A cylindrical blood vessel is partially blocked by the buildup of plaque. At one point, the plaque decreases the diameter of the vessel by 52.0%. The blood approaching the blocked portion has speed v0. Just as the blood enters the blocked portion of the vessel, what is its speed v, expressed as a multiple of v0?
To determine the speed of the blood, we can use the principle of conservation of mass and Bernoulli's equation, which relates the pressure and speed of a fluid.
Since the blood vessel is partially blocked by plaque, the decrease in diameter will cause an increase in flow speed to maintain a constant flow rate.
According to the principle of conservation of mass, the product of the fluid's speed and its cross-sectional area is constant throughout the blood vessel.
Let's denote the initial cross-sectional area of the blood vessel as A0 and the reduced cross-sectional area at the blocked portion as A.
Using the conservation of mass, we can write the equation:
A0 * v0 = A * v
Now let's consider Bernoulli's equation, which relates pressure and speed:
P0 + 1/2 * ρ * v0^2 = P + 1/2 * ρ * v^2
Where P0 is the initial pressure, P is the pressure within the blocked portion, and ρ is the density of the blood (assumed constant).
Since the plaque does not cause any sudden pressure changes, we can assume P0 = P.
Using these equations, we can solve for v/v0, the ratio of the blood's speed as it enters the blocked portion to its initial speed:
A0 * v0 = A * v
1/2 * ρ * v0^2 = 1/2 * ρ * v^2
Canceling ρ/2, we get:
v^2 = v0^2 * A0 / A
Taking the square root of both sides:
v = v0 * sqrt(A0 / A)
To find the value of A0 / A, we can use the fact that the plaque decreases the diameter of the vessel by 52.0%, which corresponds to a decrease in cross-sectional area of (0.52)^2 = 0.2704.
Therefore, A0 / A = 1 / 0.2704 = 3.694.
Substituting this value back into the equation for v:
v = v0 * sqrt(3.694)
So, the speed of the blood as it enters the blocked portion, v, is approximately 1.921 times the initial speed, v0.
to maintain the volumetric flow rate
... the blood must go faster in a narrower vessel
v = v0 * √(1 / .520)
oops ...
v = v0 * √(1 - .520)