**1. Introduction**

Nanofluid, characterized by a significant increase in a number of properties compared to conventional engineered fluid [1], is found to serve in many practical applications, for example, petroleum engineering [2–5], power industry [6,7], and medical science [8,9], which has drawn particular attentions for cancer treatments in recent years. Cancer covers a huge group of diseases

which can damage any portion of the body. Nowadays, cancer is a main cause of death all over the world, around 70% of cancer deaths materialize in middle- and low-income countries. There are many treatments to cure cancer, such as surgery, radiation and chemotherapy, but these procedures may harm the normal tissue. Hanahan and Weinberg [10] have explained six cancer hallmarks, helping to differentiate features between the tumor and normal tissue, and maybe come up with better alternative therapies. These hallmarks include inducing activating invasion and metastasis, resisting cell death, angiogenesis, enabling replicative immortality, sustaining proliferative signaling, and evading growth suppressors. Based on these cancer hallmarks, latest therapies for cancer treatment have been introduced. Nowadays, nanomedicine (nanomedicine is a branch of nanotechnology, or utilization of materials less than 100 nm, applied to medicine and health sciences) is the prominent procedure for treating cancer. The nanocarriers' properties, including their targeting modifications, favorable drug release profiles, high surface-to-volume ratios, and nanoscale sizes, may authorize them to reach and target the tissue of a tumor and the deliverance of drugs in a stable and controlled manner. For cancer research, nanomaterials are available in modified shape, because to treat specific tumors, size and surface features are crucial. The size of the nanoparticle is a key attribute, which travel across the bloodstream, ensuring delivery of nanocarriers to tumor tissue. The small-scale nanoparticles can stockpile comfortably in the physiological tumor vessels and also extravagate into normal tissue. In view of many nanoparticle applications in bio-fluid flows, many researchers have concentrated their work in the field of bio-nanofluids. For instance, Prakash et al. [11] have presented the study of nanofluids which is relevant to bio-inspired nanofluid smart pump designs, which may also be exploited in smart-drug delivery. Abbas et al. [12] have provided mathematical modelling to describe the peristaltic transport of blood (blood is treated as nanofluid) and analyzed the entropy analysis. They concluded that such a study can help in analyzing blood flow in small blood vessels with elastic walls. Abdelsalam and Bhatti [13] have given a theoretical model to describe the effect of sundry variables on the feature of blood flows in the presence of nanoparticles, and suggested that Brownian motion and chemical reaction exhibit dual variation of nanoparticles' volume fraction. Shah et al. [14] have presented the theoretical study, which is applicable to the drug-delivery system, as the micro-polar nanoparticles of gold are proficient drug-delivery and drug-carrying mediums. Bhatti et al. [15] studied the two-phase flow under the effects of coagulation with peristaltic pumping through the Prandtl stress model, with magnetic field and porous medium terms. They analyzed that friction forces flourish with the altitude of clot height and particle concentration, on the other hand they are minimized with other involving factors in the problem.

It is extensively known that biological liquids, such as gastric fluids and blood, generally behave as non-Newtonian fluids. Many researchers have considered Jeffrey/viscoelastic fluid as biological (synovial, blood, gastric, chyme, and saliva) fluid. To delineate the stress relaxation effects of real fluids, the viscoelastic fluid model is appropriately competent. These effects cannot describe the usual Newtonian fluid model. In addition to this, the Jeffrey fluid model can also describe the characteristic of memory-time scale. Kahshan et al. [16] have described the Creeping flow of a viscoelastic fluid in a channel with an application to flow in a flat-plate hemodialyzer. Pandey and Tripathi [17] have explored the viscoelastic fluid flow by peristalsis in a channel in order to apply the model to the swallowing of food-bolus through the esophagus. Ellahi et al. [18] used Jeffrey fluid as bio-fluid and studied the problem of the peristaltic flow of Jeffrey fluid in a non-uniform rectangular duct, which may be applicable to modern drug delivery systems with great utility. Ramesh et al. [19] have considered Jeffery's viscoelastic formulation, which is employed in the rheology of blood. Some more studies can be seen through [20–25].

In thermodynamics, entropy is a measure of the number of specific modes in which a thermodynamic system can be organized, often called a measure of impedance or a measure of progress toward thermodynamic equilibrium. Pakdemirli and Yilbas [26] have analyzed the entropy generation mechanism of non-Newtonian fluid through a pipe. According to them, the entropy Brinkman number causes an increase in entropy generation. Entropy generation in a peristaltic

pumping problem has been presented by Souidi et al. [27]. Heat and fluid flow causing entropy generation in backward-facing step flow is suggested by Abu-Nada [28]. More studies on entropy generation in peristaltic transports are reported in [29–31], but none of these established the entropy analysis of viscoelastic nanofluids in eccentric cylinders having a peristaltic outer surface.

Keeping in mind the physiological applications of a peristaltic propulsion of viscoelastic fluids, the investigators focused on the entropy generation and Bejan number during the peristaltic transport of viscoelastic nanofluid in the annulus region of two eccentric cylinders. The equations of governing the flow are considered in the cylindrical geometry. The concerned non-dimensional system of equations is solved using optimal homotropy perturbation technique under the long wavelength and low Reynolds number assumptions. The effects of involved parameters on the pressure rise, velocity, temperature, nanoparticle concentration, entropy generation, and Bejan number are shown through graphical illustrations.
