Effects of Barodiffusion on the Transportation of a Compound near a Bifurcation

Andrés Chavarría
Research Center Juelich,Germany
a.chavarria@fz-juelich.de

 

Introduction

A normal transport/diffusion equation has a constant solution for constant Dirichlet-Boundary conditions. Still so measurements show a non constant solution near e.g. a bifurcation. This has as a consequence a cleaning effect on the sovent, specially when the bifurcation is not symmetrical. The normal transport/diffusion equation is thus not enough to describe the concentration near such a bifurcation. It turns out that barodiffusion, a type of diffusion which happens when there is a gradient of pressure, could be the driving force of such cleaning effects. This sort of diffusion is normally neglected because its effects are small, but specially when the flow is within capillaries, vessels which have a small ratio of thickness to length, it becomes of importance.
 

Equations

The problem at hand is described by the flow equations, normally a simplification of the Navier-Stokes equations, and a modified transport/diffusion equation containing a barodiffusion contribution. Only one compound, which was assumed to have a small concentration, was considered.

Figure 1. shows the type of domain that was contemplated. In this simplified example it is composed from one inlet $\Gamma_{in}$ and two outlets$\Gamma_{out}$. The boundary condition on the natural boundary $\Gamma_{sw}$ models that no compound can diffuse through it, while on the inlet and on the outlet a constant profile is prescribed.
Type of domain and its boundaries

Figure 1.: Type of domain considered and its corresponding boundaries. The natural boundary is indexed with "sw", while the inlet and the outlets are indexed with "in" and "out".
 

By using the smallness of the concentration and assuming that the variations are also small, the barodiffusion contribution reduces to a source term proportional two the laplacian of the pressure. Hence only the non-harmonic portion of the pressure produces barodiffusion. Also an incompressible flow was assumed. This is cannot completely describe correctly the problem, becuase the density depends on the concentration and hence on the space coordinates. Still so, becuase the concentration variations are taken to be small, the aberration from the incompressibility is small.
As a whole, the equation system describing the problem is given byEquations

whereas:

$\Omega$ is the domain, $\Gamma_{sw}$ the natural boundary, $\Gamma_{in}$ and$\Gamma_{out}$ the inlets and outlets, and $\mathbf{n}$ is outer normal vector of the corresponding boundary;
$\mathbf{v}$ is the velocity, $p$ the pressure, and $c$ the concentration;
$\mathbf{h}$ is a Poiseuille-profile, and $h_c$ a constant;
$\nu$ is the kinematic viscosity, $D$ the diffusion constant, and $\kappa$ the barodiffusion constant.


As known the Poisseuille solution of the flow in a pipe has a harmonic pressure, hence a non-curved and non-branching pipe will show no barodiffusion. The transport term in the Navier-Stokes equation is responsible for the non-harmonic fraction of the pressure (to see this build the divergence of Navier-Stokes and take the incompressibility into account). Therefor at locations where the transport term is of importance e.g. at bifurcations, the biggest impact of barodiffusion should be found. This can be seen very well in the numerical simulations.
 

Numerical Simulation

The above equations were solved numerical using the toolkit Gascoigne for two simplified domains. On account of memory usage and difficulty of modeling, the domains are two dimensional. One of them represents a symmetrical bifurcation, while the other an unsymmetrical one.

For the calculation water was used as a solvent, hence $\nu$=0.01 cm2/s and density 1 g/cm3. The diffusion and barodiffusion constants were chosen to be $D$ =0.01 cm2/s and $\kappa$ =10 - 4 cm3 s/g. The ratio thickness to lentgth was chosen to be 0.1, which controls the importance of the transport term in the compound equation. A small ratio means that the transport term will have a small effect. The barodiffusion and diffusion constants were chosen to be very big, but of importance is the ratio between them $\kappa$/$D$ = 0.01, which has a fairly normal magnitude.

Figure 2. shows the velocity components of the flow near a symmetrical bifurcation, while Figure 3. shows the pressure and concentration. Only the solutions near the bifurcation are shown. To enhance the effects the Poiseuille-Solution of the pressure was substracted, i.e. the non-harmonic fraction is shown. The concentration is normalized (this means 0<=c<=1). The effects produced by the barodiffusion are evident. Both bifurcating branches get the same amount of compound. A dilution right at the bifurcation is found, as was expected because at such a location the transport term of the Navier-Stokes equation has a high impact.

Figure 4. and 5. show instead the velocity components, pressure and concentration for an unsymmetrical bifurcation. As in the symmetric case dilution is found right at the bifurcation, but also accumulation where the small branch starts. Of special interest is that the smaller branch gets less compound than the other. It is here evident that a cleaning effect of the solvent appears.
velocity components for symmetrical bifurcation

Figure 2.: Velocity componentf of the flow in a symmetrical bifurcation.
 
 
 

pressure and concentration at symmetrical bifurcation

Figure 3.: Pressure and concentration in a symmetrical bifurcation. Only the non-harmonic fraction of the pressure is shown. Locations with dilution are clearly found.
 
 
 

velocity components for unsymmetrical bifurcation

Figure 4.: Velocity componentf of the flow in an unsymmetrical bifurcation.
 
 
 

pressure and concentration for unsymmetrical bifurcation

Figure 5.: Pressure and concentration in an unsymmetrical bifurcation. like in the symmetrical one, only the non-harmonic fraction of the pressure is shown and locations with dilution are present. Also location with accumulation are found and the small branch gets less compound than the other one, hence a cleaning effect is present.
 

Literature

Literature