Introduction:

1. Surface Tension model: • The surface tension was modeled using Continuum Surface Force (CSF). • Surface tension was reformulated as a body force in the Navier-Stokes equation. • For this purpose for each computational cell, curvature k was calculated. t = 0ms t = 0ms t = 0.5ms t = 0.5ms t = 1.5ms t = 1.5ms V t = 3.5ms t = 3.5ms t = 7.5ms t = 7.5ms t = 9ms t = 9ms H d D t = 10.5ms t = 10.5ms t =13ms t =13ms t = 15.5ms t = 15.5ms 2α t = 17ms t = 17ms t = 18.5ms t = 18.5ms t = 25ms t = 25ms L/D= 2.128 t= 6.8ms σ= 0.146 N/m ( 2 ×Water surface tension) L/D= 2.93 t= 10.2ms σ= 0.073 N/m (Water surface tension) L/D= 3.75 t= 13.4ms σ= 0.0365 N/m (1/2 ×Water surface tension) t=0ms t=1ms t=0ms t=4ms t=8ms t=13.5ms t=2.5ms t=0ms t=18ms t=30ms t=8.5ms t=10.2ms By increasing inertia of the drop, changing the velocity to 1.5 m/sec from 1 m/sec in previous case, drop moves through the nozzle. Drop starts to oscillate freely to get its initial shape as a spherical drop. Initial velocity is equal to 1m/sec. The inertia of the drop is not enough to overcome the opposite forces from the wall, and effects of the surface tension. 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). • The surface cells are defined as the cell with 0<f<1. • 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 θ, and the obstacle volume fraction is defined as 1-θ. • The internal obstacle is characterized as a fluid with infinite density and zero velocity. • θis independent of time. • θ= 1, not an obstacle, open to the flow.. • θ=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 2α 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 cell’s 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 8.95×10-5 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 Laboratoryhttp://www.mie.utoronto.ca/labs/mfl