter equal to unity, refer to Dutta and Beskok [47, p. 5098], and for simplicity assumed an exponentially decaying potential as in Meisel and Ehrhard [25] rather than more complex potential functions as in Afonso et al. [48] and Wang et al. [49]). The details of the formulation for the Stokes layer case, i.e. for \(u^*=0\) and \(E_\mathrm{z}^*=E_{\mathrm{z},0}^*\mathfrak {R} \big ( \mathrm {e}^{\mathrm{i} \omega ^* t^*} \big )\), are found in Dutta and Beskok [47].$$\begin{aligned} \frac{\partial w^*}{\partial t^*} + u^* \frac{\partial w^*}{\partial x^*} = \nu ^* \left( \frac{\partial ^2 w^*}{\partial x^{*2}} + \frac{\partial ^2 w^*}{\partial y^{*2}} \right)  \frac{\overline{\varepsilon }^* \zeta ^* E_\mathrm{z}^*(x^*,t^*)}{\ell _\mathrm{d}^{*2}\rho ^*} \mathrm {e}^{y^*/\ell _\mathrm{d}^*}, \end{aligned}$$(34)The above problem can be conveniently simplified under the assumption that Debye–Hückel layer \(\ell _\mathrm{d}^*\) is much thinner than the channel height and the viscous layers studied in Sect. 5. As also discussed by Qiao and Aluru [50], this hypothesis is amply verified as \(\ell _\mathrm{d}^*\) is very small, i.e. of the order of \(100~\hbox {nm}\) [47, 51], therefore about three orders of magnitude smaller than the channel height and two orders smaller than the viscous layers generated by the travelling waves. This means that the electric potential is confined in this very thin nearwall Debye–Hückel layer, while the bulk flow is electrically neutral and driven underneath by the electroosmotic motion of the Debye–Hückel layer. The scaled form of (34) iswhere \(\varPi _\mathrm{z}=\overline{\varepsilon }^* \zeta ^* E_\mathrm{z}^* \lambda ^*/(\mu ^* h^* U^*)=O(1)\), \(\mu ^*\) is the dynamic viscosity of the fluid, \(\delta _\mathrm{d}=\ell _\mathrm{d}^*/h^*\ll 1\), and \(\lambda ,U,R=O(1)\). Note that here \(w=w^*/U_\mathrm{b}^*\) as \(A^*\) cannot be used for scaling like in Sect. 2 because it is not defined. It is found in the following through asymptotic matching. By defining the Debye–Hückellayer coordinate \(Y_\mathrm{d}=y/\delta _\mathrm{d}\) and velocity \(W_\mathrm{d}=w\), the Debye–Hückellayer equation is found at leading order$$\begin{aligned} \frac{\partial w}{\partial t} + \frac{u}{U} \frac{\partial w}{\partial x} = \frac{1}{\lambda R U} \frac{\partial ^2 w}{\partial x^2} + \frac{\lambda }{R U} \frac{\partial ^2 w}{\partial y^2}  \frac{\varPi _\mathrm{z}}{R \delta _\mathrm{d}^2} \mathrm {e}^{y/\delta _\mathrm{d}}, \end{aligned}$$(35)whose solution is$$\begin{aligned} \frac{\partial ^2 W_\mathrm{d}}{\partial Y_\mathrm{d}^2}=\frac{U \varPi _\mathrm{z}}{\lambda } \mathrm {e}^{Y_\mathrm{d}}, \end{aligned}$$(36)obtained by use of the boundary conditions \(W_\mathrm{d}(0)=0\) and \(W_\mathrm{d}'(\infty )=0\). The Helmholtz–Smoluchowski velocity is obtained as follows:$$\begin{aligned} W_\mathrm{d}(x,Y_\mathrm{d},t)=\frac{U \varPi _\mathrm{z}}{\lambda }\Big ( \mathrm {e}^{Y_\mathrm{d}}  1 \Big ), \end{aligned}$$(37)In dimensional form, (38) becomes \(W_{\mathrm{hs}}^*=\big (\overline{\varepsilon }^* \zeta ^*E_{\mathrm{z},0}^*/\mu ^*\big )\mathfrak {R} \big [\mathrm {e}^{2 \pi \mathrm{i} \left( x^*  U^* t^* \right) /\lambda ^*} \big ]\). This is the velocity that drives the bulk electrically neutral flow which we have studied in the previous sections. We can now quantify the amplitude of the wall travelling waves defined in (3), i.e. \(A^*=\overline{\varepsilon }^* \zeta ^*E_{\mathrm{z},0}^*/\mu ^*\). In summary, the bulk spanwise flow, which is relevant for mixing, is thus described by Eq. (4) and driven by the unsteady and streamwisemodulated Helmholtz–Smoluchowski velocity \(W_{\mathrm{hs}}\) in (38), while the spanwise velocity in the very thin Debye–Hückel layer is brought to zero at the wall through the noslip condition. The composite solution is \(w_\mathrm{c}(x,y,t)=w(x,y,t)+W_\mathrm{d}(x,Y_\mathrm{d},t)W_{\mathrm{hs}}(x,t)\).$$\begin{aligned} W_{\mathrm{hs}}(x,t) = \lim _{Y_\mathrm{d} \rightarrow \infty }W_\mathrm{d}(x,Y_\mathrm{d},t)=\frac{U \varPi _\mathrm{z}}{\lambda }. \end{aligned}$$(38)As for the SAW mixer, the passive scalar Eq. (33) is to be solved. Similar electroosmotic microfluidic mixers have been studied by Sasaki et al. [52] and Huang et al. [53]. Sasaki et al. [52] employed meandering electrodes to mix two microstreams and stress the importance of obtaining analytical results for the fluid flow in order to optimize the mixing performance, while Huang et al. [53] generated inplane microvortices to prove that up to 30fold mixing enhancement can be achieved compared to mixing due to diffusion only. The spanwise spatial pattern displayed in Fig. 11(right) can be achieved by utilizing thin strips of different glass coatings and spatially modulated electric fields [18, 20].
