# Influence of Stress Jump Condition at the Interface Region of a Two-Layer Nanofluid Flow in a Microchannel with EDL Effects

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

_{2}O

_{3}nanoparticles, whereas Region II has porous media saturated with TiO

_{2}. Table 1 lists the physical parameters of the fluid and nanoparticles. The Buongiorno model is used to simulate nanofluid flow in Region I. The Brinkmann-extended Darcy law is employed to illustrate the flow of nanofluids in a porous layer region. The steady-state laminar flow is considered to be one-dimensional, owing to the significant presence of an electric field. due to the presence of an electric double layer (EDL) and the applied pressure.

- Region I:

#### 2.1. Problem Statement and Assumptions

- The direction of the flow is assumed to be along the x-axis.
- The flow velocity in the $\overline{z}$-direction is negligible, since the length of microchannel L is much larger than its height H. Hence ${\overline{w}}_{i}\approx 0$,
- The velocity component in the $\overline{y}$-direction is considered to be zero, i.e., ${\overline{v}}_{i}=0$,
- The flow is assumed to uni-directional along the $\overline{x}$-axis but its properties changes with respect to the $\overline{z}$-axis, hence ${\mathbf{V}}_{\mathbf{i}}=({\overline{u}}_{i}\left(\overline{z}\right),0,0)$,
- The body force, ${\mathbf{F}}_{\mathbf{i}}={\mathrm{\ae}}_{\mathbf{ei}}\mathbf{E}+{J}_{i}\times \mathbf{B}$, represents the sum of electro-osmosis and the electromagnetic forces, where $\mathbf{E}=({E}_{x},{E}_{y},0)$ is the electric field, $\mathbf{B}=(0,0,{B}_{0})$ is the applied magnetic field, and ${J}_{i}={\sigma}_{i}(\mathbf{E}+{\mathbf{V}}_{i}\times B)$ is the current density of the ion.
- The inertial effects in the porous region of the microchannel (Region II) are negligible.
- Region I of the channel is filled with nanofluid, while the channel’s Region II is filled with the porous medium saturated with nanofluid, having uniform permeability only.
- Proceeding from the analysis presented in [26], the stress jump condition is utilized at the interface. Simultaneously, the electric potential, temperature, nanoparticle concentration, and flux at the interface are presumed to be continuous. Finally, the no-slip condition is applied to the velocity boundaries, while the temperature and nanoparticle concentration are assumed to have a constant distribution on the boundaries.

- Region I: ($-{H}_{1}\le \overline{z}\le 0$)

- when $z=-{H}_{1}$:

#### 2.2. Problem Non-Dimensionalization

- Region I $(-{h}_{1}\le \eta \le 0)$:

- when $\eta =-{h}_{1}$:

#### 2.3. Skin Friction Coefficient and Nusselt Number

## 3. Problem Solution

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${\overline{C}}_{1},{\overline{C}}_{2}$ | volumetric fractions of nanoparticles; |

${C}_{f1},{C}_{f2}$ | local skin friction coefficients; |

${D}_{{B}_{1}},{D}_{{B}_{2}}$ | Brownian diffusion coefficients; |

${D}_{{T}_{1}},{D}_{{T}_{2}}$ | thermophoretic diffusion coefficients; |

$\overline{P}$ | pressure, Pa; |

${\overline{T}}_{1},{\overline{T}}_{2}$ | non-dimensional nanofluid temperatures in two regions, K; |

${\mathbf{V}}_{\mathbf{1}},{\mathbf{V}}_{\mathbf{2}}$ | non-dimensional velocities of the fluid, $m/s$; |

$B{r}_{1},B{r}_{2}$ | Brinkman numbers; |

${B}_{0}$ | magnetic field in z-direction; |

$Da$ | Darcy number; |

${\left({c}_{p}\right)}_{f},{\left({c}_{p}\right)}_{s}$ | fluid and nanoparticle specific heats; |

