Numerical analysis of thermophoretic particle deposition in a magneto-Marangoni convective dusty tangent hyperbolic nano ﬂ uid ﬂ ow – Thermal and magnetic features

: In the current study, we focus on the Magneto-Marangoni convective ﬂ ow of dusty tangent hyperbolic nano ﬂ uid (TiO 2 – kerosene oil) over a sheet in the presence of thermophoresis particles deposition and gyrotactic micro-organisms. Along with activation energy, heat source, variable viscosity, and thermal conductivity, the Dufour-Soret e ﬀ ects are taken into consideration. Variable surface tension gradients are used to identify Marangoni convection. Melting of drying wafers, coating ﬂ ow technology, wielding, crystals, soap ﬁ lm stabilization, and micro ﬂ uidics all depend on Marangoni driven ﬂ ow. This study ’ s major objective is to ascertain the thermal mobility of nanoparticles in a ﬂ uid with a kerosene oil base. To improve mass transfer phenomena, we inserted microorganisms into the base ﬂ uid. By using similarity transformations, the resulting system of nonlinear partial di ﬀ erential equations is converted into nonlinear ordinary di ﬀ erential equations. Using a shooting technique based on RKF-45th order, the numerical answers are obtained. For various values of the physical parameters, the local density of motile microorganisms, Nusselt number, skin friction, and Sherwood number are calculated. The ﬁ ndings demonstrated that as the Marangoni convection parameter is raised, the velocity pro ﬁ les of the dust and ﬂ uid phases increase, but the microorganisms, concentration, and temperature pro ﬁ les degrade in both phases.

By using similarity transformations, the resulting system of nonlinear partial differential equations is converted into nonlinear ordinary differential equations.Using a shooting technique based on RKF-45th order, the numerical answers are obtained.For various values of the physical parameters, the local density of motile microorganisms, Nusselt number, skin friction, and Sherwood number are calculated.The findings demonstrated that as the Marangoni convection parameter is raised, the velocity profiles of the dust and fluid phases increase, but the microorganisms, concentration, and temperature profiles degrade in both phases.

Du
Dufour number E a activation energy coefficient ( ) k f thermal conductivity of the fluid ( )  The nanofluid principle is created by the integration of nanoparticles (1-100 nm) with base liquids.Nanoparticles are usually recommended for enhancing the heating rate in various industrial and technical systems.Choi [5] provided the first analysis of nanofluids using experimental assumptions and data.Numerous academics have researched the flow of nanofluids over different geometries [6][7][8][9].
Many complicated engineering issues, including combustion, rain erosion, waste water treatment, lunar ash flows, paint spraying, corrosive particles in motor oil flow, nuclear reactors, polymer technologies, etc., involve the phenomenon of fluid flow including millimeter-sized dust particles.Saffman [10] provided an inquiry on fluid particle suspension and the stability of laminar flow of dusty fluid.Agranat [11] explored how pressure gradient impacts the rate of heat transmission in a fluid containing dust particles.The safety of nuclear reactors, gas cleaning, micro contamination management, and heat exchanger corrosion are only a few applications of the thermophoresis phenomenon in industry and micro-engineering.This phenomenon happens when a mixture of several movable particle kinds is exposed to a temperature variation.
Different particle kinds respond in different ways.Thermophoresis allows microparticles to move away from warm surfaces and deposit on cool surfaces.The thermophoretic force is the force that the temperature difference has on the suspended particles.The thermophoretic velocity of a particle is its rate of motion.Thermophoresis particle deposition on a wedge-shaped forced convective heat and mass transfer flow in two dimensions with variable viscosity was analyzed by Rahman et al. [12].Abbas et al. investigated the deposition of thermophoretic particles in Carreau-Yasuda fluid on a chemically reactive Riga plate [13].According to the studies [14][15][16], particle deposition has a considerable impact on liquid flow.
The word "bioconvection" refers to a phenomenon brought on by microorganisms.These bacteria have a propensity to accumulate at the upper section of the fluid, which becomes unstable, as a result of the strong density stratification.When exposed to an external stimulus, moveable microorganisms in the base fluid move in a certain direction, increasing the density of the base fluid.Mobile microorganisms boost the mass transfer rate of species in the solution and have industrial uses in enzyme biosensors, chemical processing, polymer sheets, and biotechnological research.The radiative flow of the Casson fluid via a rotating wedge containing gyrotactic microorganisms was studied by Raju et al. [17].For more information, check previous literature [9,[18][19][20][21][22][23].
The Marangoni convective transport mechanism commonly manifests when the liquid-liquid or liquid-air interface surface tension varies on the concentration or the temperature distribution.The study of mass and heat transfer in this phenomenon has garnered a lot of interest due to its numerous applications in the fields of nanotechnology, welding processes, atomic reactors, silicon wafers, thin film stretching, soap films, melting, semiconductor processing, crystal growth, and materials sciences.Kairi et al. [24] investigated the effect of the thermosolutal Marangoni on bioconvection in suspension of gyrotactic microorganisms over an inclined stretched sheet.Roy et al. [25] studied a non-Newtonian nanofluid thermosolutal Marangoni bioconvection in a stratified environment.The role that Marangoni convective flow plays in the passage of mass and heat into diverse systems was carefully explored in the previous literature [26][27][28][29][30][31].
The innovative aspect of the current study is the examination of the importance of Marangoni convective flow of magnetized dusty tangent hyperbolic nanofluid over sheet in the presence of thermophoresis particle deposition and gyrotactic microorganisms.Due to the inspiration provided by the aforementioned investigations and uses, the activation energy, heat source, and Soret and Dufour effects have also been discussed.According to the material mentioned above, the current test is brand-new and has not yet been studied.With the help of the RKF-45th method, the resulting problem is numerically solved, and the effects of the pertinent parameters on the distributions of temperature, solutal, velocity, microbes, local skin friction, Sherwood number, and Nusselt number have been carefully analyzed.In order to provide details, the current study addresses the following inquiries: • What impact do the Weissenberg number and power law index parameter have on the temperature and velocity profiles?• How do the temperature, microbe concentration, and velocity profiles for the fluid (phase-I) and particle (phase-II) phases change as a result of Marangoni convection?• What impact does the parameter for nanoparticle volume fraction have on the thermal and velocity profiles?• What effects do thermophoretic and chemical reaction parameters have on concentration profiles?
What effects do Dufour and Soret numbers have on the profiles of temperature and concentration?

