Dynamics of Non-Isothermal Pressure-Driven Flow of Generalized Viscoelastic-Fluid-Based Nanofluids in a Channel

The investigation considers numerical analysis and computational solution of unsteady, pressure-driven channel ﬂow of a generalized viscoelastic-ﬂuid-based nanoﬂuid (GVFBN) subject to exothermic reactions. Temperature-dependent ﬂuid thermal conductivity is considered, and the ﬂow is subject to convective cooling at the walls. The non-isothermal generalized Giesekus constitutive model is employed for the GVFBN. A Carreau model is used to describe the shear-rate dependence of ﬂuid viscosity, and exothermic reactions are assumed to follow Arrhenius kinetics. An eﬃcient semi-implicit numerical technique based on the ﬁnite-diﬀerence method is applied to obtain computational solutions to the model equations. The computational methodologies are built into the MATLAB software. The eﬀects of various ﬂuid and ﬂow parameters, speciﬁcally the nanoparticle volume fraction, are explored. The results demonstrate that those parameters which only directly couple to the energy equation (but are otherwise indirectly coupled to momentum and stress-constitutive equations, say via the temperature-dependent viscosities and relaxation times) would only show prominent eﬀects on ﬂuid temperature but not on the ﬂuid velocity or the polymer stresses. The results also demonstrate, as in the literature on exothermic ﬂows, that the values of exothermic-reaction parameter must be carefully controlled as large values would lead to thermal runway phenomena. The illustrated results are consistent with the existing literature and additionally add novel new contributions to non-isothermal and pressure-driven channel ﬂow of GVFBN under convective cooling conditions.


Introduction
A mixture of metallic (silver, copper, etc.) nanometre-sized particles suspended in a conventional base fluid (water, oil, etc.) is referred to as a nanofluid. For a comprehensive description of nanofluids and their utility, say, in the heattransfer-rate (HTR) characteristics, the reader is referred to, say, [1].
Optimal material conditions for fluid viscosity and thermal conductivity may be enhanced by simultaneously using several types of nanoparticles (say three different types) of various shapes, sizes, density, etc. in the same nanofluid mixture (see, for example, [2,3]). e present research focuses on the effects of shear-dependent viscosity and fluid elasticity and hence will use homogeneous nanoparticles of one kind for illustrative purposes.
Similar investigations, as in [1], on thermodynamic properties and effects of various types of nanofluids have been conducted, say in .
e importance of such investigations cannot be overstated, especially considering the novel applications of nanofluids to thermal conductivity and HTR improvement, microfluidics, fuel-cell development, chemotherapy, thermal storage, electronic cooling and heating, etc.
A vast majority of the research on nanofluid flow has been conducted with Newtonian base fluids. e recent developments in non-isothermal constitutive models for viscoelastic fluid flow and their related applications (see, for example, [1,[25][26][27][28][29][30]) have made it possible for the extension of the Newtonian-based fluid models to more general non-Newtonian (and specifically viscoelastic)-fluid-based nanofluids. e current work represents such an extension to viscoelastic nanofluidics, specifically using GVFBN. e extensions to more general non-Newtonian-fluidbased nanofluids are not simply a theoretical and mathematical nicety but are rather driven by the vast and contemporary relevance to industrial, technological, and medical applications of non-Newtonian fluids in general and viscoelastic fluids in particular (see [31]). e work in [1] focused on novel development and analysis of GVFBN in shear-driven isobaric flows. e current work extends this analysis to non-isobaric (pressure-driven) channel flow of GVFBN. Additionally, instead of the isothermal wall boundary conditions employed in [1], the current work explores symmetrical convective cooling on the channel walls. e paper is organized in the following sequence. Section 2 summarizes the model and governing equations. Section 3 gives the numerical algorithms and computational methodologies for the model problem as well as preliminary test results on temporal and spatial convergence. e exploration and discussion of the effects of the various embedded parameters are detailed in Section 4. Concluding remarks are given in Section 5.

