Published by the University of Cape Town (UCT) under the non-exclusive license granted to UCT by the author. Published by the University of Cape Town (UCT) under the non-exclusive license granted to UCT by the author.

## Nanofluid

In fact, the thermal conductivity of the nanoparticles is one or more order(s) higher than that of conventional heat transfer fluids, even at low concentrations. 5] stated that addition of less than 1% volume fraction of nanoparticles increases the thermal conductivity of the base fluids by approximately two times.

## An overview on the thermophysical properties of nanofluids

*Thermal conductivity ( κ n f )**Theoretical models of ( κ n f )**Experimental Models of ( κ n f )**Viscosity ( η n f )**Theoretical based Models of viscosity ( η n f )**Experimental based Models of viscosity ( η n f )**Density**Specific heat capacity**Single phase**Homogeneous Model**Thermal dispersion model**Buongiorno model**Two phase*

According to Yu et al.[18], the layered molecules are in an intermediate physical state between a base liquid and nanoparticle. Considering thermophoresis and Brownian effects, and to avoid repetition, we discuss only the energy, we discuss only the energy and concentration equations. the energy equation in the homogeneous model is converted to [13],.

## Methods of heat flows

### Conduction

The direct transfer of heat from one substance to another due to the collision of atoms and molecules is known as conduction.

### Convection

*Force convection**Natural or free convection**Mixed convection*

This is the combination of natural convection and forced convection phenomena, or it takes place when both buoyancy and pressure forces interact.

Radiation

## Rheology and compelx fluids

*Newtonian Fluids**Non-Newtonian Fluids**Time independent**Shear thickening**Shear thinning**Bingham plastics**Time dependent**Rheopectic fluids**Thixotropic fluids**Viscoelastic Fluids*

It can also be stated as "fluids that deviate from Newton's law of viscosity are non-Newtonian fluids". Dilatant fluids” and exemplified by suspensions of sand and cornstarch, water and cornmeal, etc.

Thermal runway

## Shear banding

In addition, certain viscoelastic fluids show a non-monotonic relationship between shear stress and shear rate for some parameter values. In addition, there are many constitutive models (such as Johnson-Segalman ROly-Poly, [88]) that exhibit shear band.

## The Finite Difference Method (FDM)

Implicit scheme: when∂∂tu is approximated using the backdifference method at time steptn+1 and approximating the spatial derivative by implicating the central difference method at point xi. This scheme uses the central difference method at timesn+1/2 and iterates the central difference at the spatial point xi.

## Outline of the thesis

Explicit scheme: when ∂∂ut is approximated using the forward difference technique at time step tn and approximating the spatial derivative while using the central method at point xi. Crank-Nicolson Scheme: This scheme is the midpoint and average of the two techniques discussed earlier in the above.

## Introduction

In the single-phase approach, as adopted in this paper, the fluid dynamics equations are fitted to include a volume fraction function for the homogeneously mixed nanoparticles. These viscosity-altering effects of the nanoparticles must be included in the viscosity-constitutive modelling.

## Problem Formulation

### Dimensionless equations

The dimensionless parameters of interest are the; Reynolds number (Re), Brinkman number (Br), Deborah number (De), activation energy parameter (α), Prandtl number (Pr), Peclet number (Pe= Re Pr, Frank-Kamenetskii parameter (δ1), and the ratio of the polymer to the total viscosity (β).Introducing the non-dimensional variables and dimensionless parameters into the governing equations yields the dimensionless equations listed below. ε is the Giesekus dimensionless nonlinear parameter (reducing ε = 0 to the Oldroyd -B model), is the dimensionless total stress tensor, .

The shear rate viscosity parameter, m, gives the ratio of zero shear rate to the infinite shear rate viscosities.

Initial and boundary conditions

Numerical Solution

## Results

### Devolvement of steady solutions

Likewise, the steady-state solutions for the velocity reproduce the same linear plot and will be omitted in subsequent analysis.

Temporal and spatial convergence

### Parameter dependence of solutions

The polymer stress components exhibit opposite behavior of temperature and thermal conductivity and decrease with increasing nanoparticle volume fraction, see fig. subjected to exothermic reaction in shear flow. The exothermic reaction parameter (δ1) should be carefully controlled to mitigate against thermal runaway. 2-16) provides an illustration of the thermal path for the different fluid types. -23) shows that the temperature increases with increasing Br. This is physically realistic because viscous heating of the fluid particles in the flow channel causes the temperature to increase with increasing Br, while the fluid viscoelasticity shows an opposite behavior, see fig.

The behavior of the solutions with different values of n is similar to the behavior for m, with differences only in the magnitude of the increase/decrease of the quantities.

## Concluding Remarks

The investigation considers the numerical analysis and computational solution of unsteady, pressure-driven channel flow of a generalized viscoelastic-fluid-based nanofluid (GVFBN) undergoing exothermic reactions. The thermal conductivity of the fluid depending on the temperature is considered and the flow is subjected to convective cooling at the walls. A Carreau model is used to describe the shear rate dependence of fluid viscosity, and exothermic reactions are assumed to follow Arrhenius kinetics.

The illustrated results are consistent with the existing literature and additionally add new novel contributions to non-isothermal and pressure-driven channel flow of GVFBN under convective cooling conditions.

## Introduction

The exploration and discussion of the effects of the various nested parameters is detailed in Section 3.4.

## Problem Formulation

Initial and boundary conditions

## Numerical Solution

Given the symmetrical flow geometry for the pressure-driven flow, it is sufficient to consider only the upper half channel-like∈[0,1] instead of the entire channel-like∈[−1,1].

