Magnetic dipole effects on unsteady flow of Casson-Williamson nanofluid propelled by stretching slippery curved melting sheet with buoyancy force

In particular, the Cattaneo-Christov heat flux model and buoyancy effect have been taken into account in the numerical simulation of time-based unsteady flow of Casson-Williamson nanofluid carried over a magnetic dipole enabled curved stretching sheet with thermal radiation, Joule heating, an exponential heat source, homo-heterogenic reactions, slip, and melting heat peripheral conditions. The specified flow's partial differential equations are converted to straightforward ordinary differential equations using similarity transformations. The Runge–Kutta–Fehlberg 4-5th order tool has been used to generate solution graphs for the problem under consideration. Other parameters are simultaneously set to their default settings while displaying the solution graphs for all flow defining profiles with the specific parameters. Each produced graph has been the subject of an extensive debate. Here, the analysis shows that the thermal buoyancy component boosts the velocity regime. The investigation also revealed that the melting parameter and radiation parameter had counterintuitive effects on the thermal profile. The velocity distribution of nanofluid flow is also slowed down by the ferrohydrodynamic interaction parameter. The surface drag has decreased as the unsteadiness parameter has increased, while the rate of heat transfer has increased. To further demonstrate the flow and heat distribution, graphical representations of streamlines and isotherms have been offered.

the velocities are known to improve when the parameter related to rotating fluid is enlarged.Further several authors 37,38 carried out the study for the nanofluid and hybrid nanofluid flow in presence of Cattaneo-Christov heat flux.Ahmad et al 39 discussed double diffusion for the flow Eyring-Powell -liquid.Wang et al 40 deliberated the Cattaneo-Christov heat-mass transfer for a third-grade fluid flow over a stretched surface and homotopy scheme was implemented to obtain the convergent solutions.
An examination of the above recent literature reveals that the present investigation which models the unsteady flow of a Casson-Williamson nanofluid, is innovative and an advancement in the field since combination of flow of two different non-Newtonian fluid models has many applications in the manufacturing sector.The modelling has been done in curvilinear coordinates, under the circumstances of the buoyancy effect, the Cattaneo-Christov heat flux model, thermal radiation, homo-heterogenic processes, Joule heating, exponential heat production, and the magnetic dipole moment.The slip and melting heat conditions are considered at the boundary of the stretching surface.When it comes to curved shape, considerable effects, and optimal peripheral circumstances, the innovative combination of Casson nanofluid and Williamson nanofluid in the current work is relatively worth analysing.The results of the present study are significant and welcome the further research in the field due to the showcase of lucid behavioural changes of all flow profiles for imperative parameters of engineering interest.