Problem Formulation
e schematic of the model problem is displayed in Figure 1. An incompressible GVFBN is subjected to unsteady, laminar, and pressure-driven flow in a channel formed by two infinitely long parallel walls. e GVFBN is assumed to have both variable shear viscosity and variable thermal conductivity and is susceptible to exothermic reactions following Arrhenius theory. e x * -axis is considered parallel to the flow direction and the y * -axis is perpendicular to it.
Following the model developments in [1], the appropriate dimensionless parameters of current interest are the Reynolds number (Re), ambient-temperature parameter (θ a ), Brinkman number (Br), Deborah number (De), activation-energy parameter (α), Prandtl number (Pr), Peclet number (Pe � Re · Pr), Frank-Kamenetskii parameter (δ 1 ), Biot number (Bi), and the ratio of the polymer-to-the-total viscosity (β): e nanofluid quantities (nf) are obtained from linear combinations of the volume fractions (φ) of the base fluid (f) and of the solid nanoparticles (s), e.g., e appropriate dimensionless governing equations are 2 Mathematical Problems in Engineering e dimensionless dissipation term for the single-mode non-isothermal Giesekus model is where the polymeric-stress tensor (τ) and conformation tensor (b) are related by e dimensionless temperature-dependent viscosities, thermal conductivity, and relaxation times are with η � η s (T) + η p ( _ c, T), In equations (7) and (8), m ≥ 1 is the shear-rate-viscosity parameter, 0 ≤ n ≤ 1 is the shear-thinning parameter, and ℵ is the empirical shape factor (where ℵ � 3 for nanoparticles of spherical shape) [1]. e shear-rate viscosity parameter, m, gives the ratio of zero shear rate to the infinite shear-rate viscosities. e value m � 1 indicates that the viscosity is independent of shear rates. is would be the case, say, for Newtonian fluids/nanofluids and for ordinary viscoelastic fluids/nanofluids. e generalized Newtonian-fluid-based nanofluid (GNFBN) model is obtained by taking β � 0 (and m ≠ 1) and De � 0. e choice De � β � 0 and m � 1 represents a Newtonian nanofluid (refer to [1,25]).

Initial and Boundary Conditions.
Given the symmetric flow geometry for the pressure-driven flow, it is sufficient to consider only the upper half channel y ∈ [0, 1] instead of the full channel y ∈ [− 1, 1]. e following initial and boundary conditions therefore apply:

Numerical Solution
e semi-implicit finite-difference numerical and computational methodologies are based on the ideas described in [1,25]. e velocity equation is discretized as where G � (zp/zx) � 1 denotes the constant pressure gradient in the flow direction. e velocity updates in time via the scheme Mathematical Problems in Engineering where e resultant system of algebraic equations represents a diagonally dominant tridiagonal linear system. e semiimplicit finite-difference method for temperature equation follows similarly: Specifically, the temperature updates in time via the scheme where e equations for the extra stress tensor, τ (3), are solved analogously: � explicit terms. (16) e solutions for the various tensor components therefore follow directly: e explicit terms for τ 11 , τ 12 , and τ 22 are, respectively,

Results
Unless otherwise indicated, the following list of values, for the embedded variables and material parameters, will be assumed: e above will constitute the default variable and parameter values in this study. In the subsequent graphical results, it will be assumed and understood that, where variable and parameter values are not explicitly stated, they will be given by the default values.

Convergence in Time and Space.
It is important to validate the utility of a numerical and computational algorithm before deploying it to solving physical problems. Figures 2-5 show, as required, that the computational algorithms are independent of both time step and mesh size-specifically, the algorithms give the same results for a large range of time steps and mesh sizes.

Transient Development of Solutions to Steady State.
Given the focus on unsteady flow, it is equally important (in addition to the demonstration of time step and mesh size convergence) to demonstrate that the algorithms are capable of capturing the transient (time) development of steady solutions, including the capabilities to also illustrate the final steady-state solutions. Figures 6 and 7 show, as required, the development of solutions in time until steady states are reached. Figure 8 gives an illustration of the variation of flow quantities with φ, the volume fraction of the embedded nanoparticles. e results show an increase in the flow quantities with increasing φ. e increase in fluid temperature with increasing φ is obvious, given that the increased volume of heat conducting nanoparticles would directly also increase fluid thermal conductivity (see Figure 9). e increase in fluid temperature naturally also decreases the fluid viscosity and hence also diminishes the viscous drag effects, hence the increase in fluid velocity with increasing φ as observed in Figure 8.

