# Thermosolutal Marangoni Convection for Hybrid Nanofluid Models: An Analytical Approach

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}O as the base fluid and fractions of TiO

_{2}–Ag nanoparticles. The mathematical approach given here employs the similarity transformation, in order to transform the leading partial differential equation (PDE) into a set of nonlinear ordinary differential equations (ODEs). The derived equations are solved analytically by using Cardon’s method and the confluent hypergeometric function. The solutions are further graphically analyzed, taking into account parameters such as mass transpiration, chemical reaction coefficient, thermal radiation, Schmidt number, Marangoni number, and inverse Darcy number. According to our findings, adding TiO

_{2}–Ag nanoparticles into conventional fluids can greatly enhance heat transfer. In addition, the mixture of TiO

_{2}–Ag with H

_{2}O gives higher heat energy compared to the mixture of only TiO

_{2}with H

_{2}O.

## 1. Introduction

_{2}-AG on H

_{2}O base) on the MHD TS-MC with chemically radiative Newtonian fluid flow, in the presence of heat sources/sinks, where the analytical solution is obtained by applying Cardon’s method and confluent hypergeometric functions. A number of parameters affecting the flow is further discussed, such as the impact of volume fraction, inverse Darcy number, Marangoni number, magnetic field, heat source/sink, thermal radiation, Schmidt number, chemical reaction coefficient, and mass transpiration parameter. In the following Sections, the theoretical model is presented, along with the mathematical solutions, and results are discussed.

## 2. Mathematical Model for the Flow Problem

_{0}is applied along the y-axis. It is also believed that the hybrid nanofluid is electrically conductive and has a low magnetic Reynolds number, hence the induced magnetic field is neglected. The fluid concentration and ambient temperature are ${C}_{\infty}$ and$\text{}{T}_{\infty}\text{}$and the constant mass transfer velocity ${v}_{0}$ together with heat and mass transfer in a stationary fluid.

^{4}is expanded using Taylor’s series (see paper by Sneha et al. [38]),

#### 2.1. Expressions and Thermophysical Properties of the HNF

_{2}(index 1) and Ag (index 2), respectively, and ${C}_{p}$ is the specific heat capacity.

_{2}O and TiO

_{2}and Ag nanoparticles.

#### 2.2. Similarity Transformation

#### 2.3. Exact Solution for Momentum Equation

#### 2.4. Exact Solution for Temperature and Concentration

## 3. Results

#### 3.1. Velocity Profiles

_{2}-Ag, H

_{2}O) solution, while dotted lines (green, orange, and black) refer to the (TiO

_{2}, H

_{2}O) solution. In Figure 2a–c, one observes that the physical solutions are obtained for ${a}_{1}$ and ${a}_{2}$ roots and non-physical solutions are obtained for ${a}_{3}$ roots. Furthermore, the physical solutions are directly affected by ${V}_{c}$ values. When porosity $D{a}^{-1}$ increases from 0 to 5, one observes that at ${V}_{c}=1$ (Figure 2a—suction case) the physical flow solutions show that the (TiO

_{2}-Ag, H

_{2}0) HNF presents higher velocity values compared to the (TiO

_{2}, H

_{2}O). At ${V}_{c}=0$ (Figure 2b—impermeable case), the physical flow solutions of (TiO

_{2}-Ag, H

_{2}O) and (TiO

_{2}, H

_{2}O) HNF are identical. In Figure 2c, at ${V}_{c}=-1$ (injection case), the physical flow solutions of the (TiO

_{2}, H

_{2}O) HNF present higher velocity values compared to the (TiO

_{2}-Ag, H

_{2}O).

_{2}-Ag, H

_{2}O) compared to (TiO

_{2}, H

_{2}O); in Figure 2e for ${V}_{c}=0$ (impermeably case), the physical flow solutions of (TiO

_{2}-Ag, H

_{2}O) are similar to (TiO

_{2}, H

_{2}O), while in Figure 2f, for ${V}_{c}=-1$ (injection case) we obtain smaller velocities for (TiO

_{2}-Ag, H

_{2}O) compared to (TiO

_{2}, H

_{2}O).

_{c}, $D{a}^{-1}$, and Q. The range of physical and non-physical surface velocity corresponds to the positive and negative roots, respectively. Let us point out that the physical solutions are obtained for ${a}_{1}$ and ${a}_{2}$ roots and non-physical solutions are obtained for the ${a}_{3}$ root in Figure 3a–c. While the porosity number $D{a}^{-1}$increases from 2 to 5 and Q = 1, it is observed that for all three cases of mass transpiration (${V}_{c}=1$, suction, ${V}_{c}=0$, impermeable, ${V}_{c}=-1$, injection), the physical flow solutions for the (TiO

_{2}, H

_{2}O) mixture lead to an increase in surface velocity compared to the (TiO

_{2}-Ag, H

_{2}O) HNF. By increasing the magnetic field $Q$ from 1 to 2, in Figure 4d–f, we observe that the surface velocity is higher for the (TiO

_{2}, H

_{2}O) compared to the (TiO

_{2}-Ag, H

_{2}O) for all three ${V}_{c}$ cases.

_{2}, H

_{2}O) compared to the (TiO

_{2}-Ag, H

_{2}O) HNF. We attribute this behavior to the higher density of the (TiO

_{2}-Ag, H

_{2}O) HNF, which imposes obstacles in fluid motion, compared to the lower density mixture of (TiO

_{2}, H

_{2}O).

_{2}, H

_{2}O) compared to (TiO

_{2}-Ag, H

_{2}O).

_{2}, H

_{2}O) compared to (TiO

_{2}-Ag, H

_{2}O).

_{1}the volume fraction of TiO

_{2}and as φ

_{2}the volume fraction of Ag nanoparticles. It is noted that by increasing φ

_{1}and φ

_{2}at the same time, the velocity values of (TiO

_{2}-Ag, H

_{2}O) are decreased in the respective fluid mixtures. By only increasing the value φ

_{1}, the velocity values of (TiO

_{2}, H

_{2}O) are decreased in the respective fluid mixtures. As also shown in previous cases (Figure 3, Figure 4, Figure 5 and Figure 6), higher velocities are observed for (TiO

_{2}, H

_{2}O) compared to (TiO

_{2}-Ag, H

_{2}O).

#### 3.2. Temperature Profiles

_{2}and Ag in water solution $\left({\phi}_{1},{\phi}_{2}\right)$ are investigated next. Starting from the ${N}_{r}$ effect, as it increases in the range {0.5, 1.0, 1.5, 2.0}, in Figure 9a–c, one obtains greater thickness in the thermal boundary. As long as the ${N}_{I}$ effect is concerned, as it decreases in the range ${N}_{I}=0,-10,-30$, in Figure 10a–c, there is decreasing thickness in the thermal boundary layer. The thermal boundary layer increases when $D{a}^{-1}$ increases (Figure 11a–c). Finally, in Figure 12a–c, one observes that the thermal boundary layer increases while increasing the volume fraction of TiO

_{2}and Ag.

_{2}-Ag, H

_{2}O) compared to (TiO

_{2}, H

_{2}O), the mixing of two nanoparticles TiO

_{2}and Ag in H

_{2}O results in greater heat energy than the single nanoparticle TiO

_{2}in H

_{2}O.

#### 3.3. Concentration Profiles

_{2}-Ag, H

_{2}O) mixture present higher values compared to the (TiO

_{2}, H

_{2}O) mixture, and this is evidence that the mixing of two nanoparticles TiO

_{2}and Ag in H

_{2}O results in greater chemical energy than the single nanoparticle TiO

_{2}in H

_{2}O.