## Results

Convergence in time and space

Transient development of solutions to steady state

### Parameter dependence of solutions

The response of the flow quantities to changes in the Prandtl number, Pr, is shown in the figure. For these reasons, small changes in Pr have no discernible effect on the fluid velocity and polymer stress components, Fig. The behavior of the liquid temperature with variations in the exothermic reaction parameter δ1 is similar to that of Br in terms of the connections of these parameters with heat sources, see Fig.

The respective behavior of flow quantities with variations in the viscoelastic parameters, γ and De, is illustrated in Fig.

## Concluding Remarks

The polymeric (viscoelastic) behavior of VFBN is modeled via the constitutive Giesekus equation with appropriate adjustments to incorporate both the non-isothermal and nanoparticle effects. Nahme type laws are used to describe the temperature dependence of the VFBN viscosities and relaxation times. VFBN is modeled as a single-phase homogeneous mixture, and the effects of the nanoparticles are therefore based on the volume-fraction parameter.

First, under shear band conditions of the Giesekus-type VFBN model, we observe remarkable HTR and Therm-C enhancement in VFBN compared to NFBN.

Introduction

## Problem Formulation

### Model assumptions

However, none of these considerations detract from the primary goal of investigating the broader effects of nanoparticles on HTR and Therm-C enhancement.

### Dimensionless governing equations

The size, shape, distribution, orientation, etc. of the nanoparticles still represent a large open field with respect to exploring the optimal conditions for HTR and Therm-C enhancement. Specifically, p is the pressure field, ε is the Giesekus nonlinear parameter, σ is the total stress tensor, Sis is the deformation tensor, τ is the polymer stress tensor, (ηs)n f is the solvent viscosity for the nanofluid, (ηp)nf is the polymer viscosity for the nanofluid, (η)nf is the total viscosity for the nanofluid, ()nf is the thermal conductivity for the nanofluid, α is the activation energy parameter, β is the ratio between polymer and total viscosity, Br is the Brinkman number, δ1 is the Frank-Kamenetskii parameter, De is the Deborah number, Pr is the Prandtl number, Pe is the Peclet number, Re is the Reynolds number, and the subscript ()nf represents nanofluid.

Initial and boundary conditions

Numerical and computational algorithms

## Graphical and qualitative results

*Time development of steady smooth solutions**Mesh-size and time-step and convergence**Development of Shear-banding**Thermal runway*

As mentioned in the introduction, shear banding phenomena in the shear flow of viscoelastic fluids represent observable and physical discontinuities in the shear velocity and velocity profiles of the flow. The Rolie-Poly viscoelastic constitutive model was developed to explain shear banding phenomena through flow-induced inhomogeneities. Viscoelastic constitutive models of Johnson-Segalman and Giesekus allow the mechanisms of shear banding phenomena through constitutive instabilities, if certain values of viscoelastic material parameters are taken.

The occurrence of shear bands via constitutive instabilities is extensively investigated in [86] using the Johnson-Segalman viscoelastic constitutive model.

## Parameter dependence of solutions under shear-banding conditions

4-16 shows the dependence of the VFBN velocity and temperature on the Brinkman number, Br. 4-16 also shows, as already expected, that the temperature of the nanofluid increases with increasing Br but decreases with increasing polymer viscosity, β. 4-16 shows that the temperature increase with respect to Br is linear for viscoelastic flow compared to the exponential increase observed with respect to δ1.

This therefore means that increases in Br are not expected to lead to thermal runaway in shear flow of VFBN.

## Concluding Remarks

The paper investigates the gravity flow of viscoelastic fluid-based nanofluids (VFBN) along an inclined plane under non-isothermal conditions and convective cooling at a free surface. The responses of flow variables to variations of various fundamental flow parameters are explored graphically and discussed qualitatively. Specifically, new responses (VFBN velocities, VFBN temperatures, VFBN thermal conductivities, and VFBN polymer stresses) to variations in the volume fraction of incorporated nanoparticles are shown.

Such novel responses of VFBN flow variables to variations in nanoparticle volume fraction provide the main framework for the fundamental contributions of this study.

Introduction

## Problem Formulation

### Initial and boundary conditions

The dimensionless initial and boundary conditions are,. 5-16) Due to the hyperbolic structure of the equations for polymeric stresses, their corresponding boundary conditions are reconstructed from the mainstream.

Numerical Solution

## Results

*Time devolvement of steady solutions**Time-step and mesh-size convergence**Code validation**Sensitivity of solutions to embedded parameters*

This expected increase of the VFBN rate for variations in φ is also illustrated in Fig. Similar responses (as withφ) of the VFBN thermal conductivity to variations in both the thermal conductivity parameter, A2 and the activation energy parameter, α are respectively illustrated in Figs. The VFBN thermal conductivity (and thus also the VFBN temperature) is expected to decrease with increasing Pr.

We notice that an increase in the Deborah number, De, increases the elastic effects in the fluid, therefore the VFBN relaxation time increases with increasing De.

## Concluding Remarks

Pramuanjaroenkij, "Review of convective heat transfer enhancement with nanofluids," International journal of heat and mass transfer, vol. Vafai, "A critical synthesis of thermophysical properties of nanofluids," International journal of heat and mass transfer, vol. Das, "Entropy generation due to flow and heat transfer in nanofluids,” International Journal of Heat and Mass Transfer , vol.

Roetzel, “Conceptions for heat transfer correlatie of nanofluids”, International Journal of heat and Mass transfer, vol.