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

Authors: Idrees Khan; Tiri Chinyoka; Andrew Gill
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

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.


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## Body

## 1. 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 heat-transfer-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]). The 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 [4–24]. The 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. The recent developments in non-isothermal constitutive models for viscoelastic fluid flow and their related applications (see, for example, [1, 25–30]) have made it possible for the extension of the Newtonian-based fluid models to more general non-Newtonian (and specifically viscoelastic)-fluid-based nanofluids. The current work represents such an extension to viscoelastic nanofluidics, specifically using GVFBN.The extensions to more general non-Newtonian-fluid-based 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]). The work in [1] focused on novel development and analysis of GVFBN in shear-driven isobaric flows. The 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.The paper is organized in the following sequence. Section2 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. The exploration and discussion of the effects of the various embedded parameters are detailed in Section 4. Concluding remarks are given in Section 5.


## 2. Problem Formulation

The schematic of the model problem is displayed in Figure1.Figure 1
Geometry of the model problem.An incompressible GVFBN is subjected to unsteady, laminar, and pressure-driven flow in a channel formed by two infinitely long parallel walls. The GVFBN is assumed to have both variable shear viscosity and variable thermal conductivity and is susceptible to exothermic reactions following Arrhenius theory. Thex∗-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 θa, Brinkman number Br, Deborah number De, activation-energy parameter α, Prandtl number Pr, Peclet number Pe=Re⋅Pr, Frank-Kamenetskii parameter δ1, Biot number Bi, and the ratio of the polymer-to-the-total viscosity β:(1)β=ηp∗η∗,De=λ0∗U0∗h∗,Re=ρf∗U0∗a∗η∗,Pr=cpf∗η∗κf∗,θa=T∗−T0∗αT0∗,Br=η∗U0∗2κf∗αT0∗,Bi=h∗a∗k0∗,Renf=ρnfρfRe,Penf=ρcpnfρcpfPe,α=R∗T0∗E∗,δ1=Q∗A∗C∗a∗2κf∗R∗TW∗2exp−1α.The nanofluid quantitiesnf are obtained from linear combinations of the volume fractions φ of the base fluid f and of the solid nanoparticles s, e.g.,(2)ρnf=φρs+1−φρf,ρcpnf=φρcps+1−φρcpf.The appropriate dimensionless governing equations are(3)∇⋅u=0,RenfDuDt=−Renf∇p+∇⋅σ,PenfDTDt=−∇⋅κnf∇T+BrQD+δ1expT1+αT,τ+ετ2+Deλ¯Tτ∇−τDDtln1+αT=ηpT1−φ5S,with(4)σ=τ+ηsT1−φ5S,S=∇u+∇uT,τ∇=DτDt−∇u⋅τ−τ⋅∇uT.The dimensionless dissipation term for the single-mode non-isothermal Giesekus model is(5)QD=ηsT1−φ5S:∇u+γS:τ+1−γG^2Deλ¯TI1+Trb−1−6,where the polymeric-stress tensor τ and conformation tensor b are related by(6)τ=G^b−I.The dimensionless temperature-dependent viscosities, thermal conductivity, and relaxation times are(7)ηsT=1−βexp−αT,ηsnf=ηsT1−φ5,ηpγ˙,T=β+m−11+λγ˙2n−1/2e−αT,ηpnf=ηpγ˙,T1−φ5,with(8)η=ηsT+ηpγ˙,T,ηnf=η1−φ5,λ¯T=11+αTexp−αT,κnf=κs+1−ℵκf−ℵ−1φκf−κsκs+1−ℵκf+φκf−κs1+αA2T.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]. The shear-rate viscosity parameter, m, gives the ratio of zero shear rate to the infinite shear-rate viscosities. The value m=1 indicates that the viscosity is independent of shear rates. This would be the case, say, for Newtonian fluids/nanofluids and for ordinary viscoelastic fluids/nanofluids. The generalized Newtonian-fluid-based nanofluid (GNFBN) model is obtained by taking β=0andm≠1 and De=0. The choice De=β=0 and m=1 represents a Newtonian nanofluid (refer to [1, 25]).


### 2.1. Initial and Boundary Conditions

Given the symmetric flow geometry for the pressure-driven flow, it is sufficient to consider only the upper half channely∈0,1 instead of the full channel y∈−1,1. The following initial and boundary conditions therefore apply:(9)u0,y=0,T0,y=0,τ0,y=0,for0≤y≤1,∂∂yut,0=0,∂∂yTt,0=0,fort>0ut,1=0,∂∂yTt,1=−BiTt,1−θa,fort>0.


## 2.1. Initial and Boundary Conditions

Given the symmetric flow geometry for the pressure-driven flow, it is sufficient to consider only the upper half channely∈0,1 instead of the full channel y∈−1,1. The following initial and boundary conditions therefore apply:(9)u0,y=0,T0,y=0,τ0,y=0,for0≤y≤1,∂∂yut,0=0,∂∂yTt,0=0,fort>0ut,1=0,∂∂yTt,1=−BiTt,1−θa,fort>0.


