Analysis of power law fluid for nutritional transport in the human capillary

This paper accords with the theoretical analysis of blood flow in human capillaries, focusing on nutrient exchange from the tissue to the capillary region. The proposed mathematical model employs the Krogh capillary-tissue exchange system, where the pressure gradient at the capillary entrance is the driving mechanism. The power-law fluid model characterizes blood rheology. The formulated partial differential equations are simplified using lubrication approximation theory and then solved for exact solutions. The diffusion equation is solved through numerical computation in PYTHON using finite difference method. The Darcy number Da, pressure gradient at the capillary channel entrance ξ, vertical velocity in the porous region V_w, slip parameter σ, reference pressure P₀, fluid behavior index n, diffusion coefficient Λ, the rate of nutrient production/consumption M, and partition coefficient η control the dynamics of the flow. The increase in ξ enhances velocity and causes a diminishment in pressure in the capillary region. ξ only plays a crucial role in driving flow in the capillary region while the concentration of nutrients in the tissue region is independent of it. The velocity, pressure and concentration of nutrients decrease with an increase in n. The concentration of nutrients increases with increasing M and η while it decreases with increasing Λ. When blood is characterized by the shear-thinning power-law fluid, it transports more of nutrients from tissue regions and flows at a higher velocity, necessitating a higher level of pressure to maintain the flow.