Turbulent boundary layer modification in a channel flow using injection of microbubbles as a means to achieve drag reduction was studied. The physical mechanism of this phenomenon is not yet fully understood. To obtain some information related to this phenomenon, single-phase (pure water) flow and two-phase (water and microbubbles) channel flow measurements are taken. The void fraction conditions were varied while maintaining a Reynolds number of 5128 based on the half channel height. The study indicates that the presence of microbubbles within the boundary layer modifies the turbulence structure such that variations in time and space turbulent scales are observed, as well as ejection and sweep phenomena.

The main goal of this manuscript is to generalize Darcy’s law from conventional calculus to fractal calculus in order to quantify the fluid flow in subterranean heterogeneous reservoirs. For this purpose, the inherent features of fractal sets are scrutinized. A set of fractal dimensions is incorporated to describe the geometry, morphology, and fractal topology of the domain under study. These characteristics are known through their Hausdorff, chemical, shortest path, and elastic backbone dimensions. Afterward, fractal continuum Darcy’s law is suggested based on the mapping of the fractal reservoir domain given in Cartesian coordinates $x_i$ into the corresponding fractal continuum domain expressed in fractal coordinates ${\xi }_i$ by applying the relationship ${\xi }_i={ϵ}_0{(x_i/{ϵ}_0)}^{{\alpha }_i-1}$, which possesses local fractional differential operators used in the fractal continuum calculus framework. This generalized version of Darcy’s law describes the relationship between the hydraulic gradient and flow velocity in fractal porous media at any scale including their geometry and fractal topology using the ${\alpha }_i$-parameter as the Hausdorff dimension in the fractal directions ${\xi }_i$, so the model captures the fractal heterogeneity and anisotropy. The equation can easily collapse to the classical Darcy’s law once we select the value of 1 for the alpha parameter. Several flow velocities are plotted to show the nonlinearity of the flow when the generalized Darcy’s law is used. These results are compared with the experimental data documented in the literature that show a good agreement in both high-velocity and low-velocity fractal Darcian flow with values of alpha equal to $0<{\alpha }_1<1$ and $1<{\alpha }_1<2$, respectively, whereas ${\alpha }_1=1$ represents the standard Darcy’s law. In that way, the alpha parameter describes the expected flow behavior which depends on two fractal dimensions: the Hausdorff dimension of a porous matrix and the fractal dimension of a cross-section area given by the intersection between the fractal matrix and a two-dimensional Cartesian plane. Also, some physical implications are discussed.