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- Abstract
- 1 Introduction
- 2 Problem Formulation
- 3 Non-dimensionalization and time scales
- 4 Numerical Method
- 5 Validation
- 6 Results
- 7 Conclusions
- References

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J Fluid Mech. Author manuscript; available in PMC 2011 January 1.

Published in final edited form as:

J Fluid Mech. 2010 January 1; 642: 509–539.

doi: 10.1017/S0022112009992692PMCID: PMC2841450

NIHMSID: NIHMS153759

T. N. Swaminathan: ude.nnepu.saes@nimawsnt

See other articles in PMC that cite the published article.

We present detailed results for the motion of a finite sized gas bubble in a blood vessel. The bubble (dispersed phase) size is taken to be such as to nearly occlude the vessel. The bulk medium is treated as a shear thinning Casson fluid and contains a soluble surfactant that adsorbs and desorbs from the interface. Three different vessel sizes, corresponding to a small artery, a large arteriole, and a small arteriole, in normal humans, are considered. The hematocrit (volume fraction of RBCs) has been taken to be 0.45. For arteriolar flow, where relevant, the Fahraeus-Lindqvist effect is taken into account. Bubble motion cause temporal and spatial gradients of shear stress at the cell surface lining the vessel wall as the bubble approaches the cell, moves over it and passes it by. Rapid reversals occur in the sign of the shear stress imparted to the cell surface during this motion. Shear stress gradients together with sign reversals are associated with a recirculation vortex at the rear of the moving bubble. The presence of the surfactant reduces the level of the shear stress gradients imparted to the cell surface as compared to an equivalent surfactant-free system. Our numerical results for bubble shapes and wall shear stresses may help explain phenomena observed in experimental studies related to gas embolism, a significant problem in cardiac surgery and decompression sickness.

An understanding of the motion of a gas bubble through a blood vessel is of fundamental importance in gas embolism studies related to illnesses such as decompression sickness. Gas bubbles may enter the blood stream during cardiopulmonary bypass for cardiac operations or in vascular procedures. This may also happen due to decompression from hyperbaric exposures, such as in diving or extravehicular activity during space exploration. The intravascular gas emboli may deposit into organs, like the heart or the brain, and as a result, cause permanent injury.

A gas bubble moving in a blood vessel may cause damage to the cells lining the interior wall of the vessel (endothelial cells). The mechanisms for the injury appear to be related to the shear stress level imparted and the stress reversals that occur at the cell surface during the bubble motion (Mukundakrishnan *et al.*, 2008a). Independently, experiments in our laboratory have shown that such an injury to the cell may be mitigated by the addition of a soluble surfactant to the continuous phase (Suzuki *et al.*, 2004). The surfactant in the blood stream appears to significantly change the dynamics of the bubble-cell interaction (Branger & Eckmann, 2002). Furthermore, a non-toxic, soluble, inert surfactant introduced into the bulk phase opportunistically adsorbs onto the gas-liquid interface and modifies the interfacial tension. This results in a change of the bubble shape and its residence time adjacent to the cell surface as compared to an equivalent surfactant-free system. The adsorbed surfactant molecules may also shield biological moieties in the blood stream from encountering the adsorptive surface. The actual mechanisms responsible for this mitigation are not understood, and in this context, an understanding of the role of the surfactant is warranted (Eckmann *et al.*, 2006).

In earlier studies, we have experimentally and numerically investigated the motion of a bubble through a surfactant-laden Newtonian fluid and through a blood-like liquid medium (Casson fluid) without the presence of a surfactant (Cavanagh & Eckmann, 1999; Eckmann & Lomivorotov, 2003; Mukundakrishnan *et al.*, 2007, 2008a; Zhang *et al.*, 2006; Mukundakrishnan *et al.*, 2009, 2008b). Here, through advanced numerical modeling, we investigate the effect of the presence of a soluble surfactant on the motion of a finite sized gas bubble in a blood vessel.

A number of numerical techniques for treating the motion of a fluid particle or bubble through a surfactant laden fluid, either soluble (Muradoglu & Tryggvason, 2008; Johnson & Borhan, 2003; Ghadiali & Gaver, 2003) or insoluble (Tsai & Miksis, 1994; Li & Pozrikidis, 1997; Yon & Pozrikidis, 1998; Eggleton *et al.*, 1999, 2001) have been described in the literature. These are complemented by experimental studies (Rodrigue *et al.*, 1996; Branger & Eckmann, 1999) and some analyses (Oguz & Sadhal, 1988; Park, 1992; Milliken & Leal, 1994; Rodrigue *et al.*, 1999). The numerical methods for the study of free-surface flows include, front-tracking or immersed-boundary (Shin & Juric, 2002; Unverdi & Tryggvason, 1992; Torres & Brackbill, 2000), level set (Osher & Fedkiw, 2001; Sussman *et al.*, 1999), phase-field (Jacqmin, 1999), volume-of-fluid (Scardovelli & Zaleski, 1999; James & Lowengrub, 2004), coupled level-set and volume-of-fluid (Sussman & Puckett, 2000), immersed interface (Lee & Leveque, 2003), ghost-fluid methods (Udaykumar *et al.*, 2003), and moving mesh interface tracking (Quan & Schmidt, 2007). Combined experimental and numerical studies examining the rise of a bubble in a surfactant laden fluid have also been described (Palaparthi *et al.*, 2006).

In this paper, a numerical procedure for evaluating the axisymmetric motion of a nearly occluding gas bubble (the topology of interest to the health sciences) in a surfactant laden Casson fluid in a cylindrical tube been described and implemented. The interface of the bubble is tracked using a modified front-tracking method. Flow and species discontinuities are smoothed and the surface tension force is distributed over a thin layer near the interface to become a equivalent volume force. The Navier-Stokes and continuous phase mass-transfer equations are solved on a fixed Eulerian mesh. A level contour reconstruction procedure is employed for the periodic redistribution of the front points and reconstruction of a new front. An explicit background mesh of interconnected marker points is used to represent the interface. Surface convection-diffusion equations are solved on this interface with suitable coupling to the bulk species concentrations to account for interfacial adsorption and desorption.

The properties of the bulk fluid include both the shear-thinning effects of blood as well as the Fahraeus and Fahraeus-Lindqvist effects (see Ayyaswamy (2008)), while the non-dimensional parameters associated with the surfactant have been broadly chosen from Somasundaran (2006) and Chang & Franses (1995). Results are also provided for a specific case of C_{12}E_{6} whose properties are provided in Palaparthi *et al.* (2006). The surfactant properties, in general, are similar to those of ethoxylated surfactants and follow the Langmuir isotherm. The specific case of C_{12}E_{6}, however, follows the follows the Frumkin isotherm. Three different vessel sizes, corresponding to a small artery, a large arteriole, and a small arteriole, in normal humans (Fung, 1997), have been considered. The corresponding representative flow Reynolds numbers are 200, 2 and 0.2, respectively. The degree of bubble occlusion is characterized by the ratio of the equivalent-volume bubble radius to vessel radius (aspect ratio), *λ*, in the range 0.9≤ *λ* ≤ 1.05. For non-spherical bubbles, the applicable radius of the bubble is taken to be that of a spherically shaped bubble which has the same volume as the non-spherical one (equivalent-volume spherical radius). The initial bulk flow corresponds to a fully established Casson flow in a cylindrical vessel.

