ANALYSIS OF CASSON FLUID MODEL FOR BLOOD FLOW IN EXTERNALLY APPLIED MAGNETIC FIELD - Theoretical and Computational Research in the 21st Century - page 244

⎛ ⎞

πτ

R

3

1 6 τ 4 τ

Q =

R

+

H

−

H

⎜ ⎟

⎜ ⎟

2

4 9 τ 7 τ

⎝ ⎠

2 R R

τ 16

1/ 2

2 ττ = Q 4

H

+ τ

(17)

RH H

3

π R 7

Equations (16) and (17) give the volumetric flow rate

2

1/ 2

⎡ ⎤

⎛

23 2

⎞

2

4 πτ R 16 R τ Rp ∂ ∂ B 2 π R τ

⎛ ⎞

(18)

Q = ⎢

H

+ .

H

− τ R − − m +

H

⎥

⎜ ⎟

⎜ ⎟

H 1

m

2

7 m

2

4 ∂ z ∂ z m π R

⎢ ⎝ ⎠ ⎥

⎝ ⎠

⎣ ⎦

2 2 2

Therefore, from Eqs. (18) and (14), the velocity of flow is

2

⎡

1/ 2

⎤

⎛

23 2

⎞

2

1 4 πτ R 16 R τ Rp ⎛ ∂ ∂ B ⎞ 2 π R τ

u = ⎢

H

+ .

H

− τ R − − m +

H

⎥

(19)

⎜ ⎜ ⎟ ⎟ H 1

π R

2

m

2

7 m

2

4 ∂ z ∂ z mR π

⎢ ⎝ ⎠ ⎥

⎝ ⎠

⎣ ⎦

2 2 2

τ

Shear stress is considered as ,

∂
∂

u
r

τ

Thus, = -k ( ) r = R (20)

where k = μ

∂
∂

u
r

τ

Therefore, = − μ ( ) r = R (21)

Differentiating Eq. (19) with respect to r and substituting the value in
Eq. (22), we obtain shear stress

⎡

1/ 2

⎤

⎛ 4 πτ

23

R 16 R τ Rp

2

∂ ∂ B ⎞ 4 τ 72 τ Rp ∂ ∂ B

⎛ ⎞ ⎛ ⎞

(22) τμ =− ⎢

H

+ .

H

− τ R − − m +

H

+

H

− τ − − m ⎥

⎜ ⎟

⎜ ⎟ ⎜ ⎟

H 1 H 1

m

2

7 m

2

4 ∂ z ∂ z m π R

3

7 m

2

2 ∂ z ∂ z

⎝ ⎠ ⎝ ⎠

⎢ ⎝ ⎠ ⎥

⎣ ⎦

2 2 2 2