*2.2. Other Solutions Driven by a Gradient Change*

Ito [18], in addition to extending the RGS solution of Equation (9), has developed the solution for the case when the pressure gradient given to a pipe begins to increase or decrease in proportion to the time (gradient change scenario from Figure 2c):

$$\begin{cases} \quad t < 0 \text{ then } -\frac{1}{\rho} \frac{\Delta p}{L} = G\_0\\ t \ge 0 \text{ then } -\frac{1}{\rho} \frac{\Delta p}{L} = \varepsilon t + G\_0 \end{cases} \tag{10}$$

The final analytical solution in the Ito (I) case is:

$$v\_I = \frac{\left(\varepsilon t + G\_0\right)\left(R^2 - r^2\right)}{4\nu} - \frac{\varepsilon \left(R^2 - r^2\right)\left(3R^2 - r^2\right)}{64\nu^2} + \frac{2R^4\varepsilon}{\nu^2} \sum\_{n=1}^{\infty} \frac{J\_0\left(\lambda\_n\frac{f}{R}\right)}{\lambda\_n\frac{5}{4}J\_1\left(\alpha\_n\right)} e^{-\lambda\_n\frac{2}{R^2}t}.\tag{11}$$

There is a problem with the above equation regarding how the value of *ε* function shall be calculated. The analysis of this equation shows that the lower the value of the *ε* coefficient (representing jerk as its unit is [m/s3]), the smaller the angle of inclination of the increasing gradient time curve, which will physically mean a larger time required to obtain the selected reference average value of the flow velocity.

The review of analytical solutions in this group revealed that among all derived analytical solutions, there is not one in which the ramp pressure gradient change is gradual, as shown in Figure 2d. Such a solution seems desirable because in practice there are no technical possibilities to change the pressure gradient in an instant way (for example, the valve opening time in the work of van de Sande et al. was about 0.1 s). In real systems, the change of pressure gradient will be strongly related to the valve opening time.

Avula [53] and Avula and Young [54] indicated that changes in velocity during accelerating flows in real systems occur differently than described by the above classical RGS theory. During their research, they experimentally recorded the pressure gradient histories (for selected values of Reynolds numbers—see Figure 5), then they described their mathematical form in an approximate way:

$$f(t\_A) = \frac{1}{2}\frac{dp^\*}{d\p} = a\_1(1 - e^{-a\_2t\_A}) + a\_3t\_A^{\
u\_4}e^{-a\_5t\_A}.\tag{12}$$

where *a*1, ... , *a*<sup>5</sup> are constants calibrated with the reference to experimental results and *x*ˆ, *p*∗ and *tA* are the normalized axial coordinate, pressure and time, respectively.

**Figure 5.** Exemplary variation of pressure gradient with time *Re* ≈ 1600.

The final modified semi-analytical Avula's solution for the velocity profile is:

$$\begin{split} \upsilon^\* = 2 \sum\_{n=1}^{\infty} \frac{l\_0(\eta \circ \frac{\mathfrak{z}}{\mathfrak{X}})}{\eta\_n^3 \cdot l\_1(\eta\_n)} \left\{ \frac{a\_1 \mathrm{Re}}{2 \eta\_n^2} \left( 1 - \exp \left( -2 \frac{\eta\_n^2}{\mathrm{Re}} t\_A \right) \right) + \frac{a\_1 \mathrm{Re}}{a\_2 \mathrm{Re} - 2 \eta\_n^2} \left( e^{-a\_2 t\_A} - e^{-2 \frac{\eta\_n^2}{\mathrm{Re}} t\_A} \right) + \\\ a\_3 e^{-2 \frac{\eta\_n^2}{\mathrm{Re}} t\_A} \int\_0^{t\_A} u^{a\_4} e^{-(a\_5 u + 2 \frac{\eta\_n^2}{\mathrm{Re}} u)} du \right\}, \end{split} \tag{13}$$

where dimensionless time *tA* = *t vm*,∞ *<sup>R</sup>* , *vm*,<sup>∞</sup> <sup>=</sup> <sup>−</sup>*R*<sup>2</sup> 8*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>8</sup>*<sup>μ</sup>* is the final steady-state mean velocity and *η<sup>n</sup>* is a consecutive zero of a Bessel function of zero order *J*<sup>0</sup> <sup>√</sup>*sRe*/2 = 0.

As the third right-hand side term of this solution (Equation (13)) cannot be evaluated explicitly, this solution needed to be evaluated partially numerically. This numerical solution significantly limits its practical application. A different mathematical form of the function *f*(*tA*) to describe changes in the pressure gradient over time should be found. Such a new function should be integrable to give a complete analytical solution for a wide range of final Reynolds numbers. Illustrative comparisons of the results obtained with the classical RGS model and Avula's solution are shown in Figures 6 and 7.

**Figure 6.** Velocity profiles comparison of RGS and Avula solutions for relatively: (**a**) small times; (**b**) large times (adapted from [53]).

