*2.1. Rapid Instantaneous Increment of Pressure Gradient (Roiti-Gromeka-Szyma ´nski Solution)*

As mentioned in the introduction, the accelerated flow of liquid in pipes (Figure 1) has been the subject of many studies. All the authors who successfully derived the final solution started from the dynamic equation of motion first derived by Navier [1]:

$$\frac{\partial v}{\partial t} = -\frac{1}{\rho} \frac{\partial p}{\partial x} + \nu \left( \frac{\partial^2 v}{\partial r^2} + \frac{1}{r} \frac{\partial v}{\partial r} \right) \to \frac{\partial v}{\partial t} = G + \frac{\nu}{r} \frac{\partial}{\partial r} \left( r \frac{\partial v}{\partial r} \right) \tag{1}$$

They assumed the incompressibility of the liquid and that the velocity field is only dependent on the pipe radius and time *v* = *v*(*r*, *t*). This means that the problem is simplified to isothermal flow in long, constant inner diameter pipelines, where the entrance effects can be neglected. Other boundary conditions are:

(a) flow starts initially from rest:

$$v(r,0) = 0,\tag{2}$$

(b) no-slip condition as viscous fluid in contact with a rigid wall will adhere to the wall due to the effects of viscosity [43]. In the analyzed problem, the solid pipe wall boundary velocity is assumed to be equal to zero:

$$v(\mathbb{R}, t) = 0,\tag{3}$$

(c) final velocity profile consistent with the parabolic Hagen–Poiseuille equation:

$$v(r,\infty) = v\_{\max} \left( 1 - \left(\frac{r}{R}\right)^2 \right) \tag{4}$$

where: *vmax* <sup>=</sup> <sup>−</sup>*R*<sup>2</sup> 4*μ ∂p <sup>∂</sup><sup>x</sup>* <sup>=</sup> <sup>Δ</sup>*<sup>p</sup> <sup>L</sup> <sup>R</sup>*<sup>2</sup> <sup>4</sup>*<sup>μ</sup>* <sup>=</sup> *<sup>G</sup> <sup>R</sup>*<sup>2</sup> <sup>4</sup>*<sup>ν</sup>* ; Δ*p* = *po* − *pi*, *po*—outlet pressure, *pi*—inlet pressure. Maximal velocity occurs at the pipe axis (*r* = 0).

(d) sudden imposition of a pressure gradient (Figure 2a):

$$\begin{cases} \text{ } G = 0 \text{ } for \, t < 0\\ G = G\_{\infty} = -\frac{1}{\rho} \frac{\Delta p}{L} \, for \, t \ge 0 \end{cases} \tag{5}$$

The literature review in the Introduction indicates that the final solution was found a relatively long time ago by at least three authors. Roiti [5] studied theoretically and experimentally the unsteady flow of liquid inside a vertically fixed cylindrical pipe, with its gravitational outflow to the atmosphere. He carried out the integration of general formulas using the same method by which his senior colleague Prof. Betti [6] determined the temperature distribution acting on the cylinder. Roiti was the only researcher who, in addition to formulas for the flow velocity, also derived formulas for the displacement of fluid elements during this movement.

**Figure 2.** Changes of pressure gradients: (**a**) step change from rest, (**b**) step change from initial to final value, (**c**) linear increase (**d**) ramp change.

Another researcher dealing with this phenomenon, albeit excluding gravitational forces (flow in the horizontal duct), was Gromeka [7]. The work in which he derived the final solution for this type of flow was his doctoral thesis. Gromeka also based his derivation on the analogy existing between some problems related to the movement of viscous fluids and those known from the theory of heat conduction. He carefully used the method of solving heat distributions in circular cylinders proposed by Poisson in his dissertations on "Mémoire sur la distribution de la chaleur dans les corps solides" [44]. In addition to deriving the final formula for the velocity profile (for step gradient change Figure 2a), Gromeka proved that the flow which is forced by a pulsating pressure gradient also for large times tends towards the Hagen–Poiseuille flow. Gromeka also derived formulas for accelerated flow with possible slippage of liquid elements on the walls of the pipe. Both Gromeka's and Roiti's solutions remained unnoticed by the world of science for a long time, likely due to a combination of being published in scientific journals by the researchers' home universities and the slow movement of information at this time. Over a hundred years ago, there was no internet, and materials of this type were not widely available (in contrast to today's open-access papers). Interestingly, Roiti's solution was not quoted in

the work of Allievi [45] or that of other Italian (and non-Italian) researchers of unsteady flows, e.g., Fasso [46], Aresti [47].

Fifty years after the publication of Gromeka's thesis, and sixty after the publication of Roiti's work, this interesting topic was addressed by a young Polish researcher, Piotr Szyma ´nski, who had just defended his doctoral thesis. Szyma ´nski, unaware of the work of his predecessors, during a one-year training course in Paris in 1928, applied the theory of the Fourier–Bessel series to find an analytical solution to this problem. In addition, after deriving the final formula, he conducted a number of mathematical studies on the continuity of its derivatives (using Abel transformations). It is worth mentioning that Szyma ´nski in his work [13] analyzed, similarly to Gromeka, the pulsating nature of the flow, and this took place long before Womersley or Uchida's work. In the case of pulsating flow, Szyma ´nski analyzed the derivatives of the presented general solution of this problem with the help of the Fourier series theory as well as on the basis of M. Lebesque's theorem. Like Gromeka, however, he did not define the final solution to this problem, which is known today. It is also worth adding here that Szyma ´nski's main work was preceded by the publication of a conference paper of the 3rd International Congress of Applied Mechanics held in Stockholm in 1930 [48]. Among the distinguished participants of this congress was Theodore von Kármán.

