Heat transfer analysis with temperature-dependent viscosity for the peristaltic flow of nano fluid with shape factor over heated tube
Introduction
Research and advancements in the area of nano-science have received a lot of attention since last two decades. Nanofluid dynamics is a part of nano-science that deals with the study of concentration, volume fraction and energy transport of nanoparticle with base fluids. In last few decades, the work on nanofluiddynamics is much attraction for the reader and industry people. Das et al. [1] reported a review report on heat transfer in nanofluids where they discussed the effects in thermal conductivity by suspension of small particles of micrometer sized and suggested the directions for future developments. In continuation of review report on nanofluids, Wang and Mujumdar [2] presented the heat transfer characteristics of nanofluids; Trisaksri, and Wongwises [3] further elaborated the report on nanofluids; Murshed et al. [4] studied the thermo physical and electro kinetic properties of nanofluids; Wang and Mujumdar [5] extend his review report for theoretical and numerical studies on nanofluids; Kakac and Pramuanjaroenkij [6] reviewed the convective heat transfer enhancement with nanofluids; Wen et al. [7] focused his review for nanofluids applications; Yu and Xie [8]presented preparation, stability mechanisms, and applications of nanofluids in their review report; Mahian et al. [9] discussed the applications of nanofluids for solar energy; Taylor et al. [10] extended the application of nanofluids for Small particles, big impacts; Younes et al. [11] studied the thermal conductivity of nanofluids; Sarkar et al. [12] presented another review report on hybrid nanofluids and its applications; Kakac and Pramuanjaroenkij [13] most recently presented a new review report where they studied single-phase and two-phase treatments of convective heat transfer enhancement with nanofluids. They have compared Nusselt number and friction factor correlations of nanofluids, nanofluid characters in common phenomena's such as boiling, radiation, entropy generation and jet flows, experimental consistency, chemical nature, mass diffusion coefficients and properties of novel nanofluids and noted that in most of the works the results shown common agreement. Most recently some interesting works [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24] on nanofluids and its application in different domain have been incorporated.
Peristaltic transport is a biological mechanism which entails the conveyance of material induced by a progressive wave of contraction or expansion along the length of a distensible vessel (tube). This effectively mixes and propels the fluid in the direction of the wave propagation. Peristaltic flows of non-Newtonian viscous fluids are encountered in many complex physiological systems including urine transport from the kidney to the bladder, chyme motion in the gastrointestinal tract, movement of ovum in the female fallopian tube, vasomotion of small blood vessels, transport of spermatozoa, and swallowing food through the esophagus (and other biomedical applications which are summarized in Fung [25]) and also phloem trans-location in plants as described by Thaine [26] and Thompson [27]. In a mathematical context, peristaltic flows fall in the category of moving boundary value problems. They have as a result mobilized considerable interest in recent years. Many analytical investigations of peristaltic propulsion have therefore been communicated and these have addressed a diverse range of geometries under various assumptions such as large wavelength, small amplitude ratio, small wave number, small Deborah number, low Reynolds number and creeping flow, etc. Representative works in this regard include Ellahi [28]; Hameed and Nadeem, [29]; Tan and Masuoka, [30], [31], [32]; Mahomed and Hayat, [33]; Fetecau and Fetecau, [34]; Malik et al., [35]; Dehghan and Shakeri, [36]. Some relevant studies on the topic can be found from the list of references (Nadeem and Akbar, [37], [38]) and several references therein.
Non-Newtonian models are extremely diverse and include viscoelastic, viscoplastic, micro-continuum and other formulations. Variable-viscosity models are also an important sub-section of rheological liquids. They have therefore also been investigated in the context of peristaltic flows, since viscosity variation (e.g. with temperature) is an important characteristic of certain physiological (and industrial) materials. Peristaltic transport of a power-law fluid with variable consistency was examined by Shukla and Gupta [39]. They observed that for zero pressure drops, flow rate flux is elevated for greater amplitude of the peristaltic wave whereas it is suppressed with increasing pseudo-plastic nature of the fluid. They further noted that wall friction is reduced as the consistency decreases. Srivastava et al. [40] studied the peristaltic transport of a fluid with variable viscosity through a non-uniform tube. They showed that the pressure rise is markedly lowered as the fluid viscosity decreases at zero flow rates but is infect independent of viscosity variation at a certain value of flow rate. However beyond this critical flow rate, the pressure rise is enhanced with greater viscosity. Abd El Hakeem et al. [41] studied using perturbation expansions, the influence of variable viscosity and an inserted endoscope on peristaltic viscous flow. They employed an exponential decay model for viscosity and observed that pressure rise is decreased with increasing viscosity ratio whereas it is enhanced with increasing wave number, amplitude ratio and radius ratio. Further studies of variable-viscosity peristaltic flow include Abd El Hakeem et al. [42] for magneto hydrodynamic fluids, Khan et al. [43] for inclined pumping of non-Newtonian fluids and Akbar [44] present nanoparticle volume fraction for phase model.
