#### 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.

#### Answer

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

**Given:**

The law of laminar flow is as given below.

v(r)=P4ηl(R2−r2) (1)

The length of the blood vessel is given below.

l=3 cm

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

P=3000 dynes/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(R2−r2)

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

v(0)=30004×0.027×3(0.012−0)=9259.25×0.0001

v(0)=0.925 cm/s

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

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

v(r)=P4ηl(R2−r2)

Substitute 3 cm for l, 0.005 cm for r, 3000 dynes/cm2 for P, 0.027 for η and 0.01 cm for R in the equation (1).

v(0.005)=30004×0.027×3(0.012−0.0052)=9259.259×0.000075=0.694 cm/s

v(0)=0.694 cm/s

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

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

v(r)=P4ηl(R2−r2)

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

v(0.01)=30004×0.027×3(0.012−0.012)=0

v(0.01)=0

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

#### To determine

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

#### Answer

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

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*, 3000 dynes/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*, 3000 dynes/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*, 3000 dynes/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.

#### To determine

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

#### Answer

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

#### Explanation

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.