Flow through a systemic capillary diameter D 8 mu m is driv

Flow through a systemic capillary (diameter D = 8 mu m) is driven by pressure gradient - partial differential P/partial differential z = 7000Pa/cm. The capillary is cylindrical. Blood has viscosity mu = 3.2 centipoise (1 cP = 0.001 Pa-s). There is no need to re-derive the Poiseuille equation developed in class. a. What is the flow rate Q (m^3/s) through the capillary? b. What is the resistance R to flow through the vessel (Pa-s/m^4)? c. What shear stress tau_W does blood apply to the vessel wall (Pa)?

Solution

Answer a.

Flow rate Q through capillary as per Hagen–Poiseuille equation is = ×R⁴/8µ×p/z

Here, R = 4 × 10^-6 m

        µ = 3.2× 0.001 = 3.2 × 10^-3 Pa-s

        p/z = 7000 pa/cm = 700000 pa/m

       Flow rate (Q) = ×(4 × 10^-6)⁴/[8×3.2 × 10^-3]×700000

                           = 2.199 × 10^-15 m³/s (Answer)

Answer b.

Resistance to flow through the vessel = µL/R⁴           

Here, µ = 3.2 × 10^-3 Pa-s

Length of vessel (L) = Not given

R= 4 × 10^-6 m

Resistance= 3.2 × 10^-3 × L/ (4 × 10^-6)⁴

               = 1.25 × 10¹⁹ × L (Answer)

Answer c.

Shear stress applied by blood on the vessel = 32 µ×Q/ ×d³

                   Here, µ = 3.2 × 10^-3 Pa-s

                             Q = 2.199 × 10^-15 m³/s

                             d= 8 µm = 8×10^-6 m

        Shear stress applied by blood on the vessel = 32×3.2 × 10^-3 × 2.199 × 10^-15 / (8×10^-6 )³          

                                                                              = 28.15 Pa (Answer)