Effect of a surface tension gradient on the slip flow along a superhydrophobic air-water interface

Superhydrophobic surfaces have been shown to produce significant drag reduction in both laminar and turbulent flows by introducing an apparent slip velocity along an air-water interface trapped within the surface roughness. In the experiments presented within this study, we demonstrate the existence of a surface tension gradient associated with the resultant Marangoni flow along an air-water interface that causes the slip velocity and slip length to be significantly reduced. In this study, the slip velocity along a millimeter-sized air-water interface was investigated experimentally. This large-scale air-water interface facilitated a detailed investigation of the interfacial velocity profiles as the flow rate, interfacial curvature, and interface geometry were varied. For the air-water interfaces supported above continuous grooves (concentric rings within a torsional shear flow) where no surface tension gradient exists, a slip velocity as high as 30% of the bulk velocity was observed. However, for the air-water interfaces supported above discontinuous grooves (rectangular channels in a Poiseuille flow), the presence of a surface tension gradient reduced the slip velocity and in some cases resulted in an interfacial velocity that was opposite to the main flow direction. The curvature of the air-water interface in the spanwise direction was found to dictate the details of the interfacial flow profile with reverse flow in the center of the interface for concave surfaces and along the outside of the interface for convex surfaces. The deflection of the air-water interface was also found to greatly affect the magnitude of the slip. Numerical simulations imposed with a relatively small surface tension gradient along the air-water interface were able to predict both the reduced slip velocity and back flow along the air-water interface.

Figure 1

Schematic diagram of the flow channels used in experiments. For the channel shown in (a), a horizontal groove was introduced at the center of the sidewall which was then covered by the vertical air-water interface after water is injected. The channel in (a) is 5.5 mm wide ($W$), 3.3 mm high ($H$), and 58 mm long ($L$), with the groove size being 1.0 mm wide $\left(W_{\mathrm{g}}\right)$, 5.0 mm long $\left(L_{\mathrm{g}}\right)$, and 5.0 mm deep $\left(D_{\mathrm{g}}\right)$. The channel in (b) contains a vertical groove centered at the bottom of the channel which is covered by a horizontal air-water interface. The channel in (b) is 5.0 mm wide ($W$), 2.0 mm high ($H$), and 56 mm long ($L$), with the groove size being 1.0 mm wide $\left(W_{\mathrm{g}}\right)$, 15.0 mm long $\left(L_{\mathrm{g}}\right)$, and 2.0 mm deep $\left(D_{\mathrm{g}}\right)$. The geometry in (c) is a torsional shear flow. The bottom plate in (c), with diameter $D=40\mathrm{mm}$, contained an annular groove which supports an air-water interface, with radius of centerline $r_{\mathrm{c}}=8\mathrm{mm}$. The width of the annular groove ranges between $0.94\mathrm{mm}<W_{\mathrm{g}}<1.83\mathrm{mm}$. The top plate was rotated and the gap between the top and bottom plate ranges between 1.0 mm < $H<4.0$ mm. The slices highlighted in green in (a) and (b) demonstrate the plane within the focal plane of the microscope.

Figure 2

(a) Schematic diagram of the cross section of the channel showing the geometry of the air-water interface as it deflected out of the groove (convex). (b) Representational experimental image of the convex air-water interface showing the particle image velocimetry tracer particles and a reflection of the illuminating laser light sheet from the air-water interface.

Figure 3

Measurements of the velocity profile normal to the air-water interface. (a) An image of fluorescent particles near the air-water interface with the velocity vector field superimposed for the flow in the channel shown in Fig. 1 with $U_{\infty }=0.56\mathrm{mm}/\mathrm{s}$. (b) The velocity profile along the line normal to the interface. The velocity was time averaged over 500 frames. The solid symbols $(■▶▴•)$ in (b) indicate the experimental data at different bulk velocities provided in the legend. The open symbols $(—\square —,—○—)$ in (b) represent the results of numerical simulations with a shear-free boundary condition imposed on the interface while the solid line $(—)$ represent the simulation results for an interface with a no-slip boundary condition. The velocity and position have been normalized by bulk velocity $\left(U_{\infty }\right)$ and channel width ($W$), respectively. In this experiment, $1.0<\mathrm{Re}<1.8$ and $0.41\times {10}^{-5}<\mathrm{Ca}<0.76\times {10}^{-5}$.