Bubble motion causes temporal and spatial gradients of shear stress at the cell surface lining the blood vessel wall as the bubble approaches the cell, passes over it and departs. Rapid reversals occur in the sign of the shear stress imparted to the cell surface during this motion. Shear stress gradients together with sign reversals are associated with a recirculation vortex at the rear of the bubble. Presence of the surfactant in the bulk and the interface reduces the level of the shear stresses and the strengths of the vortical motions as compared to an equivalent clean system. Furthermore, the shape of the bubble, and consequently the gap between the bubble and the vessel wall changes in the presence of the surfactant resulting in a change in the bubble residence time near the proximity of a cell surface. The net effect is to reduce the level or eliminate the mechanically driven injury to the cell surface. The model predictions concur with our experimental findings and are thought to seriously impact gas embolism studies.

The organization of this paper is as follows. The problem formulation is followed by a description of the numerical methodology. The numerical procedure is validated. Detailed computational results are described for the surfactant problem and where necessary comparisons with the clean system study are presented. The final two sections deal with the major results and the important conclusions therein.

The axisymmetric motion of a nearly occluding gas bubble of density *ρ _{g}* and viscosity

Both the gas and the liquid phases are modeled as incompressible fluids governed by the equations of motion (Tryggvason *et al.*, 2001; Zhang *et al.*, 2006):

$$\nabla \cdot \mathbf{\text{u}}=\text{0}$$

(1)

$$\begin{array}{c}\rho \left(\frac{\partial \mathbf{\text{u}}}{\partial t}+\mathbf{\text{u}}\cdot \nabla \mathbf{\text{u}}\right)=-\nabla p+\nabla \cdot \mu \left(\nabla \mathbf{\text{u}}+{\nabla}^{T}\mathbf{\text{u}}\right)+\\ {\int}_{S(t)}\left[\sigma \left(\mathrm{\Gamma}\left(s\right)\right)\mathcal{K}{\mathbf{\text{n}}}_{f}+\frac{\partial \sigma \left(\mathrm{\Gamma}\left(s\right)\right)}{\partial s}{\mathbf{\text{t}}}_{f}\right]\phantom{\rule{0.2em}{0ex}}\widehat{\delta}\left(\mathbf{\text{x}}-{\mathbf{\text{x}}}_{f}\right)dS+\rho \mathbf{\text{g}}.\end{array}$$

(2)

Here, **u** (*v _{r}, v_{z}*), is the fluid velocity with

$$\rho \left(r,z,t\right)={\rho}_{l}\mathcal{H}\left(r,z,t\right)+{\rho}_{g}\left(1-\mathcal{H}\left(r,z,t\right)\right),$$

(3)

$$\mu \left(r,z,t\right)={\mu}_{l}\mathcal{H}\left(r,z,t\right)+{\mu}_{g}\left(1-\mathcal{H}\left(r,z,t\right)\right).$$

(4)

Here, *t* is time, and (*r, z, t*) is a Heaviside function such that,

$$\mathcal{H}\left(r,z,t\right)=\{\begin{array}{ll}1& \left(r,z,t\right)\phantom{\rule{0.2em}{0ex}}\text{in the bulk},\\ 0& \left(r,z,t\right)\phantom{\rule{0.2em}{0ex}}\text{in the bubble}\end{array}\cdot $$

(5)

The bulk liquid has been modeled as a two-layer fluid of density *ρ _{l}* consisting of a cylindrical core of concentrated red blood cell (RBC) suspension, surrounded by a cell-free layer. The cell-free layer is modeled as a Newtonian fluid of constant viscosity

$${\mu}_{l}=\{\begin{array}{ll}{\mu}_{c},& 0\le r\le \left(1-\delta \right)\phantom{\rule{0.2em}{0ex}}R\\ {\mu}_{\mathit{\text{layer}}},& \left(1-\delta \right)R<r\le R\end{array},$$

(6)

where *δ* is the non-dimensional thickness of the cell-free layer normalized with *R*. The evaluation of this thickness and the two viscosities *μ _{layer}* and

$${\mu}_{c}={\left[\sqrt{{\mu}_{\infty}}+\sqrt{\frac{{\tau}_{y}}{\left|\gamma \right|}}\left(1-{e}^{-\sqrt{m\left|\gamma \right|}}\right)\right]}^{2},$$

(7)

where *μ*_{∞} is the asymptotic Newtonian viscosity, *τ _{y}* is the yield stress,

The surface tension of the interface, separating the continuous and dispersed phases, varies with the surfactant concentration, Γ, on the interface, according to the equation of state given by the Langmuir isotherm (Levich, 1962):

$$\sigma ={\sigma}_{s}+\Re T{\mathrm{\Gamma}}_{\infty}ln\left(1-\frac{\mathrm{\Gamma}}{{\mathrm{\Gamma}}_{\infty}}\right),$$

(8)

where *σ _{s}* is the surface tension of a surfactant-free clean interface, is the ideal gas constant,

$$\frac{\partial \mathrm{\Gamma}}{\partial t}+{\nabla}_{s}\cdot \left({\mathbf{\text{u}}}_{\mathrm{s}}\mathrm{\Gamma}\right)+\mathrm{\Gamma}\left({\nabla}_{s}\cdot {\mathbf{\text{n}}}_{f}\right)\left(\mathbf{\text{u}}\cdot {\mathbf{\text{n}}}_{\mathbf{\text{f}}}\right)={D}_{s}{\nabla}_{s}^{2}\mathrm{\Gamma}+{j}_{n},$$

(9)

with *j _{n}* being the diffusional flux at the interface given by Fick's law:

$${D}_{l}\left({\mathbf{\text{n}}}_{f}\phantom{\rule{0.2em}{0ex}}\cdot \nabla \right)C={k}_{a}{C}_{s}\left(1-\frac{\mathrm{\Gamma}}{{\mathrm{\Gamma}}_{\infty}}\right)-{k}_{d}\mathrm{\Gamma},$$

(10)

in which *k _{a}* and

$$\frac{\partial C}{\partial t}+\left(\mathbf{\text{u}}\cdot \nabla \right)C=\nabla \cdot \left(D\nabla C\right)-\underset{S(t)}{\int}\left[{k}_{a}{C}_{s}\left(1-\frac{\mathrm{\Gamma}}{{\mathrm{\Gamma}}_{\infty}}\right)-{k}_{d}\mathrm{\Gamma}\right]\widehat{\delta}(\mathbf{\text{x}}-{\mathbf{\text{x}}}_{f})dS,$$

(11)

where the diffusion coefficient *D* may be written as.

$$D={D}_{l}\mathcal{H}.$$

(12)

These equations are coupled with suitable initial and boundary conditions. At the inlet boundary *z* = 0, the velocity profile is a fully developed two-layer Casson profile. This velocity profile and the associated pressure gradient have to be numerically evaluated for each vessel size using the Casson two layer model (equations 6,7) and applicable boundary conditions (see, Mukundakrishnan *et al.* (2008a) for details). These evaluations are based on the maximum flow velocity at the centerline of a bubble-free blood vessel of any given size (Berger *et al.*, 1996). The velocity profile and the associated pressure gradient so calculated are used as inlet boundary conditions. In vivo experimental and clinical literature regarding multifocal arterial gas embolization as occurs in cardiac surgery and in decompression sickness indicates that bubble passage and trapping effects are important within the microcirculatory arteriolar vessels. In this size range, the pulsatility effects of the cardiac cycle on blood flow are much less predominant, so that an approximation of steady flow is reasonable. At the outlet boundary, *z* = *L*, the outflow condition,

$$\frac{\partial \mathbf{\text{u}}}{\partial z}\phantom{\rule{0.2em}{0ex}}(r,L,t)=0,$$

(13)

is used. The reflective boundary conditions, at *r* = 0 are given as:

$$\begin{array}{l}{v}_{r}\phantom{\rule{0.2em}{0ex}}(0,z,t)=0,\\ \frac{\partial {v}_{z}}{\partial r}\phantom{\rule{0.2em}{0ex}}(0,z,t)=0,\\ \frac{\partial C}{\partial r}\phantom{\rule{0.2em}{0ex}}(0,z,t)=0.\end{array}$$

(14)

The wall no-slip, no flux boundary conditions at *r* = *R* are,

$$\begin{array}{l}\mathbf{\text{u}}\phantom{\rule{0.2em}{0ex}}(R,z,t)=0,\\ \frac{\partial C}{\partial r}\phantom{\rule{0.2em}{0ex}}(R,z,t)=0.\end{array}$$

(15)

The initial surfactant concentration in the bulk is taken to be

$$C(r,z,0)={C}_{0}\mathcal{H}\phantom{\rule{0.2em}{0ex}}(r,z,0).$$

(16)

Consider the non-dimensional quantities (denoted by superscripted *): *s** = *s/d, r** = *r/d, z** = *z/d, ρ** = *ρ/ρ _{l}*,

$$\frac{\partial \mathbf{\text{u}}}{\partial t}+\nabla \mathbf{\text{u}}\cdot \mathbf{\text{u}}=-\nabla p+\frac{1}{Re}\nabla \cdot \mu (\nabla \mathbf{\text{u}}+{\nabla}^{T}\mathbf{\text{u}})+\frac{1}{\text{We}}{\int}_{S}\left[\sigma \mathcal{K}{\mathbf{\text{n}}}_{f}+\frac{\partial \sigma}{\partial s}{t}_{f}\right]\phantom{\rule{0.2em}{0ex}}\widehat{\delta}(\mathbf{\text{x}}-{\mathbf{\text{x}}}_{f})ds+\frac{1}{{\text{Fr}}^{2}}\rho \widehat{\mathrm{g}},$$

(17)

$$\frac{\partial \mathrm{\Gamma}}{\partial t}+{\nabla}_{s}\cdot ({\mathbf{\text{u}}}_{\mathrm{s}}\mathrm{\Gamma})+\mathrm{\Gamma}({\nabla}_{s}\cdot {\mathbf{\text{n}}}_{f})(\mathbf{\text{u}}\cdot {\mathbf{\text{n}}}_{f})=\frac{1}{{\text{Pe}}_{s}}{\nabla}_{s}^{2}\mathrm{\Gamma}+\frac{1}{\alpha \phantom{\rule{0.2em}{0ex}}\text{Pe}}({\text{n}}_{f}\phantom{\rule{0.2em}{0ex}}\cdot \nabla )\phantom{\rule{0.2em}{0ex}}C,$$

(18)

$$\frac{\partial C}{\partial t}+(\mathbf{\text{u}}\cdot \nabla )\phantom{\rule{0.2em}{0ex}}C=\frac{1}{\text{Pe}}\nabla \cdot (D\nabla C)-\underset{S}{\int}[{\text{St}}_{a}\phantom{\rule{0.2em}{0ex}}{C}_{s}\phantom{\rule{0.2em}{0ex}}(1-\mathrm{\Gamma})-\alpha \phantom{\rule{0.2em}{0ex}}{\text{St}}_{d}\mathrm{\Gamma}]\widehat{\delta}\phantom{\rule{0.2em}{0ex}}(\mathbf{\text{x}}-{\mathbf{\text{x}}}_{f})\phantom{\rule{0.2em}{0ex}}ds,$$

(19)

in which the relevant non-dimensional parameters are defined as, Reynolds number Re = *ρU _{max}d/μ*, Weber number
$\text{We}={\rho}_{l}{U}_{\mathit{\text{max}}}^{2}d/\sigma $, Froude number
$\text{Fr}={U}_{\mathit{\text{max}}}/\sqrt{gd}$, surface Péclet number Pe

Based on these non-dimensional numbers, the important time scales relevant to this problem are: convection time scale *d/U _{max}*, momentum diffusion time scale

Equations 1-12, subject to the appropriate initial and boundary conditions (equations 13-16) are solved using a front-tracking method coupled with a level contour reconstruction procedure. Briefly, the procedure consists of,

- Given a particular interface position, and bulk and interface concentration distributions, calculate the surface tension forces and distribute them as body forces to the fixed Eulerian grid.
- Solve the fluid flow equations with the given conditions to obtain the velocity and pressure fields.
- Use the computed velocity field to update the interface and bulk surfactant values.
- Update the position of the interface using the velocity field in a Lagrangian fashion.
- Reconstruct the interface using the level contour reconstruction procedure if the bubble volume changes by more than 0.5% or periodically after every few time steps.
- If the interface is reconstructed, project the surface concentration profile to the new interface in such a way as to conserve the amount of surfactant on the interface.
- Repeat these steps until either steady state is reached or the bubble reaches the outlet of the computational domain.

A brief description of the procedure follows.

The surface tension force, **f*** _{st}*, acting on a segment of the interface between two points A and B can be derived from equation 2 as,

$${\mathbf{\text{f}}}_{\mathit{\text{st}}}=\underset{A}{\overset{B}{\int}}\left(\sigma \mathcal{K}{\mathbf{\text{n}}}_{f}+\frac{\partial \sigma}{\partial s}{\mathbf{\text{t}}}_{f}\right)\phantom{\rule{0.2em}{0ex}}ds.$$

(20)

Following the use of the Frenet formulae for the unit vectors in the *r* − *z* plane as given in James & Lowengrub (2004), this reduces to,

$${\mathbf{\text{f}}}_{\mathit{\text{st}}}=\underset{A}{\overset{B}{\int}}\phantom{\rule{0.2em}{0ex}}\left[\frac{\partial}{\partial s}(\sigma {\mathbf{\text{t}}}_{f})-\sigma \phantom{\rule{0.2em}{0ex}}\left(\frac{\partial {\mathbf{\text{t}}}_{f}}{\partial s}+\frac{{\mathbf{\text{n}}}_{fr}}{r}\right)\right]\phantom{\rule{0.2em}{0ex}}ds,$$

(21)

**n*** _{fr}* is a vector in the

$${\mathbf{\text{f}}}_{\mathit{\text{st}}}={\sigma}_{A}{\mathbf{\text{t}}}_{A}-{\sigma}_{B}{\mathbf{\text{t}}}_{B}-{\mathbf{\text{e}}}_{r}\underset{A}{\overset{B}{\int}}\sigma ds,$$

(22)

where, **t*** _{A}* and

$${\mathbf{\text{F}}}_{st(i,j)}=\frac{{\sum}_{e}{\rho}_{i,j}{\mathbf{\text{f}}}_{\mathit{\text{ste}}}{D}_{i,j}(\mathbf{\text{x}}-{\mathbf{\text{x}}}_{m})}{{\sum}_{e}{\rho}_{i,j}{D}_{i,j}(\mathbf{\text{x}}-{\mathbf{\text{x}}}_{m})},$$

