Roughness Induced Boundary Slip in Microchannel Flows

Surface roughness becomes relevant if typical length scales of the system are comparable to the variations as it is the case in microfluidic setups. Here, an apparent slip is often detected which can have its origin in the misleading assumption of perfectly smooth boundaries. We investigate the problem by means of lattice Boltzmann simulations and introduce an “effective no-slip plane” at an intermediate position between peaks and valleys of the surface. Our simulations agree with analytical results for sinusoidal boundaries, but can be extended to arbitrary geometries and experimentally obtained data. We find that the apparent slip is independent of the detailed boundary shape, but only given by the distribution of surface heights. Further, we show that slip diverges as the amplitude of the roughness increases which highlights the importance of a proper treatment of surface variations in very confined geometries.

Figure 1

(Color online) The effective boundary at $h_{\mathrm{eff}}$ is found between deepest valleys ($h_{\min }$) and highest peaks ($h_{\max }$).

Figure 2

Periodic surfaces: (a) cosines given by $h(x)=h_{\max }/2+h_{\max }/2\cos (qx)$. (b) squares with height and separation given by $h_{\max }$. (c) triangles, $h_{\max }$ high and $2h_{\max }$ wide.

Figure 3

(Color online) Effective height $h_{\mathrm{eff}}$ over average roughness $R_a$ for a cosine geometry and different variations $k$. Symbols denote numerical data and lines are given by Eq. (1). The inset shows $\beta (k)/h_{\max }$ according to Eq. (1). For $k>1$ the slope becomes negative, demonstrating that the theory fails for more complex surface structures.

Figure 4

(Color online) (a) Effective height $h_{\mathrm{eff}}$ versus $R_a$ for triangles, blocks (see Fig. 2), and an equally distributed random roughness. (b) $h_{\mathrm{eff}}$ versus $R_a$ for triangles with $h_{\max }=5$ and 10. The distance between triangles $a$ is varied to obtain the given $R_a$. Values of $h_{\max }=5$ are scaled by a factor of 2.

Figure 5

(Color online) Simulated $h_{\mathrm{eff}}$ versus $R_a$ for gold coated glass and a randomly generated surface with Gaussian distributed heights. The background image shows the gold surface (left) and the artifically generated structure (right).

Figure 6

(Color online) $\beta$ versus $R_a$ for water and randomly distributed roughness. By assuming $h_{\mathrm{eff}}=h_{\max }$, $\beta$ is in the range of $h_{\max }-h_{\mathrm{eff}}$ for small $R_a$, but diverges for large $R_a$.