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You can obviously move a solid at a different speed along the surface of another solid, so how come the velocity of the fluid at the fluid-solid interface must be equal to that of the solid? What physical property dictates that the no-slip is valid for fluids but not solids? The Wikipedia page has the following:

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 brings down the fluid velocity to zero.

which didn't make sense to me since I'm not sure how they prove that this is true for any fluid-solid interface.

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    $\begingroup$ No-slip condition is only an assumption, and it does fail for some special interfaces. $\endgroup$ – user99917 Jan 4 '16 at 0:01
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    $\begingroup$ No-slip is only reasonable in viscous non-rarefied fluids; once this is no longer the case (i.e. when $\mathrm{Kn}>1$ in e.g. microflows) then there may be slip between the fluid particles and the solid boundary. $\endgroup$ – nluigi Jan 4 '16 at 10:14
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I am no expert on this, but i'll try to figure it out.

Well, you can have a no-slip condition for a solid too. It is the reason we can write with a chalk on a blackboard. But the effect is not as pronounced in solids because the cohesive force is really strong as compared to an adhesive force. So there is an energy imbalance in favor of the solid.

Another way you can look at it, is that fluid tends to displace air as it settles on a surface, creating a vacuum between the fluid and the solid which is hard to break. In a solid, since you have an irregular surface usually, the vacuum is hard to create. If you grind and polish two solid surface to a very, very high degree of smoothness, you can see the same effect, i.e. the working principle of slip gauges.

This is pretty much in layman's terms, but if you wish to learn more, do read about friction and the surface energy of materials. It will give you a stronger theoretical background into the mathematics, but is essentially useless if you are interested in CFD modelling.

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Since you linked the Wikipedia article, you should have seen that the no-slip condition really is an approximation, even for dense fluids (see the contact line example in the article). The short answer is thus that the fluid layer nearest the wall can conform to the wall velocity, and will do so to a decent approximation, since the molecules above can (relatively) easily slide past it. For a solid, the particles of a solid moving over a surface are (pretty much) stuck to that solid, and will move with it once a critical force is exceeded.

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