Effect of patterned slip on micro- and nanofluidic flows

We consider the flow of a Newtonian fluid in a nano- or microchannel with walls that have patterned variations in slip length. We formulate a set of equations to describe the effects on an incompressible Newtonian flow of small variations in slip and solve these equations for slow flows. We test these equations using molecular dynamics simulations of flow between two walls which have patterned variations in wettability. Good qualitative agreement and a reasonable degree of quantitative agreement is found between the theory and molecular dynamics simulations. The results of both analyses show that patterned wettability can be used to induce complex variations in flow. Finally we discuss the implications of our results for the design of microfluidic mixers using slip.

Figure 1

The two-dimensional channel geometry.

Figure 2

The function $kh\left(y\right)\sim v_1$ is shown in a channel for values of $kw=0.1,1,10$ respectively. We have taken $\delta ∕w=1$.

Figure 3

The function $h^{\prime }\left(y\right)\sim u_1$ is shown in a channel for values of $kw=0.1,1,10$ respectively. We have taken $\delta ∕w=1$.

Figure 4

Contour plot showing $\Delta \delta \left(x\right)∕\delta =\cos \left(kx\right)$ and the corresponding variations in $v_1(x,y)$ and $u_1(x,y)$ in a channel for $kw=1$. Regions with dark shading indicate negative velocity and regions with light shading indicate positive velocities. We have taken $\delta ∕w=1$ and the channel length is $L=20w$.

Figure 5

Vector plot showing $\Delta \delta \left(x\right)∕\delta =\cos \left(kx\right)$ and the corresponding velocity vector $\mathbf{(}{u}_1(x,y),v_1(x,y)\mathbf{)}$ in a channel for $kw=1$. We have taken $\delta ∕w=1$ and the channel length is $L=\pi w$.

Figure 6

Contour plot showing a square wave $\Delta \delta \left(x\right)∕\delta$ and the corresponding variations in $v_1(x,y)$ and $u_1(x,y)$ in a channel for $kw=1$. Regions with dark shading indicate negative velocity and regions with light shading indicate positive velocities. We have taken $\delta ∕w=1$ and the channel length is $L=20w$.

Figure 7

Plot of the change in velocity down the channel for $U∕w^2{k}^3\nu =15$ (solid line) and for $U∕w^2{k}^3\nu =0$ (dashed line) for $kw=0.1$. We have taken $\delta ∕w=1$. The effect of this term is to cause a phase lag in the velocity corrections away from the walls (effectively shifting these changes downstream) and to increase the magnitude of these corrections.

Figure 8

This plot shows the time-averaged longitudinal velocity $u$ across the channel at $x=5\sigma$ (where $c_{fs}=0.9$) and at $x=-5\sigma$ (where $c_{fs}=0.5$). We calculate the effective slip length by fitting $U(1+\delta ∕w-y^2∕w^2)$ to the profiles (fits are shown). In the solvophilic region ($x>0$—i.e., where $c_{fs}=0.9$) we calculated an effective slip length of $\delta =9.1\sigma$ and in the the solvophobic region $(x=-0.5,c_{fs}=0.5)$ we calculated an effective slip length $\delta =13.0\sigma$.

Figure 9

A plot showing the square-wave $c_{fs}\left(x\right)$ boundary condition with $kw=\pi$ imposed on the walls of the molecular dynamics simulation, the effective slip lengths ${\delta }_{\mathrm{eff}}$ induced by $c_{fs}$, and a corresonding contour plot showing the variations in $v(x,y)=v_1(x,y)$. The channel width is $2w=20\sigma$ with periodic boundary conditions applied at $x=\pm 10\sigma$. The peak longitudinal flow velocity is $U=0.4{(ϵ∕m)}^{1∕2}$ and the peak transverse velocity is $V=0.03{(ϵ∕m)}^{1∕2}$. Regions with light shading indicate flow in the $y$ direction and regions with light shading indicate flow in the negative $y$ direction. Note that the variations in $v(x,y)$ are 90° out of phase with the variations in $c_{fs}$.

Figure 10

A comparison of the transverse velocity $v$ at $x=0$ across the channel from the simulation in Fig. 9 and from theory. The theory has been fitted to the simulation data by calculating an effective slip length $\delta =6.7\sigma$ across the solvophilic region ($x>0$—i.e., where $c_{fs}=0.9$) and an effective slip length $\delta =3.6\sigma$ across the solvophobic region $(x<0,c_{fs}=0.5)$. Thus $\delta =5.2\sigma$ and $\alpha =0.3$. It is seen from the comparison that the theory underestimates the peak values of $v$ by a factor of 2–3.

Figure 11

A plot showing the square wave $c_{fs}\left(x\right)$ boundary condition with $kw=\pi$ imposed on the walls of the molecular dynamics simulation, the effective slip lengths ${\delta }_{\mathrm{eff}}$ induced by $c_{fs}$, and the corresponding variations in $v(x,y)=v_1(x,y)$ in a channel. The channel width is $2w=20\sigma$ with periodic boundary conditions applied at $x=\pm 10\sigma$. The peak flow velocity is $U=1.30{(ϵ∕m)}^{1∕2}$ and the peak transverse velocity is $V=0.060{(ϵ∕m)}^{1∕2}$. Regions with light shading indicate flow in the $y$ direction and regions with light shading indicate flow in the negative $y$ direction. Note the downstream phase shift in the variations in $v(x,y)$ with respect to the variations in $c_{fs}$, especially in comparion with Fig. 9.

Figure 12

Suggested designs for mixing devices. The light regions would be coated in such a way as to induce a large slip length (say, with a superhydrophobic coating), while the dark regions would be coated to induce a small slip length or no slip (say, with a superhydrophilic coating). More complicated patterns may enhance the mixing, provided the patterns are on a length scale comparable to the channel width.