Stokes Flow Inside Topographically Patterned Microchannel Using Boundary Element Method

Abstract

Abstarct This study focuses on the investigation of two-dimensional steady Stokes flow inside topographically patterned microchannel. Boundary element method (BEM) is used to solve the Stokes equation and obtain the streamline profiles. The velocity field and pressure gradients are obtained by taking the appropriate spatial derivatives of the stream function and vorticity variables. We restrict ourselves to rectangular stepped geometries and study the effect of variation of step width, step height and step frequency. Interestingly, ‘crown-shaped’ patterns in the horizontal velocity profiles are formed when a sudden contraction is met in the flow region. Pressure gradients, together with the velocity and streamline profiles are analyzed to gain a wholesome understanding of the flow physics.

Effect of magnetic field on viscous flow through porous wavy channel using boundary element method

“…When uniform elements are involved in the calculation, integrals can be determined analytically. The influence coefficients A jj and B jj are obtained and presented as follows [35,36]…”

“…where l j denotes the length of the j th element. To determine the values of C jj , D jj , E jj , and F jj , we follow Nishad et al [36] and deduce the expression for computing the respective integrals analytically. The fundamental solutions namely, G L , G MH , and G C are given by…”

“…When uniform elements are involved in the calculation, integrals can be determined analytically. The influence coefficients $${{\mathrm{A}}}_{{\mathrm{jj}}}$$ and $${{\mathrm{B}}}_{{\mathrm{jj}}}$$ are obtained and presented as follows [35, 36] $$$$\begin{equation}\ {A}_{jj} = \ - {\Lambda }_j,\ {B}_{jj} = \ - \frac{{{l}_j}}{{2\pi }}\left( {\ln \left( {\frac{{{l}_j}}{2}} \right) - 1} \right)\end{equation}$$$$where $${{\mathrm{l}}}_{\mathrm{j}}$$ denotes the length of the $${{\mathrm{j}}}^{{\mathrm{th}}}$$ element. To determine the values of $${{\mathrm{C}}}_{{\mathrm{jj}}},{\mathrm{\ D}}_{{\mathrm{jj}}},{\mathrm{\ E}}_{{\mathrm{jj}}},{\mathrm{\ and\ F}}_{{\mathrm{jj}}}$$, we follow Nishad et al.…”

“…To determine the values of $${{\mathrm{C}}}_{{\mathrm{jj}}},{\mathrm{\ D}}_{{\mathrm{jj}}},{\mathrm{\ E}}_{{\mathrm{jj}}},{\mathrm{\ and\ F}}_{{\mathrm{jj}}}$$, we follow Nishad et al. [36] and deduce the expression for computing the respective integrals analytically. The fundamental solutions namely, $${{\mathrm{G}}}^{\mathrm{L}},{\mathrm{\ G}}^{{\mathrm{MH}}},{\mathrm{\ and\ G}}^{\mathrm{C}}$$ are given by $$$$\begin{equation}{G}^L = \frac{1}{{2\pi }}\ \ln \overset{\scriptscriptstyle\smile}{r},\ \frac{{\partial {G}^L}}{{\partial {\bm{n}}}} = \frac{1}{{2\pi }}\ \frac{{\partial \overset{\scriptscriptstyle\smile}{r}}}{{\partial {\bm{n}}}} = \frac{{{\bm{r}}.$$…”