The blood speed in a normal segment of a horizontal artery is 0.111 m/s. An abnormal segment of the artery is narrowed down by an arteriosclerotic plaque to 1/5 of its normal cross-sectional area. What is the difference in blood pressure between the normal and constricted segments of the artery?

To determine the difference in blood pressure between the normal and constricted segments of the artery, we need to apply the principles of fluid dynamics and Bernoulli's equation.

Bernoulli's equation states that in a steady, ideal fluid flow with no elevation difference, the sum of the pressure, kinetic energy, and potential energy per unit volume is constant. In this case, we can neglect elevation and potential energy since the artery is horizontal.

The equation can be expressed as follows:

P1 + 1/2ρv1^2 = P2 + 1/2ρv2^2

Where:
P1 and P2 are the pressures at the normal and constricted segments of the artery, respectively.
v1 and v2 are the velocities at the normal and constricted segments of the artery, respectively.
ρ is the density of blood.

Given that the blood speed in the normal segment (v1) is 0.111 m/s and the constricted segment has 1/5 of the normal cross-sectional area, we can determine the blood speed in the constricted segment (v2).

Since the cross-sectional area is reduced by 1/5, the velocity must increase by a factor of 5 to maintain the same flow rate of blood. Therefore, v2 = 5v1 = 5 * 0.111 m/s = 0.555 m/s.

Now we can substitute the values into Bernoulli's equation:

P1 + 1/2ρv1^2 = P2 + 1/2ρv2^2

P1 + 1/2ρ(0.111 m/s)^2 = P2 + 1/2ρ(0.555 m/s)^2

Simplifying the equation, we can see that the density of blood (ρ) cancels out from both sides, so we don't need its value. We only require the density to be the same for the equation to hold.

P1 + 1/2(0.111 m/s)^2 = P2 + 1/2(0.555 m/s)^2

Calculating the right-hand side:

P1 + 0.00617 m^2/s^2 = P2 + 0.1539 m^2/s^2

Subtracting 0.00617 m^2/s^2 from both sides:

P1 = P2 + 0.1539 m^2/s^2 - 0.00617 m^2/s^2

P1 = P2 + 0.14773 m^2/s^2

The difference in blood pressure between the normal and constricted segments of the artery is approximately 0.14773 m^2/s^2.