   Chapter 3.7, Problem 29E

Chapter
Section
Textbook Problem

Refer to the law of laminar flow given in Example 7. Consider a blood vessel with radius 0.01 cm, length 3 cm, pressure difference 3000 dynes/cm2, and viscosity η = 0.027.(a) Find the velocity of the blood along the center-line r = 0, at radius r = 0.005 cm, and at the wall r = R = 0.01 cm.(b) Find the velocity gradient at r = 0, r = 0.005, and r = 0.01.(c) Where is the velocity the greatest? Where is the velocity changing most?EXAMPLE 7  (a)

To determine

To find: The velocity of the blood along the center line r=0, at radius r=0.005cm and at the wall r=R=0.01cm.

Explanation

Given:

The law of laminar flow is as given below.

v(r)=P4ηl(R2r2) (1)

The length of the blood vessel is given below.

l=3cm

The pressure difference between the ends of the vessel is given below.

P=3000dynes/cm2

The viscosity of the blood is given below.

η=0.027

Calculation:

Calculate the velocity of the blood along the center line using the equation (1).

v(r)=P4ηl(R2r2)

Substitute 3cm for l, 0 for r, 3000dynes/cm2 for P, 0.027 for η and 0.01cm for R in the equation (1).

v(0)=30004×0.027×3(0.0120)=9259.25×0.0001

v(0)=0.925cm/s

Thus, the Velocity of the blood at r=0cm is 0.925cm/s.

Calculate the velocity of the blood at r=0.005cm using the equation (1).

v(r)=P4ηl(R2r2)

Substitute 3 cm for l, 0

(b)

To determine

To find: The velocity gradient at r=0, r=0.005 and r=0.01.

(c)

To determine

To find: Where is the velocity the greatest and where it changes the most.

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