Wall Slip

True slip versus apparent slip

A fundamental distinction is drawn between true (microscopic) slip and apparent (macroscopic) slip.^[1][5]

True slip occurs when the fluid velocity at the wall surface is nonzero relative to the wall at the molecular scale, requiring a genuine velocity discontinuity at the solid-fluid interface.

Apparent slip occurs when a thin depletion or lubrication layer of low viscosity forms adjacent to the wall due to migration of the dispersed phase (particles, droplets, macromolecules) away from the wall. The bulk fluid appears to slip because the high-viscosity bulk is separated from the wall by a thin low-viscosity film, even though no-slip holds at the film-wall contact line.^[1][6]

For concentrated suspensions and emulsions, apparent slip dominates.^[6] For entangled polymer melts, both mechanisms may operate in distinct stress regimes.^[2][4]

Mechanisms in polymer melts

For entangled linear polymer melts, two molecular mechanisms have been identified:^[2][4]

Weak slip occurs at a first critical wall shear stress ${\displaystyle \tau _{c1}}$, where flow-induced forces cause adsorbed polymer chains to detach or desorb from the wall surface. The resulting slip velocity is modest and approximately proportional to the applied shear stress, placing this regime within the Navier slip framework.

Strong slip occurs at a second, higher critical wall shear stress ${\displaystyle \tau _{c2}}$, where chains in the bulk melt disentangle from the adsorbed surface monolayer. The slip velocity rises sharply and is a nonlinear, often power-law function of the wall shear stress. In extreme cases this produces plug flow, where virtually all velocity gradient is localized in a nanometer-scale interfacial layer.^[24]

Molecular theories for these transitions have been proposed by Brochard-Wyart and de Gennes (1992),^[27] who predicted that the slip length scales strongly with molecular weight as ${\displaystyle b\sim M^{3.4}}$ for entangled melts, a scaling confirmed experimentally for polyethylene and polybutadiene.^[2] Leger, Hervet, Massey, and Durliat (1997)^[28] developed a more complete theory using near-field laser velocimetry. Mhetar and Archer (1998)^[29] characterized slip in entangled polymer solutions using particle tracking velocimetry.

Mechanisms in suspensions and pastes

For concentrated suspensions, the dominant mechanism is the formation of a particle-depleted layer adjacent to the wall.^[1][6] If the layer has thickness ${\displaystyle \delta }$ and viscosity ${\displaystyle \eta _{f}}$ much lower than the bulk viscosity ${\displaystyle \eta _{b}}$, the apparent slip velocity is

${\displaystyle v_{s}={\frac {\delta \,\eta _{b}}{\eta _{f}}}\,{\dot {\gamma }}_{w}}$

Even a nanometer-thick depletion layer can produce macroscopically measurable apparent slip when ${\displaystyle \eta _{b}\gg \eta _{f}}$.^[6] Meeker, Bonnecaze, and Cloitre (2004) developed a micromechanical model for slip and flow in soft particle pastes.^[30] Isa, Besseling, and Poon (2007) used confocal microscopy to image shear zones and wall slip in capillary flow of concentrated colloidal suspensions.^[31] Ballesta, Besseling, Isa, Petekidis, and Poon (2008) measured slip and flow in colloidal hard-sphere glasses.^[32]

Poiseuille flow with slip

For the fully developed, isothermal, laminar flow of an incompressible fluid in a circular capillary of radius ${\displaystyle R}$ and length ${\displaystyle L}$, driven by a pressure drop ${\displaystyle \Delta P}$, the wall shear stress is^[11][8]

${\displaystyle \tau _{w}={\frac {R\,\Delta P}{2L}}}$

The apparent shear rate (assuming no slip) is

${\displaystyle {\dot {\gamma }}_{a}={\frac {4Q}{\pi R^{3}}}}$

where ${\displaystyle Q}$ is the volumetric flow rate. When wall slip occurs with velocity ${\displaystyle v_{s}}$, the volumetric flow rate decomposes as^[9]