(23)

where **x** = (*i*Δ*r, j*Δ*z*), **x*** _{m}* = (

$${D}_{i,j}(\mathbf{\text{x}}-{\mathbf{\text{x}}}_{m})=\frac{\widehat{\delta}({r}_{m}/\mathrm{\Delta}r-i)\phantom{\rule{0.2em}{0ex}}\widehat{\delta}({z}_{m}/\mathrm{\Delta}z-j)}{2\pi r\mathrm{\Delta}r\mathrm{\Delta}z}.$$

(24)

The one-dimensional delta function in the above is numerically approximated as (Griffith & Peskin, 2005),

$$\widehat{\delta}(\widehat{d})=\{\begin{array}{ll}{\widehat{\delta}}_{1}(\widehat{d}),& \phantom{\rule{1em}{0ex}}|\widehat{d}|\phantom{\rule{0.2em}{0ex}}\le 1,\\ 1/2-{\widehat{\delta}}_{1}(2\phantom{\rule{0.2em}{0ex}}-|\widehat{d}|),& \phantom{\rule{1em}{0ex}}1<\phantom{\rule{0.2em}{0ex}}|\widehat{d}|\phantom{\rule{0.2em}{0ex}}<2,\\ 0,& \phantom{\rule{1em}{0ex}}|\widehat{d}|\phantom{\rule{0.2em}{0ex}}\ge 2,\end{array}$$

(25)

and
${\widehat{\delta}}_{1}(\widehat{d})=\left(3-2|\widehat{d}|+\sqrt{1+4|\widehat{d}|-\phantom{\rule{0.2em}{0ex}}4{\widehat{d}}^{2}}\right)/8$. Here, *d^* denotes the distance from the origin of the source (the front position).

The unsteady mass and momentum equations are discretized using a second-order finite-difference based variable density projection method (Sussman & Puckett, 2000). The velocity, density, and viscosity are all located at cell centers. The time stepping procedure is based on the second-order Crank-Nicholson method. An intermediate velocity field is obtained using a semi-implicit viscous solver. In this, a second-order predictor-corrector method based on the unsplit Godunov method (Colella, 1990) is used to evaluate the advective terms while a standard second order central finite difference is used to evaluate the diffusion terms. For the diffusion term, the spatial distribution of viscosity is computed explicitly using the second invariant of strain rate tensor evaluated at the *n ^{th}* time step. Such an evaluation does not impose any additional stability criteria as noted in Radl

The surfactant exchange between the bulk and the interface is modeled as occurring in a thin layer adjacent to the interface (see Muradoglu & Tryggvason (2008)). This technique is different from that described in an earlier paper by Zhang *et al.* (2006) wherein this exchange occurs exactly on the interface.

The interface surfactant evolution equation (9) is used to update the concentration explicitly. First the diffusion flux *j _{n}*, given in equation 10, is computed in an explicit manner using the previous time step values. The computed diffusion flux is then used in equation 9 to update the interface concentration. This update is done on the Lagrangian grid. The bulk surfactant evolution equation (11), is solved on the Eulerian grid. The diffusion flux on the interface is converted to a bulk source term. This is done using a method similar to that used for the distribution of surface tension (see equation 23). The bulk source term

$${J}_{i,j}=\frac{\text{\u2211}_{e}{\mathcal{H}}_{i,j}{D}_{i,j}\phantom{\rule{0.2em}{0ex}}(\mathbf{\text{x}}-{\mathbf{\text{x}}}_{m})\phantom{\rule{0.2em}{0ex}}\frac{{j}_{ne}{r}_{e}{l}_{e}}{{r}_{i,j}\mathrm{\Delta}r\mathrm{\Delta}z}}{\text{\u2211}_{e}{\mathcal{H}}_{i,j}{D}_{i,j}\phantom{\rule{0.2em}{0ex}}(\mathbf{\text{x}}-{\mathbf{\text{x}}}_{m})},$$

(26)

where *D _{i,j}* (

The redistribution of the front points and the conservation of the bubble volume are simultaneously achieved using a level contour reconstruction procedure (Shin & Juric, 2002; Shin *et al.*, 2005). As the bubble moves, and especially if the interface significantly deforms, it becomes necessary to recompute the front points defining the interface. Here, this is done in such a way that the final mass (volume) of the bubble, based on the calculation of a new interface, is within 0.5% of the original volume of the bubble while at the same time minimizing the deviation from the computed advected boundary (see, Mukundakrishnan *et al.* (2007)). Upon reconstruction, the new front points are assigned using the intersection of the computed boundary with the Eulerian grid.

Subsequent to the reconstruction of the interface, the surfactant concentration on the new front points must be recomputed to be consistent with the previous interface points. The surfactant mass on the interface Γ* _{m}* must be conserved. The total interface surfactant mass may be written as

$${\mathrm{\Gamma}}_{m}{|}_{\mathit{\text{total}}}\phantom{\rule{0.2em}{0ex}}\approx 2\pi \sum _{i=1}^{n-1}{\mathrm{\Gamma}}_{i}\phantom{\rule{0.2em}{0ex}}\left(\frac{{r}_{i}+{r}_{i+1}}{2}\right)\phantom{\rule{0.2em}{0ex}}({s}_{i+1}-{s}_{i}),$$

(27)

where, *r* is the radial distance in a cylindrical coordinate system measured from the axis of the tube and s* _{i}* is the arc length to the

$${\mathrm{\Gamma}}_{m}{|}_{A}^{B}\phantom{\rule{0.2em}{0ex}}\approx 2\pi \sum _{i=1}^{n-1}{\mathrm{\Gamma}}_{i}\phantom{\rule{0.2em}{0ex}}\left(\frac{{r}_{i}+{r}_{i+1}}{2}\right)\phantom{\rule{0.2em}{0ex}}(({s}_{i+1}-{s}_{i})\cap {s}_{AB}).$$

(28)

Here *s _{AB}* represents the arc length between the two points. Since the total length of the interface itself may change during reconstruction, the equivalent point for

$${s}_{A\prime 0}={s}_{A0}\frac{{s}_{\mathit{\text{total\_new}}}}{{s}_{\mathit{\text{total\_old}}}},$$

(29)

with a corresponding interface point *B′* corresponding to *B*. We conserve the mass locally such that

$${\mathrm{\Gamma}}_{m}{|}_{A\prime}^{B\prime}\phantom{\rule{0.2em}{0ex}}={\mathrm{\Gamma}}_{m}{|}_{A}^{B}\phantom{\rule{0.2em}{0ex}},$$

(30)

wherein *A, B, A′* and *B′* in general do not coincide with the interface points *i*. This is achieved by projecting the surfactant mass from each segment of the old interface into the appropriate, scaled new interface segment(s). This mass conservation will automatically satisfy the total mass conservation limit given in equation 27 to within computational accuracy, which is verified upon each projection.

Validation for the surfactant-free cases are given in Mukundakrishnan *et al.* (2007, 2008a) and will not be repeated here. We have now validated the code in the presence of the surfactant, the details of which are presented in Appendix A.

Here, we examine, for otherwise identical conditions, the time histories of an initially clean bubble and an initially contaminated bubble as they both attain steady states in the surfactant-laden bulk medium. Figures 2(a) and 2(b) show that the steady state, the interface concentrations are independent of the imposed initial conditions.

