No-slip condition
In fluid dynamics, the no-slip condition is a boundary condition which enforces that at a solid boundary, a viscous fluid attains zero bulk velocity. This boundary condition was first proposed by Osborne Reynolds, who observed this behaviour while performing his influential pipe flow experiments. [1] The form of this boundary condition is an example of a Dirichlet boundary condition.
In the majority of fluid flows relevant to fluids engineering, the no-slip condition is generally utilised at solid boundaries. [2] This condition often fails for systems which exhibit non-Newtonian behaviour. Fluids which this condition fails includes common food-stuffs which contain a high fat content, such as mayonnaise or melted cheese. [3]
Physical justification
Particles close to a surface do not move along with a flow when adhesion is stronger than cohesion. At the fluid-solid interface, the force of attraction between the fluid particles and solid particles (adhesive forces) is greater than that between the fluid particles (cohesive forces). This force imbalance causes the fluid velocity to be zero adjacent to the solid surface.
The no-slip condition is only defined for viscous flows and where the continuum concept is valid.
Slip behaviour
As the no-slip condition was an empirical observation, there are physical scenarios in which it fails. For sufficiently rarefied flows, including flows of high altitude atmospheric gases [4] and for microscale flows, the no-slip condition is inaccurate.[5] For such examples, this change is driven by an increasing Knudsen number, which implies increasing rarefaction, and gradual failure of the continuum approximation. The first-order expression, which is often used to model fluid slip, is expressed as [math]\displaystyle{ u - u_\text{Wall} = C\ell \frac{\partial u}{\partial n} }[/math] where [math]\displaystyle{ n }[/math] is the coordinate normal to the wall, [math]\displaystyle{ \ell }[/math] is the mean free path and [math]\displaystyle{ C }[/math] is some constant known as the slip coefficient, which is approximately of order 1. Alternatively, one may introduce [math]\displaystyle{ \beta = C\ell }[/math] as the slip length. [6] Some highly hydrophobic surfaces, such as carbon nanotubes with added radicals, have also been observed to have a nonzero but nanoscale slip length. [7] Similarly, some researchers have investigated this slip condition, modelling the cause as due to the high smoothness of highly-ordered nanoscale surfaces. [8]
While the no-slip condition is used almost universally in modeling of viscous flows, it is sometimes neglected in favor of the 'no-penetration condition' (where the fluid velocity normal to the wall is set to the wall velocity in this direction, but the fluid velocity parallel to the wall is unrestricted) in elementary analyses of inviscid flow, where the effect of boundary layers is neglected.
The no-slip condition poses a problem in viscous flow theory at contact lines: places where an interface between two fluids meets a solid boundary. Here, the no-slip boundary condition implies that the position of the contact line does not move, which is not observed in reality. Analysis of a moving contact line with the no slip condition results in infinite stresses that can't be integrated over. The rate of movement of the contact line is believed to be dependent on the angle the contact line makes with the solid boundary, but the mechanism behind this is not yet fully understood.
See also
References
- ↑ Reynolds, Osbourne. (1876). "I. On the force caused by the communication of heat between a surface and a gas, and on a new photometer". Proceedings of the Royal Society of London 24 (164): 387-391.
- ↑ Day, Michael A. (2004). "The no-slip condition of fluid dynamics". Erkenntnis 33 (3): 285–296. doi:10.1007/BF00717588.
- ↑ Campanella, O. H.; Peleg, M. (1987). "Squeezing Flow Viscosimetry of Peanut Butter". Journal of Food Science 52: 180–184. doi:10.1111/j.1365-2621.1987.tb14000.x. https://doi.org/10.1111/j.1365-2621.1987.tb14000.x.
- ↑ Schamberg, R. (1947). The fundamental differential equations and the boundary conditions for high speed slip-flow, and their application to several specific problems (Thesis).
- ↑ Arkilic, E.B.; Breuer, K.S.; Schmidt, M.A. (2001). "Mass flow and tangential momentum accommodation in silicon micromachined channels". Journal of Fluid Mechanics 437: 29-43. doi:10.1111/j.1365-2621.1987.tb14000.x. https://doi.org/10.1111/j.1365-2621.1987.tb14000.x.
- ↑ David L. Morris; Lawrence Hannon; Alejandro L. Garcia (1992). "Slip length in a dilute gas". Physical Review A 46 (8): 5279–5281. doi:10.1103/PhysRevA.46.5279. PMID 9908755. Bibcode: 1992PhRvA..46.5279M. https://scholarworks.sjsu.edu/physics_astron_pub/86.
- ↑ Kim Kristiansen; Signe Kjelstrup (2021). "Particle flow through a hydrophobic nanopore: Effect of long-ranged wall–fluid repulsion on transport coefficients.". Physics of Fluids 33 (10).
- ↑ M. Kratzer; S. K. Bhatia; A. Y. Klimenko (2023). "Knudsen layer behaviour and momentum accommodation from surface roughness modelling". Journal of Statistical Physics 190 (3).
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