Question 9.4: The so-called skin friction coefficient cf is defined as the...

The dynamic pressure is defined as \rho \overline{\nu }^{2}/2, where \overline{\nu } is a scalar measure of the mean velocity [i.e., the speed; see Bernoulli’s equation (8.80)]. For the case of a tube flow, therefore, the mean dynamic pressure is Some Exact Solutions

where the mean velocity is given by Eq. (9.50). Hence, in this case (Fung 1993),

                  \overline{\nu }=\frac{Q}{A}=\left(-\frac{\pi a^{4}}{8\mu }\frac{dp}{dz} \right)\left(\frac{1}{\pi a^{2}} \right)=-\frac{a^{2}}{8\mu }\frac{dp}{dz}.        (9.50)

which, via Eq. (9.54), can be written as

\frac{dp}{dz}=-\frac{8\mu Q}{\pi a^{4}}     or    \frac{dp}{dz}=-\frac{8\mu \overline{\nu } }{a^{2}};            (9.54)

where the Reynolds’ number is

with d=2a the diameter of the tube. Re is a very important nondimensional parameter in fluid mechanics; it will be discussed in greater depth in Chap. 10. In summary, the wall shear stress in this tube flow can also be written as

For example, in the human aorta,

\overline{\nu }=0.15  m/s,            d=0.03  m
\rho =1,060   kg/m^{3},           \mu =3.3\times 10^{-3}  Ns/m^{2};

hence,

where 1   kg   m/s^{2}=1 N. This value is less than that expected for turbulent flow (Re\gt 2,100); hence, our laminar assumption applies. Note, too, that the asso-ciated value of \tau _{w}=0.13  kg/m  s^{2}=0.13   Pa, which is within the reported range albeit lower than that which is usually reported (1.5 Pa).