*3.2. Other Flow Rate Solutions and Comments*

Das and Arakeri [36] prepared and made experiments in which the motion was generated by a piston. In this case, the pressure is unknown and determined indirectly by the piston motion. Assuming incompressible flow, the piston motion is felt immediately at each cross-section of the pipe and the volume flux at any cross-section corresponds to the volume flux due to piston motion:

$$2\pi \int\_0^R rv dr = v\_p(t)\pi R^2\tag{34}$$

where *vp* is a piston valve. Similarly, as in Anderson and Tiseth and pressure-gradientdriven flows, the main equation is the Navier momentum equation defined by Equation (1). Using the above assumption, the solution obtained for the reverse flow that was firstly accelerated from the rest, then was constant and finally decelerated to rest, as presented

in Figure 11, will be investigated below. During the piston acceleration (0 ≤ *t* ≤ *t*0) the solution is:

$$\begin{split} v\_{DA} &= \frac{v\_p}{t\_0} \Bigg[ 2t \left( 1 - \left( \frac{r}{R} \right)^2 \right) + \underbrace{\frac{R^2}{\nu} \left( \frac{1}{8} \left( 1 - \left( \frac{r}{R} \right)^4 \right) - \frac{1}{6} \left( 1 - \left( \frac{r}{R} \right)^2 \right) \right)}\_{T\_1} \\ &+ 2v\_p \frac{R^2}{\nu} \sum\_{n=1}^{\infty} \frac{I\_0(a\_n) - I\_0(a\_n \frac{r}{R})}{a\_n^3 I\_1(a\_n)} \cdot \underbrace{\exp \left( -a\_n^2 \frac{\nu}{R^2} t \right)}\_{T\_2} \end{split} \tag{35}$$

where: *vp*—piston velocity during the time *t*<sup>0</sup> to *t*1.

**Figure 11.** Typical trapezoidal variation of piston velocity (motion) with time.

For a constant piston velocity (*t*<sup>0</sup> ≤ *t* ≤ *t*1):

$$w\_{DA} = 2v\_p \left(1 - \left(\frac{r}{R}\right)^2\right) + v\_p \frac{2R^2}{\nu} \sum\_{n=1}^{\infty} \frac{J\_0(a\_n) - J\_0(a\_n \frac{r}{R})}{a\_n^3 J\_1(a\_n)} \left(\underbrace{T\_2 - \underbrace{\exp\left(-a\_n^2 \frac{\nu}{R^2}(t - t\_0)\right)}\_{T\_3}}\_{T\_3}\right) \tag{36}$$

During piston deceleration (*t*<sup>1</sup> ≤ *t* ≤ *t*2):

$$w\_{DA} = 2v\_p \frac{R^2}{\nu} \sum\_{n=1}^{\infty} \frac{J\_0(a\_n) - J\_0(a\_n \frac{r}{R})}{a\_n^3 j\_1(a\_n)} \left( \underbrace{T\_3 - \frac{\exp\left(-a\_n^2 \frac{\nu}{R^2} (t - t\_1)\right)}{t\_2 - t\_1}}\_{T\_4} \right) + 2v\_p \frac{t\_2 - t}{t\_2 - t\_1} \left(1 - \left(\frac{r}{R}\right)^2\right) - \frac{R^2}{\nu} \frac{v\_p}{t\_2 - t\_1} T\_1 \tag{37}$$

and after the piston has stopped (*t*<sup>2</sup> ≤ *t* ≤ ∞):

$$v\_{DA} = 2v\_p \frac{R^2}{\nu} \sum\_{n=1}^{\infty} \frac{f\_0(a\_n) - f\_0\left(a\_n \frac{r}{R}\right)}{a\_n^3 f\_1(a\_n)} \left(T\_4 + \frac{\exp\left(-a\_n^2 \frac{\nu}{R^2}(t - t\_2)\right)}{(t\_2 - t\_1)}\right) \tag{38}$$

where: *αn*—zeros of the second-order Bessel function *J*2(*αn*), *vp* is piston velocity, *t*<sup>0</sup> is the time of acceleration, *t*<sup>1</sup> is the time when a stepper motor is switched off and *t*<sup>2</sup> is the time when piston motion stops.