## 3. Numerical Solution

The semi-implicit finite-difference numerical and computational methodologies are based on the ideas described in [1, 25]. The velocity equation is discretized as(10)Renfun+1−unΔt=RenfG+∂∂yτ12n+ηsnfn∂2∂y2un+ξ+∂∂yun∂∂yηsnfn,where G=∂p/∂x=1 denotes the constant pressure gradient in the flow direction. The velocity updates in time via the scheme(11)−r1uj−1n+1+Renf+2r1ujn+1−r1uj+1n+1=Renfujn+RenfG+1−ξΔtηsnfn∂2∂y2un+Δt∂∂yτ12n+1−βΔt∂∂yun∂∂yηsnfn,where(12)r1=ξηsnfnΔtΔy2.The resultant system of algebraic equations represents a diagonally dominant tridiagonal linear system. The semi-implicit finite-difference method for temperature equation follows similarly:(13)Penf∂T∂t=κnfn∂2∂y2Tn+ξ+∂∂yTn∂∂yκnfn+BrQDn+δ1expTn1+αTn.Specifically, the temperature updates in time via the scheme(14)−r2Tj−1n+1+Penf+2r2Tjn+1−r2Tj+1n+1=PenfTjn+1−ξΔtκnf∂2∂y2Tn+Δt∂∂yTn∂∂yκnfn+ΔtBrηsnfn∂∂yun2+δ1expTn1+αTn+ΔtBrγτ12n∂∂yun+1−γΔtBrG^2Deλ¯TI1+Trb−1−6,where(15)r2=ξκnfΔtΔy2.The equations for the extra stress tensor,τ (3), are solved analogously:(16)τn+ξ+ετ2n+Deλ¯nτn+1−τnΔt=explicit terms.The solutions for the various tensor components therefore follow directly:(17)Deλ¯n+ξΔtτn+1=explicit terms.The explicit terms forτ11, τ12, and τ22 are, respectively,(18)Deλ¯n−1−ξΔtτ11n+ΔtDeλ¯nτ12n∂∂yun+τ11n∂∂ylog1+αTn−εΔtτ112+τ122,Deλ¯n−1−ξΔtτ12n+ΔtDeλ¯nτ22n∂∂yun+τ12n∂∂ylog1+αTn+Δtηpnf∂∂yun−εΔtτ11τ12+τ222,Deλ¯n−1−ξΔtτ22n+ΔtDeλ¯nτ22n∂∂ylog1+αTn−εΔtτ122+τ−12τ22.


## 4. Results

Unless otherwise indicated, the following list of values, for the embedded variables and material parameters, will be assumed:(19)α=0.1,Br=1,Re=1,Pr=1,De=2,γ=0.5,β=0.2,Δy=0.01,Δt=0.1,t=50,G^=10−3,m=1.2,δ1=0.1,λ=0.01,n=0.5,ζ=1,φ=0.04,ε=0,A2=0.2,Bi=1,θa=0.1,ℵ=3.The 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.


### 4.1. 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. Figures2–5 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.Figure 2
Dependence of solutions on time step size.Figure 3
Dependence of solutions on mesh size.Figure 4
Dependence of solutions on time step size.Figure 5
Dependence of solutions on mesh size.


### 4.2. 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. Figures6 and 7 show, as required, the development of solutions in time until steady states are reached.Figure 6
Transient development to steady-state profiles,Δt=0.05.Figure 7
Transient development of steady maximum flow quantities.


### 4.3. Parameter Dependence of Solutions

Figure8 gives an illustration of the variation of flow quantities with φ, the volume fraction of the embedded nanoparticles. The results show an increase in the flow quantities with increasing φ. The 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).Figure 8
Response of flow quantities to variations inφ.Figure 9
Response of nanofluid thermal conductivity to variations inφ.The 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 increasingφ as observed in Figure 8.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. The polymeric-stress component τ11, on the other hand, decreases with increasing Br.Figure 10
Response of flow quantities to variations inRe.Figure 11
Response of flow quantities to variations inBr.Given their opposing influence to heat sources in the energy equation, the fluid temperature behaves oppositely with increasing Prandtl numberPr (see Figure 12), as it would with Br.Figure 12
Response of flow quantities to variations inPr.The Prandtl number indirectly enters the momentum and polymeric-stress equations via the temperature dependence of the viscosity. For these reasons, small variations inPr have no discernible influence on the fluid velocity and polymeric-stress components (Figure 12). The activation-energy parameter, α, has similar effects to those of Br (see Figure 13). The increase of fluid temperature with increasing α as illustrated in Figure 13 can be directly linked to the increase of nanofluid thermal conductivity, κnf, with increasing α (see Figure 14).Figure 13
Response of flow quantities to variations inα, where δ1=0.001.Figure 14
Response of nanofluid thermal conductivity,κnf, to variations in α.Figure15 illustrates the required increase in fluid velocity with increased pressure driving force.Figure 15
Effects of variations in pressure driving force on fluid velocity.The behaviour of fluid temperature with variations in the exothermic-reaction parameterδ1 is similar to that with respect to Br given the linkages of these parameters to the heat sources (see Figure 16). The major difference is that the fluid temperature increases linearly with increasing Br, whereas the fluid temperature increases exponentially with increasing δ1 (see [25]). The possibility of thermal runway phenomena therefore looms large with regard to increases in δ1 due to exponential temperature growth. Indeed, for large values of the exothermic-reaction parameter δ1, the steady-state solutions would not be attainable—the solutions would blow up in finite time (see Figure 17).Figure 16
Response of flow quantities to variations inδ1.Figure 17
Finite time blow-up of fluid temperature for large value ofδ1.The respective behaviour of flow quantities with variations in the viscoelastic parameters,γ and De, is illustrated in Figures 18 and 19, respectively.Figure 18
Response of flow quantities to variations inγ.Figure 19
Response of flow quantities to variations inDe.As expected, the polymeric (viscoelastic) stresses increase with increasingDe. It is also noticed, as expected, that the fluid temperature increases with increasing γ given the dominance of entropic heat dissipation (over energetic heat storage) for larger values of γ.The heat exchange between the ambient environment, the channels walls, and the nanofluid is related to the Biot numberBi (see (10)). Enhanced convective cooling at the channel walls is directly related to higher Bi. The 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.Figure 20
Response of flow quantities to variations inBi.The 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.Figure 21
Response of flow quantities to variations inm.


