Question 13.3: Imagine that we are making a model of blood flow through an ...
Imagine that we are making a model of blood flow through an artery; however, we could only use water as our fluid flowing through the model artery. Calculate what the angular frequency of the pulsatile waveform and the initial inlet velocity should be if the characteristic length (diameter) of the blood vessel is 10 cm, the heartbeat is 72 beats/min (angular frequency is 5.24 rad/sec), and the inlet velocity is 50 cm/s.
Using the Reynolds number, we can calculate the initial inlet velocity of the model:
Re_{blood}= Re_{model}
\frac{\rho_{b} \upsilon_{b} d_{b}}{\mu_{b}} = \frac{\rho_{m} \upsilon_{m} d_{m}}{\mu_{m}}
\upsilon_{m}= \frac{\rho_{b} \upsilon_{b} d_{b} \mu_{m}}{\mu_{b} \rho_{m} d_{m}}= \frac{1050 kg/m^{3}(50 cm/s)(10 cm)(1 cP)}{1000 kg/m^{3}(10 cm)(3.5 cP)}= 15 cm/s
Using the Womersley number, we can calculate the angular frequency of the model:
\alpha_{blood}= \alpha_{model}
d_{b} \left(\frac{\omega_{b}}{\nu_{b}}\right)^{1/2}= d_{m} \left(\frac{\omega_{m}}{\nu_{m}}\right)^{1/2}
d_{b} \left(\frac{\rho_{b} \omega_{b}}{\mu_{b}}\right)^{1/2}= d_{m} \left(\frac{\rho_{m} \omega_{m}}{\mu_{m}}\right)^{1/2}
\omega_{m}= \frac{\mu_{m}}{\rho_{m}}\left[\frac{d_{b}}{d_{m}}\left(\frac{\rho_{b} \omega_{b}}{\mu_{b}} \right)^{1/2} \right]^{2} = \frac{1 cP}{1000 kg/m^{3}} \left[\left(\frac{10 cm}{10 cm} \right) \left(\frac{(1050 kg/m^{3}) (72 beats/min)}{3.5 cP} \right)^{1/2} \right]^{2}
= 21.6 beats/min