Parameter Dependence of Solutions.
Viscous effects are inversely proportional to the Reynolds number, Re. Alternatively, high speed flows are synonymous with high Reynolds numbers. Figure 10 therefore demonstrates, as expected, an increase in velocity with increasing Re. Figure 11 shows the dependence of flow quantities with Brinkman number, Br. Since Br is directly connected to the strength of the heat sources, the fluid temperature (and hence also the fluid velocity) is expected to increase with increasing Br as illustrated in Figure 11. e polymeric-stress component τ 11 , on the other hand, decreases with increasing Br.
Given their opposing influence to heat sources in the energy equation, the fluid temperature behaves oppositely with increasing Prandtl number Pr (see Figure 12), as it would with Br.
e Prandtl number indirectly enters the momentum and polymeric-stress equations via the temperature dependence of the viscosity. For these reasons, small variations in Pr have no discernible influence on the fluid velocity and polymeric-stress components ( Figure 12). e activationenergy parameter, α, has similar effects to those of Br (see Figure 13). e increase of fluid temperature with increasing α as illustrated in Figure 13 can be directly linked to the increase of nanofluid thermal conductivity, κ nf , with increasing α (see Figure 14). Figure 15 illustrates the required increase in fluid velocity with increased pressure driving force. e behaviour of fluid temperature with variations in the exothermic-reaction parameter δ 1 is similar to that with respect to Br given the linkages of these parameters to the heat sources (see Figure 16). e major difference is that the fluid temperature increases linearly with increasing Br, whereas the fluid temperature increases exponentially with increasing δ 1 (see [25]). e possibility of thermal runway phenomena therefore looms large with regard to increases in δ 1 due to exponential temperature growth. Indeed, for large values of the exothermic-reaction parameter δ 1 , the steadystate solutions would not be attainable-the solutions would blow up in finite time (see Figure 17). e respective behaviour of flow quantities with variations in the viscoelastic parameters, c and De, is illustrated in Figures 18 and 19, respectively.
As expected, the polymeric (viscoelastic) stresses increase with increasing De. It is also noticed, as expected, that the fluid temperature increases with increasing c given the dominance of entropic heat dissipation (over energetic heat storage) for larger values of c.
e heat exchange between the ambient environment, the channels walls, and the nanofluid is related to the Biot number Bi (see (10)). Enhanced convective cooling at the channel walls is directly related to higher Bi. e cooled walls in turn lead to lower nanofluid temperatures in the bulk flow. Figure 20 therefore illustrates, as expected, the decrease in (bulk) nanofluid temperature with increasing Bi.
Mathematical Problems in Engineering e nanofluid viscosity (at constant shear rates) increases with increasing shear-thinning parameter, m. An increase in m therefore increases resistance to flow (due to the higher viscosity) and hence leads to a decrease in fluid velocity as shown in Figure 21. A decrease in both the fluid temperature and the polymeric stresses with increasing m is also noticed.

Concluding Remarks
e investigation employed efficient and convergent semiimplicit numerical and computational algorithms based on the finite-difference methods to analyze the non-isothermal, pressure-driven, channel flow of a generalized viscoelasticfluid-based nanofluid (GVFBN) subject to convective cooling at the channel boundaries. e assumed mathematical model is based on a single-phase (constant concentration) nanofluid mixture in which the nanoparticles are homogeneously mixed with the generalized viscoelastic-fluid base.
e research results are explored in qualitative (graphical) detail illustrating the effects of the variations of the fundamental embedded parameters on the flow quantities, specifically, the fluid velocity, fluid temperature, and polymer stresses. e results demonstrate that those parameters which only directly couple to the energy equation (but are otherwise indirectly coupled to the momentum and stress-constitutive equations, say via the viscosities and relaxation times) would only show prominent effects on nanofluid temperature but not on the fluid velocity or the polymer stresses. e results also demonstrate, as in the literature on exothermic flows, that the values of exothermic-reaction parameter must be carefully controlled as large values would lead to thermal runway phenomena.
e present results are consistent with the existing literature and additionally add novel new contributions to non-isothermal and pressure-driven channel flow of GVFBN under convective cooling conditions.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.