*2.3. Universal Solution for Arbitrary Pressure Gradient*

The analysis of a number of scientific papers for the purposes of this review revealed that there is a universal solution that can be used to determine the velocity profile for any change in the pressure gradient. Information about this solution will be the topic of this subsection.

Fan in his Ph.D. thesis [61] derived analytical solutions for: (a) impulsive pressure gradient (here the pressure gradient is a function of Dirac delta function); (b) flow under a constant pressure gradient (analogous to RGS solution); (c) flow under a harmonically oscillating pressure gradient; (d) a general solution by superposition integrals and (e) a general solution by the transformation of the governing equation of motion to a homogeneous equation. The latter helped Fan to derive a final analytical solution for an accelerated flow in a pipe with a rectangular cross-section, which will not be discussed due to limited practical applications. The most important solution according to this subsection title is the general solution defined by superposition integral. Since the considered problem is linear, Fan noticed that the solution for any arbitrary *f*(*t*) can be obtained by one of the following superposition integrals:

$$v(r,t) = -\int\_0^t f(u)\overline{v}\_1(r,t-u)du = -\int\_0^t f(t-u)\overline{v}\_1(r,u)du,\tag{18}$$

where: <sup>1</sup> *ρ ∂p <sup>∂</sup><sup>x</sup>* = *f*(*t*) and *v*<sup>1</sup> *r*ˆ, ˆ*t* = 2 ∑<sup>∞</sup> *n*=1 *J*0(*r*ˆ*λn*) *<sup>λ</sup><sup>n</sup> <sup>J</sup>*1(*λn*)*e*−*λ*<sup>2</sup> *<sup>n</sup>* <sup>ˆ</sup>*<sup>t</sup>* is the solution that Fan obtained for impulsive pressure gradient, or:

$$v(r,t) = -f(0)\overline{v}\_2(r,t) - \int\_0^t f'^{(u)}\overline{v}\_2(r,t-u)du = -\int\_0^t f(u)\frac{\partial \overline{v}\_2(r,t-u)}{\partial t}du,\tag{19}$$

where *v*2(*r*, *t*) is the RGS solution *v*2(*r*, *t*) = <sup>1</sup> 4*ν <sup>R</sup>*<sup>2</sup> − *<sup>r</sup>*<sup>2</sup> <sup>−</sup> <sup>8</sup>*R*<sup>2</sup> <sup>∑</sup><sup>∞</sup> *n*=1 *J*0(*λ<sup>n</sup> <sup>r</sup> R* ) *λ*3 *<sup>n</sup>*·*J*1(*λn*) *e* <sup>−</sup>*λ*<sup>2</sup> *n ν R*2 *t* .

The First Fan integral solution Equation (18) is of simple mathematical structure. The second solution (known as Duhamel's integrals) is based on the more demanding RGS solution Equation (19). Both superposition solutions described by Equations (18) and (19) were derived for the case with the initial flow at rest (mean velocity equal to zero). A solution that takes into account the initial flow was derived by Daneshyar [62]. Daneshyar gives a solution for an arbitrary *f*(*t*), using the theory of integral transforms to solve the main Equation (1). He proved after Sneddon [63] that if *ξ<sup>n</sup>* is chosen to be a root of:

$$J\_0(R\xi\_n^x) = 0\tag{20}$$

then

$$\int\_0^R r \left( \frac{\partial^2 v}{\partial r^2} + \frac{1}{r} \frac{\partial v}{\partial r} \right) l\_0(r \xi\_n) dr = -\xi\_n^2 \overline{v} \tag{21}$$

where

$$
\overline{v} = \int\_0^R r v(r, t) J\_0(r \xi\_n) dr \tag{22}
$$

is the finite Hankel transform of *v*. When both sides of Equation (1) are multiplied by *r J*0(*rξn*) and integrated (within the limits 0 to *R*), the following ordinary differential equation is obtained for *v*:

$$\frac{d\overline{\upsilon}}{dt} + \nu \mathfrak{J}\_n^2 \overline{\upsilon} = f(t) \int\_0^R r \mathfrak{J}\_0(r \mathfrak{J}\_n) dr = f(t) \frac{R}{\mathfrak{J}\_n} \mathfrak{J}\_1(R \mathfrak{J}\_n) \tag{23}$$

