Gravitational forces resulting from microgravity, take off and landing of spacecraft are experienced by individual cells in the living organism.

1
Numerical Simulation of Deformation and Shape
Recovery of Drops Passing Through a Capillary
Amirreza Golpaygan, Ali Jafari Nasser
Ashgriz Department of Mechanical and Industrial
Engineering University of Toronto

• Numerical technique
• Full Navier-stokes and continuity equations for
  an incompressible and Newtonian fluid are solved
  numerically.
• To solve the flow equations within the drop, the
  numerical model needs to track the location of
  the liquid interface.

• Introduction
• Gravitational forces resulting from
  microgravity, take off and landing of spacecraft
  are experienced by individual cells in the living
  organism.
• Such stresses alter cell shape, cytoskeleton
  organization and internal pre-stresses in the
  cell tissue matrix.
• Spaceflight is associated with a significant
  increase in the number of circulating blood cells
  including leukocyte, B cells and T-helper cells
  and their motion through capillaries.
• Prior studies have shown that the stresses due
  to the spaceflight lead to a sympathetic nervous
  system-mediated redistribution of circulating
  leukocytes.
• In addition, study of the cell migration is
  relevant to several other biological processes
  such as embryogenesis, and cell division.
• Obtaining the properties of human blood cell is
  necessary to have a better understanding of the
  deformability of human cells, in particular the
  leukocytes, under various stress conditions such
  as those in a spaceflight and microgravity.
• Properties of a drop, surface tension and
  viscosity can be determined based on the
  dynamical behavior and shape deformation during
  motion through a nozzle.

• Interface Tracking model (Volume-Of-Fluid)
• For each cell a volumetric function f defined,
  representing the amount of the fluid present in
  that cell.

• Characteristic length of the drop is defined as
  the elongated length of the drop (L) after
  deformation over its in initial diameter (D).

s(N/m) t (ms) L/D
0.146 6.8 2.128
0.073 10.2 2.93
0.0365 13.4 3.75

• The surface cells are defined as the cell with
  0ltflt1.
• Properties used in the Navier-stokes equation
  for the surface cell are calculated based on the
  value of f.
• A teach time step the unit normal vectors are
  calculated and the function f is advected

• Internal obstacle modeling
• Internal obstacles are modeled as a special case
  of two phase flow.
• The fluid volume fraction is defined as ?, and
  the obstacle volume fraction is defined as 1-?.
• The internal obstacle is characterized as a
  fluid with infinite density and zero velocity.
• ? is independent of time.
• ? 1, not an obstacle, open to the flow..
• ? 0, is an obstacle, close to the flow.

Cell shape is the most critical determinant of
cell function.

• Proposed Model
• In order to study the cell cytoskeleton
  deformation during the cell migration, cell is
  modeled as viscous liquid drop with interfacial
  tension moving through a controlled surface
  environment.
• The viscous liquid drop represents the cell
  which has been forced to migrate through a nozzle
  representing capillaries in the tissue of human
  body.
• The morphological changes in the drop shape
  represent changes in the cytoskeleton of the
  cell.
• The viscosity of liquid drop is representative
  of the resistance of the cytoskeleton to the
  shape deformation.
• A drop with the diameter D and initial velocity
  of V moving toward a nozzle with the conic angle
  of 2a and the diameter d at its outlet.
• Inertia, surface tension, viscosity, and wall
  effects are the parameters which determine the
  dynamics of the drop and its shape.

The velocity vectors for the drop with initial
velocity of 1 m/sec. The viscous effect and wall
effects damp the inertia, therefore the drop
oscillates inside the nozzle.
The velocity vectors for the drop with initial
velocity of 1.5 m/sec. After the nozzle, the drop
continues oscillation to gain its initial shape.

• The Navier- stokes equations are modified and
  solved based on considering the obstacle

• Conclusion
• A 3-dimensional computational model for a cell
  migrating through a channel with the shape of
  nozzle is presented. The cell is modeled as a
  viscous drop. For the liquid viscous drop, full
  Navier-stokes equations considering surface
  tension and internal obstacle are solved.
• The results of simulation for the shape
  deformation and recovery are presented.
• The work is in progess to obtain a correlation
  for the changes in the cell viscosity with the
  changes in the cells cytoskeletal structure in
  order to gain a qualitative description of the
  cytoskeletal deformation process of the cell.

• F represent present body force, surface tension.

• Results
• The following figures represent a drop with
  radius of 1.15mm simulating the cell moving
  toward a passage.
• The nozzle has a conic angle of 35.5. The outer
  diameter of the nozzle is equal to 0.86 mm.
• The drop properties are surface tension 0.073
  N/m, and kinematics viscosity of 8.9510-5 m/s2.

• The outcome is determined based on the balance
  of the forces.
• The inertia of the drop forces it against the
  resistance from the wall resisting its forward
  motion, and the resistance from the surface
  tension against deformation.
• The viscosity of the drop acts as the internal
  friction which is another barrier against the
  inertia.

Multiphase Flow and Spray System Laboratory
http//www.mie.utoronto.ca/labs/mfl