The No Slip Condition (in Poiseuille Flow)




In the previous article, the Poiseuille velocity profile and pressure-flow relationship was derived for a Newtonian fluid in a straight cylindrical tube.  At some point in the derivation, we invoke the "No Slip" condition which states that the velocity of the fluid at the wall must be equal to the velocity of the wall, i.e. 0.  Is that justified?  What's the basis for it?

I'm going to try to explain this in a much more general way that has to do with the "continuum hypothesis" of how materials behave, both fluids and solids. We all know that mattered is made up of particles. In most cases however, you would have to look very closely – at a very small length scale to determine that this is the case.  Materials behave grossly as if they are continuous, not particulate. Correlates of this theory include the ability to treat all of the properties of the matter as continuous -- properties like temperature, concentration of a chemical within a fluid, density, and velocity e.g. if the material is moving or deforming.

The next figure is intended to depict a short segment of a circular tube in which Poiseuille flow is present as suggested by the parabolic velocity profile ( blue vectors).  Suppose you could somehow label the fluid – superimpose or literally embed a grid into the fluid suggested by the gray lines.  If it were a solid material, we could consider literally painting the grid lines onto the material and I've witnessed this trick done with fluids using colored dye. You may be familiar with Tag MRI labeling of the myocardium and certainly know about speckle tracking in echocardiography wherein we attempt to follow an intrinsic labeling pattern of echogenicity embedded in the muscle.



The video below shows the consequence of tracking the above grid as the flow is allowed to occur according to the Poiseuille velocity profile.  We find  that fluid elements touching the wall remain touching the wall at their original locations, but something much more profound also.  Every fluid element remains in contact with same fluid elements it was originally in contact with. 

A moment's consideration of how this video was constructed will allow a better understanding of what it is to be of fluid ( or any other piece of material).  Consider the boundary between ANY 2 fluid elements, however large and however small down to the limit of the continuum hypothesis.  At every point on the boundary there is only one velocity.  Hence the boundary cannot separate from the material element that lies on either side of it.  ( This type of "hand-waving" argument can be formalized with mathematical precision and correctness also. )  This is true no matter how we divide or subdivide the whole into smaller elements; an element can never escape from the one adjacent to it.  Hence the No-Slip condition is just a microcosm of a condition that is grossly true throughout the material - material elements remain touching the ones they have always touched!! 

Now certainly that can't remain true forever, there are practical limitations to this description.  Wait around long enough and most of the fluid that started out on the figure gets pretty far downstream.  Nevertheless, the fluid touching the wall remains there -- to the extent to which a fluid is continuous.  We all know also that materials can break - separate into more than one piece.  In mathematical terms I believe this necessitates a discontinuity in the properties of the material; we actually have a surface inside the material where TWO velocities exist at the same location -- that's the only way for the material on one side of the boundary to "escape" from the material on the other side.  For a solid this gets into the realm of fracture mechanics.  

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