Velocity Profile for Immiscible Viscous Fluids

This Demonstration shows the behavior of three immiscible, incompressible fluids in laminar flow. The fluids are layered vertically and subjected to steady-state Couette flow; the top plate moves to the right at a constant velocity, and the bottom plate is stationary. At each height, the velocities represented by the black arrows are relative to the top plate velocity. You can change the viscosities and heights of fluids 2 and 3 with sliders. The velocity distribution shows the impact of fluid viscosity on the fluid's velocity gradient. The velocity versus height plot indicates the relative slopes (or velocity gradients) in each fluid. All numbers are dimensionless.

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DETAILS

Couette flow is fluid flow through parallel plates where one plate is stationary and the other is moving at a constant velocity. In this Demonstration, the flow is at steady-state and fully developed. The Navier–Stokes equations in the and directions are used:
,
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Since flow is steady-state and there is no flow in the direction, these equations simplify to:
,
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Flow is in the direction, so drops out:
,
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Since the flow is fully developed the velocity in the direction of flow does not change. It is assumed that the pressure is hydrostatic so pressure does not vary in the direction:
,
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For the boundary conditions at and at , the simplified Navier–Stokes equation is:
,
where is density, is gravity, is pressure, is viscosity, is fluid velocity, is plate velocity, is time and is the distance between plates.
Reference
[1] B. R. Munson, T. H. Okiishi, and W. W. Huebsch, Fundamentals of Fluid Mechanics, 6th ed., Hoboken, NJ: John Wiley & Sons, 2010.

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