**Figure 7.** Time variation of discharge.

It can be seen from the comparisons in Figures 6 and 7 that a large discrepancy in the profile dynamics is noticeable for small values of the dimensionless time *tA* < 10, while for larger values *tA* > 200, the Avula solution begins to catch up with the classical RGS solution.

A similar solution to the above was analyzed by Smith [55], who in his derivation referred to some of the solutions discussed in the classical textbook "Hydrodynamics" by Dryden et al. [56]. Smith's solution was obtained for the following pressure gradient:

$$\frac{\partial p}{\partial \mathbf{x}} = b\mu k^2 e^{-\nu k^2 t},\tag{14}$$

where *k* and *b* are positive constants and *μ* and *ν* are the dynamic and kinematic viscosities, respectively. The final Smith's (SM) solution is as follows:

$$w(r,t)\_{SM} = b \left\{ 1 - \frac{J\_0(kr)}{J\_0(kR)} \right\} e^{-\nu k^2 t} + 2bk^2 \sum\_{n=1}^{\infty} \frac{J\_0\left(\lambda\_n \frac{r}{R}\right) e^{-\lambda\_n^2 \frac{r}{R^2} t}}{\lambda\_n \left(\frac{\lambda\_n^2}{R^2} - k^2\right) J\_1(\lambda\_n)}.\tag{15}$$

When deriving the above solution, it was assumed that the pressure gradient appears at time *t* = 0 and then gradually decays exponentially to zero (and not to a constant value). Such a flow cannot be treated strictly as accelerated as two different periods occur. Firstly, liquid accelerates until the fluid reaches the maximal mean velocity, while after that it starts to decelerate until it comes back to rest again. That is why it is not a subject of the present review. However, this analytical solution is presented and discussed shortly, mainly because it has an interesting feature when the forced decay rate is equal to one of the natural decay rates. It is analogous to the feature experienced in the classical problem regarding oscillations of a linear pendulum in the case when the forcing frequency is equal to the natural frequency (there are two distinct singularities in this final solution).

Other problems with the accelerated pipe flow solution of the RGS type have been reported by Otis [35]. This author noticed that, usually, the pressure gradient does not remain constant but rather it is the total head *Hi*, which is schematically presented in Figure 8.

**Figure 8.** Otis reference drawing indicating time variation of energy grade line (adapted from [35]).

During a flow acceleration, some portion of this head is utilized to establish the kinetic head of the flow. The above is illustrated in Figure 8 in which there are three energy grade lines: (a) just after the instantaneous valve opening, (b) at a later time and (c) after reaching steady-state flow (Hagen–Poiseuille steady state). During flow development, the pressure gradient along the pipe is diminished because of the observed velocity head development. With the above in mind, the main Navier equation becomes non-linear and it is necessary to use numerical methods to model it. Otis [35] presented experimental results (of accelerated flow) of an unnamed researcher and a realistic numerical example of a start-up for SAE 10W-30 oil driven by a head of 1.053 m in a 10-cm diameter pipe with a length of 6.25 m. The main conclusions were that the final steady-flow will occur in less than half the time predicted by the RGS solution. In the same paper, Otis developed a start-up parameter in the following form:

$$M \equiv \frac{G\_0 D^4}{2048\nu^2 \rho L'} \tag{16}$$

where *G*<sup>0</sup> = *γHi*/*L*. Otis concluded that analysis of this parameter revealed that when *M* < 1, the wall shear stress rises monotonically with time, while for *M* > 1, the shear stress overshoots the steady-state value. In addition, Otis stated: "Such behavior resulting from the fact that the time constant for flow start-up decreases with *M*, whereas the time constant for boundary layer growth is independent of *M*." Otis's correction was verified numerically by Singh [57], who derived a corrected equation that he did not solve analytically but led to a numerical form suitable for the creation of a computer program and succeed numerical calculations. Patience and Mehrota [58] wrote the Otis start-up parameter in a much simpler form:

$$M \equiv \frac{Re \cdot D}{64L} \tag{17}$$

They also noted that both Szyma ´nski [13] and Otis [35] overlooked the hydrodynamic developing region effect in their works. Additionally, Patience and Mehrota proved that in the analyzed comparison start-up cases by Otis, the flow was not fully developed. Their conclusion was based with help of the Fargie and Martin *ξ* coefficient [59]. Andersson and Kristoffersen also dealt with this problem. In [34], they demonstrated that the start-up flow Otis parameter *M* defined by Equation (16) is the crucial parameter also when the entrance effects are taken into account. For long pipes (*M* = 0), the Anderson and Kristoffersen numerical solution differs by less than 1% from the exact analytic solution obtained by RGS. For shorter pipes, these authors observed that the start-up period and the resulting steadystate flow rate are significantly reduced due to the entrance region effects. When *M* = 0.5 the start-up time and ultimate flow rate are reduced by about 55% and 40%, respectively, compared with the classical solution of RGS derived for long pipes. In view of the above

comments and discussed works, Patience and Mehrota [60] introduced a correction of the method proposed by Otis.
