In this paper, we will be interested in the viscous flow fluid over a rough wall which has a periodic roughness and small amplitude. The Reynolds number for the flow over a rough wall is low and the creeping flow equations apply. The no-slip boundary condition on the rough wall applies. By using an asymptotic expansion, the analytic expressions are obtained for the pressure and the components f velocity of the flow of second order due to the roughness, then concluding the expressions of pressure and velocity of the flow on the rough wall.

Based on the assumption of low Reynolds number, the flow around a spherical particle settling towards a corrugated wall in a fluid at rest is described by Stokes equations. In the case of the small amplitude of wall roughness, the asymptotic expansion coupled with the Lorentz reciprocal theorem are used to derive analytical expressions of the hydrodynamic effects due to wall roughness. The evolution of these effects in terms of roughness parameters and also the sphere-wall distance are discussed.

Separation of particles in a fluid domain is relevant in various industrial applications. The effect due to the roughness is preponderant compared with that due to fluid inertia so that the Reynolds number is low and the creeping flow equations apply. The wall roughness is assumed to be rigid and periodic, varied in one direction. The trajectories of freely moving particles in a shear flow are calculated.

In this paper we are interested in the approximation of the integral \[I_0(f,\omega)=\int_0^\infty f(t)\,e^{-t}\,J_0(\omega t)\,dt\] for fairly large $\omega$ values. This singular integral comes from the Hankel transformation of order $0$, $f(x)$ is a function with which the integral is convergent. For fairly large values of $\omega$, the classical quadrature methods are not appropriate, on the other side, these methods are applicable for relatively small values of $\omega$. Moreover, all quadrature methods are reduced to the evaluation of the function to be integrated into the nodes of the subdivision of the integration interval, hence the obligation to evaluate the exponential function and the Bessel function at rather large nodes of the interval $]0,+\infty[$. The idea is to have the value of $I_0(f,\omega)$ with great precision for large $\omega$ without having to improve the numerical method of calculation of the integrals, just by studying the behavior of the function $I_0(f,\omega)$ and extrapolating it. We will use two approaches to extrapolation of $I_0(f,\omega)$. The first one is the Padé approximant of $I_0(f,\omega)$ and the second one is the rational interpolation.

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