Cxy zð Þ ¼ exp
l x, y

ð

Þ
z

ð

Þ

ð16:13Þ

Where l x, y

ð

Þ ¼ l þ δ cos x

ð Þ cos y

ð Þ

ð16:14Þ

where δ is the delta function with the property that it is nonzero at a given x and

y and zero at all other values of x and y which defines step like surface giving a

structure of rectangular roughness to the wall as shown in Fig. 16.19. The product

cos(x) cos(y) in Eq. (16.14) represents sinusoidal roughness illustrated by

Fig. 16.20. With this representation of roughness, the diffusion coefficient thus

calculated from Eq. (16.7) has a behavior represented by Fig. 16.21 drawn as a

function of z.

5. Effect of Elastic Confinement: The arteries have elastic walls which can be further

incorporated in this microscopic model by considering compression represented

by “d” and recovery represented by “b” due to elastic wall as shown in Fig. 16.22.

Thus the modified frequency takes the form

Ω ¼ ^π

2t1 ^¼

πω

2 sin
¹ 1
d=A þ b=A

ð

Þ

ð16:15Þ

Corresponding velocity autocorrelation function and diffusion coefficient have

been calculated using Eq. (16.7), and corresponding results obtained are displayed in

Fig. 16.23 and Figs. 16.24 and 16.25, respectively. Figure 16.24 shows how the

diffusion goes to a non-zero minimum before rising again for an elastic wall where

the diffusion completely dies down if the artery walls are hard in nature. Figure 16.25

depicts diffusion behavior of liquid pertaining to walls of different elastic strength.

The walls with more elastic strength reached a comparatively higher minimum as

compared to the other wall with lower elastic strength. Thus, more elasticity of wall

would lead to reduced freezing of the confined liquid. In analogy, one can take the

example of flow of blood in arteries which are referred to as microtubes/nanotubes,

Fig. 16.19 Rectangular Roughness. Step like surface with δe ¼ 1

294

K. Tankeshwar and S. Srivastava