## 1 Introduction  
A variety of functional applications are based on the premise that the flow is fully developed. It is assumed that after a certain time the fluid has travelled a certain length (development length--\(L\)) along the channel, after which the flow no longer changes in the direction of flow. This is used, for example, in extrusion dies, lab-on-a-ship, etc. [1, 2, 3]. The development length of Newtonian flows in channels and pipes (see Figure 1) is well understood [4]. [PERSON] et al. [4] developed two correlations between \(L\) (the distance the fluid travels to become fully developed) and the Reynolds number, \(Re=\frac{\rho U_{H}}{\eta}\), where \(U\) is the imposed mean inlet velocity, \(\rho\) is the fluid density, \(H\) is the width of the channel (for pipes, one  
Figure 1: Schematic of the channel and pipe geometries used for the study of the development length. (**a**) Channel flow. (**b**) Pipe flow. should replace \(H\) by \(D\)-diameter), and \(\eta\) is the Newtonian viscosity. These correlations are given by,  
\[\frac{L}{H}=[(0.631)^{1.6}+(0.0442 Re)^{1.6}]^{1/1.6}, \tag{1}\]  
\[\frac{L}{D}=[(0.619)^{1.6}+(0.0567 Re)^{1.6}]^{1/1.6}, \tag{2}\]  
and predict well the development length for channel and pipe flows, respectively. For other works on the development length of Newtonian fluids please consult the following references [5; 6; 7; 8; 9; 10]. For generalised Newtonian fluids (with varying viscosity), several works have been proposed in the literature [11; 12; 13; 14; 15; 16; 17; 18; 19]. We would like to highlight the works of [PERSON] et al. [19] and [PERSON] [18], in which they presented two correlations for the development length in channel and pipe flows of power-law fluids (the viscosity is a function of the second invariant of the deformation at tensor, \(\hat{\gamma}\) (for simple flows, \(\hat{\gamma}\) is simply the shear rate). The viscosity is then given by \(\eta=k\gamma^{n-1}\)). The correlations are given by,  
\[\frac{L}{H}=[(f(n)-\exp{(15.706-4.002)})^{1.6}+(0.0444 Regen)^{-0.209n^{2}+0.645 n+1.225}]^{1/1.6}, \tag{3}\]  
\[\frac{L}{D}=[(0.246n^{2}-0.675n+1.03)^{1.6}+(0.0567 Re_{MR})^{1.6}]^{1/1.6}, \tag{4}\]  
for channel and pipe flows, respectively. Here, \(Re_{gen}=\frac{6\rho L^{2-n}H^{n}}{k}{\left(\frac{n}{4n+2}\right)}^{n}\), \(Re_{MR}=\frac{8\rho L^{2-n}D^{n}}{k}{\left(\frac{n}{6n+2}\right)}^{n}\), and \(f(n)=\frac{-0.355}{1+2\exp{(0.553-4.273n)}}\). Note the increasing complexity in the correlations when going from a Newtonian to a power-law fluid. In the case of viscoelastic fluids, the number of papers on this topic is smaller. This is due to the complexity of viscoelastic flows, such as the presence of singularities at the entrance of the channel, overshoots in the velocity profile, and the high Weissenberg number problem. We would like to highlight the work of [PERSON] and [PERSON] [20] in which they perform numerical simulations to determine the development length of an Oldroyd-B fluid and conclude that the development length (for a fixed \(Re\)) increases slightly with the Weissenberg number, \(Wi=\frac{\lambda Lt}{H}\) (where \(\lambda\) is the relaxation time of the fluid in seconds), but is more strongly affected by \(Re\). [PERSON] [1] proposed a theoretical work for the development length of viscoelastic fluids entering an extrusion die. They presented an expression for estimating the length of the entrance region, which has applications in the extrusion industry. In the work by [PERSON] et al. [10] the authors present a study on the flow development of a Bingham plastic fluid in tubes and channels considering the Papanastasiou regularisation and the finite element method. They considered the [PERSON]'s slip law at the wall and concluded that as slip increases, the development length initially increases exhibiting a global maximum before vanishing rapidly above the critical point corresponding to sliding flow. More recently, [PERSON] et al.