INVESTIGATION OF THERMAL RADIATIVE TANGENT HYPERBOLIC NANOFLUID FLOW DUE TO STRETCHED SHEET †

The current study illuminates the enactment of tangent hyperbolic nanofluid past a bidirectional stretchable surface. The phenomena of heat and mass transfer with joule heating, chemical reaction and thermal radiation have been debated. For motivation of problem convective boundary conditions and heat source are part of this study. The modeled partial differential equations are mended into ordinary differential equations with the help of appropriate self-similarity transformations. Furthermore, the resulting system of ODEs is numerically handled by using well-established shooting scheme and acquired numerical outcomes are compared with ND Solve command of Mathematica. The Effects of prominent parameters on velocity, temperature and volumetric concentration distribution are inspected through graphs. The influence of emerging parameters involved in this study on flow and heat removal features are deliberated in detail. As we are increasing the values of power-law index 𝑛 , Prandtl number 𝑃𝑟 and Magnetic parameter 𝑀 , outcomes increment in skin friction coefficient while decline in the Nusselt number is seen.


INTRODUCTION
Due to wider applications of non-Newtonian fluid, many researchers paid their attention, in the last few years.The applications of non-Newtonian fluids contain food products, personal protective equipment braking and damping devices, printing technology and drag reducing agents.Shampoo, melted butter, blood, paint, cornstarch, starch suspensions, toothpaste, custard and Ketchup are examples of such type of fluids [1].
One of the most significant kinds of non-Newtonian fluid is Tangent hyperbolic.Kumar et al. [2] reported that the Tangent hyperbolic fluid has capability to illustrate the behavior of shear thinning.Rehman et al. [3] examined the interplay between the stratification of tangent hyperbolic fluid and the combined effects of thermal radiation, and concluded that with the greater value of solar radiation, there are escalations in the fluid temperature.Shafiq et al. [4] discovered the magnetic flow of bio convective tangent hyperbolic fluid containing swimming microorganism with thermal effects and they reported that the temperature field surge for higher value of thermophoresis parameter.Salahuddin et al. [5] examined the tangent hyperbolic due to stretched cylinder across the stagnation point.Naseer et al. [6] exposed the concept of tangent fluid for buoyancy and thermal effects and reported that the temperature function decreased with increasing value of Prandtl number.Prabhakar et al. [7] scrutinized the flow tangent hyperbolic fluid with influence of inclined Lorentz forces due to stretched surface.
Nanofluid is an important category of non-Newtonian fluids having nanometer-sized particles.propylene glycol, water, ethylene glycol etc., are base fluids, to boost the thermal conductivity of base fluid's nanoparticles are used.Nanofluids are the homogeneous typical combination of nanoparticles are usually made by metals (Cu, Zn, Al), nonmetals (nanotubes, boron, carbon) and carbides (SiC, Fe3C, CaC 2 ) with base fluids (glycol, water and ethylene glycol) [8].The inclusion of a modest number of nanoparticles improved the thermal conductivity of heat transfer fluids, according to research by Choi et al. [9].The concept of increasing the heat conductivity by using nanoparticles was first proposed by Choi and Eastman [10].Izadi et al. [11] inspected the behavior of nanofluid with laminar forced flow.As noted by Wong et al. [12], they have numerous engineering and biomedical uses, including as in microelectronics, nuclear reactors, process industries, and cancer therapy.Nayak et al. [13] explored the influence of solar radiation on electrical conducting nanofluid dur to vertical sheet.
The study of squeezing flows in several commercial and practical applications in unusual domains including pharmaceutical manufacturing, energy production, polymer exclusion, energy in space technology, nuclear reactors, chemical reactions, and solar energy in space technology aim to slow down and employed other technologies.In industrial applications, squeezing flows are used in lubrication, bearings and motors.Lin et al. [14] exhibited the qualities of electrical conducting squeezed fluid flow among annular vertical plates.Thermal radiation and heat transfer has numerous uses in production procedures, semiconductors, chemical processes, engineering, and other several areas of technology.The micro convection thermal effect in two phase mixtures is, according to Sohn and Chen [15], more potent than it is in a single-phase EEJP. 3 (2023) Muhammad Jawad, et al. fluid.Makinde [16] deliberated the radiative flow of heat transfer free convection in the presence of porous sheet.Hussain et al. [17] premeditated the heat transfer phenomena in squeezing flow with thermal radiation and bio convection between two equivalent plates.Hayat et al. [18] debated the influence of solar radiation in Jeffery squeezing fluid flow.Additional study on the influence of heat transfer process and thermal radiation in fluid flow can be considered in [19][20][21].
The study of electrically conducting fluids with magnetic field is known as magneto hydrodynamics MHD.Magneto-hydrodynamic flow has many presentations in numerous fields of science engineering like MHD accelerator, power generator, heat exchangers and cooling of reactors as deliberated by Hari et al. [22].Rashidi et al. [23] determined the electrical conducting flow of nanofluid with nonlinear radiation due to vertical stretched surface.Zhang et al. [24] inspected the radiative flow of MHD nanofluid with heat flux, variable surface and chemical reaction.
The illustration of this investigation is to study the numerical outcomes of electrical conducting fluid flow of hyperbolic nanofluid fluid joule heating and chemical diffusion due to stretching sheet using convective boundary conditions.The couple of nonlinear PDEs of governing model are transformed into a set ODEs by use of appropriate revolution called similarity function.Operating the shooting scheme numerous mathematical outcomes are validated.Furthermore, impression of effective parameters is talk about through graphs in detail.

