Spatial Decay Bounds for the Brinkman Fluid Equations in Double-Diffusive Convection

In this paper, we consider the Brinkman equations pipe flow, which includes the salinity and the temperature. Assuming that the fluid satisfies nonlinear boundary conditions at the finite end of the cylinder, using the symmetry of differential inequalities and the energy analysis methods, we establish the exponential decay estimates for homogeneous Brinkman equations. That is to prove that the solutions of the equation decay exponentially with the distance from the finite end of the cylinder. To make the estimate of decay explicit, the bound for the total energy is also derived.

The Flow in Periciliary Layer in Human Lungs with Navier-Stokes-Brinkman Equations

In the human respiratory tract, air breathed in is often contaminated with strange particles such as dust and chemical spray, which may cause people respiratory diseases. However, the human body has an innate immune system that helps to trap the debris by secreting mucus to catch the foreign particles, which are removed from the body by the movement of tiny hairs lining on the surface of the epithelial cells in the immune system. The layer containing the tiny hairs or cilia is called Periciliary Layer (PCL). In this research, we find the velocity of the fluid in the PCL moved by a ciliary beating by using the Navier-Stokes-Brinkman equations. We apply the Galerkin finite element method to determine numerical solutions. For the steady linear case of the equation, the numerical result is in good agreement with an exact solution. Including the time derivative and nonlinear terms, we show that the velocity of the liquid is affected by the velocity of the solid, which follows the physical meaning of the fluid flow. The result can be applied as a bottom boundary condition of the mucous layer to be able to find the velocity of mucus in the human lungs.

Darcy Brinkman Equations for Hybrid Dusty Nanofluid Flow with Heat Transfer and Mass Transpiration

In the current work, we have investigated the flow past a semi-infinite porous solid media, after presenting a similarity transformation, governing equations mapped to a system of non-linear PDE. The flow of a dusty fluid and heat transfer through a porous medium have few applications, viz., the polymer processing unit of a geophysical, allied area, and chemical engineering plant. Further, we had the option to get an exact analytical solution for the velocity to the equation that is non-linear. The highlight of the current work is the flow of hybrid dusty nanofluid due to Darcy porous media through linear thermal radiation with the assistance of an analytical process. The hybrid dusty nanofluid has significant features improving the heat transfer process and is extensively developed in manufacturing industrial uses. It was found that the basic similarity equations admit two phases for both stretching/shrinking surfaces. The existence of computation on velocity and temperature profile is presented graphically for different estimations of various physical parameters.

Fluid Flow in Composite Regions Past a Solid Sphere

A steady, 2-D, incompressible, viscous fluid flow past a stationary solid sphere of radius 'a' has been considered. The flow of fluid occurs in 3 regions, namely fluid, porous and fluid regions. The governing equations for fluid flow in the clear and porous regions are Stokes and Brinkman equations, respectively. These governing equations are written in terms of stream function in the spherical coordinate system and solved using the similarity transformation method. The variation in flow patterns by means of streamlines has been analyzed for the obtained exact solution. The nature of the streamlines and the corresponding tangential and normal velocity profiles are observed graphically for the different values of porous parameter 'σ'. From the obtained results, it is noticed that an increase in porous parameters suppresses the fluid flow in the porous region due to less permeability; as a result, the fluid moves away from the solid sphere. It also decreases the velocity of the fluid in the porous region due to the suppression of the fluid as 'σ' increases. Hence the parabolic velocity profile is noticed near the solid sphere.

On the Cauchy problem associated with the Brinkman flow in

In this study, we continue our study of the Cauchy problem associated with the Brinkman equations [see (1.1) and (1.2) below] which model fluid flow in certain types of porous media. Here, we will consider the flow in the upper half-space \[ \mathbb{R}_{+}^{3}=\left\{\left(x,y,z\right) \in\mathbb{R}^{3}\left\vert z\geqslant 0\right.\right\}, \] under the assumption that the plane $z=0$ is impenetrable to the fluid. This means that we will have to introduce boundary conditions that must be attached to the Brinkman equations. We study local and global well-posedness in appropriate Sobolev spaces introduced below, using Kato's theory for quasilinear equations, parabolic regularization and a comparison principle for the solutions of the problem.