Question

Poiseuille's law remains valid as long as the fluid flow is laminar. For sufficiently high speed, however, the flow becomes turbulent, even if the fluid is moving through a smooth pipe with no restrictions. It is found experimentally that the flow is laminar as long as the Reynolds Number Re is less than about 2000: Re = 2v Normal 0 false false false IN X-NONE X-NONE MicrosoftInternetExplorer4 rhoR Normal 0 false false false IN X-NONE X-NONE MicrosoftInternetExplorer4 /η. Here Normal 0 false false false IN X-NONE X-NONE MicrosoftInternetExplorer4 v, rho, and η are, respectively, the average speed, density, and viscosity of fluid, and R is the radius of the pipe. Calculate the highest average speed that blood (rho = 1060 kg/m3, η = 4.0 x 10-3 Pa.s) could have and still remain in laminar flow when it flows through the aorta (R = 8.0 x 10-3 m)


Normal 0 false false false IN X-NONE X-NONE MicrosoftInternetExplorer4

Answer 1

Answer: 0.471 m/s

Explanation:

We are given the followin equation:

$Re=\frac{D v \rho}{\eta}$ (1)

Where:

$Re$ is the Reynolds Number, which is adimensional and indicates if the flow is laminar or turbulent

When $Re<2100$ we have a laminar flow

When $Re>4000$ we have a turbulent flow

When $2100<Re<4000$ the flow is in the transition region

$D=2R$ is the diameter of the pipe. If the pipe ha a radius $R=8(10)^{-3} m$ its diameter is $D=2(8(10)^{-3} m)=0.016 m$

$v$ is the average speed of the fluid

$\rho=1060 kg/m^{3}$ is the density of the fluid

$\eta=4(10)^{-3} Pa.s$ is the viscosity of the fluid

Isolating $v$:

$v=\frac{Re \eta}{D \rho}$ (2)

Solving for $Re=2000$

$v=\frac{(2000)(4(10)^{-3} Pa.s)}{(0.016 m)(1060 kg/m^{3})}$ (3)

Finally:

$v=0.471 m/s$