Due to the fact that the analyzed solution, as can be seen from the above discussion, was derived independently by three researchers, it is suggested to define this solution with the abbreviation RGS derived from the first letters of the surnames of the authors. Leaving aside the tedious details of the derivations, the final solution of the system of Equation (1), assuming a step change in pressure, is the following function:

$$v(r,t)\_{RGS} = v\_{\max} \left[ \left[ 1 - \left(\frac{r}{R}\right)^2 \right] - 8 \sum\_{n=1}^{\infty} \frac{J\_0\left(\lambda\_n \frac{r}{R}\right)}{\lambda\_n^3 \cdot J\_1\left(\lambda\_n\right)} e^{-\lambda\_n^2 \frac{v}{R^2}t} \right],\tag{6}$$

where *λ<sup>n</sup>* are the nth zero, or root, of a Bessel function *J*0(*λn*).

An exemplary flow that develops according to the RGS solution described by Equation (6) is presented in Figure 3.

**Figure 3.** Development of velocity profile in RGS-accelerated flow driven by a step change of pressure gradient.

Gerbes in the 1950s [17] developed the same accelerated solution with help of the Laplace technique; in his work, he also presented a solution for decelerated flow as well as a solution for pulsating flow. The generalization of Equation (6) for the vertical (upward and downward) as well as arbitrary sloping pipes was presented recently by

Urbanowicz et al. [49]. The maximal velocity for this case depends on the pipe slope angle *β* as follows (Figure 4):

$$
\sigma\_{\max} = \frac{R^2}{4\mu L} (\Delta p - g\rho L \sin \beta). \tag{7}
$$

From Equation (7), it follows that in the case of upward vertical flow where *β* = 90◦ : *vmax* = *<sup>R</sup>*<sup>2</sup> <sup>4</sup>*μ<sup>L</sup>* (Δ*p* − *gρL*), while in the case of downward vertical flow where *β* = −90◦, the maximal final velocity has the largest possible value equal to *vmax* = *<sup>R</sup>*<sup>2</sup> <sup>4</sup>*μ<sup>L</sup>* (Δ*p* + *gρL* sin *β*).

**Figure 4.** Definition of pipe slope angle (adapted from [49]).

Knowing the solution for velocity distribution at the pipe cross–section (Equation (6)) enables us to calculate the following important parameters describing unsteady accelerated pipe flow:


The final form for the RGS-type accelerated flow is [49]:

$$\begin{cases} \upsilon\_{m,RGS} = \upsilon\_{\infty} \left( 1 - 32 \sum\_{n=1}^{\infty} \frac{\varepsilon^{-\lambda\_n^2 \ell}}{\lambda\_n^4} \right) \\ \qquad \tau\_{w,RGS} = \frac{4\mu \tau\_{\infty}}{R} \left( 1 - 4 \sum\_{n=1}^{\infty} \frac{\varepsilon^{-\lambda\_n^2 \ell}}{\lambda\_n^2} \right) \\ \quad f\_{RGS} = \frac{64}{R\varepsilon\_{\infty}} \left[ 1 - 4 \sum\_{n=1}^{\infty} \frac{\varepsilon^{-\lambda\_n^2 \ell}}{\lambda\_n^2} \right] \cdot \left[ 1 - 32 \sum\_{n=1}^{\infty} \frac{\varepsilon^{-\lambda\_n^2 \ell}}{\lambda\_n^4} \right]^{-2} \end{cases} \tag{8}$$

Ito [18] generalized the above solution for the case where the initial velocity of the fluid is not zero, which means a steady Hagen–Poiseuille flow is a starting point, and next an instantaneous pressure gradient change takes place from an initial value equal *G*<sup>0</sup> to a new final value *G*∞ (Figure 2b). In this case, the final solution is steady-state Hagen–Poiseuille flow but represented with a higher maximal velocity (in the axial position). The analytical solution obtained for this extended case is:

$$v(r,t)\_{RGSI} = \frac{G\_{\infty}(R^2 - r^2)}{4\nu} - \frac{2R^2(G\_{\infty} - G\_0)}{\nu} \sum\_{n=1}^{\infty} \frac{J\_0(\lambda\_n \frac{r}{R})}{\lambda\_n^3 \cdot J\_1(\lambda\_n)} e^{-\lambda\_n^2 \frac{\nu}{R^2}t} \tag{9}$$

The RGS analytical solution was initially verified experimentally by Letelier and Leutheusser [50] and in more detail by van de Sande et al. [38], Baibikov et al. [8] and Lefebvre and White [39]. In all the mentioned experimental works, good agreement between the RGS solution and the experimental results was indicated. As it is known from the standard textbooks on steady pipe flow, the critical Reynolds number is about 2320 [14]. Below this value laminar flow takes place. For Reynolds numbers of up to 3000, transitional flow occurs, and above that value, turbulent flow can be assumed. Lefebvre and White [39] noticed that the laminar flow persisted in their experimental runs up to very high values of Reynolds numbers in the range between 2·105 and 5·105. In another study by van de Sande et al. [38], a lower value (Re = 57,500) was found. In both cases, however, it can be seen that the values of the critical Reynolds number exceeded the critical one in steady-state flow. From a paper that is a continuation of the Lefebvre and White study [51], a transitional Reynolds number formula has been proposed *Ret* ≈ <sup>450</sup>*D a*/*ν*<sup>2</sup> 1/3, in which *a* is acceleration, which was constant during experiments. A different formula based on the Knisely et al. experiments [52] is *Ret* ≈ 1.33*D a*/*ν*<sup>2</sup> 1/31.86 .