e | charge of a proton; |

${C}_{0}$ | reference volume fraction for nanoparticles; |

${C}_{w}$ | volume fraction for nanoparticles on the microchannel walls; |

${E}_{s}$ | non-dimensional external electric field parameter; |

${E}_{x},{E}_{y}$ | electric field in x− and y−directions, respectively; |

${F}_{1},{F}_{2}$ | body forces caused by uniform electromagnetic field; |

H | channel height; |

${H}_{1},{H}_{2}$ | channel height of two regions; |

${h}_{1},{h}_{2}$ | non-dimensional heights of two regions; |

$H{a}_{1},H{a}_{2}$ | Hartman numbers; |

${k}_{B}$ | Boltzmann constant; |

${k}_{f1},{k}_{f2}$ | fluid’s thermal conductivity in two regions; |

${k}_{nf}$ | the ratio of the fluid’s thermal conductivities; |

L | microchannel length; |

${n}_{0}$ | bulk ionic concentration; |

${N}_{B1},{N}_{B2}$ | Brownian motion parameters; |

${N}_{T1},{N}_{T2}$ | thermophoresis parameters; |

$N{u}_{1},N{u}_{2}$ | local Nusselt numbers; |

P | non-dimensional pressure gradient; |

${q}_{w1},{q}_{w2}$ | heat flux on the channel walls; |

$R{e}_{1},R{e}_{2}$ | Reynolds numbers; |

${S}_{e1},{S}_{e2}$ | lateral direction electric field strengths; |

${U}_{a1},{U}_{a2}$ | average velocities of the fluid; |

${u}_{1},{u}_{2}$ | velocity of fluid in two regions; |

W | microchannel’s width; |

$x,y,z$ | Cartesian coordinates; |

$\widehat{z}$ | ion valency. |

$\mathbf{Greek}\mathbf{Letters}$ | |

${\alpha}_{1},{\alpha}_{2}$ | thermal diffusivity of the nanofluid, ${m}^{2}{s}^{-1}$; |

$\u03f5$ | porosity of the porous region; |

${\epsilon}_{0}$ | permittivity of vacuum, $mkgs$; |

${\epsilon}_{{R}_{1}},{\epsilon}_{{R}_{2}}$ | medium’s dielectric constants; |

$\epsilon $, | medium’s dielectric constant ratio; |

$\kappa $ | permeability of the porous region; |

$\gamma $ | constant coefficient; |

$\beta $ | the adjustable stress jump coefficient; |

${\overline{\psi}}_{1},{\overline{\psi}}_{2}$ | dimensional electrostatic potential, V; |

${\left({\rho}_{n}\right)}_{s}$ | nanoparticles density, $kg{m}^{-3}$; |

${\left({\rho}_{n}\right)}_{f}$ | nanofluid’s density, $kg{m}^{-3}$; |

${\overline{\rho}}_{{e}_{1}},{\overline{\rho}}_{{e}_{2}}$ | densities of charges, $C{m}^{-3}$; |

$\eta $ | non-dimensional spatial variable; |

${\mathrm{\Gamma}}_{1},{\mathrm{\Gamma}}_{2}$ | non-dimensional pressure gradient parameters; |

${k}_{1},{k}_{2}$ | electro-osmotic parameters; |

${\lambda}_{N}$ | the ratio of the any physical quantity N, where $N\in \epsilon ,nf,\mu ,\sigma ,{D}_{B},{D}_{T},\alpha ,\tau ,\rho $; |

${\mu}_{1},{\mu}_{2}$ | fluid’s dynamic viscosity in two regions; |

${\mu}_{eff}$ | the ratio of viscosity to porosity in Region II; |

${\theta}_{0}$ | dimensionless reference temperature; |

${\theta}_{1},{\theta}_{2}$ | temperature distributions (non-dimensional); |

${\Psi}_{1},{\Psi}_{2}$ | non-dimensional nano-particle volume fractions; |

${\overline{\rho}}_{e1},{\overline{\rho}}_{e2}$ | densities of the charges; |

${\tau}_{w1},{\tau}_{w2}$ | shear stresses on channel’s opposite walls; |

${\mathrm{\psi}}_{1},{\mathrm{\psi}}_{2}$ | non-dimensional nano-particle volume fractions; |

${\overline{\zeta}}_{1},{\overline{\zeta}}_{2}$ | zeta potentials (dimensional). |

${\zeta}_{1},{\zeta}_{2}$ | zeta potentials (non-dimensional); |

${\mathrm{\Phi}}_{1},{\mathrm{\Phi}}_{2}$ | viscous dissipation factors; |

$\mathbf{Subscripts}\mathbf{Indices}$ | |

$1,2$ | indices for regions I and II; |

$s,f$ | subscript notations for solids and fluids; |

w | indicate the quantities on walls of the channel. |

$\mathbf{Abbreviations}$ | |

$EDL$ | electric double layer; |

$FDM$ | finite difference method. |

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**Figure 2.**Comparison of solutions for (

