## 1 Introduction  
Phenomena associated with nonlinear dynamic instability are prevalent in nature [1, 2, 3, 4, 5, 6]. Hydrodynamic fingering is a type of instability that occurs when one fluid is injected into another due to the change in fluid properties at bulk or interfacial regions, such as Saffman-Taylor instability, which is driven by viscosity difference [7, 8, 9, 10, 11]; Rayleigh-Taylor instability, which is driven by density difference [12, 13, 14]; and Marangoni instability, which is driven by interfacial tension difference [15, 16, 17]. Hydrodynamic fingering also occurs when a gel-like material forms at the interface of fluid pairs. [PERSON] et al. [18] studied fingering instabilities that appear when two aqueous solutions of identical viscosities react together. A variety of fingering instabilities were observed owing to viscoelastic micellar formation at the interface between the two fluids. Some of the observed displacement patterns were similar to the instabilities of growing filamentary organisms. The growth of filamentary organisms was influenced significantly by a stretchable elastic membrane, resulting in a finger-like growth pattern [19]. Moreover, the growth of bacterial colonies through diffusion-limited processes in a nutrient concentration field results in the formation of random fractals, which have properties identical to the fingering pattern induced by gel formation [20; 21; 22; 23]. The fingering instability induced by gel formation also shares mechanisms with precipitation patterns in confined chemical gardens, which are plant-like mineral structures originating from a precipitation reaction between a metallic salt and an aqueous alkaline solution. Precipitation is concentrated at the interface of reacted substances, forming a semipermeable membrane, out of which fingers grow, depending on the reactant concentrations. The spatial precipitate patterns depend on the injection direction, owing to the reaction-diffusion fronts that are asymmetrical [24; 25; 26]. Fluid displacement with gel formation is also essential in various engineering applications. [PERSON] et al. [27] used a polyacrylamide gel with chromium as a crosslinker to study the use of polymer gel solutions for the remediation of potential CO\({}_{2}\) leakage from storage reservoirs. A similar study by [PERSON] et al. [28] showed that injecting a silicate solution into a porous medium saturated with CO\({}_{2}\) results in the precipitation of amorphous silica that solidifies into a gel-like structure. This gel formation led to a 99% reduction in permeability and, thus, prevented CO\({}_{2}\) leakage. Furthermore, the gel blobs produced from a chemical reaction play a dominant role in improving oil recovery in the enhanced heavy oil recovery method based on alkali flooding. The gel blobs stick in the pores, blocking the flow path of the injected fluid. The blocked path causes the fluid to divert and push the trapped oil, thus, improving the oil recovery [29; 30]. Hydrodynamic fingering is generally studied in Hele-Shaw cells, which are two parallel glass plates separated by a small gap that makes it possible to observe a two-dimensional flow pattern [31; 32]. Moreover, the two-dimensional system is more accessible for numerical calculations. Flows in the Hele-Shaw cell can be modeled mathematically based upon [PERSON]'s law [8]. Recently, a numerical simulation based on Computational Fluid Dynamics, micro-Particle Image Velocimetry, and Stereoscopic Particle Image Velocimetry have also been used to study the entrance phenomena, i.e., flow detachment and vortex generation in Hele-Shaw cells [33; 34]. Although many experimental and theoretical studies have been conducted to study hydrodynamic fingering, quantitative studies on hydrodynamic fingering induced by gel formation, such as ones in which the effects of the sequential growth of the gel film thickness on fingering instability are analyzed, have not been performed and modeled mathematically.