**1. Introduction**

Numerous applications of heat transfer liquids or coolants can be found in a variety of fields, such as automobiles, industry, electronics, and cooling processes. In all such industrial applications, cooling by liquids has been used for years. The process of cooling by fluids may be the single phase (where there is no phase change in the coolant) or two-phase (where coolant liquid will experience a phase change). In the latter, latent heat influences the cooling efficiency [1]. Several coolants, such as water, ethylene glycol, blend of water and glycol, propylene glycol and amalgamation of water, and propylene glycol, are used as coolants in automobiles and industrial cooling processes. In the last two decades, several researchers have devoted their efforts to increasing thermal conductivity of coolants, thereby improving heat transfer capabilities. The pioneering work of Choi et al. [2] introduced nanofluids by insertion of solid nanoparticles into liquids, thus enhancing the thermal properties of these liquids. This pioneering work has remarkably revolutionized modern engineering and the industrial world.

Nanofluids are an amalgamation of suspended solid material particles and customary liquids (ethylene glycol, water). This new type of advanced material possesses amazing capabilities that trigger the process of heat transfer and augments the thermal conductivity of the base fluid. Enhancement in the thermal conductivity and heat transfer is visualized once ferrite nanoparticles are added into the base liquid. Several examples featuring heat transfer can be quoted, including chemicals, cooling and heating system of buildings, and avionics cooling systems. Nanofluids exhibit potentially exceptional features in comparison to macrometer-sized particles. This is because nanoparticles have sufficiently larger surface area compared to micrometer-sized particles; this is the reason nanofluids possess incomparable capabilities of heat transfer [3].

In several electromagnetic applications with high permeability, e.g., electromagnetic wave absorbers and inductors, usage of nickel–zinc ferrite can be noticed. To minimize energy losses related to bulk powders, usage of nickel–zinc nanoparticles has been recommended by a number of researchers [4–6]. In addition, a majority of electronic gadgets require such materials to be compressed into outsized shapes with the required thickness, which is reasonably challenging if the size of these particles is large enough. Several methods have been proposed to get nickel–zinc ferrite, including ball milling, precipitation, and hydrothermal. Ferrofluids are colloidal fluids comprising ferromagnetic or ferrimagnetic nanoparticles suspended in an electrically insulated hauler fluid. In the current examination, ethylene glycol (C2H6O2) was taken as a carrier fluid. The assumed ferrite nanoparticle was nickel–zinc ferrite (NiZnFe2O4) crystallize in the normal spinal structure. Typically, at room temperature, the inverted spinals are ferromagnetic and normal spinals are paramagnetic. Moreover, zinc ferrites act like antiferromagnetic in nature at low temperature. This feature makes ferromagnetic nanofluids more relevant in different real-world applications [7,8]. The ferrofluid's flow with the effect of thermal gradients and the magnetic field was discussed by Neuringer [9]. Majeed et al. [10] demonstrated the heat transfer investigation in a ferromagnetic fluid flow.

The subject of fluid flow past stretched surfaces has diverse engineering and industrial applications, including paper production, glass blowing, crystal growing, hot rolling, manufacturing of rubber sheets, annealing of copper wires, etc. The coined work of Crane [11] discussing the flow past a linearly stretching surface urged fellow researchers to discover more avenues in this exciting and interesting subject. This was followed by the remarkable work of Gupta and Gupta [12] who pondered on the flow past a spongy surface. Then, Chakrabarti and Gupta [13] examined the hydromagnetic flow past a stretched surface. Andersson et al. [14] considered the flow of power-law fluid past a surface, which was linearly stretched under the influence of magnetic forces. The flow of an Oldroyd-B fluid with the impact of generation/absorption was deliberated by Hayat et al. [15]. Muhammad et al. [16] discussed the effect of thermal stratification in the ferromagnetic fluid on a stretching sheet. Ramzan and Yousaf [17] demonstrated that the elastic viscous nanofluid finished a bi-directional stretching surface in view of Newtonian heating. Hussain et al. [18] utilized the exponentially stretching sheet to scrutinize the flow of Jeffrey nanofluid with radiation effects. Some recent explorations highlighting various fluid flows past stretched surfaces with coatings can be found in references [19–22].

In today's cutting-edge engineering technology, curved stretching has a broad relevance because of its different uses in industry, for example, in transportation and electronics. Sanni et al. [23] attained a numerical solution for the viscous fluid flow on a curved stretched channel. Sajid et al. [24] inspected the ferrofluid (Fe3O4) flow on a curved sheet with effects of Joule heating and magnetic forces. Rosca and Pop [25] studied time-dependent flow along a spongy curved surface. Imtiaz et al. [26] introduced the effect of homogeneous/heterogeneous reactions in ferrofluid embedded in a stretching surface. Naveed et al. [27] calculated heat transfer and used the micropolar fluid to analyze the effects over a curved surface with thermal radiation.

A literature review has specified that copious studies are available relating to nanofluids with linear/nonlinear/exponential stretching surfaces but comparatively less research work is available highlighting curved stretched surfaces. This gets even narrower when we talk about the study of hybrid nanoliquid with entropy optimization past curved surfaces. Therefore, our task here is to discuss hybrid nanoliquid flow comprising ferromagnetic nanoparticle, i.e., nickel–zinc ferrite (NiZnFe2O4), and the base fluid, i.e., ethylene glycol (C2H6O2), over a curved surface with entropy optimization coating. The whole analysis was performed with added impressions of nonlinear thermal radiation with entropy optimization coatings. The analysis was supported by the convective heat and mass flux boundary conditions. Numerical solution of the envisioned model was obtained by utilizing bvp4c from MATLAB. The traits of the sundry parameters on involved distributions were thoroughly discussed keeping their physical justification in mind.