Interface concentration profiles at various times for two cases starting from different initial prescribed concentrations. (a) Variation of the interface surfactant concentration for various various times. (b) Variation with time of the total interface **...**

In figure 2(a), the interface concentration profile for three different times is shown for both the cases, and with increasing time, both the profiles converge to the same profile. In figure 2(b), we observe that the total interface surfactant mass is converging to the same steady state value starting from the two different initial conditions.

The feature that the eventual steady state interface concentration is independent from the starting condition is particularly noteworthy. Our numerical experimentation has shown that the computational time to obtain the steady state starting from a clean bubble is considerably larger than that starting with an initially contaminated bubble. Where the transient history is not of concern, for computation it is therefore convenient to start with a slightly contaminated bubble, with an initial shape identical to that of a clean bubble, uniformly coated with the surfactant. This feature becomes particularly advantageous at lower Péclet numbers. In this paper, results for Re=0.2 have been developed using initially contaminated bubbles thus saving considerable computational time. The characteristics shown in figure 2(a) are further discussed in detail in section 6.1).

Results for bubble motion in the presence of a soluble surfactant in the bulk phase have been compared with those for an equivalent clean bubble (Mukundakrishnan *et al.*, 2008a). Fluid flow direction is from the bottom of the vertical vessel towards the top and occurs at a fixed pressure gradient. A soluble surfactant with a far-field concentration of *C*_{0} is assumed to be present in the bulk phase. The bubble rises in the vessel under the action of body forces (due to gravity) and surface forces (fluid motion), and deforms as it moves.

For all the cases considered here, excepting for the C_{12}E_{6} surfactant, *C*_{0} is assumed to be the same. This leads to different values of *α* (see table 1). Table 1 also lists the parameter values for Ca, Pe and St_{a}/St_{d}. The clean interface Ca values are provided in table 1 (Mukundakrishnan *et al.*, 2008a). All the other properties of the surfactant, including adsorption and desorption coefficients (*K _{a}* and

Three aspect ratios, *λ*, have been considered, and are 0.9, 1.0 and 1.05. For all the cases studied, a thin annular film is assumed to be present between the bubble and the vessel wall at bubble introduction. For *λ* ≥ 1.0, the initial shape of the bubble is assumed to be elongated with a hemispherical cap at each end. The shape, the velocity and the surface concentration of the bubble eventually attain steady states. In this study, we choose to discuss phenomena after the attainment of the steady state.

Figure 3 shows the streamlines for flow over a clean bubble in a reference frame attached to the bubble for various Reynolds numbers and *λ* = 1.0. The bubbles are generally deformed with a bulge at the rear primarily due to the imposed pressure gradient. At a certain location along the interface, a very narrow layer of fluid separates the bubble from the wall. This proximity of the bubble to the vessel wall is of particular interest in this problem as shall be seen later. The drainage flow squeezes through this narrow gap and is mainly responsible for the surface mobility of the bubble. This in turn gives rise to internal vortical motions. Secondary internal vortices are also noted and the vortex strengths increase at higher values of Re. The sense of rotation of the secondary vortices is opposite to that of the primary vortex. Two distinct recirculatory bolus formations may be discerned, one at the front region of the bubble and the other at the rear. Two stagnation rings, one convergent ring at the rear and the other a divergent ring at the front, exist on the bubble surface where the back flow adjacent to the wall (drainage flow) meets the recirculating boluses. For the convergent stagnation ring (zone), the streamlines move radially outward from near the axis of the tube towards this ring. For the divergent stagnation ring, the streamlines are directed inward towards the axis of the tube. More details for the clean bubble case are available in Mukundakrishnan *et al.* (2008a).

Streamlines for flow over a clean bubble as viewed in a reference frame moving with the steady state velocity of the bubble for three representative Reynolds numbers. The aspect ratio (λ) is 1.0.

Next, we consider bubble motion in a soluble surfactant laden flow through the vessel. Upon the introduction of the bubble, diffusive and convective mechanisms in the bulk phase will bring the surfactant from the bulk medium to the gas-liquid interface. Here, adsorption and desorption mechanisms will be initiated. Although the vortical motions inside the bubble are essentially similar in their dynamic characteristics to the clean bubble case, they are considerably weaker in strength. Diffusion and advection of the adsorbed surfactant occur at the interface, the latter due to surface mobility. Consistent with the bulk concentration and interfacial capacity for adsorption and desorption of the surfactant, a steady state mass transfer profile is eventually established displaying a non-uniform distribution of the surfactant at the interface.

We now discuss the steady state concentration profiles at the interface. In an unbounded stationary medium containing a soluble surfactant, the motion of a gas bubble would sweep the surfactant on the interface towards the rear, and the front region will be of higher surface tension (see Sadhal *et al.* (1997), Chapter 3). However, for a nearly occluding bubble moving inside a circular tube, the distribution of the surfactant at the interface is complex. In figure 4, we display the steady state interface concentrations of the surfactant as a function of arc length for various Reynolds numbers. The arc length along the interface is measured starting from the trailing edge of the bubble. In view of the deformed shape of the bubble, the arc length measured is normalized with respect to the total arc length from the trailing edge to the leading edge. At every Reynolds number, the highest and the lowest surfactant concentrations on the interface occur in the vicinity of the convergent and divergent stagnation rings, respectively, accompanied by a complicated distribution elsewhere along the interface.

Interface concentration for the three cases of Reynolds numbers with identical far-field concentration. The aspect ratio (λ) is 1.0. The normalized arc length along the surface of the bubble is measured from the rear stagnation point.

The attainment of the steady state concentration profiles may be explained using figure 5 which is a schematic of the streamlines inside and outside the bubble in the reference frame of the bubble. Points A and D in this figure are the rear and front points located on the axis of the bubble, respectively. Points B and C show the approximate locations of the convergent and divergent stagnation rings. Upon the introduction of the bubble, there will be increased accumulation of the surfactant in the vicinity of A relative to that near D. The prevailing interface convection will sweep the surfactant from A towards B. On the bubble front, surfactant will be swept by the incoming flow towards C. From C, interface convection moves the surfactant toward D and B. The drainage flow will sweep the surfactant deposited between B and C towards B. Thus at B, there is increased accumulation due to contributions from both A and C. The accumulation of surfactant at B and depletion at C will set up interface concentration gradients. This leads to surface diffusion (measured by the surface Péclet number Pe* _{s}*) which will tend to equalize the surfactant distribution. The final concentration profile is thus determined by the interplay between these various mechanisms. The surfactant boundary layer in this problem is very small compared to the bubble diameter for all three Reynolds numbers. This is consistent with the high values of bulk Péclet number, Pe (see table 1). There is negligible variation in the value of the bulk surfactant concentration level near the vessel wall. Our computations show this variation to be less than 1%.

A major effect of the presence of the surfactant on the bubble surface is the reduction in surface mobility. As the bubble moves, an internal vortex is formed (somewhat similar to a Hill's spherical vortex (Happel & Brenner, 1983), or more generally a toroidal vortex). A surfactant coating changes the properties (shape and strength) of this vortex (Sadhal & Johnson, 1983; Johnson & Sadhal, 1983). The streamlines for a clean and a surfactant coated bubble as viewed in an inertial frame of reference, for *λ* = 1.0, are shown in figure 6 for various Reynolds numbers. Presence of the surfactant causes the recirculation vortex noted here to be weaker as evident from the increased distortion of the streamlines for the clean bubble case.

