Plane Poiseuille Flow of Two Superposed Fluids

velocity profiles

d

2

/

d

1

vs.

Q

2

/

Q

1

d

2

/

d

1

vs.

μ

2

/

μ

1

Q

2

/

Q

1

1.1

μ

2

/

μ

1

4.45

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

μ

1

and

μ

2

and two volumetric flow rates

Q

1

and

Q

2

. Conservation of mass then determines the location of the liquid-liquid interface in the channel, which can be expressed as a thickness ratio

d

2

/

d

1

.

The velocity field in each layer is given by

U

i

=1+

a

i

y+

b

i

2

y

,

i=1,2

. The velocity is dimensionless with the interfacial velocity

U

0

, and the coordinate

y

is dimensionless with the thickness of the upper layer

d

1

. The parameters

a

i

and

b

i

are given by:

a

1

=(m-

2

n

)/(

2

n

+n),

b

1

=-(m+n)/(

2

n

+n)

,

a

2

=

a

1

/m

,

b

2

=

b

1

m

.

The subscripts 1 and 2 denote the upper and lower fluid, respectively;

m=

μ

2

/

μ

1

is the viscosity ratio; and

n=

d

2

/

d

1

is the thickness ratio. The origin of the vertical coordinate

y

is located at the interface such that the range of

y

is given by

-n≤y≤1

[1].

Vary the flow rate and viscosity ratios to see their effect on the velocity profiles for the superposed flow. For

m≠1

, the velocity gradient is not continuous across the interface, but the shear stress is necessarily continuous. The value of the thickness ratio

n=

d

2

/

d

1

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.