The solution that satisfies the initial condition *v*(*ξn*, 0) = *v*<sup>0</sup> is:

$$\overline{\sigma} = \int\_0^t f(u) \frac{R}{\tilde{\xi}\_n} f\_1(R\tilde{\xi}\_n) \exp\left[-\nu \tilde{\xi}\_n^2 (t-u)\right] du + \overline{\sigma}\_0 \tag{24}$$

And, next, with the inversion theorem of finite Hankel transforms, a general solution is found to be:

$$\nabla v = v\_0 + \frac{2}{R^2} \sum\_{n=1}^{\infty} \frac{J\_0(r\_{5,n}^{\pi})}{\left[f\_1(R\_{5,n}^{\pi})\right]^2} \overline{v} = v\_0 + \frac{2}{R} \sum\_{n=1}^{\infty} \frac{J\_0(r\_{5,n}^{\pi})}{\overline{\xi}\_n I\_1(R\_{5,n}^{\pi})} \int\_0^t f(u) \exp\left[-\nu \overline{\xi}\_n^2 (t-u)\right] du \tag{25}$$

that is valid for an arbitrary pressure gradient (hidden under *f*(*t*) function).

It can be now noticed that substituting for *Rξ<sup>n</sup>* = *λ<sup>n</sup>* one gets *ξ<sup>n</sup>* = *λn*/*R*, which, when inserted into the above formula, will reduce this solution to the following form:

$$v = v\_0 + 2\sum\_{n=1}^{\infty} \frac{l\_0\left(\frac{r}{R}\lambda\_n\right)}{\lambda\_n f\_1(\lambda\_n)} \int\_0^t f(u)e^{-\lambda\_n^2(t-u)} du,\tag{26}$$

In the special case of the above solution, *v*<sup>0</sup> can be treated as an initiation laminar flow profile of steady Hagen–Poiseuille type flow.

A careful literature review reveals that universal solutions of this type were derived many times. Chronologically, the earliest form of this solution was presented by Vogepohl in a short note published in 1933 [30]. Then in 1956, Roller, in his master's thesis [64], noticed (similarly to Daneshyar about 15 years later) that the solution of the Navier momentum equation can be obtained with the use of the analogy to the temperature distribution in a cylindrical rod (defining the analogous boundary and initial condition) [63]. Zielke in his Ph.D. thesis [65] rederived Fan's solution in the form of Equation (19) using Laplace transforms. Hersey and Song by using Laplace transforms [66], and Avula with the help of the Cauchy residue and convolution integral theorem [53], derived the same solution but based on different normalized quantities. These solutions were again derived and used with the help of the eigenfunction method in the work of Xiu et al. [67], where the starting flow was analyzed, and in the Sun and Wang paper [68], where the water hammer case (starting from steady Hagen–Poiseuille flow) was examined:

$$\psi(\mathbf{\hat{r}},\mathbf{\hat{x}},\mathbf{\hat{t}}) = \sum\_{n=1}^{\infty} \frac{16f\_0(\mathbf{\hat{r}}\lambda\_n)}{\lambda\_n^3 f\_1(\lambda\_n)} e^{-\lambda\_n^2 \mathbf{\hat{r}}} + \sum\_{n=1}^{\infty} \int\_0^{\frac{\pi}{2}} \frac{2f\_0(\mathbf{\hat{r}}\lambda\_n)}{\lambda\_n f\_1(\lambda\_n)} \left[ -\frac{\partial \hat{H}(\mathbf{\hat{x}},\mathbf{u})}{\partial \mathbf{\hat{x}}} \right] e^{-\lambda\_n^2 (\mathbf{\hat{t}} - \mathbf{u})} d\mathbf{u} \tag{27}$$

where: *<sup>x</sup>*<sup>ˆ</sup> <sup>=</sup> *<sup>x</sup>*/*L*; *<sup>r</sup>*<sup>ˆ</sup> <sup>=</sup> *<sup>r</sup>*/*R*; *<sup>v</sup>*<sup>ˆ</sup> <sup>=</sup> *<sup>v</sup>*/*v*<sup>0</sup> and *<sup>H</sup>*<sup>ˆ</sup> <sup>=</sup> <sup>Δ</sup>*pR*<sup>2</sup> *<sup>L</sup>μv*<sup>0</sup> .

