Figure 1. shows the type of domain that was contemplated. In this simplified
example it is composed from one inlet
and two outlets.
The boundary condition on the natural boundary
models that no compound can diffuse through it, while on the inlet and
on the outlet a constant profile is prescribed.
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 by
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.
For the calculation water was used as a solvent, hence =0.01 cm2/s and density 1 g/cm3. The diffusion and barodiffusion constants were chosen to be =0.01 cm2/s and =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 / = 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.
Figure 2.: Velocity componentf of the flow in a 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.
Figure 4.: Velocity componentf of the flow in an unsymmetrical
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.