**1. Introduction**

Fluids that are often magnetized by the existence of an exterior magnetic field are known as ferrofluids, which is an abbreviation for fluid and ferromagnetic particles. These fluids are made up of colloidal fluids formed of nanosized ferromagnetic or ferrimagnetic particles that have been stopped inside the fluid transporter. Brownian motion causes particle suspension and must not start moving under normal conditions. Besides that, to avoid clogging, each ferromagnetic particle is encased in a solvent, and the nano-scaled ferromagnetic particles have a weak magnetic attraction whenever the surfactant's Van der Waals force adequately stopped aggregation or clustering. Numerous applications of ferromagnetic fluids have emerged in a variety of fields. Heat transfer agents, angular

**Citation:** Usman, A.H.; Shah, Z.; Kumam, P.; Khan, W.; Humphries, U.W. Nanomechanical Concepts in Magnetically Guided Systems to Investigate the Magnetic Dipole Effect on Ferromagnetic Flow Past a Vertical Cone Surface. *Coatings* **2021**, *11*, 1129. https://doi.org/10.3390/ coatings11091129

Academic Editor: Eduardo Guzmán

Received: 16 August 2021 Accepted: 12 September 2021 Published: 16 September 2021

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momentum changers, friction reducers, and so on are used in electronic equipment, analytical techniques, and medical science; some examples can be found in the references [1–3]. Because of these numerous applications, many researchers and scientists have been focused on this subject. Andersson and Vanes [4] first investigated the influence caused by magnetic dipoles on ferrofluids. Zeeshan et al. [5] investigated the convective heat transfer flow of ferromagnetic fluids with partial slip effects using a stretching sheet. Hayat et al. [6] reported on radiation and magnetic dipole effects of Williamson ferromagnetic fluid flow across a stretched surface.

A nanofluid is a nanometer-sized particle suspended in a fluid. Choi [7] established the basic extension of "nanofluid", and scientific results verified that heat transfer can be significantly enhanced through the mixture of tiny metallic nanomaterials with the working fluids. A few studies in particular on nanofluids have been conducted. Ellahi [8] performed an analytical study and concluded that the temperature variable and viscosity affects MHD non-Newtonian nanofluid flow in a pipe. Ellahi et al. [9] presented peristaltic nanofluid flow with entropy generation via a medium of porosity. Hayat et al. [10] investigated the flow of third-grade nanofluids caused by a rotating stretchable disk containing a heat source and a chemical reaction. Awais et al. [11] explored the effects of magnetohydrodynamics on peristaltic ciliary-induced flow coatings to rheological hybrid nanofluids. Reddy et al. [12] studied the boundary layer naturally convective MHD nanofluid flow along a vertical cone under the influence of chemical reaction and heat suction/injection.

Due to disorganized frameworks and destabilization, low-density microorganisms remain on the surface of a fluid, causing bioconvection. Because nanoparticles move differently than motile microorganisms, the cumulative importance of nanomaterials and bioconvection is such that they play a vital role in microfluidic devices. Bioconvection is a novel manufacturing and fluid mechanic with a biological phenomenon involving gyrotactic microorganisms. As a result, it becomes an interesting field of research to which many researchers continue to pay attention. Alsaedi et al. [13] investigated stratified magnetohydrodynamic nanofluid flow, causing bioconvection in gyrotactic microorganisms. Hayat et al. [14] researched the magnetohydrodynamic (MHD) nonlinear radiative nanofluid flow with gyrotactic microorganisms. Nadeem et al. [15] reported on the Rosseland assessment for ferromagnetic fluid with involvement of magnetic dipoles and gyrotactic microorganisms. Bhatti and Michaelides [16] researched thermo-bioconvection nanofluid flows across a Riga plate as a function of Arrhenius activation energy. Waqas et al. [17] have also numerically simulated the magnetized non-Newtonian bioconvection nanofluid flow along stretching cylinders/plates.

Combining mass and heat fluxes in liquid saturated porous media is crucial among a wide range of engineering procedures such as heating systems, oil and gas reservoirs, and chemical catalytic reactor designs [18,19]. The dragging force, the Darcy–Forchheimer technique, is a widely popular method for simulating fluid passed through a porous medium with high velocity. In the literature, flow through a cone in Darcy–Forchheimer porous media has already been analyzed by many researchers. Kumar et al. [20] investigated the non-Darcy MHD viscoelastic fluid flow through a flat plate and a vertical cone. Chamkha et al. [21] explored the non-Newtonian natural convective nanofluid flow over a saturated cone in a non-Darcy porous medium with uniform volume fraction and heat fluxes. Mallikarjuna et al. [22] researched the impacts of radiation, thermophoresis and transpiration on convective non-Darcy flow via a rotating cone. Durairaj et al. [23] investigated the chemically reacting Casson fluid of a non-Darcy porous medium flow through a flat plate and a vertical cone saturated with heat generating/absorbing. Patrulescu et al. [24] investigated a convection flow due to a vertical plate embedded in a bi-disperse non-Darcy porous medium.

According to a recent literature review, despite important applications in extrusion systems, geothermics, organic compounds, geophysics, improved manufacturing techniques, material processing, and improved energy generation, research on viscous ferrofluid flows via a linear vertical cone with consideration of Darcy–Forchheimer porous media has

been studied by very few researchers in the past. The aim of this research is to use the Forchheimer-extended Darcy law to explore the effect of magnetic dipole and porosity relations in the boundary layer of a ferromagnetic nanofluid flow via a vertical cone surface. The study of the effect of the magnetic dipole on ferromagnetic nanofluid flow via the vertical cone surface makes this work different from the existing literature. A nonlinear ordinary differential equation replaces the governing equations and is solved using HAM techniques. Initially, Liao [25–27] presented a homotopy analysis method with HAM. The method has fast convergent solutions with many advantages over some existing methods. Various researchers have been drawn to it as a result of its rapid convergence [28–31]. The results collected for all associated parameters on all profiles are shown graphically. The validation of the results by comparing them to previously published material in the literature is an important feature of the presented model. In this regard, illustrious coherence has been attained.
