Blood from an animal is placed in a bottle 1.3 m above a 3.6 cm long needle, of inside diameter 0.35 mm , from which it flows at a rate of 2.1 cm3/min .
What is the viscosity of this blood? Assume ρblood=1.05×103kg/m3.
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To find the viscosity of the blood, we can use the Hagen-Poiseuille equation, which relates the flow rate of a liquid through a narrow tube to its viscosity and other parameters.
The Hagen-Poiseuille equation is given as follows:
Q = (π * ΔP * r^4) / (8 * η * L)
Where:
Q = Flow rate of the fluid
ΔP = Pressure difference
r = Radius of the tube
η = Viscosity of the fluid
L = Length of the tube
In this case, we are given the following information:
Flow rate (Q) = 2.1 cm^3/min = (2.1 * 10^-6) m^3/s (since 1 cm^3 = 10^-6 m^3)
Radius (r) = 0.35 mm / 2 = (0.35 * 10^-3 / 2) m
Length (L) = 3.6 cm = (3.6 * 10^-2) m
Density of blood (ρ_blood) = 1.05 * 10^3 kg/m^3
First, let's calculate the pressure difference (ΔP).
ΔP = ρ * g * h
Where:
ρ = Density of blood
g = Acceleration due to gravity
h = Height of liquid column
In this case, h = 1.3 m.
ΔP = (1.05 * 10^3) kg/m^3 * 9.8 m/s^2 * 1.3 m
Now, let's substitute the known values into the Hagen-Poiseuille equation and solve for viscosity (η).
(2.1 * 10^-6) m^3/s = (π * ΔP * (0.35 * 10^-3 / 2)^4) / (8 * η * (3.6 * 10^-2) m)
Rearranging the equation to solve for η:
η = (π * ΔP * (0.35 * 10^-3 / 2)^4) / [(8 * (3.6 * 10^-2) m * (2.1 * 10^-6) m^3/s)]
Now, substitute the calculated value of ΔP and the known values of the other parameters into the equation to find η.
η = (π * ΔP * (0.35 * 10^-3 / 2)^4) / [(8 * (3.6 * 10^-2) m * (2.1 * 10^-6) m^3/s)]
η = (π * [(1.05 * 10^3) kg/m^3 * 9.8 m/s^2 * 1.3 m] * (0.35 * 10^-3 / 2)^4) / [(8 * (3.6 * 10^-2) m * (2.1 * 10^-6) m^3/s)]
Evaluating this expression will give us the value of viscosity (η) for the blood.