MAGNETO HYDRODYNAMIC EFFECTS ON AN ATHEROSCLEROTIC ARTERY—A THEORETICAL STUDY - Theoretical and Computational Research in the 21st Century - page 68

The volumetric fl ow fl ux Q is thus calculated as

RZ ()

∫

Q = 2 π rudr

0

(18)

{}

2

QRz = π () u

Therefore, Eqs. (17) and (18) reduces the form

⎡

n + 1 n + 1

⎤

2

π R ⎧∂ ⎛ p ∂ B ⎞ ⎫ ⎧∂ ⎛ p ∂ B ⎞ ⎫

Q =− − m r − 2 τ − − m R − 2 τ

⎢ ⎥

⎨ ⎬ ⎨ ⎬

⎜ ⎟ ⎜ ⎟

(19)

1 H 1 H

∂ p ∂ B

⎛ ⎞ ∂ z ∂ z ∂ z ∂ z

⎩ ⎝ ⎠ ⎭ ⎩ ⎝ ⎠ ⎭

⎢ ⎥

n

⎣ ⎦

2 m − m

⎜ ⎟

2 1

∂ z ∂ z

⎝ ⎠

Again, Eqs. (15) and (19) reduces the form

⎡

n + 1 n + 1

⎤

π R

2

⎧∂ p ∂ B ⎫ ⎧∂ p ∂ B ⎫

⎛ ⎞ ⎛ ⎞

Q =− ⎢ − m r − 2 τ − − m R − 2 τ ⎥

⎨ ⎬ ⎨ ⎬

⎜ ⎟ ⎜ ⎟

⎝

1

⎠

H

⎝

1

⎠

H

⎛ ∂ p ∂ B ⎞ ∂ z ∂ z ∂ z ∂ z

⎩ ⎭ ⎩ ⎭

⎢ ⎥

n

⎣ ⎦

2 m − m

⎜ ⎟

2

⎝

1

⎠

∂ z ∂ z

1

1

⎡

n + 1 n + 1

⎤

n

{( k n + 3)} ⎧ ∂ ⎛ p ∂ B ⎞ ⎫ ⎧ ∂ ⎛ p ∂ B ⎞ ⎫ ⎛ n + 3 ⎞

n

1

(20)

τ =− (1) ⎢ − mr − 2 τ − − mR − 2 τ ⎥ + τ

n

⎨ ⎬ ⎨ ⎬

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

R 1 H 1 H H

∂ p ∂ B ⎝ ∂ z ∂ z ⎠ ⎝ ∂ z ∂ z ⎠ ⎝ n + 2 ⎠

⎛ ⎞

⎢ ⎩ ⎭ ⎩ ⎭ ⎥

1

⎣ ⎦

2{ Rm − m }

⎜ ⎟

n

2

⎝

1

⎠

∂ z ∂ z

Shear stress is considered as ,

τ

∂
∂

u
r

τ

Thus, = − k ( ) r = R (21)

where k = μ

∂
∂

u
r

τ

Therefore, = − μ ( ) r = R (22)

Differentiating Eq. (17) with respect to r and substituting the value in
Eq. (22) we obtain shear stress,