#### 3.4. Validation

_{2}–Ag with H

_{2}O gives higher heat energy compared to the mixture of only TiO

_{2}with H

_{2}O for Newtonian radiative flow at the thermosolutal Marangoni boundary over a porous medium, under the effect of magnetic field and mass transpiration in fluid velocity, and obtained an exact analytical solution in terms of hypergeometric functions. In the absence of the HNF and Q$=0,\text{}D{a}^{-1}=0$, this agrees to the results obtained by Magyari et al. [8]. When $Q=0,$ the absence of HNF leads to the results of Mahabaleshwar et al. [9], while, when also considering the unsteady case, the results agree to the results by Hassan [10]. The results of all these studies along with the results of the present paper are summarized in Table 3.

## 4. Conclusions

_{c}, thermal radiation, ${N}_{r}$, heat source and sink, ${N}_{I}$, the inverse Darcy number, $D{a}^{-1}$, the volume fraction of nanoparticles in water, ${\phi}_{1},{\phi}_{2}$, the magnetic field, Q, the chemical reaction coefficient, $K$, and Schmidt number, ${S}_{c}$.

- the physical solution’s effect is directly determined by V
_{c}, $D{a}^{-1}$, and Q; - V
_{c}has a direct impact on surface velocity and Marangoni number, M; - by increasing the values of the magnetic field, Q, and porosity, $D{a}^{-1}$, the fluid velocity decreases;
- on the other hand, by increasing the Marangoni number, M, the fluid velocity increases;
- the velocity and thermal boundary layer decrease by increasing the volume fraction of TiO
_{2}and Ag within H_{2}O; - furthermore, the (TiO
_{2}, H_{2}O) mixture presents higher velocity values, but less heat and chemical energy compared to the (TiO_{2}-Ag, H_{2}O); - the thermal boundary layers increase when ${N}_{r}$ increases and decrease when ${N}_{I}$ increases;
- the thermal and chemical boundary layers increase by increasing the value of $D{a}^{-1}$;
- the concentration profile decreases when S
_{c}and K increase.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Latin symbols | |

${A}_{1},{A}_{2},{A}_{3},{A}_{4},{A}_{5}$ | constants |

${B}_{0}$ | applied magnetic field |

$C$ | dimensional concentration |

${C}_{P}$ | specific heat, constant pressure |

$D$ | Mass diffusivity |

${D}_{1},{D}_{2}$ | constants |

$D{a}^{-1}$ | inverse Darcy number |

$F$ | velocity similarity |

${F}_{\infty}$ | Constant |

$F\left(\eta \right)$ | $\mathrm{axial}\text{}\mathrm{velocity}\text{}$ |

${F}^{\prime}\left(\eta \right)$ | transverse velocity |

$H$ | confluent hypergeometric function |

$k$ | permeability |

K | chemical reaction coefficient |

$L$ | characteristic/reference length |

M | Marangoni number |

$m,n$ | constants |

${N}_{r}$ | radiation parameter |

${N}_{I}$ | heat source and sink parameter |

Pr | Prandtl number |

${q}_{r}$ | radiative heat flux |

${q}_{w}$ | local heat flux at the wall |

Q | magnetic field |

${S}_{1},{S}_{2}$ | constants |

${S}_{c}$ | Schmidt number |

T | temperature |

T_{0} | constant |

${V}_{c}$ | mass transformation |

${V}_{c}>0$ | suction condition |

${V}_{c}=0$ | impermeability condition |

${V}_{c}<0$ | injection condition |

$\left(x,y\right)$ | axes |

$\left(u,v\right)$ | velocities along x- and y-directions |

Greek symbols | |

$\alpha $ | thermal diffusivity |

$\gamma $ | coefficient |

$\Delta $ | discriminates |

η | similarity variable |

$\kappa $ | thermal conductivity |

${\mu}_{f}$ | dynamic viscosity |

$\nu $ | kinematic viscosity |

$\rho $ | density |

$\sigma $ | electrical conductivity |

${\sigma}_{1}$ | surface tension |

${\sigma}_{0}$ | equilibrium surface tension |

${\sigma}_{s1},{\sigma}_{s2}$ | electrical conductivities, respectively, of TiO_{2} and Ag nanoparticles |

