Nonlinear correction to Darcy’s law for channels with wavy walls

Abstract

For low Reynolds numbers \({\mathcal{R}}\), the flow of a viscous fluid through a channel is described by the well-known Darcy’s law which corresponds to a linear relation between the pressure gradient \({\overline{\nabla p}}\) and the average velocity \({\overline{u}}\). When the channel is not straight and when the Reynolds number is not negligible, additional terms appear in this relation. Some previous authors investigated the first three coefficients in the expansion of \({|\overline{\nabla p}|}\) in the powers of \({\overline{u}}\) and they showed that the coefficient of \({\overline{u}^2}\) vanishes for moderate \({\mathcal{R}}\). Other authors demonstrated that this coefficient can be non-zero. This question is addressed and solved. It is demonstrated that both cases occur; Forchheimer’s law has a cubic correction for small \({\mathcal{R}}\) and a quadratic one for large \({\mathcal{R}}\). Two analytical–numerical algorithms are constructed to prove this property. These algorithms are applied to the Navier–Stokes equations in three-dimensional channels enclosed by two wavy walls whose amplitude is proportional to \({b{\varepsilon}}\), where 2b is the mean clearance of the channels and \({\varepsilon}\) is a small dimensionless parameter. The first algorithm is applied for small \({\mathcal{R}}\) by representing the velocity and the pressure in terms of a double Taylor series in \({\mathcal{R}}\) and \({\varepsilon}\). The accuracy \({O(\mathcal{R}^2)}\) and \({O(\varepsilon^6)}\) following Pade approximations yield analytical approximate formulae for Forchheimer’s law. The first algorithm is applied to symmetric channels on the theoretical level (all terms on \({\mathcal{R}}\) and \({\varepsilon}\) are taken into account) to show that \({|\overline{\nabla p}|}\) is an odd function of \({\overline{u}}\). This observation yields, in particular, a cubic correction to Darcy’s law. Numerical examples for non-symmetrical channels yield the same cubic correction. The second algorithm is based on the analytical–numerical solution to the Navier–Stokes equations for arbitrary \({\mathcal{R}}\) up to \({O(\varepsilon^{3})}\). This algorithm yields, in particular, a quadratic correction to Darcy’s law for higher \({\mathcal{R}}\).