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REVIEW ARTICLES

Moving boundaries in micro-scale biofluid dynamics

[+] Author and Article Information
W Shyy, M Francois, N N’dri, R Tran-Son-Tay

Department of Aerospace Engineering, Mechanics, and Engineering Science, University of Florida, Gainesville FL 32611

HS Udaykumar

Department of Mechanical Engineering, University of Iowa, Iowa City IA 52242

Appl. Mech. Rev 54(5), 405-454 (Sep 01, 2001) (50 pages) doi:10.1115/1.1403025 History:
Copyright © 2001 by ASME
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Figures

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Comparison of Lagrangian and Eulerian methods for interface tracking
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Illustration of the effect of the geometric conservation law on computational accuracy.
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Illustration of geometric conservation law in a moving boundary computation: a) Schematic of a moving boundary test problem with prescribed boundary velocity, b) Mass conservation residual plot for implicit and trapezoid schemes
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Illustration of a computational domain having an immersed boundary.
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Marker points considered for the estimation of the force at point P.
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Fluid points considered for the evaluation of the velocity at marker point X.
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Illustration of one level local grid refinement.
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Interfacial cell and bulk cell classification on a grid with interface passing through it. Also shown are interfacial cell properties.
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Schematic of computational domain with immersed boundaries: a) boundary cells with immersed boundary located south of cell center, b) boundary cells with immersed boundary located west of cell center, c) typical reshaped trapezoidal boundary cells corresponding to case a, d) typical boundary cells corresponding to case b.
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Schematic of interpolation for cell face values and derivative at boundary cells; a) various fluxes required for trapezoidal boundary cell, b) trapezoidal region and stencil used in computing fsw.
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Trapezoidal region and stencil used in computing fe.
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Stencil for calculation of interface flux: a) stencil for calculating, b) stencil for calculating ∂ϕ/∂x.
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Illustration of the parameters in the analysis: a) The definition of model parameters, b) computational set-up and boundary conditions
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The steady-state behavior of the simple and compound drops when subjected to uniaxial extensional flow: a) DvsCa for simple (λ2=1,γ2=1) and compound drops (λ2=1,λ3=1,γ2=1,γ3=1,R3=0.5), •: simple drop of Stone and Leal 113114, ---: simple drop (present results), o: compound drop of Stone and Leal 107, —: compound drop of Kan etal61. Shapes corresponding to the capillary numbers I to IV are shown; b) Velocity vectors at steady state for Ca=0.1.
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Effect of the capillary number or imposed flow strength on the deformation (L2=3) and subsequent recovery of the compound drop (γ2=1,γ3=1,λ2=1,λ3=100): a) Recovery shapes for the compound drop deformed at low strain rate (Ca=0.14),b) Recovery shapes for the drop deformed at higher strain rate (Ca=0.35). The sequences of figures in a and b correspond to identical extension lengths.
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The recovery behavior of the compound drop compared to several averaged simple drops: a) Lvs unscaled recovery time, b) Lvs scaled recovery time, c) Scaled rate of deformation vs scaled recovery time, d) Lvs scaled recovery time. All viscosities and scaling factors are indicated in the figures.
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Velocity vectors during recovery for the core and shell with disparate time scales (γ2=1,γ3=1,λ2=40,λ3=4000) for different initial core deformations but the same overall drop deformation. Scaled times at which the vector are shown are indicated alongside each case: a) For L3=0.618,L2=2.1, the scaling factor λapp for time is 285, b) for L3=1 and L2=2.1, the scaling factor λapp for time is 220.
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Recovery paths for compound drops with different initial core deformation and a highly viscous core (γ2=1,γ3=1,λ2=40,λ3=4000)
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Velocity vectors during recovery for the core with a time scale comparable with that of the shell (γ2=1,γ3=1,λ2=40,λ3=400) for different initial shapes. Scaled times ts during recovery are indicated in each case: a) for L3=0.618, the scaling factor λapp for times is 165; b) for L3=1, the scaling factor λapp for time is 70.
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Recovery curves for the case (γ2=1,γ3=1,λ2=40,λ3=400) when core deformation time scale is comparable with the outer layer time scale: a) Lvs scaled time ts, scaling factors obtained to match the long-time recovery asymptote of the simple drop curve are shown in legend; b) Lvsts scaled to match the initial recovery stages obtained with scaling factors shown in the legend.