${\sigma}^{*}$ | Stefan-Boltzmann constant |

${\phi}_{1},{\phi}_{2}$ | nanoparticle volume fractions of TiO_{2} and Ag, respectively |

$\varphi $ | concentration similarity variable |

$\psi $ | stream function |

Subscripts | |

$C$ | solutal quantity |

$T$ | thermal quantity |

$bf$ | base fluid |

$nf$ | Nanofluid |

$hnf$ | hybrid nanofluid |

${f}^{\prime},{f}^{\prime},{f}^{\u2034}$ | First, second and third order derivatives with respect to $\eta $ |

Abbreviations | |

Ag | silver |

BC | boundary condition |

BLF | boundary layer flow |

CNT | carbon nanotube |

EHD | electrohydrodynamics |

H_{2}O | water |

HNF | hybrid nanofluid |

MC | Marangoni convection |

MHD | magnetohydrodynamics |

ODE | ordinary differential equation |

PDE | partial differantial equation |

TiO_{2} | titanium dioxide |

TS | thermosolutal |

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**Figure 1.**The mathematical model of HNF with boundary condition. Concentration, C, velocity, V, and temperature, T, profiles in (x-y) space. B

_{0}is the magnetic field with the direction shown by arrows.

**Figure 2.**The behavior of the roots ${a}_{1},{a}_{2},\mathrm{and}{a}_{3}$ versus Marangoni number, M, and various values for the inverse Darcy number, Da

^{−1}, mass transpiration parameter, V

_{c}, and magnetic field, Q. (

**a**) V

_{c}= 1, Q = 1, and Da

^{−1}= 1 and 5; (

**b**) V

_{c}= 0, Q = 1, and Da

^{−1}= 0 and 5; (

**c**) V

_{c}= −1, Q = 1, and Da

^{−1}= 0 and 5; (

**d**) V

_{c}= 1, Q = 1 and 5, and Da

^{−1}= 0; (

**e**) V

_{c}= 0, Q = 1 and 5, and Da

^{−1}= 0; (

**f**) V

_{c}= −1, Q = 1 and 5, and Da

^{−1}= 0.

**Figure 3.**The behavior of similar surface velocity, ${F}^{\prime}\left(0\right)$, versus Marangoni number, M, and various values for the inverse Darcy number, Da

^{−1}, magnetic field, Q and mass transpiration parameter, V

_{c}. (

**a**) V

_{c}= 1, Q = 1, Da

^{−1}= 2 and 5, (

**b**) V

_{c}= 0, Q = 1, Da

^{−1}= 2 and 5, (

**c**) V

_{c}= −1, Q = 1, Da

^{−1}= 2 and 5, (

**d**) V

_{c}= 1, Q = 1 and 2, Da

^{−1}= 0, (

**e**) V

_{c}= 0, Q = 1 and 2, Da

^{−1}= 0, and (

**f**) V

_{c}= −1, Q = 1 and 2, Da

^{−1}= 0.

**Figure 4.**The axial, $F\left(\eta \right),$ and transverse, ${F}^{\prime}\left(\eta \right),$ velocities versus the similarity variable, $\eta $, for various values of Da

^{−1}and $Q=M=1$. (

**a**) F(n), for V

_{c}= 1, (

**b**) F(n), for V

_{c}= 0, (

**c**) F(n), for V

_{c}= −1, (

**d**) ${F}^{\prime}\left(n\right)$, for V

_{c}= 1, (

**e**) ${F}^{\prime}\left(n\right)$, for V

_{c}= 0, and (

**f**) ${F}^{\prime}\left(n\right)$, for V

_{c}= −1. The orange solid line refers to the (TiO

_{2}-Ag, H

_{2}O) HNF and dotted line to (TiO

_{2}, H

_{2}O).

