Governing equations

Consider steady flow of blood through a capillary of radius ${\displaystyle R}$. The capillary cross section can be divided into a core region and cell-free plasma region near the wall. The governing equations for both regions can be given by the following equations:^[2]

${\displaystyle {\frac {-\Delta P}{L}}={\frac {1}{r}}{\frac {d}{dr}}(\mu _{c}r{\frac {du_{c}}{dr}});}$ ${\displaystyle 0\leq r\ \leq R-\delta \,}$

${\displaystyle {\frac {-\Delta P}{L}}={\frac {1}{r}}{\frac {d}{dr}}(\mu _{p}r{\frac {du_{p}}{dr}});}$ ${\displaystyle R-\delta \leq r\ \leq R\ \,}$

where:

${\displaystyle \Delta P}$ is the pressure drop across the capillary

${\displaystyle L}$ is the length of capillary

${\displaystyle u_{c}}$ is velocity in core region

${\displaystyle u_{p}}$ is velocity of plasma in cell-free region

${\displaystyle \mu _{c}}$ is viscosity in core region

${\displaystyle \mu _{p}}$ is viscosity of plasma in cell-free region

${\displaystyle \delta }$ is the cell-free plasma layer thickness

Boundary conditions

The boundary conditions to obtain the solution for the two differential equations presented above are that the velocity gradient is zero in the tube center, no slip occurs at the tube wall and the velocity and the shear stress are continuous at the interface between the two zones. These boundary conditions can be expressed mathematically as:

• ${\displaystyle \left.{\frac {du_{c}}{dr}}\right|_{r=0}=0}$
• ${\displaystyle \left.u_{p}\right|_{r=R}=0}$
• ${\displaystyle \left.u_{p}\right|_{r=R-\delta }=\left.u_{c}\right|_{r=R-\delta }}$
• ${\displaystyle \left.\tau _{p}\right|_{r=R-\delta }=\left.\tau _{c}\right|_{r=R-\delta }}$

Velocity profiles

Integrating governing equations with respect to r and applying the above discussed boundary conditions will result in:

${\displaystyle u_{c}={\frac {\Delta PR^{2}}{4\mu _{p}L}}[1-({\frac {R-\delta }{R}})^{2}-{\frac {\mu _{p}}{\mu _{c}}}({\frac {r}{R}})^{2}+{\frac {\mu _{p}}{\mu _{c}}}({\frac {R-\delta }{R}})^{2}]}$

${\displaystyle u_{p}={\frac {\Delta PR^{2}}{4\mu _{p}L}}[1-({\frac {r}{R}})^{2}]}$

Volumetric flow rate for cell-free and core regions

${\displaystyle Q_{p}=\int \limits _{R-\delta }^{R}2\pi *u_{p}rdr={\frac {\pi \Delta P}{8\mu _{p}L}}(R^{2}-(R-\delta )^{2})^{2}}$

${\displaystyle Q_{c}=\int \limits _{0}^{R-\delta }2\pi *u_{c}rdr={\frac {\pi \Delta P*(R-\delta )^{2}}{8L}}[{\frac {(R-\delta )^{2}}{\mu _{c}}}+{\frac {2(R^{2}-(R-\delta )^{2})}{8\mu _{p}}}]}$

Total volumetric flow rate is the algebraic sum of the flow rates in core and plasma region. The expression for the total volumetric flow rate can be written as:

${\displaystyle Q=Q_{c}+Q_{p}={\frac {\pi \Delta PR^{4}}{8\mu _{p}L}}[1-(1-{\frac {\delta }{R}})^{4}(1-{\frac {\mu _{p}}{\mu _{c}}})]}$

Comparison with the viscosity which applies in the Poiseuille flow yields effective viscosity, ${\displaystyle \mu _{e}}$ as:

${\displaystyle \mu _{e}={\frac {\mu _{p}}{[1-(1-{\frac {\delta }{R}})^{4}(1-{\frac {\mu _{p}}{\mu _{c}}})]}}}$

It can be realized when the radius of the blood vessel is much larger than the thickness of the cell-free plasma layer, the effective viscosity is equal to bulk blood viscosity ${\displaystyle \mu _{c}}$ at high shear rates (Newtonian fluid).

Relation between hematocrit and apparent/effective viscosity

Conservation of Mass Requires:

${\displaystyle QH_{D}=Q_{c}H_{c}}$

${\displaystyle {\frac {H_{T}}{H_{C}}}=\sigma ^{2}}$

${\displaystyle H_{T}}$ = Average Red Blood Cell (RBC) volume fraction in small capillary

${\displaystyle H_{D}}$= Average RBC volume fraction in the core layer

${\displaystyle {\frac {H_{T}}{H_{D}}}={\frac {Q}{Q_{c}}}\sigma ^{2}}$, ${\displaystyle \sigma ={\frac {R-\delta }{R}}}$

${\displaystyle u_{e}={\frac {\pi \Delta PR^{4}}{8Q}}}$

${\displaystyle {\frac {u_{p}}{u_{e}}}=1+\sigma ^{4}[{\frac {u_{a}}{u_{c}}}-1]}$

Blood viscosity as a fraction of hematocrit:

${\displaystyle {\frac {u_{e}}{u}}=1-\alpha H}$