Slippery questions about complex fluids flowing past solids

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Original entry: Alexander Epstein, APPHY 226, Spring 2009

Slippery questions about complex fluids flowing past solids

Authors: Steve Granick, Yingxi Zhu and Hyungjung Lee

Soft matter keywords

Rough surfaces, slip control, hydrophobicity


Abstract from the original paper

Viscous flow is familiar and useful, yet the underlying physics is surprisingly subtle and complex. Recent experiments and simulations show that the textbook assumption of ‘no slip at the boundary’ can fail greatly when walls are sufficiently smooth. The reasons for this seem to involve materials chemistry interactions that can be controlled — especially wettability and the presence of trace impurities, even of dissolved gases. To discover what boundary condition is appropriate for solving continuum equations requires investigation of microscopic particulars. Here, we draw attention to unresolved topics of investigation and to the potential to capitalize on ‘slip at the wall’ for purposes of materials engineering.

Soft matters

Aimed at a general audience like any other Nature paper, this "progress article" was an enlightening read about slip of fluids flowing on solid surfaces. This is a topic near and dear to wetting.

Lack of slip in everyday life

This article made me realize the simple reason it is impossible to blow a surface clean of dust particles. The no-slip boundary condition of fluid flow past a solid surface--that is, flow velocity vanishes at the interface--means that small dust particles do not extend far enough beyond the adsorbing surface to be blown off. Figure 1 shows this familiar situation. Other familiar instances in which the no-slip condition makes life a little more difficult include washing soap off in the shower or sink and washing dishes. In both cases, it is much more effective to scrub than to simply pour water.

Fig. 1

Perhaps less trivial issues are fluid flow through small pipes (the effect in large pipes is negligible), accumulation of fatty detritus in arteries, and the pumping required in these cases.

Exceptions to the "no-slip dogma"

The no-slip condition is so central to fluid mechanics and works so well in the majority of situations that the exceptions have not been appreciated much outside a small community of engineers and engineering literature. It must be emphasized that no-slip is valid provided that certain assumptions are met: a single component fluid, a wetted surface, and low levels of shear stress. Then careful experiments imply that the fluid comes to rest within 1-2 molecular diameters of the surface. However, the assumptions are more restrictive and the slip more controllable than most people appreciate.

Fig. 2

The exceptions fall into these categories:

  • Flow of multicomponent fluids with different viscosity components
    • Suspensions , foodstuffs, and emulsions
    • Polymer melts (non-adsorbing polymers are dissolved in fluids of lower viscosity)
  • Viscous polymers (where there is a range of molecular weights)
  • Superhydrophobic lotus-leaf-type surfaces with Cassie-state trapped air
  • "Weak-link" argument: given a sufficiently high flow rate, the shear rate will cause either failure of fluid cohesion or the no-slip condition
  • Gas flowing past solids whose spacing is less that a few mean free paths (molecular scale pipes)
  • Superfluid helium

As a result, many computer simulations predict and experiments confirm micron-scale slip lengths in newtonian fluids such as water and alkanes. While these can be ignored in macroscopic channels, the impact is great for micro/nanochannels. Experimental methods range from optical tracking of fluorescent dyes to laser particle velocimetry to NMR imaging.

There are in fact two possible types of slip: true and apparent. As shown in Figure 2, true slip occurs when the fluid velocity is literally nonzero at the surface, whereas in apparent slip the velocity is zero but the velocity gradient is higher. True slip may occur on superhydrophobic Cassie surfaces, since the fluid rides almost entirely on air, or on very smooth surfaces at very high shear (flow) rates. Apparent slip occurs, for example, in a multicomponent fluid in which the low viscosity component (or dissolved gas) segregates near the surface and facilitates flow. Because the viscosity near the boundary is low, the velocity gradient there is higher and the bulk velocity profile extrapolates to zero below the surface. In both the true and apparent cases, the slip length is defined by this zero-velocity extrapolated distance below the surface.

Deviation from no-slip, quantified

The flow rate is linked directly to the slip, and the main idea of the experiments that show this is the hydrodynamic force <math>F_H</math> Two solid spheres of radius <math>R</math>, at spacing <math>D</math>, experience hydrodynamic force <math>F_H</math> as they approach or retreat from one another in a liquid due to the flow of fluid out of or into the space between.

<math>F_H = f^{*} \frac{6\pi R^2 \eta}{D}\ \frac{dD}{dt}</math>

So <math>F_H \propto \frac{dD}{dt}, R^2, \eta, D^{-1}</math>. When the <math>f^{*}</math> prefactor deviates from 1, this quantifies the deviation from classical no-slip condition.

Slip signifies that in the continuum model of low, the fluid velocity at the surface is finite, slip velocity = <math>v_s</math>, and increases linearly with distance from the surface.

<math>\eta v_s \equiv b \sigma_s</math>

where b is the slip length and <math>\sigma_s</math> is the shear stress at the surface.

Effect of roughness on slip

There are two theories to explain why no-slip works in most cases.

Fig. 3
  • Fluid molecules are stuck to solid walls by intermolecular forces to prevent a discontinuity
    • BUT, hydro/oleophobic surfaces are nonwetting, so the fluid-solid cohesion is less than fluid-fluid, so this is not always true
  • The microscopic roughness (physical and chemical) of real surfaces leads to viscous dissipation at the boundary regardless of surface energies
    • Simulations predict slip past perfectly smooth surfaces
    • BUT, realistic surfaces possess structur--see Figure 3.
Fig. 4

Any slip present on a surface decreases as surface roughness increases. Figure 4 shows the influence of wall roughness on flow past partially wetted surfaces. AFM images in a-c shows the r.m.s. roughness, from atomic to 6 nm, The corresponding deviation from no-slip and the slip length are compared in d and e for different flow rates. Controlling these two parameters allows the slip length to be increased to 400 angstroms or more by going to atomic smoothness and high flow rate. At 6 nm r.m.s. roughness (much larger than the size of the fluid molecules), higher flow rate can no longer cause slip.

Fig. 5

Figure 5 corroborates the relationship. Here the critical shear rate for onset of slip (left axis) and critical shear stress (right axis) are plotted against r.m.s. surface roughness for flow of DI water (filled circles) and tetradecane (open circles). The datum at the top right with arrows is above experimental resolution and represents only a lower limit.

Insoluble gas segregating on smooth surfaces

Data for smooth surfaces at high flow rates turn out to be consistent with a two-layer fluid model in which a layer <1 nm thick and viscosity 10-20 times less than the bulk adjoins the solid surface. An explanation by de Gennes, et al is that shear may nucleate vapor bubbles, which then grow to cover the solid surface such that the liquid now flows on a layer of gas. Importantly, this effect disappears when there is more than atomic surface roughness.

Fig. 6

Tetradecane, a non-polar liquid, saturated with poorly soluble argon showed slip, whereas when showed no slip when saturated with soluble carbon dioxide. Similarly, on a hydrophobic surface carrying water flow, a vapor phase may thermodynamically form near the wall. Shown in Figure 6, these findings show that gas-mediated slip can even occur on wetted surfaces where the solid surface energy is high, provided that the surface is sufficiently, i.e., atomically, smooth. This control parameter disappears when the surface is rough. In the figure, a slope of unity corresponds to no-slip, while horizontal slope is slip.

Riding on air: nanograss

A related strategy is to use superhydrophobic micro/nanostructured surfaces (such as in Figure 7) to maximize contact with air, which is exceedingly solvophobic. This is the "lotus effect" that we all know and love. The Cassie state of the fluid traps air underneath, such that the solid-liquid contact is very small.

Fig. 7