**Figure 5.**The axial, $F\left(\eta \right),$ and transverse, ${F}^{\prime}\left(\eta \right),$velocities versus the similarity variable, $\eta $, for various values of $M$ and $Q=D{a}^{-1}=1$. (

**a**) F(n), for V

_{c}= 1, (

**b**) F(n), for V

_{c}= 0, (

**c**) F(n), for V

_{c}= −1, (

**d**) ${F}^{\prime}\left(n\right)$, for V

_{c}= 1, (

**e**) ${F}^{\prime}\left(n\right)$, for V

_{c}= 0, and (

**f**) ${F}^{\prime}\left(n\right)$, for V

_{c}= −1. The orange solid line refers to the (TiO

_{2}-Ag, H

_{2}O) HNF and dotted line to (TiO

_{2}, H

_{2}O).

**Figure 6.**The axial, $F\left(\eta \right),$ and transverse, ${F}^{\prime}\left(\eta \right),$ velocities versus the similarity variable, $\eta $, for various values of $Q$ and $M=D{a}^{-1}=1$. (

**a**) F(n), for V

_{c}= 1, (

**b**) F(n), for V

_{c}= 0, (

**c**) F(n), for V

_{c}= −1, (

**d**) ${F}^{\prime}\left(n\right)$, for V

_{c}= 1, (

**e**) ${F}^{\prime}\left(n\right)$, for V

_{c}= 0, and (

**f**) ${F}^{\prime}\left(n\right)$, for V

_{c}= −1. The orange solid line refers to the (TiO

_{2}-Ag, H

_{2}0) HNF and dotted line to (TiO

_{2}, H

_{2}O).

**Figure 7.**The axial, $F\left(\eta \right)$, and transverse, ${F}^{\prime}\left(\eta \right),$ velocities versus the similarity variable, $\eta $, for various values of φ

_{1}and φ

_{2}and $M=Q=D{a}^{-1}=1$. (

**a**) F(n), for V

_{c}= 1, (

**b**) F(n), for V

_{c}= 0, (

**c**) F(n), for V

_{c}= −1, (

**d**) ${F}^{\prime}\left(n\right)$, for V

_{c}= 1, (

**e**) ${F}^{\prime}\left(n\right)$, for V

_{c}= 0, and (

**f**) ${F}^{\prime}\left(n\right)$, for V

_{c}= −1. The orange solid line refers to the (TiO

_{2}-Ag, H

_{2}O) HNF and dotted line to (TiO

_{2}, H

_{2}O).

**Figure 8.**(

**a**) Axial, $F\left(\eta \right)$, and (

**b**) transverse, ${F}^{\prime}\left(\eta \right)$, velocities versus the similarity variable, $\eta $. $M=Q=1$, $D{a}^{-1}=2$. The orange solid line refers to the (TiO

_{2}-Ag, H

_{2}O) HNF and dotted line to (TiO

_{2}, H

_{2}O).

**Figure 9.**Temperature profiles, $\theta \left(\eta \right)$, versus similarity variable, $\eta $, for various thermal radiation parameter, ${N}_{r},$ values, $D{a}^{-1}=0.5,{N}_{I}=0.3,\mathrm{and}M=Q=1$, and (

**a**) V

_{c}= 0.1, (

**b**) V

_{c}= 0, and (

**c**) V

_{c}= −0.1.

**Figure 10.**Temperature profiles, $\theta \left(\eta \right)$, versus similarity variable, $\eta $, for various heat source and sink parameter, ${N}_{I}$, values, $D{a}^{-1}=0.5,{N}_{r}=0.6,\mathrm{and}M=Q=1$, and (

**a**) V

_{c}= 0.1, (

**b**) V

_{c}= 0, and (

**c**) V

_{c}= −0.1.