MATHEMATICAL ANALYSIS
Consider the laminar, 2d electrical conducting tangent hyperbolic nanofluid due to vertical stretched sheet.Moreover, the influence of Joule heating and thermal radiation are part of this investigation.Convective boundary conditions are applied for motivation of problem.Where  and  are component of flow of fluid.MHD effect having strength 0 B is considered perpendicular to the sheet.The governing equation including influence of thermal radiation, browning motion, chemical reaction and thermophoresis are described as [25][26][27][28][29][30][31][32][33]: The associated boundary conditions are given

SIMILARITY TRANSFORMATION
The similarity variables are illustrated as: As a result, Eq. ( 1) is satisfied identically and equations (2-4) are transformed as: The boundary conditions are converted as: Where PHYSICAL QUANTITIES OF INTEREST The physical quantities are defined as: where: The physical quantities in non-dimensional form: Where  = is local Reynolds number.
NUMERICAL SOLUTION Above equations (7-9) are coupled and highly nonlinear therefore exact solution is not applicable.

RESULT AND DISCUSSION
The revelation of the section is to study the numerical outcomes exemplified in the form of graphs.The influence of emerging parameters like power-law index , Magnetic parameter , Prandtl number , thermophoresis parameter , thermal radiation parameter , Weissenberg number  and Brownian motion parameter  on velocity, temperature and concentration field are discussed.The influence of the power-law index for non-dimensional velocity function is depicted in Figure 1. it is noted that velocity field is decreased as an increment in power-law index .Because increased magnitude of  decreases fluid flow.The impression of Hartmann number  on velocity profile ′ and temperature profile  are presented in Figures (2)(3).-9) are communicated to imagine the influence of  on the energy and concentration distribution.In figures 8, it is observed that temperature function  establish a growth for rising values of thermophoresis parameter.Figure 9 is drawn to explore the influence of the thermophoresis parameter  on concentration field .It is cleared that for increasing value of thermophoresis parameter  the concentration distribution  declines.Basically, heated particles move away from high temperatures, increasing fluid temperature.Figure 10 characterizes that by improving the values of (seen with the variable thermal conductivity) small parameter  energy distribution  also augmented.Figure 11 represents that by increasing the value of thermal radiation parameter  fluid temperature also increased.Physically, increase in thermal Radiation parameter increases energy flow to fluids). Figure 12 is pinched to investigate the stimulus of the Schmidt  on concentration field .It is evaporated that for increasing value of Schmidt  the concentration field  declines.