${\displaystyle Q=Q_{\text{bulk}}+\pi R^{2}v_{s}}$

yielding

${\displaystyle {\dot {\gamma }}_{a}={\dot {\gamma }}_{a,{\text{true}}}(\tau _{w})+{\frac {4v_{s}(\tau _{w})}{R}}}$

The slip contribution scales as ${\displaystyle 1/R}$, while the bulk contribution is independent of ${\displaystyle R}$, forming the basis of the Mooney analysis.^[7]

The Mooney analysis

At fixed wall shear stress ${\displaystyle \tau _{w}}$, plotting ${\displaystyle {\dot {\gamma }}_{a}}$ versus ${\displaystyle 1/R}$ for capillaries of different radii (but identical ${\displaystyle L/D}$ ratio or after Bagley correction^[33]) yields a straight line with slope ${\displaystyle 4v_{s}(\tau _{w})}$ and intercept ${\displaystyle {\dot {\gamma }}_{a,{\text{true}}}}$.^[9][7] Repeating at multiple values of ${\displaystyle \tau _{w}}$ yields the complete slip function ${\displaystyle v_{s}(\tau _{w})}$.

This technique has been widely applied to polymer melts,^[20][21][24] concentrated suspensions,^[25] rubber compounds,^[34] food pastes,^[35] and other complex fluids. A regularized (Tikhonov) variant of the Mooney analysis has been developed by Owens, Corfield, and coworkers to improve robustness when data are noisy.^[36]

Corrections in capillary rheometry

The complete correction sequence for capillary rheometry is:^[7][8][37]

1. Bagley correction for entrance and exit pressure losses^[33]
2. Mooney correction for wall slip^[9]
3. Weissenberg-Rabinowitsch correction for non-Newtonian velocity profiles^[38]

yielding the true viscosity ${\displaystyle \eta ({\dot {\gamma }}_{w})=\tau _{w}/{\dot {\gamma }}_{w}}$.

Slit die and parallel plate geometries

For a slit die of width ${\displaystyle W\gg 2H}$, the Mooney analysis takes the form^[39][40]

${\displaystyle {\dot {\gamma }}_{a,{\text{slit}}}={\dot {\gamma }}_{a,{\text{true}}}(\tau _{w})+{\frac {6v_{s}(\tau _{w})}{2H}}}$

For parallel plate rheometry, Yoshimura and Prud'homme (1988)^[26] showed that the apparent shear rate at the rim is

${\displaystyle {\dot {\gamma }}_{a}^{PP}={\dot {\gamma }}_{\text{true}}+{\frac {2v_{s}(\tau _{w})}{h}}}$

where ${\displaystyle h}$ is the gap height, and two measurements at different gaps suffice to determine ${\displaystyle v_{s}}$. The same authors extended the correction to dynamic oscillatory measurements.^[41]

Navier linear slip

The simplest slip boundary condition, proposed by Navier (1823),^[16] is

${\displaystyle v_{s}=\beta \,\tau _{w}}$

where ${\displaystyle \beta }$ is the Navier slip coefficient. In terms of the slip length ${\displaystyle b=\beta \eta }$, the fluid velocity profile extrapolated linearly reaches zero a distance ${\displaystyle b}$ below the wall surface.^[42]

Power-law slip

For many polymer melts and concentrated suspensions above the first critical shear stress, the slip velocity follows a power-law relation:^[2][39]

${\displaystyle v_{s}=\beta '\,\tau _{w}^{m}}$

where ${\displaystyle m>0}$ is the slip exponent. Values of ${\displaystyle m}$ between 2 and 4 have been reported for filled polymers.^[2]

Threshold and asymptotic slip models

For entangled linear polymer melts exhibiting two critical stresses, empirical slip laws have been developed that incorporate a threshold below which no slip occurs and a saturation at high stresses.^[24][2]