Mathematical articulation
Consider an unsteady flow of Casson-Williamson nanofluid along a curved stretching sheet.The sheet is vulnerable to stretching around a semicircle of radius R by two equal and opposite pressures applied along the s-orientation while maintaining the origin stationary and the r-orientation normal to it as portrayed in Fig. 1.Allowing for u w (s) = as 1−α * t as the sheet's stretching speed, where t is time, a is the stretching rate, and α * is a constant with dimension reciprocal of time.When analyzing the fluid flow behavior, the impact of the magnetic dipole moment is taken into account.In the r direction, the magnetic field B m = B 0 √ (1−α * t) is applied.To examine the flow properties of the aforementioned nanofluid, the influence of buoyancy is taken into consideration.To study the thermal behavior of the flow, the effects of thermal radiation, Joule heating, and an exponential space dependent heat source are taken into account in the energy equation.In order to fully understand how it affects the flow, the Cattaneo-Christov heat flux model has been used.With the help of the slip condition, the melting surface's boundary is enhanced.
To explain the mass transfer operation, two chemical samples A and B at corresponding concentrations C a and C b are studied for their homo-heterogenic reactions.A + 2B → 3B with a rate of k c C a C 2 b is the homogenic reaction on the carrier surface, whereas A → B with a rate of k s C a is the heterogenic reaction.
These ideas lead to the formulation of the flow anchoring equations as 29 .
(1) www.nature.com/scientificreports/associated auxiliary conditions are 33 Here (u, v)-velocities along (s, r)-orientations, p-pressure, ρ-density, ν-kinematic viscosity, β-Casson param- eter, Ŵ-material time constant, g-acceleration, β T -coefficient of thermal expansion, µ 0 -magnetic permeability, M -magnetization, H-magnetic field, σ-electrical conductivity, B 0 -constant magnetic field, T-temperature, α-thermal diffusivity, c p -specific heat, τ-ratio of the effective heat capacity, q r -radiative heat flux, Q 0 -space dependent heat source, 1 -relaxation time of heat, D C a -diffusion coefficient of A , D C b -diffusion coefficient of B , T w -temperature of the fluid at the surface, T ∞ -ambient fluid temperature, D T -thermophoretic diffusion coefficient, (k c , k s )-rate constants, Ls-velocity slip coefficient, l -latent heat of the fluid, c s -heat capacity of the solid surface, T m -melting temperature, T 0 -temperature of the solid surface and k-thermal conductivity.
Given by the Rosseland estimation, the radiative heat flow is, where the Stefan-Boltzmann constant and the coefficient of mean absorption, respectively, are written as σ * and k * .
Magnetic dipole.The apparent magnetic dipole and its scalar strength cause the magnetic field to have the following effects on the liquid stream: where d is the distance between the dipoles and γ is the intensity of the magnetic field at the source.The char- acteristics of associated magnetic field H 41 are as follows (3) Vol.:(0123456789) www.nature.com/scientificreports/A direct variation of magnetic force is quantity of H , which is established by the following relation.
Temperature T may be estimated linearly from magnetization M as shown below.
Where K 1 is the ferromagnetic coefficient.
The following morphing catalysts are explored in order to understand the simplified form of flow steering equations, where the non-dimensional velocity, pressure, temperature, homogenic concentration, and heterogenic concentration regimes are listed in that order: f ′ (η), P(η), θ(η) , φ(η) and h(η) .In terms of the cohesive variable η , prime resembles differentiation; κ is the curvature parameter and C 0 is the constant.
Expression (1) is identically verified and equations ( 2) to ( 6) become The following are the transfused boundary conditions: where l +(T w −T 0) c s -melting heat parameter.When pressure P(η) is removed from Eqs. ( 14) and (15), it results in.
The following skin friction coefficient and the Nusselt number, are characteristics of engineering prominence.here τ w -wall shear stress and q w -wall heat flux which are given by.

Reduced form of above is
where Re = as 2 ν denotes local Reynolds number.

