# Interdisciplinary Applied Mathematics

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(3.32b)

Фь = Ф (y = L) = 2 k2£uw ; п< 0.

(3.32c)

The streamwise component of the velocity и and the shear stress т are defined as follows:

и = J СФ1о d£dy dp,

(3.33a)

т = po Ф,пФ1о d^dqdp,

(3.33b)

where / (•••) d^dydQ shows integration over the velocity space, and p0 is the mean density given by p0 = n0m, with n0 being the equilibrium number density and m being the molecular mass. By taking the Laplace transform of equation (3.32a) and the boundary conditions (3.32b, 3.32c),

we calculate the integral formulations of the velocity and shear stress given by equations (3.33a) and (3.33b). The Laplace transformed variables ф, u, and т are given as

П < 0,    (3.34a)

ф = фьexp

2k2 Zuw exp

-iv-L) ;

n

u = J ZФfo dZ dn dZ,    (3.34b)

т = po / £пФ1о dZ dn dZ,    (3.34c)

where s is the Laplace transformation variable. After eliminating ф from equations (3.34a), (3.34b), and (3.34c), we obtain integral formulations for u and т as follows:

й = —= uw exp (—ту(L — y) — K2?y 2 I dr] ,    (3.35a)

VA/ о    V П

к

т

uw n

exp

###### Ar(L-y) ~ к2ri’2 n

where uw is the transformed function of uw. Finally, the inverse Laplace transform provides u and т as functions of y and t: