A non-primitive boundary element technique for modeling flow through non-deformable porous medium using Brinkman equation

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

“…When the action of the magnetic field on viscous flow through a porous wavy channel is negligible that is, M 1 = 10 −4 , with the uniform value of pressure gradient, angle of inclination of magnetic field (θ = 0), wave amplitude (𝜖 = 0), and porosity of the porous region (𝜙 = 1), the 2D viscous flow through porous sinusoidal channel becomes fully developed at the verical cross section (Figure 3a). In Figure 3a, we have presented the dependency of the flow velocity (U) on the Darcy number (Da) and compared the BEM results with the exact solution (Nishad et al [27]) and observe that our results agree very well with Nishad et al [27]. This ensures the correctness of the BEM code.…”

“…𝜕Y 2 . Nishad et al [27] developed a non-primitive BEM to solve two-dimensional Brinkman equation. We introduce the vorticity variable 𝜔 with the utilization of Equation (11),…”

“…Nishad et al. [27] developed a non‐primitive BEM to solve two‐dimensional Brinkman equation. We introduce the vorticity variable $${{\omega}}$$ with the utilization of Equation (), $$$$\begin{equation}{\mathrm{\omega \ }} = \ \nabla \times {{\bf v}}\ = \ \nabla \times \ \left( {{\mathrm{U}},{\mathrm{V}}} \right) = \frac{{\partial {\mathrm{V}}}}{{\partial {\mathrm{X}}}}\ - \ \frac{{\partial {\mathrm{U}}}}{{\partial {\mathrm{Y}}}} = \frac{\partial }{{\partial {\mathrm{X}}}}\ \left( { - \frac{{\partial {{\psi}}}}{{\partial {\mathrm{X}}}}} \right) - \frac{\partial }{{\partial {\mathrm{Y}}}}\ \left( {\frac{{\partial {{\psi}}}}{{\partial {\mathrm{Y}}}}} \right) = \ - \ \left( {\frac{{{\partial }^2{{\psi}}}}{{\partial {{\mathrm{X}}}^2}} + \frac{{{\partial }^2{{\psi}}}}{{\partial {{\mathrm{Y}}}^2}}} \right) = \ - {\nabla }^2{{\psi}}\end{equation}$$$$…”

“…In such cases, we need an efficient numerical technique that can be implemented easily and able to produce a reliable solution. There are several numerical methods are available in literature including finite difference method (FDM) [23], finite element method (FEM) [24,25], finite volume method (FVM) [26], and boundary element method (BEM) [27,28], and so forth, to solve distinct kinds of fluid flow problems associated with distinct kind of transport parameters. But when fluid flow problems are associated with irregular/complex geometry or boundary conditions, BEM has advantages over any numerical techniques.…”

Gravity wave interaction with multiple submerged artificial reefs

“…The superscripts denote the domain, whereas the subscripts denote the boundary element of the j th domain. To evaluate the boundary integrals in equation (23), we adopt the analytical expressions derived by Kelmanson 26 and recently used by Nishad et al 27,28 The major advantage of using analytical expressions of these integrals is the large savings in computational time compared to the numerical Gauss quadrature method. Since the quadratic pressure boundary condition is applied to the porous barriers, which is non-linear in nature, therefore, obtaining a direct numerical solution is not straightforward.…”