**Figure 11.**Temperature profiles, $\theta \left(\eta \right)$, versus similarity variable, $\eta $, for various $D{a}^{-1}$ values, ${N}_{I}={N}_{r}=M=Q=1$, and (

**a**) V

_{c}= 0.1, (

**b**) V

_{c}= 0, and (

**c**) V

_{c}= −0.1.

**Figure 12.**Temperature profiles, $\theta \left(\eta \right)$, versus similarity variable, $\eta $, for various ${\phi}_{1},{\phi}_{2}$ values, $D{a}^{-1}=2$, ${N}_{I}=0.1,{N}_{r}=4,M=Q=1$, and (

**a**) V

_{c}= 0.1, (

**b**) V

_{c}= 0, and (

**c**) V

_{c}= −0.1.

**Figure 13.**Concentration profiles, $\varphi \left(\eta \right)$, versus similarity variable, $\eta $, for various chemical reaction coefficient, $K,$ values, $D{a}^{-1}={N}_{I}={N}_{r}=M=Q={S}_{c}=1$, and (

**a**) V

_{c}= 0.1, (

**b**) V

_{c}= 0, and (

**c**) V

_{c}= −0.1.

**Figure 14.**Concentration profiles, $\varphi \left(\eta \right)$, versus similarity variable, $\eta $, for various Schmidt number, ${S}_{c}$, values, $D{a}^{-1}={N}_{I}={N}_{r}=M=Q=K=1$, and (

**a**) V

_{c}= 0.1, (

**b**) V

_{c}= 0, and (

**c**) V

_{c}= −0.1.

**Figure 15.**Concentration profiles, $\varphi \left(\eta \right)$, versus similarity variable, $\eta $, for various $D{a}^{-1}$ values, ${S}_{c}={N}_{r}=M=Q=K=1$, ${N}_{I}=0.1$, and (

**a**) V

_{c}= 0.1, (

**b**) V

_{c}= 0, and (

**c**) V

_{c}= −0.1.

Term | Equivalent Property for the HNF Model |
---|---|

Dynamic viscosity | $\frac{{\mu}_{hnf}}{{\mu}_{f}}=\frac{1}{{\left(1-{\phi}_{1}\right)}^{2.5}{\left(1-{\phi}_{2}\right)}^{2.5}}$ |

Density | $\frac{{\rho}_{hnf}}{{\rho}_{f}}=\left(1-{\phi}_{2}\right)\left(1-{\phi}_{1}+{\phi}_{1}\frac{{\rho}_{s1}}{{\rho}_{f}}\right)+{\phi}_{2}\left(\frac{{\rho}_{s2}}{{\rho}_{f}}\right)$ |

Heat capacity | $\frac{{\left(\rho {C}_{p}\right)}_{hnf}}{{\left(\rho {C}_{p}\right)}_{f}}=\left(1-{\phi}_{2}\right)\left(1-{\phi}_{1}+{\phi}_{1}\left(\frac{{\left(\rho {C}_{p}\right)}_{s1}}{{\left(\rho {C}_{p}\right)}_{f}}\right)\right)+{\phi}_{2}\left(\frac{{\left(\rho {C}_{p}\right)}_{s2}}{{\left(\rho {C}_{p}\right)}_{f}}\right)$ |

Thermal conductivity for the HNF | $\frac{{\kappa}_{hnf}}{{\kappa}_{f}}=\frac{{\kappa}_{s2}+2{\kappa}_{bf}+2{\phi}_{2}\left({\kappa}_{s2}-{\kappa}_{f}\right)}{{\kappa}_{s2}+2{\kappa}_{bf}-{\phi}_{2}\left({\kappa}_{s2}-{\kappa}_{f}\right)}$ |

(to simplify the thermal conductivity for the HNF, we use the constant term ${\kappa}_{bf}$) | $\mathrm{where}\text{}{\kappa}_{bf}={\kappa}_{f}\frac{{\kappa}_{s1}+2{\kappa}_{f}+2{\phi}_{1}\left({\kappa}_{s1}-{\kappa}_{f}\right)}{{\kappa}_{s1}+2{\kappa}_{f}-{\phi}_{1}\left({\kappa}_{s1}-{\kappa}_{f}\right)}$ |