**a**) Temperature profile $\theta \left(\eta \right)$ with the variation in $B{r}_{1}$ and (

**b**) Velocity profile $u\left(\eta \right)$ with the variation in ${\kappa}_{1}$. Line: Niazi results [17]. Symbols: numerical results when $\beta =0$, $Da=1$, $\gamma =1$, ${\zeta}_{1}={\zeta}_{2}={\kappa}_{1}=1$, ${H}_{1}=1$, ${H}_{2}=2$, $H{a}_{1}={S}_{{e}_{1}}={\mathrm{\Gamma}}_{1}=1$, $B{r}_{1}=0.1$, and ${N}_{B1}={N}_{T1}=0.1$.

**Figure 3.**Velocity, $u\left(\eta \right)$, for different values of stress jump coefficient $\left(\beta \right)$ and Darcy number $\left(Da\right)$, when ${\zeta}_{1}={\zeta}_{2}={\kappa}_{1}=1,{H}_{1}=1,{H}_{2}=2,H{a}_{1}={S}_{{e}_{1}}={\mathrm{\Gamma}}_{1}=1$, $B{r}_{1}=0.1$, and ${N}_{B1}={N}_{T1}=0.1$.

**Figure 4.**Velocity field, $U\left(\eta \right)$, for different values of physical ratios ${\lambda}_{\epsilon}$ and ${\lambda}_{\mu}$, when $\beta =0.05$, $Da=0.01$, $\gamma =1$, ${\zeta}_{1}={\zeta}_{2}={\kappa}_{1}=1$, ${H}_{1}=1$, ${H}_{2}=2$, $H{a}_{1}={S}_{{e}_{1}}={\mathrm{\Gamma}}_{1}=1$, $B{r}_{1}=0.1$, and ${N}_{B1}={N}_{T1}=0.1$.

**Figure 5.**Temperature, $\theta \left(\eta \right)$, for different values of stress jump coefficient $\left(\beta \right)$ and Darcy number $\left(Da\right)$, when ${\zeta}_{1}={\zeta}_{2}={\kappa}_{1}=1$, ${H}_{1}=1,{H}_{2}=2$, $H{a}_{1}={S}_{{e}_{1}}={\mathrm{\Gamma}}_{1}=1$, $B{r}_{1}=0.1$, and ${N}_{B1}={N}_{T1}=0.1$.

**Figure 6.**Temperature, $\theta \left(\eta \right)$, for different values of physical ratios ${\lambda}_{nf}$ and ${\lambda}_{\mu}$, when $\beta =0.05$, $Da=0.01$, $\gamma =1$, ${\zeta}_{1}={\zeta}_{2}={\kappa}_{1}=1$, ${H}_{1}=1,{H}_{2}=2$, $H{a}_{1}={S}_{{e}_{1}}={\mathrm{\Gamma}}_{1}=1$, $B{r}_{1}=0.1$, and ${N}_{B1}={N}_{T1}=0.1$.

**Figure 7.**Concentration of nanoparticles, $\varphi \left(\eta \right)$, for different values of adjustable coefficient in stress jump condition $\left(\beta \right)$ and Darcy number $\left(Da\right)$, when ${\zeta}_{1}={\zeta}_{2}={\kappa}_{1}=1$, ${H}_{1}=1$, ${H}_{2}=2$, $H{a}_{1}={S}_{{e}_{1}}={\mathrm{\Gamma}}_{1}=1$, $B{r}_{1}=0.1$, and ${N}_{B1}={N}_{T1}=0.1$.

**Figure 8.**Concentration of nanoparticles, $\varphi \left(\eta \right)$, for different values of physical ratios ${\lambda}_{nf}$ and ${\lambda}_{\mu}$, when $\beta =0.05$, $Da=0.01$, $\gamma =1$, ${\zeta}_{1}={\zeta}_{2}={\kappa}_{1}=1$, ${H}_{1}=1$, ${H}_{2}=2$, $H{a}_{1}={S}_{{e}_{1}}={\mathrm{\Gamma}}_{1}=1$, $B{r}_{1}=0.1$, and ${N}_{B1}={N}_{T1}=0.1$.

