Consider an artery with diameter 10 mm which runs horizontally. Somewhere along the artery there is a 3 cm long constriction which reduces the diameter to 5.40 mm. Assume that before and after the constriction the blood flows at a rate of 0.310 m/s, and that the blood pressure before and after is 880 mmHg. Blood density is 1060 kg/m^3.The speed of blood flow within the constriction is 1.06 m/s.
a. What is the blood pressure within the constriction? (in mmHg)
To calculate the blood pressure within the constriction, we can use the principle of continuity of flow and the Bernoulli's equation. Here's how you can calculate it step by step:
Step 1: Calculate the cross-sectional area of the artery before and after the constriction.
The cross-sectional area of an artery can be calculated using the formula A = πr^2, where A is the cross-sectional area and r is the radius of the artery.
Before constriction:
Radius before = diameter before / 2 = 10 mm / 2 = 5 mm = 0.005 m
Area before = π(0.005)^2
After constriction:
Radius after = diameter after / 2 = 5.40 mm / 2 = 2.70 mm = 0.0027 m
Area after = π(0.0027)^2
Step 2: Calculate the volume flow rate before and after the constriction.
The volume flow rate is the product of the cross-sectional area and the flow velocity. We can use the principle of continuity of flow, which states that the volume flow rate before and after the constriction remains the same.
Before constriction:
Volume flow rate before = Area before * Flow velocity before
After constriction:
Volume flow rate after = Area after * Flow velocity after
Step 3: Calculate the blood pressure within the constriction using Bernoulli's equation.
Bernoulli's equation relates the pressure, velocity, and height of a fluid. Within a horizontal artery, the height difference can be neglected, so Bernoulli's equation can be simplified as:
Pressure + 1/2 * Density * (Velocity)^2 = Constant
Before constriction:
Pressure before + 1/2 * Density * (Velocity before)^2 = Constant
After constriction:
Pressure within + 1/2 * Density * (Velocity within)^2 = Constant
Since the constant remains the same, we can equate the two equations to find the pressure within the constriction.
Step 4: Rearrange the equations and solve for the blood pressure within the constriction.
Pressure within = Pressure before + 1/2 * Density * ((Velocity before)^2 - (Velocity within)^2)
Substitute the given values:
Pressure before = 880 mmHg
Density = 1060 kg/m^3
Velocity before = 0.310 m/s
Velocity within = 1.06 m/s
Now, plug in the values and calculate the blood pressure within the constriction.