Solutions for the piston acceleration Equation (35) and constant piston velocity Equation (36) when *t*<sup>1</sup> → ∞ form an analytical solution for the ramp change of velocity.

Das and Arakeri pointed out after analyzing the flow profiles for the piston deceleration phase that the velocity profiles close to the walls can be in the opposite direction to the

core flow. So, this type of unsteady flow does have inflection points and hence can become unstable at relatively low Reynolds numbers.

A comparison of the development of flow profiles in the Anderson and Tiseth solution (continuous lines) and Das and Arakeri ramp change version (the dotted lines obtained with the help of Equations (35) and (36)) is presented in Figure 12. It can be seen that in the final phase from the dimensionless time ˆ*t* = 0.05 both solutions start to approach each other. The Das and Arakeri solution (DA model) starts from the rest value, not as in the AT model from the non-physical mean velocity *v*ˆ*<sup>m</sup>* = 0.5.

**Figure 12.** Comparison of Das and Arakeri model (dotted lines; ˆ*t*<sup>0</sup> = 0.01) with Anderson and Tiseth model (solid lines): (**a**) <sup>ˆ</sup>*<sup>t</sup>* ≤ 0.01, (**b**) <sup>ˆ</sup>*<sup>t</sup>* ≥ 0.01.

Kannaiyan et al. [37] further generalized the AT and DA solutions introducing a condition of double-step changes in the flow rate:

$$Q(t) = v\_m(t)A = \begin{cases} 0; \ t \le 0\\ Q\_i = v\_{m,i} \cdot A; \ 0 < t \le t\_c\\ Q\_{\infty} = v\_{m,\infty} \cdot A; \ t\_c < t \le \infty \end{cases} \tag{39}$$

where: *vm*,*i*—initial mean velocity and *vm*,∞—final mean velocity of flow.

With this assumption, it was possible to derive the solution for acceleration from one steady state (or developing one) to another steady flow. This means that in this solution, compared to the Andersson and Tiseth, and Das and Arakeri, solutions, the flow does not need to start accelerating from rest.

The final solution for the axial velocity profile is:

$$v(r,t)\_{KVN} = 2v\_{m,\infty} \left[ \left(1 - \left(\frac{r}{R}\right)^2\right) - \sum\_{n=1}^{\infty} \frac{4}{a\_n^2} \left[1 - \frac{J\_0(a\_n r\_\mathbb{Z}) - J\_0(a\_n)}{J\_0(a\_n)}\right] \left[v\_{m,\infty} + v\_{m,i} \left(e^{-a\_n^2 \frac{\nu}{R^2} t\_i} - 1\right)\right] e^{-a\_n^2 \frac{\nu}{R^2} t}\right] \tag{40}$$

The solution for time-dependent pressure gradient in this case takes the following form:

$$\left(\frac{\partial p}{\partial \mathbf{x}}\right)\_{KVN} = \frac{4\mu\upsilon\_{m,i}}{R^2} \sum\_{n=1}^{\infty} \left[1 - e^{-\mathbf{a}\_n^2 \frac{\mathbf{x}}{R^2} t\_c}\right] e^{-\mathbf{a}\_n^2 \frac{\mathbf{x}}{R^2} t} - \frac{4\mu\upsilon\_{m,\infty}}{R^2} \left[2 + \sum\_{n=1}^{\infty} e^{-\mathbf{a}\_n^2 \frac{\mathbf{x}}{R^2} t}\right] \tag{41}$$

In a recent paper, Kannaiyan et al. [86] analyzed the stability of a laminar pipe flow subjected to a step-like increase in the flow rate.

Summing up the second group of models, in which the flow is forced by a specific change in the flow rate, it can be concluded that:


from a flat plug profile to a parabolic one over time (but maintains constant mean velocity). A similar change takes place at the entrance section from the reservoir to the pipe, as the parabolic profile is here formed along the pipe length in a similar way, which was noticed and described by Sparrow et al. [76];

