# 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/d

2

1

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:

U=1+ay+by

i

i

i

2

i=1,2

U

0

y

d

1

a

i

b

i

a=(m-n)/(n+n),b=-(m+n)/(n+n)

1

2

2

1

2

a=a/m

2

1

b=bm

2

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/μ

2

1

n=d/d

2

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/d

2

1

The dependence of the thickness ratio on the flow rate ratio and the viscosity ratio is shown in separate plots.