How Do You Calculate Shear Stress in a Coronary Artery?

In summary, in this problem we are estimating the shear stress at the wall of a coronary artery with a diameter of 2.5 mm and length of 3 cm. Using the given values for blood density and viscosity, we can calculate the shear stress using the equation τ = μ ∂u/∂y, where ∂u/∂y is the velocity gradient at the wall. Since the flow profile is parabolic in shape, we can find the velocity gradient by considering the relationship between maximum and average velocities and the radius of the artery.
  • #1
Gwozdzilla
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Homework Statement


Blood supply to the heart occurs through coronary arteries. Consider one of the arteries to be 2.5 mm in diameter and 3 cm in length. The average velocity of blood flow through that artery is 1.5 cm/s. Assuming the density of blood to be 1.056 g/cc and viscosity to be 3 cP (3x10-3 Ns/m2). Estimate the shear stress at the wall.

Homework Equations


τ = μ ∂u/∂y = shear stress = (viscosity) (d(velocity))/(dy)

The Attempt at a Solution


τ = (3cP)(1.5cm/s)

I'm not sure how to estimate ∂u/∂y. Is it equal to the average velocity? I think that when the blood reaches fully developed flow, it's shaped like a parabola and it's velocity is constant at a given y, but I'm not sure how to apply this information to understand the formula.
 
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  • #2
Indeed, the flow profile is a quadratic function of radius. See http://hyperphysics.phy-astr.gsu.edu/hbase/pfric.html#vel.
Using that formula and knowing the average flow, you should be able to write out exactly how the flow rate depends on radius for this example. From that you can find the velocity gradient at the wall.
 
  • #3
From Haruspex's link, how is the maximum velocity at the center of the artery related to the average velocity in the problem statement? Since the shape of the velocity profile is parabolic in r and the velocity is zero at the wall of the capillary, what is the equation for v(r) in terms of r, the average velocity, and the wall radius? What is the derivative of v with respect to r at the wall?

Chet
 

Related to How Do You Calculate Shear Stress in a Coronary Artery?

1. What is shear stress in blood vessels?

Shear stress in blood vessels refers to the force or pressure exerted by the flow of blood against the walls of the blood vessels. It is caused by the friction between the blood and the vessel walls as the blood moves through the vessels.

2. How is shear stress measured in blood vessels?

Shear stress in blood vessels is typically measured in units of dynes per square centimeter (dyn/cm²). This measurement takes into account the force of the blood flow and the area of the vessel wall that the blood is acting upon.

3. What factors affect shear stress in blood vessels?

The two main factors that affect shear stress in blood vessels are the velocity of blood flow and the diameter of the blood vessel. As blood flow increases or the vessel diameter decreases, shear stress also increases.

4. What are the effects of shear stress on blood vessels?

Shear stress can have both positive and negative effects on blood vessels. In small amounts, it can stimulate the production of nitric oxide, a molecule that helps to relax blood vessels and improve blood flow. However, excessive shear stress can damage the inner lining of blood vessels and contribute to the development of cardiovascular diseases.

5. How can shear stress in blood vessels be reduced?

There are several ways to reduce shear stress in blood vessels. Regular exercise can improve blood flow and reduce the amount of shear stress on the vessel walls. Eating a healthy diet and maintaining a healthy weight can also help to reduce shear stress. Additionally, managing conditions like high blood pressure and high cholesterol can help to prevent excessive shear stress in blood vessels.

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