Figure 4

Details of the interfacial velocity profiles for flow in the channel shown in Fig. 1 for an interfacial deflection of $\mathrm{\Delta }d=92\mu \mathrm{m}$ and an average flow speed of $U_{\infty }=1.06\mathrm{mm}/\mathrm{s}$. The velocity vector field in (a) just upstream of where the termination of air-water interface is shown in green with streamlines superimposed in blue. The normalized slip velocity along the center of the groove (along line 1) is shown in (b). The streamwise component $\left(x\right)$ of the slip velocity across the air-water interface (along line 2) is shown in (c). The negative value in (c) means the slip velocity is opposite to the main flow direction. In this condition, $\mathrm{Re}=2.1$ and $\mathrm{Ca}=1.3\times {10}^{-5}$.

Figure 5

Dependence of the slip velocity on the deflection of the air-water interface. The velocity vectors in (a) were at a convex interface with $\mathrm{\Delta }d=92\mu \mathrm{m}$ and the ones in (b) were at a concave interface with $\mathrm{\Delta }d=-56\mu \mathrm{m}$ at the bulk velocity of $U_{\infty }=1.06\mathrm{mm}/\mathrm{s}$. The streamwise component $\left(x\right)$ of the slip velocity along the crosswise direction $\left(y\right)$ at the center of the air-water interface is shown in (c) for a series of interfaces with curvature. The data include $\square ,○,▵,♦,◂$, and $•$ correspond to $\mathrm{\Delta }d=124$, 92, 60, 0, −28, and −56 μm, respectively, along with simulation result $(—\times —)$ based on the shear-free assumption at the air-water interface. In the experiments, $\mathrm{Re}=2.1$ and $\mathrm{Ca}=1.3\times {10}^{-5}$.

Figure 6

Simulation results of the interfacial flow at the interfaces with different deflections. (a) is the overall shear stress, (b) is the overall drag reduction, (c) is the normalized slip length, and (d) is the normalized local slip velocity. For simulations in (a)–(c), the experimental slip velocity shown in Fig. 5 was specified at the air-water interface as the user-defined slip boundary condition. The solid squares $(\blacksquare )$ and solid cycles $(•)$ in (a) respectively stand for the average shear stress at interface with the no-slip boundary condition and user-defined slip boundary condition, while the open cycles $(○)$ sand for the local shear stress at the center of the interface with the user-defined slip boundary condition. Drag reduction in (b) was calculated based on the average shear stress at the interfaces between the no-slip $(\blacksquare )$ and slip-defined $(•)$ data in (a). The solid cycles $(•)$ and open cycles $(○)$ in (c) denote the average slip length across the entire air-water interface and the local slip length at the center of interface. For simulations shown in (d), a shear stress was specified at the air-water interface as the boundary condition and the value of the stress was calculated from the numerical value shown in (a). More specifically, the shear stress applied in (d) was 3.61, 3.69, 3.79, 4.04, 4.16, and 4.29 mPa, for interfaces with a deflection of −56, −28, 0, 60, 92, and 124 μm, respectively.

Figure 7

Interfacial velocity vector at the air-water interfaces with different concentrations of particles. The number of particles is (a) $75\mathrm{m}{\mathrm{m}}^{-2}$ and (b) $1000\mathrm{m}{\mathrm{m}}^{-2}$. The streamwise component of the slip velocity along the spanwise direction in (a) and (b) is shown in (c). The maximum of the slip velocity changes as a function of the particle concentration at the air-water interface is shown in (d). The interface was kept constant with convex deflection $\mathrm{\Delta }d=60\mu \mathrm{m}$ in (d). Scale bar in (a) and (b) is 0.2 mm.

Figure 8

Measurements of slip velocity along the annular air-water interface in the torsional shear flow shown in Fig. 1. The vector field of slip velocity across the air-water interface is shown in (a). The azimuthal slip velocity is plotted as a function of radial position across the air-water interface in (b). The average of the normalized slip velocity at the air-water interface is plotted as a function of the interface width and channel height in (c) and (d), respectively. The channel height in (c) was kept constant as $H=2.0\mathrm{mm}$, while the interface width was kept constant as $W_{\mathrm{g}}=0.94\mathrm{mm}$ in (d). The solid squares in (b)–(d) are experimental data while the open circles correspond to simulation results. The solid line in (b) is the prediction of Eq. (3). The solid lines in (c) and (d) are predictions of Eq. (4) for the average slip velocity.