Typical frequencies of electroosmosis actuators are of the order of 10 Hz [23, 53] and the streamwise length of the microelectrode arrays, which define the forcing wavelength, can be in the range of 100–\(500~\upmu \hbox {m}\). The flow parameters therefore correspond to ratios \(\varepsilon =\lambda /R\) in the range of 0.1–1 and to U of order unity. The small\(\varepsilon \) and orderone\(\varepsilon \) regimes characterize these flows. The turningpoint WKBJ analysis and the Langer analysis are thus relevant for these flow regimes, where the wave speeds are positive and comparable with the bulk velocity. The streamwise wallshear stress would play a crucial role in the spanwise flow dynamics and thus for the mixing performance. The design proposed in Fig. 11(right) would generate a square streamwisetravelling wave of spanwise velocity, a microscale analogous version of the pipeflow wave employed by Auteri et al. [28]. As explained in Sect. 9, thanks to the linearity of the problem, Fourier decomposition will allow the use of our sinusoidalwave results to construct the base flow for this mixing problem.
As mentioned in the Introduction, the shearhorizontal waves have also been used for turbulent drag reduction. For this application, the phase speeds span a wide range of values from null to several times the bulk streamwise velocity. It therefore follows that the regime with \(\varepsilon =\lambda /R \ll 1\) properly describes these flows. It also occurs that \(\lambda \gg R^{1/2}\) and therefore the streamwise viscous diffusion due to the waves is negligible. We reiterate that, in fully developed turbulent channel flows, the laminar flow solutions may only be representative of the spanwiseaveraged velocity profile and only under strict conditions of the wave parameters, as amply discussed in Quadrio and Ricco [32]. It is interesting to note that the dragincrease regime discovered by Quadrio et al. [31] for a specific range of positive phase speeds is included within the range for which the turningpoint regime occurs. Further investigation is also required to generalize the stability analyses for the classical Stokes layer [54, 55] to the travelling wave case.
9 Summary
Channel flows with spanwise wall forcing consisting of inphase sinusoidal travelling waves of spanwise wall velocity have been investigated. A novel threedimensional timedependent solution of the incompressible Navier–Stokes equations is constructed, with the solution represented as a linear combination of complextocomplex parabolic cylinder functions. As reliable numerical solutions of the complextocomplex parabolic cylinder function are currently unavailable, asymptotic solution methods have been employed to investigate flow variations due to the Reynolds number R, the scaled phase speed U, and the scaled wavelength \(\lambda \). Only sinusoidal shearhorizontal wall forcing has been considered. However, flows produced by more general wall forcing can be expressed by a linear combination of our solutions with each term multiplied by its Fourier coefficient.
Asymptotic methods, i.e. the boundarylayer and the WKBJ theories, have been utilized to study the flow. The underlying flow physics, revealed by the dominant balances in the governing equation, is gained by using these asymptotic theories and cannot be obtained through the exact analytical solution (10) or the numerical solutions computed in Sect. 4. While the boundarylayer method and the WKBJ approach both produce excellent approximations to the flow, there are particular advantages for each method. The simplicity of the boundarylayer solutions compared to the WKBJ solution is noticeable, aided by the outer solutions being identically zero. The boundarylayer method readily enables the determination of the boundarylayer thickness, which is not available if the WKBJ method is employed. Furthermore, the link between the different physical effects is elucidated better utilizing the boundarylayer approach. There are also advantages of the WKBJ approach over the boundarylayer method. Without further scaling (introducing a second boundary layer close to \(y=2\)), the WKBJ method readily generates solutions which are valid across the full channel.
The WKBJ solution also identifies the turningpoint behaviour for \(0 \le U \le 3/2\), \(R^{1/2} \ll \lambda \ll R\), which is not explained by the boundarylayer method. As the standard WKBJ solution (25) is unbounded near the turning point, solutions have been found by a WKBJ composite expansion and the Langer method. While the Langer solution is simpler, the composite WKBJ expansion has the benefits of explicitly determining the turningpoint location and of quantifying the thickness of this thin layer. This is important because the physics changes there. The WKBJ theory shows that, when the streamwise diffusion effects are negligible, in the turningpoint layer the Poiseuille convection and the convection due to the waves cancel out so that the wallnormal viscous stresses are constant. The high accuracy of the asymptotic solutions is quantified by comparing them with the numerical profiles in Appendix D.
We have finally presented ideas on how to generate the travelling waves for microfluidic flow mixing via surface acoustic forcing and electroosmotic actuation. In the latter case, matched asymptotic expansion has been useful to obtain the Helmholtz–Smoluchowski velocity that drives the bulk spanwise flow.
Notes
Acknowledgements
This work was partially supported by EPSRC First Grant EP/I033173/1. This research used computing resources at the University of Aberdeen and the University of Sheffield. We would like to thank Samuele Viaro, Eva Zincone, Claudia Alvarenga, Michele Schirru, Dr Cecile Perrault, and Dr Elena Marensi for their helpful comments. We are also indebted to Georgios Tyreas and Prof. Robert DwyerJoyce for pointing out to us several interesting technological applications of the shearhorizontal waves. We thank Prof. David Wagg for the encouragement and suggestions on the revision of the paper.
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