Deformable tubes with various cross sectional shapes: Geometric and rheological properties

Pipes play crucial roles in rheology analysis, aiding us in understanding and predicting the behavior of various real-world flows. This study focuses on mathematically representable tubular shapes with non-circular cross-sections, controlled by four key parameters that define their form. By varying these parameters, we introduce a versatile collection of deformable pipes distinct from the traditional circular or elliptical ones commonly studied. Here, we derive an analytical formula that characterizes the topological nature and velocity profiles within these diverse shapes. Subsequently, we explore the streamlines and wall shears experienced along their surfaces. We then delve into a detailed analysis of crucial geometric parameters such as cross-sectional area, perimeter, volumetric flow rate, and Poiseuille number, encompassing a wide range of pipe shapes, from oval to hexagonal. Notably, a subset of the explored pipes features collapsible hyperbolic profiles, particularly relevant in engineering applications involving the transport of biochemical substances.