Streamlines as viewed in an inertial reference frame for clean and surfactant coated bubbles. The aspect ratio (λ) is 1.0.

A measure of the internal circulation in the bubble is given by the strength of the vortex tube(s) (see Batchelor (1967), section 2.6), and is *∫ ωdV*, where *ω* is the vorticity, *dV* is an elemental volume. The integral is carried out over the volume of the bubble. The ratio of the strengths of the vortex-tubes, between a clean and a surfactant coated bubble, is shown in figure 7 for various Reynolds numbers. The strength of the toroidal vortices present in the bubble (figure 3) decrease upon the addition of a surfactant. The reduction in vortex strength is the lowest for the largest Reynolds number considered here (200) where the inertial forces are dominant.

A detailed study of the steady state shapes of clean bubbles in motion has been described in Mukundakrishnan *et al.* (2008a). Briefly, at smaller Reynolds numbers (0.2 and 2), the capillary effects dominate over the inertial and viscous effects. For these two Reynolds numbers, for *λ* = 0.9, the bubble essentially remains spherical, while for *λ* ≥ 1.0, it nearly occludes the vessel. At the higher Reynolds number (200), inertial forces dominate. At *λ* = 0.9 bubbles form a flatter interface at the rear and are prolate spheroidal at the front. For *λ* ≥ 1.0, the bubble elongates in the axial direction due to the increased wall effects accompanying the inertial forces.

Figure 8 shows the variations in the shapes of surfactant-coated bubbles as compared to clean bubbles. There are three primary surface forces which affect the shape of the nearly occluding bubble. The first is the surface tension on the interface of the bubble and this tends to make the bubble spherical. The second is a result of the prevailing pressure gradient driving the flow. This force will tend to “flatten” the bubble in the rear causing it to be oblate spheroidal. The third force arises due to the presence of the wall. As the bubble gets closer to the wall, the drainage flow generates a force which acts to push the sides of the bubble radially inward, away from the wall causing it to elongate in the direction of its motion. The final shape of the bubble is the result of a balance of these three forces together with the body force (buoyancy). However, the body force does not play a major role in this problem, as shown later. Reduction in the value of the surface tension due to the surfactant results in increased departure from sphericity and the bubble shape is more responsive to fluid flow induced deformation forces.

Comparison of the steady state shape of an occluded bubble with and without interfacial surfactants. The solid line represents the shape of the clean interface bubble, while the dotted line represents a surfactant coated bubble.

For *λ* = 0.9 (see figure 8(a,d,g)), a clean bubble is nearly spherical with a larger gap between the interface and the vessel wall. The forces that mainly govern the shape of the bubble are the imposed pressure force, tending to flatten the bubble at the rear and the surface tension force which acts to preserve its sphericity. Reduction in surface tension results in an oblong shape. However, for *λ* ≥ 1.0, for a clean bubble, all the three surface forces are significant since the bubble interface is closer to the wall. The radially inward force arising from the drainage flow becomes more important in this case because the fluid is being squeezed through a narrower gap, and this force acts to elongate the bubble in the direction of its motion resulting in an elongated oblong shaped bubble. In the presence of a surfactant, the imposed pressure force flattens the rear of the bubble further while the force due to the draining flow acts to counter this. The result is that the minimum gap between the interface and the wall actually increases.

The non-sphericity index, (*η*), of the bubble may be defined by the ratio of twice the maximum radial extent of the bubble, as measured perpendicular from the axis of the vessel, to its axial length. For a spherical bubble, *η* is 1.0, although *η* = 1.0 does not always imply sphericity. For axially elongated bubbles, *η* is less than 1, while for radially flattened bubbles, *η* is greater than 1. Figure 9 shows the variation of *η* with *λ*, at various Reynolds numbers, for a clean and surfactant-coated bubble.

Variation in the non-sphericity (η) of the bubble upon introduction of a surfactant as a function of the aspect ratio for the three different Reynolds numbers. The solid line indicates a clean bubble while the dashed line indicates a surfactant **...**

From figures 8 and and9,9, for Reynolds number 200, the inertial forces are noted to be dominant for both the clean and the surfactant-coated bubble. The value of *η* decreases in the presence of the surfactant but the shape of the bubble does not change much. For Reynolds numbers 0.2 and 2 and *λ* ≥ 1.0, with a lowered surface tension, the wall effects become more pronounced and the interface is pushed radially inward causing a lowering of *η*. This indicates that the bubble has axially elongated. For *λ* = 0.9, *η* does not change appreciably. As the rear of the bubble flattens due to the imposed pressure, the gap between the bubble and the vessel wall decreases resulting in increased wall effects that counter this flattening.

The change in bubble shape may also be evaluated in terms of the minimum gap size between the bubble and the tube wall, and the length of the bubble, as shown in figures 10(a) and 10(b), respectively. This gap will impact the flow field near the wall and the magnitude of the shear stress experienced by the cells lining the interior of the vessel wall. Thus the minimum gap size is an important parameter to be considered in the gas embolism problem. Both shear and normal stresses applied to endothelial cells are known to produce membrane stretch and induce intracellular calcium (Ca2+) signaling (Sigurdson *et al.*, 1993; Oike *et al.*, 1994), as is commonly observed by one of the authors (DME) in current in vitro studies of cell/bubble interactions.

Bubble length (figure 10(b)) and its velocity determine the duration of time during which any given point located on the vessel wall will experience the effect of the bubble passing over it (bubble residence time). The variation in the steady velocity of the bubble in the presence of the surfactant is shown in figure 11. This velocity has been normalized with reference to that for a corresponding clean bubble.

Normalized steady velocity of the bubble for three different Reynolds numbers as a function of aspect ratio. The normalization has been done with the velocity for a corresponding clean bubble. The clean bubble velocities (in cm/sec) for the three Reynolds **...**

In general, a bubble moving in an infinite medium will experience increased retardation in the presence of a surfactant (Bond & Newton, 1928) primarily through the setup of tangential surface tension gradients (Marangoni forces). It will therefore move at a lower velocity compared to that of a clean bubble. In our problem of a deformed nearly occluding bubble in a confined flow, the steady state velocity will significantly dependent on its shape. For a given inlet velocity, up to *λ* = 1.0, the occlusion and the profile drag increase causing a reduction in the steady velocity. Beyond *λ* ≥ 1.0, the behavior is complicated. From figure 11, for Re = 200, the normalized bubble velocity increases, although not significantly. In the presence of the surfactant, there is very little change in shape of the bubble (form drag is reduced slightly compared to an equivalent clean bubble) at this Reynolds number (figure 9). The inertial forces dominate the Marangoni forces resulting in increased speed. For Re = 2 and *λ* < 1.0, the inertial forces are of the same order as viscous forces, and the Maragoni force slows down the bubble. But with increasing *λ*, the surfactant effect is such as to cause an elongation of the bubble in the axial direction. This increases the speed of the bubble and for *λ* > 1.0, the bubble moves faster than a corresponding clean bubble, although this increase in speed is small. For Re = 0.2 and *λ* < 0.97, viscous forces dominate. Together with Maragoni forces there is a reduction in the bubble speed. However beyond *λ* ~ 0.97, bubble elongation causes the bubble to speed up relative to a clean bubble. But, for *λ* > 1.0 there are competing effects between the shape change at the rear of the bubble (bulging) and the elongation in the axial direction. Around *λ* > 1.02 the shape change at the rear is responsible for the slowing down of the bubble compared to a clean bubble.

