## 2 Materials and Methods  
### The Law of Hagen and Poiseuille and the Richardson Effect  
Flow conditions can be characterized by the dimensionless Reynolds number, which gives a ratio between inertial and viscous forces. A flow is laminar if the Reynolds number \(Re=\frac{\rho\
abla D}{\eta}\) is below approximately 2300. Here, \(\rho\) is the density of the fluid, \(v\) is the axial fluid velocity, \(D\) is the pipe's diameter, and \(\eta\) is the dynamic viscosity. In this regime, the velocity profile across the pipe's cross-section adopts a parabolic shape, with its vertex at the pipe center, and wave dissipation is disregarded. A critical aspect of this flow scenario is the no-slip condition at the pipe wall [17], which dictates that the fluid velocity at the wall is zero. Under these conditions, the Hagen-Poiseuille law [1] governs the flow:  
\[Q=\frac{\pi R^{4}}{8\eta L}\Delta p=\frac{1}{R_{\rm H}}\Delta p \tag{1}\]  
This formula provides the volumetric flow rate \(Q\) given the pressure difference \(\Delta p\), with \(R_{\rm H}\) representing the hydraulic resistance, dependent on the pipe radius \(R\), fluid viscosity \(\eta\), and pipe length \(L\). The relevance of hydraulic resistance will be explored further in later sections.  
In 1929, [PERSON] [2] conducted experiments on unsteady airflow through a pipe generated by sinusoidal pressure fluctuations. Using a hot wire anemometer, he measured the radial velocity profile. He found that, within the laminar flow range of Reynolds numbers, increasing pressure frequencies caused the velocity profile to deviate from a parabolic shape, as shown on the left side of Figure 1. In this figure, the Womersley number \(Wo=R\sqrt{\omega/\
u}\) is the root of the normalized frequency of the harmonically oscillating pressure input. It is based on the assumption of an incompressible fluid [5], where \(R\) is the pipe radius, \(\
u\) the kinematic viscosity, and \(\omega\) the pressure variation frequency.  
As the profile changes, it transitions to a flatter, more uniform shape in the center, with peaks near the wall and a sharp drop to zero at the boundary. This unsteady profile resembles those found in steady, turbulent flows. The right side of Figure 1 illustrates the variation in the radial velocity profile over the phase of the harmonically oscillating pressure input for a Womersley number where the [PERSON] effect becomes apparent (\(Wo=30\)).  
Building on [PERSON]'s findings, [PERSON] [18] derived solutions for the instantaneous velocity profile as a function of the normalized radial position and the Womersley number. [PERSON] [5] performed the nondimensionalization of the axes in this context. The system equations developed in this paper allow for an analytical expression of the velocity profile transformation from parabolic to the one observed in [PERSON]'s experiments, thus extending the applicability of the presented equations beyond traditional laminar flow with a parabolic profile.  
Since the early 20 th century, various fluid behavior models have been developed, primarily from the fundamental Navier-Stokes equations. These models often incorporate simplifying assumptions to make them more computationally feasible [19; 20; 21; 22]. Typical assumptions include liquid incompressibility, neglecting heat transfer effects, and the assumption of uniform pressure distribution across the pipe's cross-section.  
[PERSON] reviewed and categorized these models in his work [5; 23], organizing them according to dimensionality and the effects considered. This paper's general governing equations are those of the two-dimensional viscous compressible model. The volumetric flow rate equation is derived from this general framework.