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

Step-by-step explanation:

We are given the followin equation:

`Re=(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=(Re \eta)/(D \rho)` (2)

Solving for
`Re=2000`

`v=((2000)(4(10)^(-3) Pa.s))/((0.016 m)(1060 kg/m^(3)))` (3)

Finally:

`v=0.471 m/s`