Description of the model
We have looked into the Marangoni convection-affected flow of dusty tangent hyperbolic nanofluid over a sheet at y = 0 close to a stagnation point.Given that the flow is constrained to the region y ≥ 0, the coordinates x and y are taken perpendicularly and vertically to the flow, respectively.We consider the free stream velocity to be ( ) = U x ex e .In Figure 1, the problem's configuration is shown.Along Thermophoretic particle deposition in a magneto-Marangoni convective flow  3 the y-axis, a magnetic field with constant strength B 0 is applied.TiO 2 in kerosene oil is used to determine its thermal properties and correlations.Gyrotactic bacteria and thermophoresis particle deposition are taken into consideration.It is assumed that the sphere-shaped dust and nanoparticles were evenly dispersed throughout the fluid.

Governing equations
The model equations for both phases are given below ( [32,33]): First phase (for fluid): (2) Second phase (for dust particles): ( ) The adopted conditions on and away from the surface are as follows: The Marangoni convection phenomenon can be described by equation (11).This phenomenon has prominent engineering and technology applications.
The viscosity that is almost temperature-dependent is given below [34]: The thermal conductivity that is also temperaturedependent is given below [34]:

Similarity transformations
The adopted similarity variables can be composed as follows: (20) The terminologies in the above equations are further specified as follows: where the microorganism gradient coefficients are C G and , the solute gradient coefficients are B F and , and the tem- perature gradient coefficients are A E and .Now using equa- tions ( 18)- (22) in governing equations (1)-( 14), we obtain the dimensionless equations (as given below).
First phase: Second phase: The simplified BCs are

Dimensionless parameters
The dimensionless analysis of the preeminent parameters is provided in Table 1.

Physical parameters
The physical parameters of prime interest are given below: Table 1: Dimensionless analysis of the prime parameters Density ( ) Heat capacitance ( ) The physical characteristics for nanofluids relations are provided in Tables 2-4.

Results and discussion
The analysis of the dominant impacts of the parameters are presented in this section.The parametric ranges are taken from the standard literature [33,36,37], e.g., The underlying cause of this phenomena is surface variation.The Marangoni effect causes liquid streams to pour, hence it is always followed by an accelerated velocity gradient.These figures show how, when Ma values increase, the temperature, concentration, and microorganism profiles all decrease dramatically.The higher attraction of the liquid to the particles in the geometry causes surface tension to form over the surface.As a result, temperature decreases as surface tension increases.The appearance of the surface molecules causes the thermal gradient to decrease.The temperature gradient lessens as a result.
Figures 4(a) and (b) and 5(a) and (b), respectively, show the effects of β v , β t , β , c and β m on the prescribed profiles in either case of dust or fluid phases.These figures demonstrate that the microbe, temperature, concentration, and velocity profiles for the particle phase increase considerably with the increase in levels of β v , β t , β c , and β m , respec- tively.The fluid phase across the boundary layer is affected by these phenomena quite in the opposite way.
The distribution of the transverse magnetic field will produce a Lorentz force similar to the drag force, which tends to slow the fluid flow in both phases.The momentum boundary layer thickness decreases as M increases.Figure 6(a) and (b) demonstrate, for the two phases (I and II), respectively, the effects of We on θ ξ , and ( ) θ ξ p .The power law index can be used to explain two different types of fluids: pseudoplastic fluids ( < n 1) and dila- tant fluids ( > n 1).The velocity profile is reduced as the values of n for the shear thinning phenomenon increases.It is because higher values of the power law index are associated with higher viscosities, which lead to lower fluid velocities.The Weissenberg number is one of the characteristics that also slows fluid velocity.Due to the direct relationship between the Weissenberg number and relaxation time, increasing Weissenberg numbers lengthen the relaxation times and increase resistance to fluid motion, which reduces fluid velocity.As we increase the values of Φ, its effect on velocity starts to disappear.Physically, when the nanoparticles' saturation exceeds that of the nanofluid, the outcome is a denser nanofluid, which causes the velocity to decrease.When the values of Φ are raised, the temperature of the nanofluid and dust phases also rises.Physically, resistance and temperature profiles increase as the concentration of nanoparticles in a tangent hyperbolic fluid increase.The temperature profiles get better as we increase the values of parameters Q t and Q e for the tangent hyperbolic nanofluid and dust phase.Figure 10   chemical reaction slows down because of a reduction in heat, the effect on the concentration profile is greater.When the activation energy increases, the modified Arrhenius mechanism exhibits increasing behavior.Any system's activation energy is acknowledged by the Arrhenius equation.When Rc is increased, the concentration gets decreased.
As the temperature gradient widened, a weaker concentration was seen because of an increase in particle mobility.Figure 11 5 which validates the approximate solutions.Tables 6 and 7 comprise the analysis of motile density number and skin friction, respectively.Figure 13 shows the effect of λ on the tangent hyper- bolic nanofluid and dust phases.For higher values of λ, the distribution of velocity tends to increase.Thermophoretic particle deposition in a magneto-Marangoni convective flow  13

Conclusion
Thermophoretic particle deposition and gyrotactic microorganisms are present in the magneto-Marangoni convective flow of dusty tangent hyperbolic nanofluid over a sheet, and a numerical solution is obtained.The following is the summary of the findings: • The values of the Weissenberg number lead the heat profiles to increase and the velocity profiles to decrease as we ascend.• An increase in the Marangoni convection parameter results in an increase in the velocity profiles and skin friction, while the microorganism profiles, heat profiles, and concentration profiles exhibit the opposite behavior for both phases.• Surface tension greatly depends on the Marangoni number.
A liquid's bulk attraction to the particles in the surface layer on its surface causes surface tension.• As a result, as the surface tension rises, the temperature falls and the bulk magnetism between the surface molecules increases.• The Soret number demonstrates opposite behavior to that of the nanofluid concentration profiles, which increase as chemical reaction parameter levels do.• The density of hybrid nanofluid and nanofluid motile bacteria profiles reduces for higher levels of Peclet number.The Peclet number effect causes motile bacteria to swim more quickly, which reduces the thickness of the microorganisms at the surface.• The value of skin friction increases as the Weissenberg number and the free stream parameter increase, while the magnetic parameter has the opposite effect.• The Nusselt number at the surface tends to increase with the increase in heat source parameters, but the Dufour number tends to increase in the other direction.
• The Soret number increases the Sherwood number, but the opposite is true when the chemical reaction parameter and the thermophoretic parameter are improved.