An important quantity of interest is the time variation of the shear stress *τ* at any given point on the vessel wall as the bubble approaches, moves over this point and departs. As the bubble moves, the entrained fluid surrounding it causes rapid changes in the magnitude and the sign of the shear stresses experienced by the wall. For a given point on the wall, the changes in shear stress manifest as a solitary wave (see Mukundakrishnan *et al.* (2008a) for discussions related to a clean bubble). Briefly, the shear stress variation occurs as follows. When the bubble is far upstream from the point under consideration, it has negligible effect on this point and the shear stress is at its basal value. As the bubble moves closer to the point, the local fluid velocity is increased. Both the local pressure gradient and the flow rate at the location of this point attain higher values and there is a corresponding increase in the shear stress. As the bubble moves over the point, the flow decelerates due to entrainment and the shear stress gradually reduces to zero. The local pressure gradient reverses from being favorable to adverse during this period. As the maximum bulge in the bubble passes over the point, a local flow reversal occurs due to the presence of the recirculating vortex (see figure 6). This vortex arises due to the drainage flow interacting with the forward flow. These interactive effects are ascribable due to the finite size of the bubble and the resultant physical interplay of the trailing gas-liquid interface and the presence of the wall. In particular, the sign of the shear stress becomes negative. Beyond this, as the maximum bulge of the bubble departs the location, there is fluid flow recovery and a significant increase in local fluid velocity in the direction of motion of the bubble. This causes the shear stress to become positive and attain a value higher than its basal value. As the bubble moves farther away from the point, the entrainment effects become progressively weaker and the shear stress gradually returns to the basal value staying there for the remainder of the motion of the bubble. This solitary wave has major potential physiological implications including induction of a complex pattern of endothelial cell membrane stretch and compression, activation of mechanotransduction pathways, loss of plasma membrane integrity and plasma membrane stress failure (Vlahakis & Hubmayr, 2000; Butler *et al.*, 2001; Huang *et al.*, 2004; McNeil & Steinhardt, 1997; Mendez *et al.*, 2004).

A comparison of the shear stresses experienced by a point on the vessel wall, for clean and surfactant-coated bubble motions, is shown in figure 12. The shear stress is normalized with the corresponding base value, *τ _{ref}*. As seen in figure 6, the recirculating vortex is significantly weakened by the presence of the surfactant. As a result the variations in shear stresses, although following a behavior similar to that for a clean bubble, are considerably lowered in magnitude. In figure 12, only the duration of time where there is significant deviation in the stress from its basal value is displayed, and this period corresponds to the motion of the bubble over the point. The effect of the surfactant is more pronounced at the lower two values of Reynolds numbers (0.2 and 2), while for Re=200, the profile of the solitary wave does not change much from that for a corresponding clean bubble. This can be explained by noting that a decrease in Reynolds number corresponds to a decrease in the surface mobility of the bubble. In turn, the recirculating vortex that is primarily responsible for causing the stress reversals is also weaker. Furthermore, as previously discussed, for

Variation of shear stress at a point on the vessel wall as the bubble passes over it. The normalization of the shear stress is done with the corresponding basal value. λ is 1.0.

In figure 13, the jump in the value of the shear stress, Δ*τ* = *τ _{max}* −

Thus far we have reported on the motion of a bubble in a vertical tube. In this section a comparison is provided with the results for a bubble moving under identical conditions in a horizontal tube. The representative Reynolds number is take to be 2 and the aspect ratio *λ* = 1.0. The comparisons for the shear stress level and the interface concentration are shown in figure 14.

Comparison of the shear stress level and interfacial surfactant concentration profiles between a horizontal and vertical tube. The flow Reynolds number is 2.0 and the aspect ratio is 1.0. The Froude number is ~0.71.

The results for the steady velocity of the bubble, the flow patterns and shear stress levels from the horizontal configuration are nearly identical to those for the vertical geometry. This confirms that at the levels of pressure drops considered for the forced flow, buoyancy effects are negligible in this problem.

The properties of the surfactant stated in table 1 have been chosen from a broad range of ethoxyl group surfactants given in Chang & Franses (1995). In order to demonstrate the validity of our results with a different surfactant, we now simulate the motion of a bubble in the presence of hexaethylene glycol dodecyl ether (C_{12}E_{6}). From Palaparthi *et al.* (2006), we have obtained the relevant parameter values for the surfactant C_{12}E_{6} and carried out the simulation for Re=2. All surfactant properties, except for the diffusivity (which will be higher in blood than in glycerol-water mixture), are assumed to be identical to those obtained for the 70:30 glycerol-water mixture listed therein. In this simulation, C_{0} is 0.2 mol/m^{3} (this is assumed to be less than the critical micelle concentration, reported as 0.209 mol/m^{3} in Palaparthi *et al.* (2006)), Γ_{∞} is 5 × 10^{−6} mol/m^{2}, both *D _{l}* and

$$\sigma ={\sigma}_{s}+\Re T{\mathrm{\Gamma}}_{\infty}\left(ln\left(1-\frac{\mathrm{\Gamma}}{{\mathrm{\Gamma}}_{\infty}}\right)+\frac{\xi}{2}{\left(\frac{\mathrm{\Gamma}}{{\mathrm{\Gamma}}_{\infty}}\right)}^{2}\right),$$

(31)

where *ξ* is the Frumkin interaction parameter and is set as 5.47 in our simulation (Palaparthi *et al.*, 2006). A comparison of the shear stress experienced by a point on the vessel wall, with and without the presence of the surfactant C_{12}E_{6}, is shown in figure 15.

Variation of shear stress at a point on the vessel wall as the bubble passes over it with and without the presence of the surfactant C_{12}E_{6}. The normalization of the shear stress is done with the corresponding basal value. λ = 1.0 and Re = 2.

The characteristics displayed in displayed in figure 15 show that the predictions follow the same trends that are observed with other surfactants, although there is a quantitative difference (compared with figure 12(b)). This gives us confidence in the validity of our results.

Thus far, we have examined bubble motion at a given far-field bulk surfactant concentration level. This has helped us to delineate the role of the surfactant on the bubble motion and to compare the results with an equivalent clean bubble case. The comparisons have shown that the presence of the surfactant helps in lowering the shear stress level imparted to the endothelial cell surface and prevent or minimize injury. However it is important to examine the behavior of differing bulk concentration levels of the surfactant in order to identify the most suitable concentration for application purposes. Towards this objective, we are embarked upon a systematic study of various concentration levels in the bulk phase. Herein we provide an illustrative result in figure 16 that shows the variation of the difference between the maximum and minimum shear stresses experienced by a point on the vessel wall for the three different Reynolds numbers, as a function of the dimensionless adsorption depths *α*.

Variation of the difference between the maximum and minimum shear stress experienced by a point on the vessel wall for different values of the dimensionless adsorption depths. The normalization of the shear stress is done with the corresponding base value. **...**

At a given Re, an increase in *α* denotes a decrease in *C*_{0}, the bulk concentration of the surfactant. Δ*τ* is noted to decrease with increasing *C*_{0}, and this indicates that the shear stress levels are increasingly attenuated at larger *C*_{0} values. The recirculation vortex strengths decrease with *C*_{0}. Therefore higher levels of *C*_{0} would be preferable from cell injury view point although other effects such as toxicity may have to be considered. In Swaminathan *et al.* (2009), a more detailed study will be presented.