In the above equation, an interesting identity is used (for initial time ˆ*t* = 0), that can be found in many papers among them the ones discussed in this review i.e., Szyma ´nski [13], Gerbes [17], Xiu et al. [67], etc:

$$\sum\_{n=1}^{\infty} \frac{8f\_0(\lambda\_n \mathfrak{f})}{\lambda\_n^3 f\_1(\lambda\_n)} = (1 - \mathfrak{f}) \text{ for } 0 \le \mathfrak{f} \le 1 \tag{28}$$

The most recent study in which this universal equation was re-derived and examined is Lee's paper published in 2017 in Applied Mathematical Modelling [69]. The motivation of this section is to systematize the derivation of the universal formula discussed above. The list of references is presented in Table 1.


**Table 1.** A list of works in which a universal formula was derived.

## *2.4. Comments on the Pressure Gradient Driven Flows*

As noticed, the formulas discussed above concerned pipes with circular cross-sections. Readers interested in other solutions, e.g., similar accelerated flow in ducts with different geometry, are referred to other works: the solution to this problem for ducts with a rectangular cross-section was derived in the works of Fan and Erdogan [21,61,72]; start-up flow in an annulus was developed by Müller [73]; for other cross-sections, an intensive study was carried out by Laura [22]; solution for pipes, taking into account the slip of the fluid on the wall [74]; start-up flow in a circular porous pipe [75]; development of unsteady flow at the entrance length of a circular tube starting from rest [19,20,54,76]; the effects of time-dependent viscosity [77–79]; unsteady laminar flow in tubes with a tapered wall thickness [80], etc.

Due to the need to use the zeros of the Bessel function, the presented solutions are a challenge, because it is usually necessary to write a short program in software such as Matlab or Wolfram Mathematica, hence the approximation forms of these formulas discussed by Muzychka-Yovanovich [23,24] and Urbanowicz et al. [4] are also an interesting proposition. It is also worth adding that governing equations of motion, are respectively analogous to heat conduction in a long cylinder with constant thermal conductivity [81]. So all the results presented in this paper for laminar flows are directly transformable to the solution of heat conduction in long cylinders with internal heat generation (simply replacing pressure gradient *G* by heat generation source term . *g cP*·*<sup>ρ</sup>* ; kinematic viscosity *<sup>ν</sup>* by thermal diffusivity *α* and velocity field *v*(*r*, *t*) by temperature field *T*(*r*, *t*)).

Moss developed [82] a dimensionless flow acceleration parameter that takes a zero value in the case of an impulsively begun flow and non-zero values for exponentially increasing flows. When this parameter is increased beyond a critical value (7.059), the boundary layer never merges. The method developed by Moss is suitable for the application of any stability analyses of unsteady flows, which is very useful to study the physical insights of different flow fields. Pozzi and Tognaccini [83] have extended the analysis of the present accelerating problem in pipes for the thermo-fluid dynamic field arising in an infinite pipe with a circular section when the incompressible fluid is impulsively started from rest by a sudden jump to a constant value of the axial pressure gradient. These authors derived analytical solutions for the temperature field taking additionally into account the dissipation of kinetic energy (Eckert number different from zero) in the relevant case of Prandtl number equal to one. The final solution has been obtained and discussed for four cases depending on the condition imposed on the wall: constant temperature, adiabatic wall, assigned heat loss, and assigned constant flux.

To sum up, the analysis of the presented analytical solutions of accelerated flows in pressure lines forced by a change in pressure gradient shows that:

