Abstract
This study discusses the flow of hybrid nanofluid over an infinite disk in a Darcy–Forchheimer permeable medium with variable thermal conductivity and viscosity. The objective of the current theoretical investigation is to identify the thermal energy characteristics of the nanomaterial flow resulting from thermosolutal Marangoni convection on a disc surface. By including the impacts of activation energy, heat source, thermophoretic particle deposition and microorganisms the proposed mathematical model becomes more novel. The CattaneoChristov mass and heat flux law is taken into account when examining the features of mass and heat transmission rather than the traditional Fourier and Fick heat and mass flux law. MoS_{2} and Ag nanoparticles are dispersed in the base fluid water to synthesize the hybrid nanofluid. PDEs are transformed to ODEs by using similarity transformations. The RKF45th order shooting method is used to solve the equations. With the use of appropriate graphs, the effects of a number of nondimensional parameters on velocity, concentration, microorganism, and temperature fields are addressed. The local Nusselt number, density of motile microorganisms and Sherwood number are calculated numerically and graphically to derive correlations in terms of the relevant key parameters. The findings show that as we increase the Marangoni convection parameter, skin friction, local density of motile microorganisms, Sherwood number, velocity, temperature and microorganisms profiles increase, whereas Nusselt number and concentration profile exhibit an opposite behavior. The fluid velocity is reduced as a result of enhancing the Forchheimer parameter and Darcy parameter.
Introduction
Marangoni convection is commonly defined as the edge dissipative layer between two phase fluid flows, such as liquid–liquid and gas–liquid interfaces. It is subjected to surface tension variations caused by changes in chemical concentration, temperature and applied magnetic fields. The presence of different fluids at interface which have distinct fluid characteristics may causes these gradients appear. External forces like gravitational and shear forces are activated as a result of the viscosity of interacting liquids. Due to their extensive usage in the disciplines of space processing, microgravity science and industrial manufacturing procedures, governing equations have drawn the attention of the majority of researchers who are interested in modeling these external forces. The importance of thermosolutal Marangoni convection flows in the procedure of mass and heat transfer into various schemes has been carefully examined in^{1,2,3,4,5}\(.\) Chemical reaction in the thermosolutal Marangoni convective flow across the Riga plate was studied by Shafiq et al.^{6}. The encouragements of heat radiation on Marangoni convective flow were examined by Hayat et al.^{7}\(.\) Microorganisms that are typically 5–10% denser than water swim upward in fluid flow known as bioconvection. By increasing the base fluid density in a specific direction, the selfimpelled microorganisms cause bioconvection in the flow. The motile microbes can be characterized into many types of microorganisms, such as oxytocic, gyrotactic, and negative gravitaxis microorganisms, on the basis of impellent. Negative gravity, a gradient in oxygen content, and a distinction between buoyant heat and mass all act as stimuli for these microbes. Motile microorganisms improve the concentration or mass transfer rate of species in the suspension, which has industrial uses in enzyme biosensors, polymer sheets, chemical processing, and biotechnological applications. Waqas et al.^{8} inspected the effect of bioconvection on secondgrade nanofluid microorganisms. For further details see^{9,10,11,12}\(.\)
Nanoparticles with a diameter of 1–100 nm suspended in a base fluid make up nanofluids. A fluid containing a nanoscale particle is referred to as nanoliquid. The physical features of base liquids, such as their density, thermal conductivity, viscosity, and electrical conductivity, continue to be affected by nanoparticles. Nanofluids are an important component of nanomaterials and are used in a variation of industrial processes, including smart computers, solar panels, renewable energy materials, optics, electronics, and catalysis. According to Madhukesh et al.^{13}, slip influences and the CattaneoChristov theory have an impact on the hydromagnetic micropolarcasson nanofluid flow over porous disc. Regarding the MHD thermal radiative heat transfer of nanofluid over a flat plate in a porous media in the presence of chemical reaction and variable thermal conductivity, consider Pal and Mandal^{14}\(.\) A lot of researchers^{15,16,17,18,19,20,21,22} have focused on nanofluid flow and its industrial and nuclear applications. A unique type of nanofluid known as a hybrid nanoliquid occurs when two different nanoscale particles are dispersed in the fluid in a variety of ways. In order to incorporate the positive effects of both nanomaterials into a single stable homogenous system, various nanomaterial configurations were considered. A hybrid nanofluid is a rapidly developing field. Hybrid electrical systems, modern automated cooling systems, fuel cells, car heat dissipation, biomedicine manufacturing, gas sensing, renewable electricity, domestic freezers and transistors are just a few of the applications of hybrid nanoliquids. The impacts of connective flow of Maxwell hybrid nanofluid passing through the channel were investigated by Ali et al.^{23}\(.\) The context^{24,25,26,27} highlights a few of the important and interesting studies. Figure 1 shows hybrid nanofluid and nanofluid manufacturing process.
The thermophoresis phenomenon is beneficial in numerous microengineering and industrial applications, especially for protecting cleaning gas, preventing microcontamination, nuclear reactors and preventing heat exchanger corrosion. This marvel happens when a combination of several moveable particles is exposed to a temperature variance the various particle kinds react in various ways. Thermophoresis permits small particles to deposit on cold surfaces and allows them to migrate away from heated surfaces. A thermophoretic force is the force caused by the temperature difference that the suspended particles perceive. Gradually, the concept of thermophoresis developed from Goren^{28} examination of the theoretical problem of aerosol particle thermophoresis in the laminar compressible boundary layer flow on a flat plate. Alam et al.^{29} examine the thermophoresis particle deposition on temporary forced convective flow caused by a rotating disc. Authors^{30,31,32,33,34,35,36} analyzed the flow with various geometries by taking thermophoretic deposition of particles into account. The heat generation/absorption outcome on the heat transmission is another amazing aspect to take into account in a variety of realworld problems. In the MHD flow with a heat generation, Shi et al.^{37} analyzed mass and heat transmission in the radiative Maxwell nanofluid. The heat generation properties on the Maxwell nano liquid flow through an expanded cylinder were deliberated by Irfan et al.^{38}. Basha et al.^{39} inspected how the flow of chemically reacting nanoliquid gets affected by the heat source.
When two different types of bodies have different temperatures, the heat transfer mechanism takes place. Mass and heat transfer mechanisms are used in a variation of processes, including the casting of metals, latent heat storage, the production of polyethylene and paper, crystal growth, nuclear reactor cooling and biomedical applications like tissue drug targeting. Fourier^{40} and Fick^{41} were the first to explain the processes of mass and heat transmission. They assert that parabolic equations exist for temperature and concentration distributions. Later, Cattaneo^{42} modified Fourier's law of heat conduction by incorporating the thermal relaxation factor; as a result, he examined heat transmission with a limited speed in thermal waves. In order to achieve the materialinvariant formulation, Christov^{43} proposed a novel model that substitutes Oldroyd's upperconvected derivative for the time derivative. Figure 2 displays flow chart of the problem.
The primary goal of our investigation is to explore the heat energy characteristics of Darcy–Forchheimer flow of hybrid nanofluid with thermosolutal Marangoni convection over an infinite disc with activation energy and Joule heating. The flow problem concentration and energy equations are regulated by the CattaneoChristov model. The study of the modified Fourier and Fick heat and mass flux model in the thermosolutal Marangoni convective flow of hybrid nanoparticles is the main novel idea. Visual representations of numerical solutions are discussed via graphs. The rates of local heat, mass and microorganism transfer are discussed and analyzed in the form of tables. Figure 3 shows thermophysical properties of the hybridnanofluid. The current investigation employs numerical and statistical techniques to address the following questions.

What impacts are observed on the temperature, microorganisms, velocity and concentration profiles in the presence of Marangoni convection parameter?

How does the hybrid nanofluid influence the rates of mass and heat transfer?

What effects does thermal and solutal relaxation parameter have on the profiles of concentration and temperature?

What are the influences of nanoparticles volume fraction parameters on thermal and velocity profiles?

How will the nonuniform heat source affect the temperature profile?

How does the concentration profile changes as a result of the thermophoretic and activation energy parameters?
Mathematical formulation
The applications of silver and molybdenum disulfide nanoparticles is given in Fig. 4 (a,b). We have considered surfacetensiondriven hybrid nanofluid flow over an infinite disc. Figure 5 depicts the physical model of the problem in cylindrical coordinate \((r, \phi , z)\) system. The flow is motivated by the Marangoni layer a surface tension caused by the surface temperature, and is symmetric to \(z = 0\) plane and axisymmetric around the zaxis with \(\frac{\partial }{\partial \phi }= 0\) for all variables. The fluid is considered to be incompressible and electrically conducting, and the flow is supposed to be steady, laminar and irrational. The magnetic field is applied along \(z\)direction. Hybrid nanofluid synthesizes using the silver (Ag) and molybdenum disulphide (MoS_{2}) particles with water (H_{2}O) as base fluid.
Assumptions of the model
The following presumptions and conditions are applied when analyzing the mathematical model.

Darcy–Forchheimer flow

Activation energy is analyzed

Nonuniform heat source is assumed

Variable thermal diffusivity is investigated

Viscous dissipation and Joule heating is addressed

The CattaneoChristov mass and heat flux is discussed

The size of nanoparticles is uniform whereas the shape is spherical

Gyrotactic microorganisms and themophoretic particles are used (Fig. 5).
Basic governing equations
The governing flow equations are (Mackolil et al.^{44}, Mahanthesh et al.^{45} and Basavarajappa et al.^{46}):
The concentration and thermal diffusion with mass and heat flow relaxation are described by the CattaneoChristov double diffusion theory.
The appropriate boundary conditions^{46}:
Nonuniform heat source
The term \({\text{q}}^{{\prime \prime \prime }}\) known as the nonuniform heat generation/absorption can be defined as (Obalalu et al.^{47})
Variable viscosity
The suggested temperature dependent viscosity is given as (Ghaly et al.^{48}).
Variable thermal conductivity
The thermal conductivity that is temperaturedependent can be defined as (Obalalu et al.^{47}).
Variable concentration relation
It is presumed that the diffusivity coefficient follows a linear function. This corresponds to the description of (Obalalu et al.^{47}).
Similarity transformation
Now take into consideration the aforementioned Von Karman transformations (Karman et al.^{49}):
Nondimensional model
The driving PDEs are transformed into ODEs using the similarity variables.
Boundary conditions
Expressions of parameters
The following are the nondimensional parameters: \(Pr=\frac{{\upsilon }_{f}\left(\rho {c}_{p}\right)}{{k}_{f}}\) is the Prandtl number, \(\delta = \frac{{{r}^{2}B}_{2}}{{T}_{\infty }}\) is the temperature difference parameter, \(\tau =\frac{{k}^{*}{v}_{f}{r}^{2}{B}_{1}}{{\upsilon }_{f}{\rho }_{f}{T}_{f}}\) is the Thermophoretic parameter, \(M=\frac{{\sigma }_{f}{B}_{0}^{2}}{{\Omega }^{*}{\rho }_{f}}\) is the magnetic parameter, \({F}_{r}=\frac{{rc}_{b}}{\sqrt{{k}_{1}}}\) is the Forchheimer parameter, \(Rc=\frac{{k}_{r}^{2}}{{\Omega }^{*}}\) is the chemical reaction parameter, \(E=\frac{{E}_{a}}{{{T}_{\infty }K}^{*}}\) is the activation energy parameter, \(K=\frac{{\upsilon }_{f}}{{{\Omega }^{*}k}_{1}}\) is the Inverse Darcy parameter, \(\text{Ec}=\frac{{v}_{f}{{\Omega }^{*}}^{2}}{{c}_{p}{B}_{1}}\) is the Eckert number, \(Rd=\frac{{16\sigma }^{*}{T}_{\infty }^{3}}{3{{k}^{+}k}_{f}}\) is the thermal radiation parameter, \(Sc =\frac{{\upsilon }_{f}}{{D}_{B}}\) is the Schmidt number, \(Ma=\frac{{\gamma }_{C}{B}_{2}}{{\gamma }_{T}{B}_{1}}\) is the Marangoni ratio number, \(Mn=\frac{{\sigma }_{0}{\gamma }_{T}{rB}_{1}}{{\mu }_{f} {\Omega }^{*}}\sqrt{\frac{v}{{\Omega }^{*}}}\) is the Marangoni number, \({\gamma }_{1}={{\Omega }^{*}\lambda }_{T}\) is the thermal relaxation parameter, and \({\gamma }_{2}={{\Omega }^{*}\lambda }_{c}\) is the solutal relaxation parameter, \(Pe=\frac{b{W}_{c}}{{D}_{m}}\) is the Peckelt number, \(\sigma =\frac{{N}_{\infty }}{{{r}^{2}B}_{3}}\) is the Microorganisms difference parameter and \(Sb=\frac{{\upsilon }_{f}}{{D}_{m}}\) is the bioconvection Schmidt number.
Physical quantities
The following are the local Sherwood number \((S\text{h})\), Nusselt number \((N\text{u})\), density of motile microorganisms \((N\text{n})\) and skin friction \((C{f}_{x})\) (Basavarajappa et al. ^{46}).
where \(Re=\frac{{r}^{2}{\Omega }^{*}}{v}\) is the local Reynolds number. Table 1 displays thermophysical characteristics of hybrid nanofluids. Table 2 shows the thermophysical properties of regular fluids and nanoparticles.
Numerical method
Equations (19)–(23) with boundary conditions (24) and (25) are numerically solved in MATLAB using the wellknown shooting approach with the RKF45th method. The most effective way for computing the numerical approximation of this kind of highly nonlinear problem is the shooting method. Unlike other numerical techniques, this method lacks any complex discretization and has excellent solution accuracy. This numerical accuracy is also determined to be excellent.
Boundary conditions
The unknowns \({n}_{1}\) to \({n}_{4}\) are estimated using the shooting method. The norms for convergence are \({10}^{6}\) and the numerical corroboration is used, with a maximum step size of 0.001. The solution procedure of the problem is given in Fig. 6.
Results and discussions
The presumed flow modeled equations are transformed to ODEs by selecting the appropriate similarity variables. The primary goal of this section is the physical explanation of involved parameters for discrete flow fields. Figures 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18 have been arranged and plotted. Table 2 displays the thermophysical features of carrier liquid and nanoparticles. The impact of dimensionless parameters such as Marangoni convection parameter, inverse Darcy number, Prandtl number, Schmidt number, Forchheimer parameter, temperature difference parameter, chemical reaction parameter, concentration difference parameter, reaction rate coefficient, the reference length and the activation energy are discussed. The impact of \(M\) on \(F\left(\xi \right)\) and \(H\left(\xi \right)\) is seen in Fig. 7a,b\(.\) The velocity profiles drop as a result of an rise in the values of \(M\). The transverse magnetic field application will produce a resistive force similar to the drag force, which has the tendency to slow down the velocity of the hybrid nanofluid. Figure 8a,b demonstrate the outcome of \({F}_{r}\) and \(K\) on \(F\left(\xi \right)\). The velocity gradient declines as a result of the growth in \(K\) and \({F}_{r}\) values, as shown in the graph. The plot shows that hybrid nanofluid velocity and associated boundary thickness decreases as \(K\) and \({F}_{r}\) increases. This is because an increase in porosity widens the pores in a porous medium, which causes resistive forces to operate against flow and lower velocity profiles. The impact of \(Ec\) and \(Rd\) on \(\theta \left(\xi \right)\) are shown in Fig. 9a,b\(.\) When the values of \(Ec\) are raised, the temperature rises. The Eckert number \(Ec\) describes the relationship between the enthalpy and the flow's kinetic energy. It represents the process by which kinetic energy is transformed into internal energy by effort against viscous fluid forces. The temperature of nanofluid and hybrid nanofluid increases due to the increased viscous dissipative heat. By raising \(Rd\), the temperature and the boundary layer thickness that corresponds to it rise. When we use the impacts of thermal radiation, the surface heat flux physically increases, and this becomes the key to raising temperature. The impact of \({A}^{*}\) and \({B}^{*}\) on \(\theta \left(\xi \right)\) is depicted in Figs. 10a,b and 11a,b. It is found that improving \({A}^{*}\) and \({B}^{*}\) values result in better temperature distributions for hybrid nanofluid and nanofluid. The nonuniform heat sources \({A}^{*}\) and \({B}^{*}\) are considered heat sources when they release heat energy into the fluid flow and operate as heat generators, which causes the temperature distribution to become more uniform. However, they are referred to as heat sinks when the nonuniform heat sources/sinks \({A}^{*}\) and \({B}^{*}\) are given negative values. The boundary layer's function as a heat sink reduces the temperature of the hybrid and nanofluid nanofluids. Figure 12a,b show how the temperature gradient changes for increasing volume fractions. The temperature distribution and thickness of the associated boundary layer both grow as the volume fractions \({\Phi }_{1}\) and \({\Phi }_{2}\) rise. Additionally, as the volume percentage of nanoparticles rises, more heat is produced, which improves the thermal profile and increases the thickness of the associated boundary layer. The impacts of \({\gamma }_{1}\) and \({\gamma }_{2}\) on \(\theta \left(\eta \right)\) and \(\phi \left(\eta \right)\) are described in Fig. 13a,b. The thermal \(\theta \left(\xi \right)\) and concentration \(\phi \left(\xi \right)\) profiles tend to decline with the increasing relaxation times, as demonstrate by careful examination of the aforementioned figures. These figures also show that the concentration and thermal boundary layer thicknesses for the traditional Fourier's law and Fick's law when mass and heat quickly pass throughout the material (i.e., \({\gamma }_{1}={\gamma }_{2}=0\)) are greater than for the CattaneoChristov doublediffusion model. Figure 14a,b show the bearing of the solutal profile for various values of \(E\) and \(Rc\). The concentration gradient is enhanced by the increase in \(E\) values, whereas the trend is the opposite for increasing \(Rc\) values, as shown in Fig. 14a,b\(.\) Stronger chemical reactions have a destructive outcome that causes the reactant species to deteriorate. Figure 15a illustrates how the influence of \(\tau\) is on \(\phi \left(\xi \right).\) The solutal profile \(\phi \left(\xi \right)\) drops as the values of \(\tau\) rise, as seen in Fig. 15a\(.\) When the temperature gradient grew, a weaker concentration is seen because of an growth in particle mobility. The outcome of \(Lb\) on \(\Theta \left(\xi \right)\) is depicted in Fig. 15b. It is clear that when \(Sb\) increases, the density profile of mobile organisms in hybrid and nanofluid diminish. The impact of \(\sigma\) and \(Pe\) on microorganism profile is depicted in Fig. 16a,b\(.\) Both the hybrid nanofluid and nanofluid microorganism profiles dropped when we enhanced \(Pe\) and \(\sigma\) values. Inversely proportional to \({D}_{m}\) (microorganisms diffusivity) and directly proportional to one another are the Peclet number \((Pe)\) and cell swimming speed (\({W}_{c}\)). Advection and diffusion rates are related to the Peclet number. As a result, a rise in \(Pe\) results from an increase in the rate of advective transport, this quickly increases the flux of microorganisms. The impact of \(Ma\) on \(F\left(\xi \right),\theta \left(\xi \right),\phi \left(\xi \right)\) and \(\Theta \left(\xi \right)\) are depicted in Figs. 17a,b and 18a,b, respectively. The graphs illustrate how raising \(Ma\) values enhance the microorganism, temperature and velocity profiles of hybrid and nanofluid. This phenomenon is brought on by the surface variation. A stronger Marangoni effect will almost always lead to a higher velocity gradient because it works as a pouring force for liquid streams. These graphs show that as the value of \(Ma\) increases, the concentration \(\phi \left(\xi \right)\) profile significantly decreases. The Marangoni number is physically connected to the surface tension. The bulk attraction of the liquid to the particles in the surface layer is which creates surface tension on a liquid's surface and because of this, as surface tension rises, surface molecule attraction grows stronger. Figures 19a,b and 20a,b show how \(Ma\) affects the Nusselt number \((Nu)\), local density microorganisms \((Nn)\), Sherwood number \((Sh)\) and skin friction \((C{f}_{x})\). Skin friction, local density microorganisms and Sherwood number are improved as \(Ma\) increases, whereas Nusselt number declines. Table 3 shows the impact of several parameters on Nusselt number \((Nu)\). Table 4 displays the influence of numerous parameters on Sherwood number \((Sh)\). Table 5 shows the effect of several parameters on local density of motile microorganisms \((Nn)\). Table 6 shows the shape factors of nanoparticle. Table 7 displays the comparison results of the present study to earlier published research, with the additional parameters set to zero.
Concluding remarks
Numerical analysis is done to determine the importance of the thermosolutal Marangoni 1convective flow of hybrid fluid across an infinite disc containing thermopherotic patricles, microorganisms and activation energy. The thermal energy analysis makes use of the CattaneoChristov model. Some important conclusions are drawn from this research.

The solutal and thermal layer thicknesses are increased but the velocity is decreased by the Lorentzian body strength. This is caused by the magnetic field's imposed retardation force.

Higher values of the Darcy number and the Forchheimer parameter results in a reduction in the axial velocity profile.

The temperature field is significantly improved by thermal energy modulations (Spacedependent coefficient and Temperature coefficient), which both add more heat to the hybrid nanoliquid system.

Increases in the Marangoni ratio parameter, chemical reaction parameter, thermophoretic parameter and concentration relaxation time cause the hybrid nanofluid concentration profile to fall, whereas the activation energy parameter exhibits the opposite behavior.

The hybrid nanofluids velocity, microorganisms and temperature increase due to Marangoni convection.

The Nusselt number shows increasing behavior by increasing the solid volume fractions.

By raising the chemical reaction parameter, and thermopherotic parameter, Sherwood number significantly decreased.

By raising the Schmidt number for bioconvection, the local density of motile microorganisms gets declined.

The calculations show that our results are in good accord with the previous research.
Data availability
All data generated or analysed during this study are included in this published article.
Abbreviations
 \((r, \phi , z)\) :

Cylindrical coordinate system
 \(\left(u w\right)\) :

Velocity fields of fluid (m s^{−1})
 \(C\) :

Species of the fluid concentration (m)
 \(N\) :

Density of motile microorganism
 \({D}_{B}\) :

Mass diffusivity coefficient (m^{2} s^{−1})
 \({W}_{c}\) :

Maximum cell swimming speed
 \({D}_{m}\) :

Diffusivity of microorganisms
 \({K}^{*}\) :

Chemical reaction coefficient (mol L^{−1} s^{−1})
 \({k}_{1}\) :

Permeability of the porous medium (m^{2})
 \(k\) :

Thermophoretic coefficient
 \({T}_{\text{f}}\) :

Reference temperature (k)
 \({C}_{p}\) :

Specific heat of the fluid (J kg^{−1} k^{−1})
 \({T}_{0}\) :

Constants
 \(T\) :

Fluid temperature (k)
 \(Sc\) :

Schmidt number
 \(Ec\) :

Eckert number
 \({k}_{f}\) :

Thermal conductivity of the fluid (W m^{−1} k^{−1})
 \({F}_{r}\) :

Forchheimer parameter
 \({T}_{\infty }\) :

Fluid ambient temperature (k)
 B*:

Temperature coefficient (k)
 \(Pr\) :

Prandtl number
 \(E\) :

Activation energy parameter
 Pe:

Bioconvection Peclet number
 \({B}_{0}\) :

Uniform magnetic field (kg s^{−2} A^{−1})
 q:

Normal heat flux (W m^{−2})
 J:

Normal mass flux (kg m^{−2} s^{−1})
 \({A}^{*}\) :

Spacedependent coefficient
 \(K\) :

Darcy parameter
 \(Ea\) :

Activation energy coefficient (kg m^{−2} s^{−2})
 \(Sb\) :

Bioconvection Schmidt number
 Nn:

Local density of motile microorganisms
 \(Rd\) :

Radiation parameter
 \(Ma\) :

Marangoni convection parameter
 \({k}^{+}\) :

Mean absorption coefficient (cm^{−1})
 Nu:

Nusselt number
 Sh:

Sherwood number
 \(Mn\) :

Marangoni number
 \({q}_{r}\) :

Radiative heat flux (kW m^{−2})
 \(Rc\) :

Chemical reaction parameter
 \(M\) :

Magnetic parameter
 \({\rho }_{f}\) :

Fluid density (kg m^{−3} s)
 \({\gamma }_{2}\) :

Solutal relaxation parameter
 \({\nu }_{f}\) :

Kinematic viscosity (m^{2} s^{−1})
 \({\sigma }_{1}\) :

Surface tension (N m^{−1})
 \(\tau\) :

Thermophoretic parameter
 \({\lambda }_{T}\) :

Relaxation time of heat flux (W m^{−2})
 \({\gamma }_{\text{C}}\) :

Surface tension coefficients for concentration
 \(\delta\) :

Temperature difference coefficient
 \({\Phi }_{1}\) :

Volume fraction of MoS_{2}
 \({\sigma }_{0}\) :

Constant
 \({\varepsilon }_{1}\) :

Viscosity variation exponent parameter
 k_{f} :

Thermal conductivity (W m^{−1} K^{−1})
 \({\varepsilon }_{3}\) :

Variable mass diffusivity parameter
 \(\lambda c\) :

Relaxation time of mass flux (kg m^{−2} s^{−1})
 \({\gamma }_{T}\) :

Surface tension coefficients for temperature (N m^{−1})
 \(\sigma\) :

Microorganisms concentration difference parameter
 \({\sigma }^{*}\) :

StefanBoltzmann constant (W m^{−2} K^{−4})
 \({\Phi }_{2}\) :

Volume fraction of Ag
 \({\sigma }_{f}\) :

Electrical conductivity (s m^{−1})
 \({\mu }_{f}\) :

Dynamic viscosity (kg m^{−1} s^{−1})
 \({\gamma }_{1}\) :

Thermal relaxation parameter
 \({\varepsilon }_{2}\) :

Variable thermal conductivity parameter
 ′:

Derivative with respect to \(\xi\)
 \(f\) :

Base fluid
 \(hnf\) :

Hybrid nanofluid
 \(0\) :

Surface
 \(\infty\) :

Ambient
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M.A. and N.K. conceived of the presented idea. M.A. and N.K. developed the theory and performed the computations. M.S.H. and J.Y. verified the method and results. N.K. and M.S.H. supervised the findings of this work. M.S.H. and J.Y. elaborate conclusion section. All authors discussed the results and contributed to the final manuscript.
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Abbas, M., Khan, N., Hashmi, M.S. et al. Numerically analysis of Marangoni convective flow of hybrid nanofluid over an infinite disk with thermophoresis particle deposition. Sci Rep 13, 5036 (2023). https://doi.org/10.1038/s4159802332011x
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DOI: https://doi.org/10.1038/s4159802332011x
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