Plane Poiseuille Flow of Two Superposed Fluids
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 /.
μ
1
μ
2
Q
1
Q
2
d
2
d
1
The velocity field in each layer is given by =1+y+, . 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:
U
i
a
i
b
i
2
y
i=1,2
U
0
y
d
1
a
i
b
i
a
1
2
n
2
n
b
1
2
n
a
2
a
1
b
2
b
1
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].
m=/
μ
2
μ
1
n=/
d
2
d
1
y
y
-n≤y≤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.
m≠1
n=/
d
2
d
1
The dependence of the thickness ratio on the flow rate ratio and the viscosity ratio is shown in separate plots.