Flow of Liquids in Thin Layers

COUETTE FLOW. In one form of layer flow, illustrated in Fig. 5.17, the fluid is bounded between two very large, flat, parallel plates separated by distance B. The lower plate is stationary, and the upper plate is moving to the right at a constant velocity Uq. For a newtonian fluid the velocity profile is linear, and the velocity u is zero at y = 0 and equals Uo t y = B, where y is the vertical distance measured from the lower plate. The velocity gradient is constant and equals uq/B. Considering an area A of both plates, the shear force needed to maintain the motion of the top plate is, from Eqs. (3.3) and (3.4), 

Flow under these conditions is called Couette flow. 

A nearly identical flow pattern exists in the annular space between two concentric cylinders, one rotating and the other stationary, provided the width of the annulus, B, is small compared to the diameters of the cylinders. A device that makes use of this is the Couette viscometer. By measuring the torque required to rotate one cylinder at a known speed, the viscosity may be readily calculated from Eq. (5.70). 

LAYER FLOW WITH FREE SURFACE. In another form of layer flow the liquid layer has a free surface and flows under the force of gravity over an inclined or vertical surface. Such flow is in steady state, with fully developed velocity gradients, and the thickness of the layer is assumed to be constant. Flow usually is laminar, and often there is so little drag at the free liquid surface that the shear stress there can be ignored. Under these assumptions and the further assumptions that the 

Velocities and velocity gradient in a fluid between fiat plates. 

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The flow of liquids in thin layers is frequently observed in everyday life a common example is the flow of rain water on window panes, road surfaces, and roofs. 

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