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.
To solve both questions, we can use the principles of fluid dynamics and apply the equations related to laminar flow, average velocity, flow rate, and pressure drop.
1. (a) To find the average velocity in the needle, we can use the equation for average velocity:
average velocity = flow rate / cross-sectional area
Given:
Flow rate (Q) = 1.0 x 10^-7 m^3/s
Inner radius (r) = 3.0 x 10^-4 m
First, we need to calculate the cross-sectional area of the needle:
area = π * (inner radius)^2
Using the values, plug them into the equation to find the average velocity:
average velocity = flow rate / area
(b) To find the pressure drop necessary to achieve the flow rate, we can use the Hagen-Poiseuille equation:
pressure drop = (8 * viscosity * length * flow rate) / (π * (radius)^4)
Given:
Length (L) = 0.020 m
Inner radius (r) = 3.0 x 10^-4 m
Viscosity (η) of water at 20 degrees Celsius: 0.001 Ns/m^2 (approximation)
Plug the values into the equation to calculate the pressure drop required.
2. (a) To find the maximum velocity, we can use the equation for maximum velocity in laminar flow:
maximum velocity = 2 * average velocity
Given:
Average velocity (v_avg) = 0.030 m/s
Plug the given value into the equation to find the maximum velocity.
(b) To find the flow rate, we can use the equation:
flow rate = cross-sectional area * average velocity
Given:
Inner radius (r) = 2.0 x 10^-3 m
Average velocity (v_avg) = 0.030 m/s
First, calculate the cross-sectional area of the artery, and then use the equation to find the flow rate.
(c) To find the pressure drop in 0.050 m, we can use the same Hagen-Poiseuille equation as in question 1(b). Plug the given values into the equation to calculate the pressure drop.
Remember, these calculations assume laminar flow and certain approximations for water viscosity.