**1. Introduction**

Fluids possess a significant role in the amplification of heat exchange rate in numerous engineering systems, e.g., heat exchangers, oil and petrochemical industries. Nanoparticle suspensions, pioneered by Choi [1], made thermal performance of these fluids more effective and it has become a topic of interest for many investigators [2–5]. Regardless of researchers' efforts, there has been an elementary issue with mono nanofluids that either they possess better thermal association or good rheological characteristics. For example, metal oxides such as Al2O3 show excellent chemical inertness as well as stability, whereas metallic nanoparticles including Al and Ag exhibit better thermal conductivities. Most of the authentic applications required transaction between different properties of nanofluids and thus hybridization of nanomaterials has been introduced. Hybrid nanofluids can be manufactured by dispersing nanoparticles of different materials individually or a mixture of nanoparticles in base fluid. For instance, impacts of Cu–Ag nanohybrids on velocity and thermal boundary layer transport inside the wedge have been investigated by Hassan et al. [6]. To gain the highest composite thermal conductivity, chemical inertness and stability by using a small-volume fraction of nanoparticles at lower production cost is the motivation of researchers behind their utilization of hybrid nanofluids [7–10]. Moreover, nanoparticles of TiO2 exhibit antibacterial and photocatalytic properties. Nguyen et al. [11] has studied antibacterial properties of TiO2 by adopting silver decorative technique and revealed that oxide nanoparticles of titanium did not show inhibitory impacts for bacteria whereas silver loaded TiO2 nanocomposites display efficient antibacterial characteristics at a concentration of 40 mg/mL. Ag nanoparticles are able to devastate pathogenic bacteria under ultraviolet radiation for efficient degradation of toxic pollutants as well as being easy to attach to cell membranes [12–14]. Therefore, Ag–TiO2 nanocomposite is preferred in this theoretical inspection. Moreover, platelet-shaped particles are chosen since they capacitate swift healing in skin injuries because of their innate capability to make a boundary intended for vascular walls.

A cilium is a microscopic, contractile, thin fiber-like slender appendage/protrusion that projects from surfaces of specific cells. In the adult human body, epithelial cells along with motile cilia are very prominent in specific brain sections. Due to their motility, they possess a considerable role in many physiological processes like locomotion, alimentation and respiration. Peristalsis is a spontaneous process of a symmetrical wave's expansion and contraction within flexible boundaries. Ciliary-induced peristalsis appears significantly in various biological transport processes such as in biomedicine, physiology and nuclear reactors. Recently, Awais et al. [15] examined second-law analysis for peristaltic nanofluid flow caused by ciliary action with magnetic effects. Furthermore, they studied convective peristalsis of viscous fluid by considering non-uniform viscosity [16] as well. Furthermore, the concept of peristaltic pumps, instigated by Engelman [17], has latterly been prominent in several biological functions including roller pumps and heart-lung machines etc. As peristalsis is a cutting-edge field due to physiological applications, several theoretical as well as experimental attempts have been made to incorporate nanoparticles in order to improve thermal performance in biomedical processes. Rashidi et al. [18] exemplified the application of MHD peristaltic transport of blood containing nanoparticles in drug delivery through an incompatible channel which is practically imperative in the bio-sciences. Hayat et al. [19] explicated mixed convective heat transfer in the peristalsis of nanoparticles suspended in water assuming convective boundary conditions and joule heating. Recently, Maqbool et al. [20] inspected the impacts of nanoparticles on magnetohydrodynamic tangent hyperbolic fluid transportation in a ciliated tube.

Attention to non-Newtonian fluids arises as the majority of the physiological fluids possess non-Newtonian behavior verified by experimental observations. In view of the fact that simplified Newtonian models yield somewhat ambiguous results, several investigations on rheological fluid behaviors have been carried out to obtain more realistic results. Examples include inelastic fluid models e.g., the power-law model and viscoelastic fluid models such as the Johnson–Segalman model, Oldroyd-B model and Maxwell model. Mixed convection impacts towards peristaltic transport of magnetohydrodynamic non-Newtonian nanofluid were numerically evaluated by Hayat et al. [21]. The current study examines the rheological nature of fluid by employing Ostwald-de-Waele power law model, a generalized one, in which rheological nature directly depends on power law index *n* and deals with the shear thinning for (*n* < 1) and shear thickening for (*n* > 1) behaviors of fluid [22–25].

Moreover, the peristaltic flow with influences of applied magnetic field led to significant applications in biomedical engineering problems [26–28]. In the case of large magnetic Reynolds number for an electrically conducting fluid, induction becomes more prominent than magnetic diffusion, and this made the induced magnetic field effects accountable. Shit et al. [29] examined the influence of induced magnetic field on peristalsis of a micropolar fluid assuming velocity slip. They observed

that peristaltic flow rate enlarges in an induced magnetic field which led to mechanical stimulation. So, magnetic induction is appropriate in cancer treatment and magneto therapy as predicted in literature [30–32]. Besides this, the performance of coatings with magnetic nanoparticles and heat transport is ever-present in various fields. Magnetic nanoparticles, approved by the FDA (Food and Drug Administration) [33], with coating are applicable in medical processes such as blood pressure control of a patient, pharmacotherapy, surgery and alcohol detoxification etc. Ellahi et al. [34] carried out a comparative investigation on shiny film coating on multi-fluids dispersed by nanoparticles. Akbar and Butt [35] inspected the physiological flow of Casson fluid through a plumb duct. They observed that fluid behaves as electrically conducting with a uniform magnetic field and found analytical results under small wave number and low-Reynolds number approximations. The recent related research can be read in [36–38].

Furthermore, endoscopic imaging is a precious diagnostic instrumental locating persistent access to tissues deep inside hollow organs of the body. A conventional white-light endoscope is a solid circular cylinder placed in a peristaltic tube. Fluid flow occurs in the space between the tube and endoscope, and then further diagnostic procedures can be made such as for bleeding, cancerous growths and precancerous polyps. Hayat et al. [39] have addressed the peristaltic transport of the MHD power law fluid with endoscope effects. Hayat and Ali [40] have inspected the impact of an endoscope on peristaltically induced flow of micropolar fluid. The influences of non-uniform viscosity on peristaltic motion of Newtonian fluid through an endoscope have been conducted by Akbar and Nadeem [41]. Rathod and Asha [42] have investigated endoscope effects along with a magnetic field on peristalsis of the Newtonian fluid. They concluded that stress formation in a curved structure wall augments as compared to straight walls. In view of the significance of research regarding endoscopy applications, various studies have made (see refs. [43–48]).

With several advantages, advanced endoscopes are deficient in the spatial resolution for detection and treatment of cancers and abnormalities at small scales. Ciliary walls have importance since these biological cilia are helpful to perform complex biomimetic functions and applicable in vitro and in vivo synthetic organs as well as drug-delivery applications. In these unmet requirements, the effects of hybrid nanofluid and induced magnetic field on endoscopy application inside a ciliated peristaltic tube are addressed. Mathematical modeling is performed by considering negligible inertial forces and small wave number. An analytical solution of governing model is carried out by the homotopy analysis method, and results are plotted physically against several sundry parameters via tables and graphs. The trapping phenomenon is examined with the effects of electromagnetic induction as well.
