  Steady blood flow, or Poiseuille flow, through compressed or defective blood
vessels is a critical issue in hemodynamics, particularly in cardiovascular
studies. This research explores a tube with a bipolar cross-section, simulating
the geometry of a bicuspid aortic valve (BAV) during an oval systolic opening.
The BAV, typically featuring two cusps instead of the usual three found in
normal tricuspid configurations, introduces unique hemodynamic challenges. As
the most prevalent congenital heart defect, BAV increases the risk of aortic
dilation and dissection. A bipolar cross-sectional analysis provides a more
accurate geometric approximation for modeling flow through these atypical valve
shapes, crucial for understanding the specific fluid dynamics associated with
BAV. We derived an exact solution for the governing equations of Poiseuille
flow within a bipolar cross-sectional tube, including velocity field, flow
rate, and wall shear stress (WSS). The velocity profiles for BAV show
remarkable agreement with previous studies using coherent multi-scale
simulations, consistently demonstrating a jet-like flow structure absent in
tricuspid aortic valve (TAV) scenarios. Analysis reveals that at the center of
the entrance, BAV blood flow velocity is significantly higher than TAV but
decreases more rapidly towards the vessel wall, creating a steeper vertical
velocity gradient and resulting in higher WSS for BAV. Additionally, the WSS,
inversely proportional to sin({\xi}*), where {\xi}* represents the bipolar
coordinate at the wall boundary, exceeds that found in a circular cylindrical
tube with an equivalent diameter. In cases of aortic valve stenosis, where
{\xi}* approaches {\pi}, the WSS increases rapidly. This elevated WSS, commonly
observed in BAV patients, may detrimentally impact the aortic wall in these
structurally abnormal valves, particularly within the ascending aorta.
