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Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 3.8, Problem 27E

(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.

Expert Solution

Answer to Problem 27E

Velocity of the blood along the center line is v(0)=0.925¯cm/s.

Velocity of the blood at 0.005cm as radius is v(0.005)=0.694¯cm/s.

Velocity of the blood at the wall is v(0.01)=0.

Explanation of Solution

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

v(0.005)=30004×0.027×3(0.0120.0052)=9259.259×0.000075=0.694cm/s

v(0)=0.694cm/s

Thus, the Velocity of the blood at r=0.005cm is 0.694cm/s.

Calculate the velocity of the blood at the wall using the equation (1).

v(r)=P4ηl(R2r2)

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

v(0.01)=30004×0.027×3(0.0120.012)=0

v(0.01)=0

Thus, the Velocity of the blood at r=0.01cm is 0.

(b)

To determine

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

Expert Solution

Answer to Problem 27E

Velocity gradient at r=0 is v'(0)=0.

Velocity gradient at r=0.005 is v'(0.005)=92.592¯(cm/s)/cm.

Velocity gradient at r=0.01 is v'(0.01)=185.185¯(cm/s)/cm.

Explanation of Solution

Calculate the velocity gradient at center line r=0.

Differentiate equation (1) with respect to r.

v'(r)=P4ηl(2r)=rP2ηl

v'(r)=rP2ηl (2)

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

v'(0)=rP2ηlv(0)=0

Thus, the Velocity of the blood at radius r=0 is 0.

Calculate the velocity gradient at r=0.005 using the equation (2).

v'(r)=rP2ηl

Substitute 3 cm for l, 0.005 for r, 3000dynes/cm2 for P, 0.027 for η in the equation (1).

v'(0.005)=(0.005)×30002×0.027×3=150.162v'(0.005)=92.592(cm/s)/cm

The velocity gradient at r=0.005 is 92.592(cm/s)/cm.

Calculate the velocity gradient at the wall edge r=0.01 using the equation (2).

v'(r)=rP2ηl

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

v'(0.01)=(0.01)×30002×0.027×3=300.162v'(0.01)=185.185(cm/s)/cm

The velocity gradient at r=0.01 is v'(0.01)=185.185(cm/s)/cm.

(c)

To determine

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

Expert Solution

Answer to Problem 27E

The velocity is greatest where r=0 and the velocity is changing most where r=R=0.01cm

Explanation of Solution

The velocity is greatest where r=0(at the center) and the velocity is changing most where r=R=0.01cm (at the edge), which means the velocity gradient is highest at the edge.

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