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Laminar flow

When a fluid flows through a tube, e.g. a catheter, the fluid velocity will be higher in the  center of the tube than along the wall. This is due to friction. Assuming there is only an axial movement along the tube the fluid can be modeled as a number of thin, concentric layers of fluid, where the outermost has a friction against the wall, the second outermost has a friction against the outermost layer etc. Layers can be called lamina, and this type of flow is called laminar flow.

The velocity profile with laminar flow can be proven to be quadratic using Poiseuille's law. A useful expression for the velocity profile is:


(Equation 1)


Note that if velocity is time-invariant a simple multiplication with a (constant) time will give an expression for the travelled distance for a particle distance r from the center of the tube, and assuming constant tube radius (R) a multiplication with the tube cross section will give the same expression for volume.


Implications for infusion and sampling

Now, assume the tube is filled with solvent (e.g. water) and at one end of the tube there is a solution with concentration C. The solution is infused into the tube (or withdrawn in the sampling case) causing the concentrated solution form a paraboloid in the solvent. This behaviour can be easily shown with a piece of tube, one transparent and one colored solution. A hint is to use a viscous fluid like thick syrup.

A cross section in the tube will be two coaxial circles - a concentrated in the center and an outer with zero concentration. The average concentration in a cross section can be computed as the fraction of the inner circle divided by the total cross section, i.e.

(Equation 2).


Inserted in eq. 1 with velocity replaced by volume and some rearranging leads to:

thus average concentration decreases linearly with (volumetric) distance in tube. Obviously Vmax = 2V, where V is the volume of solution (B) infused into the solvent (A) filled tube:

Now where does this lead us?

When we infuse or withdraw a solution into a tube, the concentration is 100% at the source, 50% at the volume pumped and 0% at twice the volume pumped. Opposite: with a catheter of 100 l:


  • After pumping 50 l the solution reaches the far end, starting at 0% increasing.
  • After 100 l there will be 50% at the far end
  • After 200 l there will be 75% at the far end
  • There will (theoretically) never be 100% solution at the far end of the catheter.


The following diagram describes the effect (V = Volume pumped, L = volumetric length of catheter):

Effects of radial transport

Pure laminar flow deals only with axial flow. If there is radial mixing between the solution and the solvent the tip of the paraboloid will be smeared out across the cross section, and although the fluid moves with the same speed as in laminar flow the concentration will be lower at the edge. The effect is that the transition from unconcentrated to concentrated solution becomes shorter/steeper.

For low fluid velocities and thin tubes diffusion will contribute to the radial transport. The subject is well covered in the article Dispersion of soluble matter..., G.I. Taylor 1953. Not free unfortunately.

At high velocities the fluid will be turbulent as opposed to laminar, i.e. the flow contains vortexes mixing the fluid radially. The effect will be similar to the difusion case but harder to analyze mathematically. Taylor wrote another article on the turbulent case: Dispersion of Matter in Turbulent flow.... For catheters the speed will rarely be high enough to create a turbulent flow.

In effect, instead of a linear concentration decay the concentration will form an erf-shaped transition along the tube. Here is a diagram for a 0.4 mm catheter at various speeds:

Water self diffusion coefficient is used. For larger molecules or more viscous media, coefficient of diffusion will be lower and the gradient more linear.

Note that the above diagram does not asymptotically resemble the laminar flow case. This is due to Taylors assumption that the length of te tube is much longer than the transition length.

Note also that for very low flow there will be an axial diffusion component, so there is a limit to how steep the transition can be. For infusion/sampling situations this has no practical meaning to my knowledge.


How to use this

Theory is reciprocal so the same model applies indpendent on whether we withdraw a sample or infuse a drug. The model does not consider fluids with different viscosity like blood pumped into a saline-filled catheter (blood has 3-4 times higher viscosity than saline), but from practical experiments we still have reasonably good match between theory and practice. Mixing entirely different fluids like glycerol and water doesn't look like laminar flow at all (viscosity differ by a factor 10000).

Typically we use low flows for continuous infusion into rodents and for classical microdialysis. In these cases we can estimate the concentration with a steep increase at the pumped volume.

For larger animals we may need to consider laminar flow model for continuous infusion. Wider catheters and higher flows reduce the radial mixing making the gradient slower.

For bolus dosage and faster sampling, e.g. blood sampling laminar flow model must be considered even for rodents.

Laminar flow with Taylor-dispersion (due to diffusion) is tricky since we often don't know the properties of the solution, i.e. coefficient of diffusion. Blod for example is a mixture of small molecules like NaCl, fatty acids, proteins and blood cells, each with their properties. Heavy particles like erythrocytes are also affected by gravity which isn't accounted for in the model.

On the positive side, the laminar flow model without diffusion describes the worst case scenario, and is very easy to calculate. For a better model start with an experiment and use the concentration achieved there.

Finally I would like to present two diagrams for bolus infusion (no calculations here) based on the pure laminar model. B is the bolus volume, L is the catheter volumetric length and V is the flush volume:

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