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Plane Poiseuille Flow of Two Superposed Fluids

This Demonstration analyzes plane Poiseuille flow of two superposed fluids. For a specified channel, the rectilinear flow field is defined by four parameters: two viscosities and and two volumetric flow rates and . Conservation of mass then determines the location of the liquid-liquid interface in the channel, which can be expressed as a thickness ratio .
The velocity field in each layer is given by , . The velocity is dimensionless with the interfacial velocity , and the coordinate is dimensionless with the thickness of the upper layer . The parameters and are given by:
, , .
The subscripts 1 and 2 denote the upper and lower fluid, respectively; is the viscosity ratio; and is the thickness ratio. The origin of the vertical coordinate is located at the interface such that the range of is given by [1].
Vary the flow rate and viscosity ratios to see their effect on the velocity profiles for the superposed flow. For , the velocity gradient is not continuous across the interface, but the shear stress is necessarily continuous. The value of the thickness ratio is also shown on the plot.
The dependence of the thickness ratio on the flow rate ratio and the viscosity ratio is shown in separate plots.

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Reference
[1] S. G. Yiantsios and B. G. Higgins, "Linear Stability of Plane Poiseuille Flow of Two Superposed Fluids," Physics of Fluids, 31(11), 1988 pp. 3225–3238. doi:10.1063/1.866933.
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