A numerical method for the evaluation of the axisymmetric motion of a finite sized, nearly occluding gas bubble in a blood vessel containing a soluble surfactant in the bulk medium has been described and implemented. Convective and diffusive mechanisms, both in the bulk medium and on the interface, have been considered. The surfactant adsorbs on to and desorbs from the interface as governed by interfacial thermodynamics. The effect of the presence of the surfactant in the bulk medium on the motion of the bubble has been systematically examined by comparing the characteristics of the results with those for an equivalent surfactant-free clean bubble motion. This comparison allows for a better understanding of exogenous surfactant-based intervention of gas embolism. The main observations include:

- Multiple intravital microscopy studies (Branger & Eckmann, 2002, 1999; Eckmann & Lomivorotov, 2003; Eckmann
*et al.*, 2005) have shown that microvascular gas embolism bubbles deposit with an elongated sausage-shaped geometry (i.e.,*λ*> 1) into blood vessels. We have shown that all the three surface forces are significant with this configuration, especially for clean bubbles close to the wall. A major resultant effect is the propagation of a shear stress solitary wave along the wall caused by the bubble, bulk fluid and interfacial motion. In vivo, this spatially and temporally varying shear stress wave is likely responsible for having induced the injurious or lethal membrane stretch of endothelial cells lining the vasculature observed during in vivo experiments of gas embolism (Eckmann & Armstead, 2006) and is consistent with other membrane stretch mechanisms of cell death (Liu*et al.*, 2002; McKinney*et al.*, 1996). We have herein demonstrated that the presence of a surfactant dramatically and beneficially alters the shear stress, yielding an important cellular mechanoprotective influence that may have a substantial clinical role in the prevention of gas embolism related tissue injury, a significant problem in cardiac surgery and decompression sickness. - The surfactant is shown to opportunistically adsorb onto the gas-liquid interface. This modifies the interfacial tension resulting in a change of the bubble shape and its residence time adjacent to the cell surface. Residence time of the bubble adjacent to an endothelial cell is a critical quantity in gas embolism. Experimental studies have shown that this time determines cell response.
- The pressure driven flow causes the bubble to deform with a bulge at the rear. A drainage flow squeezes through the narrow gap between the bubble and the wall. This flow is primarily responsible for the surface mobility of the bubble and vortical motions. We have shown that the strengths of the recirculation and of the vortical motions are reduced by the presence of the surfactant.
- For a given bulk concentration level of the surfactant chosen here, the effect of the surfactant is shown to be most prominent in the small arteriole, followed by the large arteriole and is least for the small artery.

This work was sponsored by the Office of Naval Research Grant No. N00014-08-1-0436. NIH Grant No. R01-HL067986 and NASA Grant No. N00014-08-1-0436. The authors also thank one of the reviewers for helpful insights.

The first validation involves the role of the diffusive terms in the surfactant concentration equation at the interface (equation 9). The algorithm is tested for a stationary spherical bubble with only the interface diffusion term enabled. Initially, a prescribed variation in the distribution of the interface concentration given by

$$\mathrm{\Gamma}(\theta ,t)=\frac{1}{2}(1-cos\theta )$$

(32)

is assumed. This surfactant is insoluble. The bubble is assumed to be in an infinite medium (bounding walls are set to be very far away from the bubble). The interface has no additional source term. The interface concentration is then evaluated as a function of time. The number of grid points along the interface and time step restrictions are chosen as in Zhang *et al.* (2006). An analytical solution to this problem is given in James & Lowengrub (2004) as

$$\mathrm{\Gamma}(\theta ,t)=\frac{1}{2}\left(1-cos\theta exp\phantom{\rule{0.2em}{0ex}}\left[\frac{-2t\ast}{P{e}_{s}}\right]\right)$$

(33)

where, for this comparison we define *Pe _{s}* =

The next validation considers an initially clean bubble of radius *a*, placed in a stationary medium with a uniform soluble surfactant concentration *C*_{0}. The interface concentration Γ as a function of time *t* for a given diffusion coefficient *D _{s}* is calculated. The analytical solution to this problem is given in Muradoglu & Tryggvason (2008) as,

$$\mathrm{\Gamma}={k}_{a}{C}_{0}\phantom{\rule{0.2em}{0ex}}\left(t-\frac{\omega h}{{\eta}^{3}}\left({\eta}^{2}t-2\eta \sqrt{t}+2ln\phantom{\rule{0.2em}{0ex}}\left(1+\eta \sqrt{t}\right)\right)\right),$$

(34)

where *k _{a}* is the adsorption coefficient,

Interface concentration as a function of time for an initially clean bubble placed in a bulk medium with uniform initial surfactant concentration. The analytical result is from Muradoglu & Tryggvason (2008).

Variation of bulk concentration as a function of distance from the center of the particle and the percentage error compared compared to the analytical results. The analytical result is from Muradoglu & Tryggvason (2008). The six non-dimensional **...**

To test for the Marangoni effect, we consider a bubble that is initially covered with a non-uniformly distributed insoluble surfactant with an interface concentration Γ given by,

$$\mathrm{\Gamma}={\mathrm{\Gamma}}_{\infty}z/L,$$

(35)

where z is the axial coordinate, Γ_{∞} is a reference concentration, and *L* is the length of the tube. The bubble has a radius *a* and is placed axi-symmetrically in the tube of radius *R* = 5*a* and length *L* = 15*a*. Initially, the bubble and the bulk medium are quiescent. The motion of the bubble is therefore solely due to Marangoni stresses. This example is derived from the thermocapillary migration study of a viscous drop by Young *et al.* (1959). A linear equation of state, relating the surface tension *σ* with Γ, Γ_{∞}, the surface tension of the clean interface *σ _{s}* and an elasticity number

$$\sigma ={\sigma}_{s}\phantom{\rule{0.2em}{0ex}}\left(1-{\beta}_{s}\frac{\mathrm{\Gamma}}{{\mathrm{\Gamma}}_{\infty}}\right),$$

(36)

is employed. The steady terminal velocity of the drop is given by Young *et al.* (1959) as:

$${V}_{\mathit{\text{YGB}}}=\frac{2{\sigma}_{s}{\beta}_{s}a}{L(6{\mu}_{0}+9{\mu}_{d})}.$$

(37)

With *ρ*_{0} = *ρ _{d}* = 0.2,

The results for an axisymmetric bubble rising in a cylindrical tube, in the presence of a surfactant, due to buoyancy has been computed by Muradoglu & Tryggvason (2008). Here we show a comparison between our computations and their results. The values for the various parameters used in this comparison are derived from figure 21 in Muradoglu & Tryggvason (2008) and are: *ρ _{g}/ρ_{l}* = 0.1,

In equation 7, the expressions for *μ*_{∞} and *τ _{y}* as functions of core hematocrit

$${\mu}_{\infty}=\frac{{\mu}_{p}}{{(1-{H}_{C})}^{{C}_{1}}};{\tau}_{y}^{1/2}={C}_{2}\phantom{\rule{0.2em}{0ex}}\left[{\left(\frac{1}{1-{H}_{C}}\right)}^{{C}_{1}/2}-1\right].$$

(38)

Here, *C*_{1}, *C*_{2} are constants whose values for human blood are given as 2.0 and 0.3315, respectively. The evaluations of the equations require a knowledge of *H _{C}, δ*, and

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