Electrical conductivity for the HNF | $\frac{{\sigma}_{hnf}}{{\sigma}_{f}}=\frac{{\sigma}_{s2}+2{\sigma}_{bf}+2{\phi}_{2}\left({\sigma}_{s2}-{\sigma}_{f}\right)}{{\sigma}_{s2}+2{\sigma}_{bf}-{\phi}_{2}\left({\sigma}_{s2}-{\sigma}_{f}\right)}$ |

(to simplify the electrical conductivity for the HNF, we use the constant term ${\sigma}_{bf}$) | $\mathrm{where}\text{}{\sigma}_{bf}={\sigma}_{f}\frac{{\sigma}_{s1}+2{\sigma}_{f}+2{\phi}_{1}\left({\sigma}_{s1}-{\sigma}_{f}\right)}{{\sigma}_{s1}+2{\sigma}_{f}-{\phi}_{1}\left({\sigma}_{s1}-{\sigma}_{f}\right)}$ |

**Table 2.**Thermophysical properties of the base fluid and HNF [32].

Physical Parameters | Fluid Phase (H_{2}O) | TiO_{2} | Ag |
---|---|---|---|

${C}_{p}\left(\mathrm{J}/\mathrm{KgK}\right)$ | 4179 | 686.2 | 235 |

$\rho \left(\mathrm{Kg}/{\mathrm{m}}^{3}\right)$ | 997.1 | 4250 | 10,500 |

$\kappa \left(\mathrm{W}/\mathrm{mK}\right)$ | 0.613 | 8.9528 | 429 |

$\sigma {\left(\mathsf{\Omega}/\mathrm{m}\right)}^{-1}$ | 0.05 | $2.6\times {10}^{6}$ | $62.1\times {10}^{6}$ |

Reference | Fluid | Method | Momentum Equation |
---|---|---|---|

Magyari et al. [8] | Newtonian fluid | Analytical solution | $u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\upsilon \frac{{\partial}^{2}u}{\partial {y}^{2}}$, |

Mahabaleshwar et al. [9] | Newtonian fluid | Analytical solution | $u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\upsilon \frac{{\partial}^{2}u}{\partial {y}^{2}}-\frac{\upsilon}{k}u,$ |

Hassan [10] | Newtonian fluid | Numerical | $\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\upsilon \frac{{\partial}^{2}u}{\partial {y}^{2}}-\frac{\upsilon}{k}u,$ Unsteady case |

Present work | Newtonian fluid | Analytical solution | $u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}={\upsilon}_{hnf}\frac{{\partial}^{2}u}{\partial {y}^{2}}-\frac{{\sigma}_{hnf}}{{\rho}_{hnf}}{B}^{2}u-\frac{{\mu}_{hnf}}{{\rho}_{hnf}k}u,$ with water TiO_{2}-Ag nanoparticle on a porous surface |

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## Share and Cite

**MDPI and ACS Style**

Mahabaleshwar, U.S.; Mahesh, R.; Sofos, F.
Thermosolutal Marangoni Convection for Hybrid Nanofluid Models: An Analytical Approach. *Physics* **2023**, *5*, 24-44.
https://doi.org/10.3390/physics5010003

**AMA Style**

Mahabaleshwar US, Mahesh R, Sofos F.
Thermosolutal Marangoni Convection for Hybrid Nanofluid Models: An Analytical Approach. *Physics*. 2023; 5(1):24-44.
https://doi.org/10.3390/physics5010003

**Chicago/Turabian Style**

Mahabaleshwar, Ulavathi Shettar, Rudraiah Mahesh, and Filippos Sofos.
2023. "Thermosolutal Marangoni Convection for Hybrid Nanofluid Models: An Analytical Approach" *Physics* 5, no. 1: 24-44.
https://doi.org/10.3390/physics5010003