## 4.1. 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. Figures2–5 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.Figure 2
Dependence of solutions on time step size.Figure 3
Dependence of solutions on mesh size.Figure 4
Dependence of solutions on time step size.Figure 5
Dependence of solutions on mesh size.


## 4.2. 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. Figures6 and 7 show, as required, the development of solutions in time until steady states are reached.Figure 6
Transient development to steady-state profiles,Δt=0.05.Figure 7
Transient development of steady maximum flow quantities.


## 4.3. Parameter Dependence of Solutions

Figure8 gives an illustration of the variation of flow quantities with φ, the volume fraction of the embedded nanoparticles. The results show an increase in the flow quantities with increasing φ. The 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).Figure 8
Response of flow quantities to variations inφ.Figure 9
Response of nanofluid thermal conductivity to variations inφ.The 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 increasingφ as observed in Figure 8.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. The polymeric-stress component τ11, on the other hand, decreases with increasing Br.Figure 10
Response of flow quantities to variations inRe.Figure 11
Response of flow quantities to variations inBr.Given their opposing influence to heat sources in the energy equation, the fluid temperature behaves oppositely with increasing Prandtl numberPr (see Figure 12), as it would with Br.Figure 12
Response of flow quantities to variations inPr.The Prandtl number indirectly enters the momentum and polymeric-stress equations via the temperature dependence of the viscosity. For these reasons, small variations inPr have no discernible influence on the fluid velocity and polymeric-stress components (Figure 12). The activation-energy parameter, α, has similar effects to those of Br (see Figure 13). The increase of fluid temperature with increasing α as illustrated in Figure 13 can be directly linked to the increase of nanofluid thermal conductivity, κnf, with increasing α (see Figure 14).Figure 13
Response of flow quantities to variations inα, where δ1=0.001.Figure 14
Response of nanofluid thermal conductivity,κnf, to variations in α.Figure15 illustrates the required increase in fluid velocity with increased pressure driving force.Figure 15
Effects of variations in pressure driving force on fluid velocity.The behaviour of fluid temperature with variations in the exothermic-reaction parameterδ1 is similar to that with respect to Br given the linkages of these parameters to the heat sources (see Figure 16). The major difference is that the fluid temperature increases linearly with increasing Br, whereas the fluid temperature increases exponentially with increasing δ1 (see [25]). The possibility of thermal runway phenomena therefore looms large with regard to increases in δ1 due to exponential temperature growth. Indeed, for large values of the exothermic-reaction parameter δ1, the steady-state solutions would not be attainable—the solutions would blow up in finite time (see Figure 17).Figure 16
Response of flow quantities to variations inδ1.Figure 17
Finite time blow-up of fluid temperature for large value ofδ1.The respective behaviour of flow quantities with variations in the viscoelastic parameters,γ and De, is illustrated in Figures 18 and 19, respectively.Figure 18
Response of flow quantities to variations inγ.Figure 19
Response of flow quantities to variations inDe.As expected, the polymeric (viscoelastic) stresses increase with increasingDe. It is also noticed, as expected, that the fluid temperature increases with increasing γ given the dominance of entropic heat dissipation (over energetic heat storage) for larger values of γ.The heat exchange between the ambient environment, the channels walls, and the nanofluid is related to the Biot numberBi (see (10)). Enhanced convective cooling at the channel walls is directly related to higher Bi. The 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.Figure 20
Response of flow quantities to variations inBi.The 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.Figure 21
Response of flow quantities to variations inm.


## 5. Concluding Remarks

The investigation employed efficient and convergent semi-implicit numerical and computational algorithms based on the finite-difference methods to analyze the non-isothermal, pressure-driven, channel flow of a generalized viscoelastic-fluid-based nanofluid (GVFBN) subject to convective cooling at the channel boundaries. The 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. The 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. The 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. 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 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.



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