Stokes’s equation in the fluid domain and Brinkman’s equation in the porous media are combined in the current study which is designated by the Stokes-Brinkman coupling. The current paper gives a theoretical analysis of the Stokes-Brinkman coupling. It has been shown that such a model is a good match for the knee joint. A flow model has been investigated in order to get a better understanding of the convective diffusion of the viscous flow along the articular surfaces between the joints. The Beavers and Joseph slip conditions which are a specific boundary condition for the synovial fluid are used to solve the governing system of partial differential equations for the synovial fluid and the results are provided here. We develop formulas for the interfacial velocity for both flow through special slip condition and analyse the link between the slip parameters $$\alpha $$ α and $$\beta $$ β . Thus, the damping force due to the porous medium naturally when we non-dimensionalize, some parameter which are controlling the structure like, $$\beta $$ β and $$\alpha $$ α . Through the development of an analytical solution and numerical simulation (using the finite volume approach) it is hoped that the mechanisms of nutritional transport into the synovial joint will be better understood. According to the data the average concentration has a negative connection with both the axial distance and the duration spent in the experiment. Many graphs have been utilized to gain understanding into the problem’s various characteristics including velocity and concentration, among others. Hyaluronate (HA) is considered to be present in porous cartilage surfaces and the viscosity of synovial fluid fluctuates in response to the amount of HA present.

The lid-driven cavity is an important fluid mechanical system that serves as a benchmark for testing numerical methods and for studying fundamental aspects of incompressible flows in confined volumes. These flows are driven by the tangential motion of a bounding wall. The lid-driven cavity serves as a benchmark for testing numerical methods and for studying fundamental aspects of incompressible flows in confined volumes. This article presents a complete study of lid-driven cavity flows, with the primary focus being placed on the development of the flow when the Reynolds number was increased. In order to fully comprehend the physics of flow, it is necessary to take into consideration not only pure two-dimensional flows but also flows that are periodic in one space direction and the whole three-dimensional flow.

Stokes’s equation in the fluid domain and Brinkman’s equation in the porous media are combined in the current study which is designated by the Stokes-Brinkman coupling. The current paper gives a theoretical analysis of the Stokes-Brinkman coupling. It has been shown that such a model is a good match for the knee joint. A flow model has been investigated in order to get a better understanding of the convective diffusion of the viscous flow along the articular surfaces between the joints. The Beavers and Joseph slip conditions which are a specific boundary condition for the synovial fluid are used to solve the governing system of partial differential equations for the synovial fluid and the results are provided here. Through the development of an analytical solution and numerical simulation (using the finite volume approach) it is hoped that the mechanisms of nutritional transport into the synovial joint will be better understood. According to the data the average concentration has a negative connection with both the axial distance and the duration spent in the experiment. Many graphs have been utilized to gain understanding into the problem’s various characteristics including velocity and concentration, among others. Hyaluronate (HA) is considered to be present in porous cartilage surfaces and the viscosity of synovial fluid fluctuates in response to the amount of HA present.

The potential of a partial differential equations model is to anticipate its computational behavior. The simulation of a transient 3D coupled convection–diffusion system using a numerical model is described. The main objective of this article is to offer effective limited contrast compact finite difference techniques for use with nonlinear coupled partial differential systems that mimic overseeing differential frameworks. The three-dimensional compact finite difference formulation serves as the model’s foundation. An analytical model has been used to validate finite difference techniques that are numerically compact. By examining the consistency and union of the arrangement, which may be seen from figures and information tables, we can evaluate the model and the suggested numerical schemes. The schemes are unconditionally stable and accurate up to two orders in time and six orders in space, according to the results of the stability and accuracy tests. The implicit method used in the algorithm’s design was examined for stability criteria. Because mesh-independent solutions for non-linear differential systems are expensive, block tridiagonal matrix structures, measured in terms of L2 and L∞ norms, which are inherent characteristics of schemes and have excellent agreement with the investigation arrangement, are created.

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