**Figure 9.**Local skin friction coefficient for different values of adjustable coefficient in stress jump condition $\left(\beta \right)$, when $Da=0.01$, ${\zeta}_{1}={\zeta}_{2}={\kappa}_{1}=1$, ${H}_{1}=1$, ${H}_{2}=2$, $H{a}_{1}={S}_{{e}_{1}}={\mathrm{\Gamma}}_{1}=1$, $B{r}_{1}=0.1$, and ${N}_{B1}={N}_{T1}=0.1$.

**Figure 10.**Local skin friction coefficient for different values of Darcy number ($Da$), when $\beta =0.05$, ${\zeta}_{1}={\zeta}_{2}={\kappa}_{1}=1$, ${H}_{1}=1,{H}_{2}=2$, $H{a}_{1}={S}_{{e}_{1}}={\mathrm{\Gamma}}_{1}=1$, $B{r}_{1}=0.1$, and ${N}_{B1}={N}_{T1}=0.1$.

**Figure 11.**Nusselt number for different values of adjustable coefficient in stress jump condition $\left(\beta \right)$, when $Da=0.01$, ${\zeta}_{1}={\zeta}_{2}={\kappa}_{1}=1,{H}_{1}=1,{H}_{2}=2,H{a}_{1}={S}_{{e}_{1}}={\mathrm{\Gamma}}_{1}=1$, $B{r}_{1}=0.1$, and ${N}_{B1}={N}_{T1}=0.1$.

**Figure 12.**Nusselt number for different values of Darcy number ($Da$), when $\beta =0.05$, ${\zeta}_{1}={\zeta}_{2}={\kappa}_{1}=1$, ${H}_{1}=1$, ${H}_{2}=2$, $H{a}_{1}={S}_{{e}_{1}}={\mathrm{\Gamma}}_{1}=1$, $B{r}_{1}=0.1$, and ${N}_{B1}={N}_{T1}=0.1$.

Physical Characteristic | ${\mathbf{H}}_{2}\mathbf{O}$ | ${\mathbf{Al}}_{2}{\mathbf{O}}_{3}$ | ${\mathbf{TiO}}_{2}$ |
---|---|---|---|

${c}_{p}\phantom{\rule{0.166667em}{0ex}}(\mathrm{J}\xb7{\mathrm{kg}}^{-1}\xb7{\mathrm{K}}^{-1})$ | 4179.0 | 765.0 | 686.2 |

$\rho \phantom{\rule{0.166667em}{0ex}}(\mathrm{kg}\xb7{\mathrm{m}}^{-3})$ | 997.1 | 3970.0 | 4250.0 |

$k\phantom{\rule{0.166667em}{0ex}}(\mathrm{W}\xb7{\mathrm{m}}^{-1}\xb7{\mathrm{K}}^{-1})$ | 0.6130 | 40.0 | 8.9538 |

$\alpha \times {10}^{-7}\phantom{\rule{0.166667em}{0ex}}({\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1})$ | 1.47 | 131.70 | 30.70 |

$\beta \times {10}^{-5}\phantom{\rule{0.166667em}{0ex}}\left({\mathrm{K}}^{-1}\right)$ | 21.00 | 0.85 | 0.90 |

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**MDPI and ACS Style**

Raees ul Haq, M.; Raees, A.; Xu, H.; Xiao, S. Influence of Stress Jump Condition at the Interface Region of a Two-Layer Nanofluid Flow in a Microchannel with EDL Effects. *Nanomaterials* **2023**, *13*, 1198.
https://doi.org/10.3390/nano13071198

**AMA Style**

Raees ul Haq M, Raees A, Xu H, Xiao S. Influence of Stress Jump Condition at the Interface Region of a Two-Layer Nanofluid Flow in a Microchannel with EDL Effects. *Nanomaterials*. 2023; 13(7):1198.
https://doi.org/10.3390/nano13071198

**Chicago/Turabian Style**

Raees ul Haq, Muhammad, Ammarah Raees, Hang Xu, and Shaozhang Xiao. 2023. "Influence of Stress Jump Condition at the Interface Region of a Two-Layer Nanofluid Flow in a Microchannel with EDL Effects" *Nanomaterials* 13, no. 7: 1198.
https://doi.org/10.3390/nano13071198