How Do You Calculate Shear Stress in a Coronary Artery?

[Gwozdzilla]

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.

 

[haruspex]

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.

 

[Chestermiller]

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