For a fully developed laminar flow of water (dynamic viscosity 0.001 Pa-s) through a pipe of radius 5 cm. the axial pressure gradient is —10 Pa/m. The magnitude or axial velocity (in m/s) at a radial location of 0.2 cm is::
Question
For a fully developed laminar flow of water (dynamic viscosity 0.001 Pa-s) through a pipe of radius 5 cm. the axial pressure gradient is —10 Pa/m. The magnitude or axial velocity (in m/s) at a radial location of 0.2 cm is::
Solution
The velocity profile for a fully developed laminar flow in a pipe can be described by the Hagen-Poiseuille equation:
v = (ΔP/4μL) * (R² - r²)
where: v = velocity at radial location r ΔP = pressure gradient μ = dynamic viscosity L = length of the pipe R = radius of the pipe r = radial location
Given: ΔP = -10 Pa/m (negative because the pressure decreases in the flow direction) μ = 0.001 Pa-s R = 5 cm = 0.05 m r = 0.2 cm = 0.002 m
We are looking for the velocity at a radial location of 0.2 cm, so we don't need the length of the pipe in this case. Substituting the given values into the equation, we get:
v = (-10 Pa/m / 4*0.001 Pa-s/m) * (0.05 m² - 0.002 m²) v = 2500 m²/s² * 0.00248 m² v = 6.2 m/s
So, the magnitude of axial velocity at a radial location of 0.2 cm is approximately 6.2 m/s.
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