b c c F velocity profile U x y a d b fluid element h Figure 24 Flow between two

# B c c f velocity profile u x y a d b fluid element h

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b c c F velocity profile U x y a d b fluid element h Figure 2.4: Flow between two horizontal, parallel plates with upper one moving at velocity U .
2.3. FLUID PROPERTIES 17 We first note the assumption that the plates are sufficiently large in lateral extent that we can consider flow in the central region between the plates to be uneffected by “edge effects.” We next consider the form of the velocity profile (the spatial distribution of velocity vectors) between the plates. At y = 0 the velocity is zero, and at y = h it is U , the speed of the upper plate. The fact that it varies linearly at points in between the plates, as indicated in the figure, is not necessarily obvious (and, in fact, is not true if U and/or h are sufficiently large) and will be demonstrated in a later lecture. This detail is not crucial for the present discussion. We will first digress briefly and consider the obvious question “Why is velocity zero at the stationary bottom plate, and equal to the speed of the moving top plate?” This is a consequence of what is called the no-slip condition for viscous fluids: it is an experimental observation that such fluids always take on the (tangential) velocity of the surfaces to which they are adjacent. This is made plausible if we consider the detailed nature of real surfaces in contrast to perfectly-smooth ideal surfaces. Figure 2.5 presents a schematic of what is considered to be the physical situation. Real surfaces are actually very jagged on microscopic scales and, in fact, on scales sufficiently Actual rough physical surface as it would appear on microscopic scales Fluid parcels trapped in surface crevices Figure 2.5: Physical situation giving rise to the no-slip condition. large to still accomodate the continuum hypothesis for typical fluids. In particular, this “surface roughness” permits parcels of fluid to be trapped and temporarily immobilized. Such a fluid parcel clearly has zero velocity with respect to the surface, but it is in this trapped state only momentarily before another fluid particle having sufficient momentum to dislodge it does so. It is then replaced by some other fluid particle, which again has zero velocity with respect to the surface, and this constant exchange of fluid parcels at the solid surface gives rise to the zero surface velocity in the tangential direction characterizing the no-slip condition. We observe that viscosity is very important in this fluid parcel “replacement” process because the probability of an incoming fluid parcel to have precisely the momentum (speed and direction) needed to dislodge another fluid particle from a crevice of the surface is low. But viscosity results in generation of shear forces that act to partially remove the stationary parcel, which might then be more easily replaced by the next one striking the surface at the chosen point. Furthermore, it is also important to recognize that even if fluid replacement at the solid surface did not occur, the

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