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Comparison of recovery shapes from simulation results and experimental observations: a) Sequent fluorescent-illuminated pictures of a lymphocyte recovery, b) Numerical simulations.
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Recovery curves for cell and nucleus vs recovery time for the lymphocyte simulations. Solid and dash lines represent computed recovery lengths for the cell and nucleus, respectively. Open circles and asterisks correspond to the experimental data for the cell and nucleus, respectively.
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Schematic illustration of the adherent (bridged) and free (unabridged) regions adjacent to the edge of the contact zone 122. The intensive forces supported by the membrane include a principal tension, Tm, that acts tangent to the plane of the membrane surface and a transverse shear Qm, that acts normal to the membrane plus the attractive stress, σn, which represents the adhesive forces. The curvilinear coordinates (s,θ) are also shown.
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Quantities expected to influence the receptor-mediated cell adhesion to a surface include receptor number, the density of complementary surface ligands, the force and torque transmitted to the cell by the passing fluid, the mobility of receptor in the plane of the membrane, and the contract area in which cell to surface bonds may form (adopted from Hammer and Lauffenburger, 129)
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The geometry of membrane-to-surface adhesion is illustrated 130. The surface to which the membrane binds is taken as coincident with the x-axis. Position along the contour of the membrane is tracked by the arc-length coordinate, s, at the free extremity (s→∞), tension (Tfx) is applied to the membrane at a particular angle (θfx) with respect to the surface. At the clamped extremity (s→+∞) the membrane is firmly attached to the surface so as to prevent lateral slippage.
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Spatial variation of Y⁁,T⁁, and A⁁b at steady state in the case of slip-bonds. Tfx=2 (adopted from Dembo et al. 130)
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Macro-Micro model. The cellular model is shown on the left, whereas the enlarged area on the right shows the micro domain for hte receptor model.
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Flow chart showing the interaction between the micro-model (receptor scale) and the macro-model (cellular scale)
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Instantaneous droplet shapes for Ca=1.0a) and Ca=0.1b), both with Re=100
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Instantaneous droplet shapes for two different Reynolds numbers Re=100 a) and Re=1.0 b) both with Ca=1.0
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Maximum length of a bond before breaking as a function of the reverse reaction rate, for different values of forward reaction rate: (▪) for kfo=10−7cm2/sec, (▴) for kfo=10−10cm2/sec, (♦) for kfo=10−12cm2/sec, (•) for kfo=10−14cm2/sec
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Effect of the ligand density (×1010 cm−2) on the peeling or debonding time (s);fσ=0.1,σts=4.5 dyne/cm, and σ=5 dyne/cm
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Effect of the slippage constant fσ on teh debonding time (s) for σ (circle) and σts (square) held constant respectively. The circle dots correspond to σ=5 dyne/cm, and σts is allowed to vary with fσ. The square dots correspond to σts=4.5 dyne/cm, and σ is allowed to vary with fσ.Nr=Nl=5×1010cm−2.
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Illustration of the instantaneous membrane shape resulting from the pressure field, based on the macroscopic model, and the bond force, based on the receptor-ligand model. The vertical axis is normalized by the bond length, and the horizontal axis by the cell radius.
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Effect of spring constant on membrane deformation
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Forward reaction rate as a function of time for two different locations of the bonds; X is the abscise location of the bond
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Reverse reaction rate as a function of time for two different locations of the bonds; X is the abscise location of the bond
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Schematic of an elastic membrane and the key variables
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Schematic of contact lens and computational model
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Schematic of the contact lens model and thickness profile
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Time history of the applied external pressure Pa=Paequil−A sin(π/40t)
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Maximum deflection versus time for L10, L100, L1000 with variable thickness
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Maximum pressure difference inside the tear film vs time
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Variation of tear fluid volume going in/out of domain with time
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Velocity vectors and pressure contours at t=5, 20, and 30 (case L10 with variable thickness); for L100 and L1000, the qualitative features are similar
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Averaged Reynolds number vs aspect ratio

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