Blood in a carotid artery carrying blood to the head is...

Blood in a carotid artery carrying blood to the head is moving at $0.15 \mathrm{m} / \mathrm{s}$ when it reaches a section where plaque has narrowed the artery to $80 \%$ of its diameter. What pressure drop occurs when the blood reaches this narrow section?

Key Concepts

Fluid Dynamics in Biological Systems

Understanding the behavior of blood flow in arteries requires applying fundamental fluid dynamics concepts, such as the relationship between pressure, velocity, and vessel cross-sectional area. This knowledge is crucial for analyzing how conditions like arterial plaque can affect blood pressure and flow, ultimately impacting circulatory health.

Continuity Equation

The continuity equation is a reflection of mass conservation in a steady, incompressible flow, asserting that the product of cross-sectional area and flow velocity remains constant along a streamline. When the diameter of the artery is reduced, the flow must accelerate to conserve mass, which in turn influences the pressure as dictated by Bernoulli's principle.

Bernoulli's Principle

This principle states that for an ideal, incompressible, and non-viscous fluid flowing along a streamline, the sum of the pressure energy, kinetic energy per unit volume, and potential energy per unit volume remains constant. In the context of blood flow, it explains how an increase in flow velocity (due to a decrease in the cross-sectional area of the artery) results in a decrease in static pressure.

Transcript

00:01 Here we have a problem of incompressible flow.

00:03 We have our blood artery, which then constricts down somewhat.

00:08 If we have a diameter d at the beginning, and this is going to be our first condition, at our second condition, we have a diameter of 0 .8 d.

00:20 Our velocity inlet is 0 .15 meters per second.

00:28 And this is blood, which is mostly water.

00:30 So we're going to assume a density of 1 ,000 kilograms per meter cubed throughout.

00:39 So the first thing we can start with is bernoulli's equation, which relates the velocity and pressure.

00:49 So p1 plus the static pressure plus the dynamic pressure is equivalent through the streamlines.

01:01 And because we have a closed circuit here, it's through any part of this duct.

01:14 So our pressure difference is going to be p2 minus p1, the traditional way to define pressure difference.

01:25 We can factor out the 1 half rho, because it's constant.

01:28 And then we are left with v1 squared minus v2 squared.

01:35 So we have everything here but v2.

01:38 And to do that, we can use the law of conservation of mass.

01:43 So because there is no blood piling up in this artery, the mass flow, m dot inlet, has to equal to the m dot outlet.

01:55 And we can solve for that.

01:57 So m dot 1 is equal to the second.

02:02 And if we take the density and multiply by the volume flow rate, or we can express the volume flow rate as the velocity times the area, we get our equation here.

02:17 And then because we know the artery is a circle, based on a given diameter, we can start simplifying.

02:26 So our densities cancel out.

02:30 The area of a circle is pi r squared, which is d squared over 2 squared, if we distribute the squared.

02:45 And then v2, so pi, so 0 .8 d squared.

02:51 We can distribute it to 0 .64 d squared over 2 squared.

03:00 Now, most of this cancels...

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