To solve these problems, we will use the Hagen-Poiseuille equation for laminar flow in a tube:
ΔP = 8ηLQ / πr^4
where ΔP is the pressure difference between the ends of the tube, η is the dynamic viscosity of the fluid, L is the length of the tube, Q is the volumetric flow rate, and r is the inner radius of the tube.
For water, the dynamic viscosity at 20°C is approximately η = 1.0 x 10^-3 Pa⋅s. For blood, the dynamic viscosity at 37°C is approximately η = 4.0 x 10^-3 Pa⋅s.
1(a). To find the average velocity in the needle, we can use the flow rate Q and the cross-sectional area A of the needle:
Q = Av
where A = πr^2, and v is the average velocity.
1.0 x 10^-7 m³/s = π(3.0 x 10^-4 m)²v
Solving for v, we get:
v ≈ 3.54 m/s
1(b). Now we can use the Hagen-Poiseuille equation to find the pressure drop:
ΔP = 8ηLQ / πr^4
ΔP = 8(1.0 x 10^-3 Pa⋅s)(0.020 m)(1.0 x 10^-7 m³/s) / π(3.0 x 10^-4 m)⁴
ΔP ≈ 5924 Pa
2(a). In laminar flow, the maximum velocity occurs at the center of the tube and is twice the average velocity:
v_max = 2 * 0.030 m/s = 0.060 m/s
2(b). We can find the flow rate using the average velocity and the cross-sectional area:
Q = Av
Q = π(2.0 x 10^-3 m)² * 0.030 m/s
Q ≈ 3.77 x 10^-7 m³/s
2(c). Finally, we can find the pressure drop using the Hagen-Poiseuille equation:
ΔP = 8ηLQ / πr^4
ΔP = 8(4.0 x 10^-3 Pa⋅s)(0.050 m)(3.77 x 10^-7 m³/s) / π(2.0 x 10^-3 m)⁴
ΔP ≈ 47 Pa
1.A hypodermic needle of lenght 0.020m and inner radius 3.0 x10^-4 is used to force water at 20 degree celcious into the air at a flow rate of 1.0 x10^-7m^2 s-1 (a) what is the average velocity in the needle?(assume laminar flow), (b) what is the pressure drop necessary to achieve the flow rate?
2. An artery has an inner radius of 2.0 x 10^-3m. if the temperature is 37 degree celcius, the average velocity of the blood is 0.030ms^-1 and the flow is laminar flow. find (a)the maximum velocity, (b) the flow rate and (c) the pressure drop in 0.050m, if the artery is horizontal.
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