CONCLUSION
In this investigation, tangent hyperbolic flow of nanofluid due to stretched sheet with joule heating, and heat source is examined.Chemical diffusion and thermal radiation are part of this study.First of all, velocity, temperature and volumetric concentration equations are transformed into the set of ODEs by expending similarity variables.The resulting system of ODEs is numerically handled by using well-established shooting scheme and acquired numerical outcomes.These outcomes are useful in the field of engineering and technology due to heat transfer.The key findings are listed below.
• By increasing the values of power-law index  and Hartmann number  velocity profile declines.
• By increasing values of thermal radiation , thermophoresis parameter  and Brownian motion parameter  temperature field  also increases.

Figure 1 .Figure 2 .Figure 3 .
Figure 1.Distribution of ′ for power-law index  Figure 2. Distribution of ′ for Hartmann number  Figure 3. Distribution of  for Hartmann number

Figure 2
Figure 2 is plotted to imagine that the flow profile is diminished for improving values of Hartmann number .In Figure 3, temperature field illustrates growing behavior for the increase of Hartmann number .Physically, Magnetic field increases Lorentz forces, reducing fluid motion.The impact of Prandtl number  on temperature distribution  and volumetric concentration distribution are exhibited in Figures (4-5).

Figure 4 .Figure 5 .Figure 6 .
Figure 4. Distribution of  for Prandtl number  Figure 5. Distribution of  for Prandtl number  Figure 6.Distribution of  for Brownian motion parameter

Figure 4
Figure 4 is drawn to explore the influence of the Prandtl number  on energy field .It is cleared that for increasing value of Prandtl number  the energy distribution  declines.Because of heat transfer rate decreases for increased  values.Figure 5, concentration field illustrates growing behavior for the increase of Prandtl number .The impact of Brownian motion parameter  on temperature field  and concentration field are exhibited in Figures (6-7).

Figure 7 .Figure 8 .
Figure 7. Distribution of  for Brownian motion parameter  Figure 8. Distribution of  for thermophoresis parameter

Figure 9 .
Figure 9. Distribution of  for thermophoresis parameter

Figure 6
Figure 6 is depicted to explore the manipulate of the Brownian motion parameter  on energy field .It is noted that for increasing value of Brownian motion parameter  the energy distribution  increases.Because of heat transfer rate decreases for increased  values.Figure 7, concentration field illustrates growing behavior for the increase of Brownian motion parameter .Figures (8-9) are communicated to imagine the influence of  on the energy and concentration distribution.In figures 8, it is observed that temperature function  establish a growth for rising values of thermophoresis parameter.Figure 9 is drawn to explore the influence of the thermophoresis parameter  on concentration field .It is cleared that for increasing value of thermophoresis parameter  the concentration distribution

Figure 7 ,
Figure 6 is depicted to explore the manipulate of the Brownian motion parameter  on energy field .It is noted that for increasing value of Brownian motion parameter  the energy distribution  increases.Because of heat transfer rate decreases for increased  values.Figure 7, concentration field illustrates growing behavior for the increase of Brownian motion parameter .Figures (8-9) are communicated to imagine the influence of  on the energy and concentration distribution.In figures 8, it is observed that temperature function  establish a growth for rising values of thermophoresis parameter.Figure 9 is drawn to explore the influence of the thermophoresis parameter  on concentration field .It is cleared that for increasing value of thermophoresis parameter  the concentration distribution

Figure 10 .Figure 11 .Figure 12 .
Figure 10.Distribution of  for small parameter  Figure 11.Distribution of  for radiation parameter  Figure 12.Distribution of  for Schmidt number

•
Strong abatement in temperature distribution is noted for raising values of .• Concentration and temperature and increase by increasing thermophoresis parameter  • Concentration decreases by increasing Schmidt number .