Motion of a Viscous Drop With a Moving Contact Line

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Motion of a Viscous Drop With a Moving Contact
Line
J. Zhang, M. J. Miksis, S. G. Bankoff
Presented by Donna Comissiong
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L
Fluid 2
d
Fluid 1
?
?
y
0
x
Figure 1 Configuration of drop motion in a
horizontal channel
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Purpose of This Study

• To develop a front tracking numerical method to
  solve the 2-D N-S equations in a two-phase
  region.
• To illustrate the effects of certain parameters
  on the dynamics of drop motion.
• To present a convergent numerical method to solve
  for the motion of an interface with a contact
  line.

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Difficulties Faced

• Boundary condition on the wall No slip condition
  introduces a non-integrable stress singularity at
  the contact line.
• The actual contact angle is not easily measured.

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Formulation of the Problem
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Formulation of the Problem
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Interfacial Conditions on yh(x,t)
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Condition at Contact Line
These are two possible models that can be used,
(the second is the limiting case of the first as
??? ). UCslip velocity of the contact point, ?
is a parameter with units of velocity, and ?S
static contact angle
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Front Tracking Method
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Front Tracking Method

• 2 grids- one grid for N-S equations, the second
  for description of the interface.
• A smearing of the interface is allowed To
  account properly for surface tension.
• The interface is tracked as part of the solution
  Allows explicit enforcement of contact angle
  conditions.

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Reformulation of the Problem
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Re-write density and viscosity as one-field
variables, then use them in the governing
equations
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Algorithm
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• First grid MAC (marker and cell) grid for
  velocity and density.
• Second grid separate grid aligned with the
  interface Treated as a set of arcs (elements)
  formed by connecting neighborhood points.
• Grid points on the interface move with the local
  flow field and maintain the desired contact angle
  at each step.

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• For each time step, all grid points (except the
  contact point) are moved by

Time step
Position vector of each grid point at time n1
Interfacial velocity vector at time n (via linear
interpolation of velocities at nearby grids on
MAC mesh)
NOTE The contact point also needs to be marched
in time. At each time step, it is moved to the
position so that the contact angle (formed by the
tangent line at the contact point and the bottom
wall) is enforced.
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Projection Method for MAC Mesh
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MAC mesh The dark lines represent the fluid
cell volumes we can divide the space into, and
the dashed lines are the superimposed MAC mesh
for the fluid-flow computation. The arrows show
the location where the fluid velocities are
determined, and the black circles show the nodes
where the pressures are determined. (note we do
not require BCs for Pressure)
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• MAC mesh used along with second-order
  centered-difference scheme in space,
  first-ordered forward Euler in time.
• Time stepping of momentum equations split into
  two pieces
• Introduction of an intermediate variable u.
• Determination of velocity at time (n1) via u
  and the pressure gradient.

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Each time step the interface is advanced by the
local velocity field, and the contact points
moved as described before. The above Poisson
equation is solved for pn and this is used to
update velocities, and then everything is
repeated at next time step.
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Solving the Poisson Equation

• The Poisson equation is solved by the conjugate
  gradient method(using NETLIB, along with a code
  matrix multiplication algorithm and
  preconditioner in SLAP column format).

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Mass Conservation Convergence Test

• Numerical results for percentage mass loss of a
  drop (half ellipse with set dimensions) were
  compared different mesh sizesfound that mass of
  the drop was well conserved.
• With mesh-refinement, mass loss decreased and
  converged.

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Results
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Results

• A systematic investigation was done on the
  effects of the physical parameters on the motion
  of the drop.
• Results model accurately the behavior- see
  handout

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Conclusion
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• Results found to be consistent with experiment.
• Computations illustrate accurately the flow field
  around the contact line.
• Inertial effects significantly affect the motion
  of the contact line.
• The effects of density, viscosity, surface
  tension, gravity and contact angle on the
  dynamics of drop motion have been illustrated.