Future work
Future research should expand on this work by taking into account thermal radiation, Newtonian heating, variable conditions, and trihybrid nanoparticles.These models will be highly helpful in the construction of furnaces, SAS turbines, gas-cooled nuclear reactors, atomic power plants, and unique driving mechanisms for aircraft, rockets, satellites, and spacecraft.In the future, the existing method might be used for a number of physical and technical obstacles [38][39][40][41][42][43][44][45][46][47].
fluids are employed more frequently in engineering and manufacturing processes than Newtonian fluids.The tangent hyperbolic model was one of the non-Newtonian models that Pop and Ingham [1] proposed.The tangent hyperbolic fluid model may well explain the shear thinning phenomenon.Blood, ketchup, paint, and other chemicals are a few examples of fluids with this property.Akbar [2] investigated the tangent hyperbolic fluid's peristaltic flow with convective boundary conditions.Naseer et al. [3] investigated the hyperbolic tangent fluid flow in a boundary layer over an exponentially extending vertical cylinder.Salahuddin et al.'s [4] examination of tangent hyperbolic fluid flow on stretched surfaces looked at the effects of heat production and absorption.

- 5 particles 1 m
dust particle's relaxation time, Thermophoretic particle deposition in a magneto-Marangoni convective flow  fluid-particle interaction parameter for bioconvection.

Figure 3
Figure3(a) and (b) illustrate how raising Ma improvises the profiles (velocities and temperatures) for both phases (I and II).The underlying cause of this phenomena is surface variation.The Marangoni effect causes liquid streams to pour, hence it is always followed by an accelerated velocity gradient.These figures show how, when Ma values increase, the temperature, concentration, and microorganism profiles all decrease dramatically.The higher attraction of the liquid to the particles in the geometry causes surface tension to form over the surface.As a result, temperature decreases as surface tension increases.The appearance of the surface molecules causes the thermal gradient to decrease.The temperature gradient lessens as a result.Figures4(a) and (b) and 5(a) and (b), respectively, show the effects of β v , β t , β , c and β m on the prescribed profiles in either case of dust or fluid phases.These figures demonstrate that the microbe, temperature, concentration, and velocity profiles for the particle phase increase considerably with the increase in levels of β v , β t , β c , and β m , respec- tively.The fluid phase across the boundary layer is affected by these phenomena quite in the opposite way.The distribution of the transverse magnetic field will produce a Lorentz force similar to the drag force, which tends to slow the fluid flow in both phases.The momentum boundary layer thickness decreases as M increases.Figure6(a) and (b) demonstrate, for the two phases (I and II),

Figure 7 (
a) and (b) depict the impression of Du and ϵ for the two stages (fluid and dust), respectively.The heat flux brought on by a concentration gradient is referred to as the Dufour effect.The temperature profile behaves

Figure 8 (
a) and (b) illustrate how Ec and Rd affect the profiles ( ) θ ξ and ( ) θ ξ p in either case of two phases, respectively.The temperatures of both phases increase as Ec values increase.The Eckert number describes the relationship between the enthalpy and kinetic energy of the flow.It represents the process through which internal energy is produced by applying pressure to a fluid's forces.The thermal boundary layer thickness increases for both phases (I&II) as a result of the increased viscous dissipative heat.As Rd rises, the temperature and thermal boundary layer thickness rise as well.

Figure 9 (
a) and (b) depict the effects of Q t and Q e on the temperatures of both phases.
(a) and (b) show how E and Rc affect the concentration.Figure 10(a) depicts that the concentration is an increasing function of E. The Arrhenius equation demonstrates mathematically that if a
(a) and (b) depict the effects of Le and Lb on the dimensionless temperature and concentration profiles.As Le and Lb increase, it was observed that the concentration and microorganism ( ( ) ξ Θ and ( )) ξ Θ p profiles decreased.The Lewis number is the reciprocal of the heat to mass diffusivity.The mass diffusivity declines and the thickness of the concentration boundary layer decreases with higher values of Le.
Figure 12(a) and (b) show the effects of Ω and Pe on the microorganisms distribution.In both phases (I and II), ( ) η Θ and ( ) η Θ p dropped when Pe and Ω values were increased.A numer- ical data comparison is depicted in Table

Table 6 :
Change in Nn x with various parameters

Table 7 :
Change in C fx with various parameters