Another significant development in medical engineering in recent years has been the emergence of nanofluids, a sub-category of nanoscale materials. Nanofluids comprise base fluids (water, air, ethylene glycol etc) with nano-size solid particles suspended in them. Nanofluids have gained much attention from investigators due to their high thermal conductivity and pioneering work in developing such fluids was first performed by Choi [45] Nanoparticle are generally synthesized from metals, oxides, carbides, or carbon nanotubes owing to high thermal conductivities associated with these materials. In a medical engineering context as refer in Ref. [45], nanoparticle have been found to achieve exceptional performance in enhancing thermal and mass diffusion properties of, for example, drugs injected into the blood stream. Biocompatibility of the selected metallic oxides is crucial for safe deployment of nanofluids in medicine. New potential applications for nanoparticle in nanoparticle blood diagnostic systems, asthma sensors, carbon nanotubes in catheters and stents and anti-bacterial treatment for wounds via peristaltic pump delivery was identified by Harris and Graffagnini [46]. Nanoparticles possess many unique attributes which make them particularly attractive for clinical applications. These include a surface to mass ratio which is much greater than that of other particles, site-specific targeting features which can be achieved by attaching targeting ligands to surface of particles (or via magnetic guidance), quantum properties, enhanced ability to absorb and carry other compounds, excellent large functional surface which can bind, adsorb and convey secondary compounds (drugs, probes and proteins). Further advantages encompass controllable deployment of particle degradation characteristics which can be successfully modulated by judicious selection of matrix constituents, and flexibility in administration methods (nasal, parenteral, intra-ocular). In neuro-pharmacological hemodynamic as refer in Ref. [47], it has been clinically verified that nanoparticle can easily penetrate the blood brain barrier (BBB) facilitating the introduction of therapeutic agents into the brain. Fullstone et al. [47] have also recently described the exceptional characteristics of nanoparticle (size, shape and surface chemistry) in assisting effective delivery of drugs within cells or tissue (achieved via modulation of immune system interactions, blood clearance profile and interaction with target cells). They have further shown that erythrocytes aid in effective nanoparticle distribution within capillaries. Further studies include Tan et al. [48]. Simulation of peristaltic flows of nanoparticle is therefore extremely relevant to improve administration of nanofluids in medicine. Representative studies in this regard include Tripathi and Bég [49] who considered analytically the thermal and nano-species buoyancy effects on heat, mass and momentum transfer in peristaltic propulsion of nanofluids in peristaltic pumping devices, employing the Buongiornio formulation which incorporates Brownian motion and thermophoresis. Ebaid and Aly [50] studied magnetic field effects on electrically-conducting nanofluid propulsion by peristaltic waves with applications in cancer therapy. Akbar et al. [51] investigated peristaltic slip nanofluid hydrodynamics in an asymmetric channel, obtaining series solutions for temperature, nano-particle concentration, stream function and pressure gradient. Akbar and Nadeem [52] considered peristaltic flow of Phan-Thien-Tanner nanofluid in a diverging conduit with the homotopy perturbation method. Further analyses include Bég and Tripathi [53] who considered double-diffusive convection of nanofluids in finite length pumping systems. These studies did not consider variable viscosity effects. Further literature can be viewed through Refs. [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64].
In the present article, we therefore consider the peristaltic propulsion of nanofluid in a vertical conduit with temperature-dependent viscosity. Basic formulation is employed for the nanofluid with a viscosity modification. Heat transfer is also considered and heat source/sink and thermal buoyancy effects featured. Various nano-particles are considered i.e. Titanium oxide-water, Copper oxide-water and Silver-water. Analytical solutions are derived to examine the effects of heat generation/absorption parameter, Grashof number, viscosity parameter and nanoparticle volume fraction on velocity, temperature, pressure gradient, pressure rise and wall shear stress variables. Streamline visualization is also computed to assess trapping hydrodynamics. The mathematical model is of potential importance in better understanding medical peristaltic pump nano-pharmacological delivery systems.
Section snippets
Mathematical formulation
Consider axisymmetric flow of a variable-viscosity nanofluid in a circular tube of finite length L. The tube walls are flexible and a sinusoidal wave propagates along the walls of the tube. Isothermal conditions are enforced at the walls which are maintained at a temperature, T0. At the center of the tube, a symmetric temperature condition is imposed. The geometric model is illustrated in Fig. 1 with respect to a cylindrical coordinate system .
The geometry of the wall surface is
Analytical solutions of the boundary value problem
Closed-form solutions are feasible for the transformed, non-dimensional boundary value problem. Solving Eqs. (9), (10), (11) together with boundary conditions (13) and (14) therefore generates the following expressions for temperature and axial velocity:where:
Results and interpretation
Numerical computations, based on the exact solutions derived in section 3, have been conducted to assess the influence of flow parameters on the peristaltic flow characteristics (see Table 1). These are depicted in Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10, Fig. 11, Fig. 12. These computations are based on nanofluid properties documented in Table 2. Further solutions are given for the velocity and temperature fields in Tables.
Fig. 2(a)-(b) Present the radial
Conclusions
A mathematical study has been conducted of peristaltic propulsion and heat transfer in a temperature-dependent variable-viscosity nanofluid propagating through a flexible tube under thermal buoyancy and heat generation effect. The transformed boundary value problem has been linearized via appropriate creeping flow and long wavelength approximations and solved exactly. Numerical evaluation of the closed-form solutions has been conducted in symbolic software to evaluate the influence of heat
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