## 1 Introduction  
The development of efficient heat-transfer-fluids (HTF) continues to attract the attention of scientists and engineers given the widespread applications. The development and enhancement of efficient HTF has been widely investigated experimentally, theoretically, and computationally, see for example [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. Such widespread investigations have been conducted with a view to the important industrial applications of HTF, say, to polymer processing, food and beverage processing, chemical processing, pharmaceutical manufacture, etc. The concept and development of nanofluids has largely arisen as a route to heat-transfer-rate (HTR) enhancement and efficient performance of HTF. Nanofluids are a dispersion of solid (metallic) nanometer-sized particles (nanoparticles) in a base-fluid. The available literature, see for example [9, 15, 16, 17, 18], has shown that nanofluids exhibit higher thermal conductivities as compared to the conventional (and corresponding) base-fluids. The literature specifically demonstrates that such higher thermal conductivities of nanofluids lead to increases in heat-transfer-rates (HTR) as compared to the corresponding base-fluids. Additionally, ongoing research has demonstrated that the attractive heat-transfer characteristics of nanofluids can be further improved via various methods including increasing the Brownian motion of the embedded nanoparticles suspended in the base-fluids [19, 20, 21], increasing volume-fraction of the embedded nanoparticles [22, 23, 24], increasing the fluid layers around the embedded nanoparticles [19, 20], etc. The present research specifically explores the effects, on HTR enhancement, of nanoparticle volume-fraction (i.e., concentration of nanoparticles) in the nanofluid. In comparative investigations on the thermal conductivities of nanofluids versus the corresponding base-fluids, [25] reported that up to 30% increase in the thermal conductivity can be observed in nanofluids generated from various nanoparticle/base-fluid combinations, e.g., \(Al_{2}O_{3}\)-water, \(SiO_{2}\)-water, and \(TiO_{2}\)-water. Similar results were also reported in [1] using \(CuO\)-water, \(Al_{2}O_{3}\)-water, and \(Co\)-oil nanofluids. The authors in [26] claimed a 100% increase in thermal conductivity for a range of volume-fractions 0.5-10% of aluminium particles in a water base. The results of [27] showed that a \(Cu\)-ethylene glycol nanofluid with a volume fraction of 0.3% produced a 40% increase in thermal conductivity. The investigation in [8] reported up to 78% rise in thermal conductivity for \(Cu\)-water nanofluids at a 75% volume fraction of nanoparticles. Akin to thermal conductivity enhancements investigations, studies have also been conducted on the effect of nanofluids on convective heat-transfer. The authors in [28], for example, investigated the performance of a nanofluid containing carbon nanotubes and observed that the convective heat-transfer rate can be as much as 3.5 times higher than that of the corresponding water base. Similarly, the authors in [29], using an \(Al_{2}O_{3}\)-water nanofluid, reported a convective heat-transfer rate rise of 40%. The investigations in [30, 31], on the convective heat-transfer coefficient of deionised water with a dispersion of \(Cu\) nanoparticles for both laminar and turbulent flows in a tube, demonstrated higher convective heat-transfer coefficients for the nanofluids than for the deionised water base. Even though nanofluids may physically never be truly homogeneous mixtures, the mathematical modelling of nanofluids is nonetheless generally divided into the two classes, namely single-phase (homogeneous) and two-phase (heterogeneous), see for example [32, 33, 34]. In the single-phase model, which is adopted in the present work, the nanofluid is treated as a homogeneous fluid mixture and the conventional fluid dynamical governing equations are modified to incorporate the volume-fractions of the embedded nanoparticles, see for example [32, 33].