In this article, we derived mathematical models representing blood momentum and the haematocrit in dimensional forms. These models were further scald into dimensionless forms using derived dimensionless quantities. The governing models were converted from partial differential equations to ordinary differential equations considering dense and fully developed flow. In addition, the Laplace method was deployed to solve the dimensionless system and the blood velocity function was obtained with some pertinent parameters such as haematocrit and pressure parameters respectively. The numerical simulations were carried out using Wolfram Mathematica, version 12, varying pertinent parameters within a specific range on the blood velocity and volumetric flow rate profiles. The results depict the effects of haematocrit and pressure parameters on blood velocity as well as flow rate. Specifically, blood velocity varied significantly at different pressure levels; however, it decreased as blood pressure increased due to increased resistance to flow and the narrowing of blood vessels (vasoconstriction), which leads to an increase in boundary layer thickness. Furthermore, the analysis showed a positive correlation between Hct and blood pressure as higher Hct results in increased blood viscosity, potentially elevating peripheral resistance and blood pressure. We also discovered that the blood velocity attained different magnitudes at different haematocrit counts, indicating an indirect relationship between the haematocrit (Hct) and the blood velocity. This indicates that the blood velocity adapts a high altitude increment for an increasing haematocrit. The volumetric flow rate is equally seen to be affected by the aforementioned parameters. This study may serve as a guide in understanding blood flow challenges and offer management advice to patients particularly when hospital data are available for both mathematicians and clinicians alike.