A Falling Sphere in a Stratified Fluid

1
A Falling Sphere in a Stratified Fluid
Joyce Lin, Roberto Camassa, Richard
McLaughlin The University of North Carolina at
Chapel Hill
(a)

(b)
(c)
(d)
Deformation of
the fluid interface at times (a) 0, (b) 0.5, (c)
5, and (d) 12 in a frame of reference moving
horizontally with the
sphere. Without Oseen corrections, the
deformation of the plane can be verified using
the stream function ? and the ?-component of the
velocity. Starting at some point r0, the fluid
particle will travel along the streamline to
another point rl after some time.
Fluid particle starting at point (r0,
0) travels along the streamline to position
(rl,?l) after a given amount of
time. This can then be inverted
using Newtons method, and the position of the
interface is verified.

• Introduction
• Consider a sphere falling under gravity in a
  two-layer, sharply stratified fluid. As the
  sphere passes through the interface, it will
  bring with it fluid from the upper, less dense
  layer.
• Entrained fluid for a sphere passing
  through the interface of a stratified fluid.
• Because of the no-slip condition on the
  surface of the sphere, the buoyancy force of the
  entrained fluid becomes important. To find this
  buoyancy force, we can numerically follow the
  deformation of the fluid interface and calculate
  the volume at each time step.

III. Stokes Approximation Note that the
center of mass passes out of the sphere. Thus, we
return to the Stokes equations to improve the
model by casting the stratified version of the
Stokes approximation to a form that allows
efficient numerical integration of the equations
of motion. IV. Experiments The first
experiment was used as a preliminary example of a
falling sphere through a stratified fluid. For
water and saltwater, Re is too high. The second
experiment uses karo to stay within the Stokes
regime. Note the deformation of the
stratification interface. The fluid becomes
entrained about the sphere, and a thin tail
connecting the sphere to the interface can be
seen. Experiment conducted
in a fresh water/salt water stratification with
bead density 1.04g/cc, bead radius 0.25cm, salt
water density 1.038g/cc, Re gt 1.
Experiment conducted in diluted karo/karo
stratification with bead density 1.4g/cc, bead
radius 0.25cm, karo density 1.36g/cc, Re lt 1.
Acknowledgements J.L. is supported by NSF
RTG DMS-0502266 R.C. is supported by NSF
DMS-0104329, NSF DMS-0509423, and NSF RTG
DMS-0502266 R.M. is supported by NSF
DMS-0308687 and NSF RTG DMS-0502266 Some
computational work using Data Tank, developed by
David Adalsteinsson, was supported by NSF
DMS-SCREMS 042241

• Entrained fluid volume, shown in gray.
  The blue line indicates the threshold slope,
  which was 20 in this calculation.
• The shaded fluid to the left of the blue line
  is considered that which affects the motion of
  the sphere. Analytical integration of the volume
  is infinity without Oseen corrections.
• II. A Crude Model
• This preliminary model takes into account the
  buoyancy of the sphere and of the entrained
  fluid, applying both to the center of the sphere
  at each time step.
• where ms and mf are the masses of the sphere
  and entrained fluid, respectively, U is the
  velocity of the sphere, Vf is the volume of the
  entrained fluid, ?2 is the density of the bottom
  fluid, and Fb is the buoyancy force of the
  sphere. Note that the buoyancy of the sphere will
  depend on where it is relative to the original
  position of the fluid interface.
• Typical numbers for water and brine
  experiments
• Sphere centered at (0,0)
• Radius 0.01
• Initial position of plane 0.011
• Top fluid density 1.0399
• Bottom fluid density 1.0405

r0
rl
Velocity (cm/s)
Velocity (cm/s)
Time (s)
Time (s)
Entrained Fluid Volume (cm3)
X-coordinate
Time (s)
Time (s)