Flow-curve mismatch

The simplest diagnostic is to measure the flow curve in geometries of two or more different dimensions. If slip is present, the flow curves will not superpose: the smaller-geometry curve will be shifted to higher apparent shear rates at the same wall shear stress.^[1][25]

Mooney plot linearity

At each ${\displaystyle \tau _{w}}$, plotting ${\displaystyle {\dot {\gamma }}_{a}}$ versus ${\displaystyle 1/R}$ tests the Mooney assumption. Linearity confirms that ${\displaystyle v_{s}=f(\tau _{w})}$ only. Non-linearity indicates geometry-dependent slip.^[24][45]

Roughened surfaces

Using roughened or textured measurement surfaces (crosshatched plates, serrated bobs, vane geometries) suppresses wall slip by eliminating the depletion layer. Comparing results with smooth and rough surfaces provides a direct measure of the slip contribution.^[14][46][47]

Velocimetry

Direct measurement techniques include nuclear magnetic resonance (NMR) velocimetry,^[48] particle image velocimetry (PIV),^[30] near-field laser velocimetry,^[22][28] confocal microscopy,^[31] and evanescent wave microscopy. These methods directly yield the slip velocity without requiring the multi-radius Mooney procedure.

Polymer extrusion

Wall slip directly affects the throughput-pressure relationship in polymer extrusion. If unrecognized, slip causes overestimation of the shear-thinning index, underestimation of true viscosity, and incorrect prediction of die pressure drops.^[4][10] The onset of flow instabilities (sharkskin, spurt, gross melt fracture) that limit production rate is closely linked to wall slip.^[43][49] The phenomenon also affects commercial production of polymer films, fibers, and injection-molded parts.^[10][20]

Concentrated suspensions

In the processing of ceramic pastes, pharmaceutical formulations, food products, and cement, wall slip may be either detrimental (if unrecognized in rheometry) or beneficial (if exploited to reduce pumping energy).^[39][25][50] Plucinski, Gupta, and Chakrabarti (1998) characterized wall slip of mayonnaise in viscometers.^[51] The problem arises similarly in drilling muds, cement slurries, and natural fluid muds in harbors and estuaries.^[52]

Yield-stress fluids

For yield-stress fluids, wall slip produces an additional complication: plug flow in the core coexists with slip at the wall. Failing to correct for wall slip can lead to grossly underestimated yield stress values.^[39][50][53]

Microfluidics

In microfluidics, channel dimensions can approach the slip length ${\displaystyle b}$. The flow rate enhancement factor relative to classical no-slip Poiseuille flow in a channel of half-height ${\displaystyle h}$ with Navier slip at both walls is^[42]

${\displaystyle {\frac {Q_{\text{slip}}}{Q_{\text{no-slip}}}}=1+{\frac {3b}{h}}}$

For ${\displaystyle b\sim h}$, slip can double or triple the flow rate at fixed pressure drop. This has practical significance for microfluidic lab-on-a-chip systems processing biopolymer solutions and colloidal suspensions.^[42]

Rubber compounds

Wall slip of rubber compounds (SBR, EPDM) in capillary and slit die flows has been studied by numerous groups. Mourniac, Agassant, and Vergnes (1992) found that the standard Mooney analysis fails for SBR compounds and developed alternative gap-dependent characterizations.^[34] Leblanc (2001) reported similar difficulties with filled rubber compounds, including negative slip velocities from the Mooney analysis.^[54] Kleinschmidt and coworkers (2023) proposed an improved correction model for filled rubber compounds.^[55]

Colloidal systems and soft matter

Wall slip has been observed in a wide variety of soft matter systems, including colloidal glasses,^[32] concentrated emulsions,^[56] foams,^[57] and microgel pastes.^[30] The phenomenon is increasingly recognized as a fundamental component of how high-solid dispersions respond to mechanical deformation, rather than merely an experimental artifact.^[6]