Sticking collision between a sphere and a textured wall in a viscous fluid

The collision of a sphere with a wettable microtextured wall in a viscous fluid is investigated experimentally, focusing on the region close to the contact with the wall. High-frequency laser interferometry is used for measuring small displacements of the sphere in that region. The wall texture consists of an a array of square micropillars, whose geometrical parameters (height, width, and spacing of the pillars) are varied. The wall texture decreases the hydrodynamic resistance, and hence the drag on the sphere, compared to the case of a smooth wall. At small Reynolds and Stokes numbers, this drag reduction is quantified in terms of an equivalent plane boundary, shifted down from the top of the pillars. The shift length depends on the geometrical parameters of the pillars array and is compared to available predictions of effective slip length for a flow over arrays of micropillars in the Wenzel state. At finite Stokes number, below the bouncing transition, the wall texture influences the relative importance of sphere inertia and drag force in the near wall region. As a result, a great diversity of sphere dynamics are obtained by varying the texture geometrical parameters. These dynamics can be captured considering a shift-length-modified drag force in the equation of motion of the sphere.

Figure 1

Schematic of the experimental setup. (a) Settling sphere interferometer, (b) side view of a sphere near an array of micropillars, and position of the equivalent smooth plane, and (c) profilometry of a micropillar array with $\mathrm{\Phi }=0.3,L=179\mu \mathrm{m},2b=106\mu \mathrm{m}$, and $e=85\mu \mathrm{m}$.

Figure 2

Electronic signal recorded as a function of time during the settling of a steel sphere of diameter $2a=12.7$ mm toward a smooth (glass) wall. The arrow indicates the arrest of the sphere on the surface.

Figure 3

Dimensionless velocity $V/V_{\mathrm{St}}$ measured as a function of dimensionless distance $h/a$ for a sphere of diameter $2a=6.98$ mm settling at small St and Re numbers towards different surfaces: smooth (glass) surface (crosses); array of micropillars with $\varphi =0.3,e=85\mu \mathrm{m},L=L_1=179\mu \mathrm{m}$ (diamonds); array of micropillars with $\varphi =0.05,e=89\mu \mathrm{m},L=L_2=240\mu \mathrm{m}$ (squares). Lines: experimental curves for a smooth surface shifted to the left along the horizontal axis with $s_e=5.9\mu \mathrm{m}$ (dashed line) and $s_e=32.8\mu \mathrm{m}$ (continuous line).

Figure 4

Symbols: Values of dimensionless shift lengths $s_e/L$ (a) and $s_0/L$ (b) obtained experimentally in the far field and in the near field, respectively, as a function of dimensionless pillar height $e/L$, for textures of different pillar fractions $\mathrm{\Phi }$ listed in Table 1. (a) Lines: dimensionless slip length $L_e/L$ as predicted in Ref. [19] for $\eta /{\eta }_g=1$, with Eqs. (4, 5, 6).

Figure 5

Dimensionless velocity $V/V_{\mathrm{St}}$ measured as a function of dimensionless distance $h/a$ for a sphere of diameter $2a=6.98$ mm settling at small St and Re numbers towards different surfaces. Diamonds: array of pillars with $\varphi =0.3,e=85\mu \mathrm{m},L=L_1=179\mu \mathrm{m}$. Crosses: smooth (glass) surface. Continuous line: Eq. (2) with $s_e=5.9\mu \mathrm{m}$. Dashed-dotted line: Eq. (3) with $s_e$ replaced by $s_0=17.8\mu \mathrm{m}$.

Figure 6

(a) Dimensionless velocity $V/V^{*}$ as a function of dimensionless sphere-to-wall distance $h/a$, for a steel sphere of diameter $2a=12.7$ mm settling near the same array of micropillars $\left(\varphi =0.05,e=21\mu \mathrm{m}\right)$ in (circles) 47V100000 silicon oil at $\text{St}\simeq 0$ with $V^{*}=V_{\text{St}}=612$ mm/s, (triangles) 47V1000 silicon oil at $\text{St}=3.8$ with $V^{*}=V_0=643$ mm/s. (b) Velocity-distance curve for a steel sphere of diameter $2a=12.7$ mm settling in 47V1000 silicon oil at St = 3.8 near a smooth surface and arrays of micropillars.

Figure 7

Dimensionless velocity $V/V_0=d\epsilon /d\tau$ as a function of dimensionless sphere-to-wall distance, $h/a=\varepsilon$, of steel spheres (a) of different diameters near a square array of micropillars of pillar height $e=21\mu \mathrm{m}$ and pillar concentrations $\mathrm{\Phi }=0.05$. Symbols: experiments. Solid lines: solution of Eq. (11) with $s_0=17\mu \mathrm{m}$. (b) of same diameter $\left(\text{St}=2.5\right)$ near square arrays of micropillars of pillar concentration $\mathrm{\Phi }=0.05$ and different pillar heights. Symbols: experiments. Solid lines: solution of Eq. (11) with $s_0=0$, 17, 43, and $59\mu \mathrm{m}$.

Figure 8

Dimensionless velocity $V/V_0=d\epsilon /d\tau$ as a function of dimensionless sphere-to-wall distance, $h/a=\varepsilon$, of a steel sphere of diameter $2a=12.7$ mm $\left(\text{St}=3.8\right)$, near different square arrays of micropillars. (a) Same pillar height $e=60\mu \mathrm{m}$ but different pillar concentrations $\mathrm{\Phi }$. Symbols: experiments. Solid lines: solution of Eq. (11) with values of $s_0$ (in $\mu \mathrm{m})$: 31, 18.7, 13.8, 0. (b) Same pillar concentration $\mathrm{\Phi }=0.15$ but different pillar heights. Symbols: experiments. Solid lines: solution of Eq. (11) with values of $s_0$ (in $\mu \mathrm{m})$: 35.5, 31, 13.8, 0.

Figure 9

Dimensionless velocity $V/V_0=d\epsilon /d\tau$ as a function of dimensionless sphere-to-wall distance, $h/a=\varepsilon$, of a carbide tungsten sphere of diameter $2a=8$ mm, settling toward different arrays of micropillars at $\text{St}=4.9$. Symbols: experiments. Solid lines: solution of Eq. (11) with values of $s_0$ (in $\mu \mathrm{m})$: 17.8, 13.8, 8.3, 0.