Energy and flow conjuncture in knee joint synovial fluid: A Brinkman-based numerical study

Abstract

In this article, we explore the modeling of the synovial membrane, which plays an essential role in regulating the flow of synovial fluid, ensuring proper lubrication, facilitating nutrient transport, and removing waste within the knee joint. The flow of synovial fluid, a non-Newtonian fluid containing large hyaluronan molecules, is intricately influenced by the properties of the synovial membrane, which acts as a porous medium. To explore this dynamic, the Brinkman equation, an extension of Darcy’s law, is utilized for the first time to study the synovial membrane, introducing a novel approach to the analysis. This equation is particularly relevant as it accounts for both viscous forces and the permeability of the membrane, allowing for a more accurate representation of fluid behavior in regions where synovial fluid interacts with the porous membrane. Additionally, the energy equation is critical in understanding how heat transfer influences synovial fluid dynamics. Within biological joints, temperature variations can occur due to metabolic processes, friction from movement, or external factors such as injury or inflammation. These temperature differences have a direct impact on the fluid’s viscosity and the membrane’s permeability, both of which are central to regulating fluid movement. When we consider the flow in the x-direction, it is largely governed by factors such as permeability, heat transfer properties, and viscous resistance within the fluid. In contrast, flow in the y-direction introduces an additional component, buoyancy forces driven by temperature gradients. These forces, characterized by the Grashof number, interact with the flow and modify its behavior in the vertical direction, where natural convection due to temperature differences may complement or oppose the flow driven by external forces. The Peclet number, derived from the energy equation, further highlights the balance between convective and diffusive heat transfer. This interplay is crucial in understanding how heat generated during joint movement, or from external sources, affects both the temperature profile and the fluid flow within the joint. By incorporating these non-dimensional numbers, the Grashof number, Darcy number, Peclet number, and Reynolds number into the modeling framework, we gain a deeper understanding of the complex mechanisms that govern synovial fluid movement and heat distribution within the knee joint. These numbers provide valuable insight into how external factors such as temperature, fluid viscosity, and joint movement interact, allowing for a more comprehensive study of both normal physiological conditions and pathological scenarios, such as inflammation or joint degeneration.