Numerical procedure
It is possible to ensure the accuracy of the IVP solution by repeating the simplified equations twice with step lengths of h and h/2.
To establish good synergy, this process must first undergo extensive simulation due to the shorter step length.One of these methods, the Runge-Kutta Fehlberg scheme, contains a protocol to determine whether the appropriate step length is being used.Every step yields two accurate approximations of the solution, which are then (20) + q r r=0 .
(27) Cf s (Re) κ between present study and previous study 42 .
κ Zhang et al. 41    www.nature.com/scientificreports/discussed.If the two answers closely synergize, the accuracy of the approximation is impaired; otherwise, the step size is reduced.If the answer settles on more digits, the step length is adjusted.Each step results in values as below:  www.nature.com/scientificreports/Then an approximation using 4 th order RK-method is.www.nature.com/scientificreports/It is noteworthy that k 2 value is not counted in the above given formula.The other value of y is known by 5 th order RK-method as: If y i+1 + y * i+1 is small enough, then the method is terminated; or else the simulation is carried on using lesser step size h .The local truncation error is y i+1 − y * (i+1) .
1, Me = 0.7 while carrying out numerical extractions for all flow fields against the relevant parameters.Table 1 compares the bvp4c methodology developed by Zhang et al. 42 with the present  www.nature.com/scientificreports/numerical method to provide validation.There appears to be a reasonable amount of consistency among the data in the table.Significantly more detail has been provided on each resulting graph.Plots of the behavioral changes in the velocity panel (f ′ (η)) for the magnetic parameter (M * ) , ferrohydrody- namic interaction parameter (β m ) , unsteady factor (δ * ) , thermal buoyancy factor ( T ) , velocity slip factor (L 1 ) and dipole distance (b) are shown in Figs. 2, 3, 4, 5, 6 and 7, respectively.The variations in velocity ( f ′ (η) ) for ascending magnetic attribute ( M * ) values are clearly shown in Fig. 2, which is declining in nature.The Lorentz force, which is present and amplified by the increase in M * , is the cause of this velocity impedance.When increas- ing values of β m are present, it is clear from Fig. 3 that the velocity panel de-escalates.The velocity distribution is decreased as a result of the dominance of the ferrohydrodynamic interaction factor and increased Lorentz force.The fluid's behavior increases as it moves farther from the sheet, as seen by the indicated behavior of the velocity distribution in Fig. 4 for expanding values of δ * .The explanation for this is because a rise in δ * causes an  increase in the reciprocity of time factor, which slows the pace at which the sheet stretches.The velocity regime for the parameter T is shown in Fig. 5.The relative variable impact of the thermal floating force on the flow of nanofluid is personified by it.As seen in Fig. 5, the increase in thermal buoyancy force causes the flow to move more quickly.Figure 6 clearly illustrates the influence of the velocity slip factor's de-escalation on the velocity distribution.In stretched sheets and fluid flows, an increase in L 1 creates heterogenic velocity, which results in a decrease in the velocity distribution.The effect of a magnetic dipole's dimensionless distance on the velocity regime is briefly explored in Fig. 7.The velocity regime is shown to steadily grow as b increases, despite the dipole distance increasing.
The fluctuations of the thermal regime are shown in Figs. 8, 9, 10, 11, 12, 13, 14 and 15 for the components of exponential heat generation (Q) , heat dissipation ( m ) , melting heat (Me) , unsteadiness (δ * ) , thermal buoyancy ( T ) , radiation (Rd) , Curie temperature (ǫ) and thermal relaxation (C H ) in sequential order.Figure 8 shows the  changes of the thermal panel for changing Q , and it thrives as a result of the heat creation within the nanofluid flow.Temperature panel changes in response to rising heat dissipation factor values.The temperature panel reduces as m grows due to an increase in the transfer of heat away from the sheet, as seen in Fig. 9.Because of the combined effects of radiation and the melting heat phenomena, which is depicted in Fig. 10, the thermal regime decreases with increasing values of Me .Figure 11 depicts the growing behaviour of the thermal distribu- tion for expanding values of δ * as a result of the cyclical behaviour of the stretching sheet.For rising levels of T , Fig. 12 shows the thermal regime's decrementing characteristic.Because there are more thermal floating forces present when the temperature is raised, the hotness of the fluid is replaced by coolness, which causes the thermal panel to sink. Figure 13 shows how the radiation parameter (Rd) affects the temperature panel.Because of the decrease in mean absorption coefficient, the thermal profile improves at higher values of Rd .Figure 14 shows the thermal regime curves with rising Curie temperature values.The thermal regime thrives as the value of it is  raised because it raises the ambient temperature close to the sheet, while the thermal regime deteriorates for the fluid farther from the sheet.Figure 15 depicts how the temperature regime changes as C H values increase.The reason for this is thought to be due to the thermal boundary layer eroding.
To identify the distinctive differences in the solutal regime for the components of the unsteady factor (δ * ) , homogenic reaction strength (k 1 ) , heterogenic reaction strength (k 2 ) and velocity slip factor (L 1 ) , Figs. 16,17,18 and 19 are successively detailed.The mass movement in the flow is impeded by the flow's enhanced unsteadiness.As a result, mass distribution slows down as δ * increases in magnitude, as seen in Fig. 16. Figure 17 shows the concentration panel's curves for the upshot values of k 1 .The concentration profile is discouraged by a rise in k 1 .The link between the concentration panel and k 2 is further explained in Fig. 18.Due to an increase in mass diffu- sion, a rising influence of k 2 deescalates the mass transfer profile.The influence of the velocity slip parameter on the mass transfer regime is shown as a result in Fig. 19.Here, when L 1 increases, the mass transfer panel decreases.The 3D skin friction coefficient is represented against the unsteadiness parameter (δ * ) in Fig. 20 for changing ferrohydrodynamic interaction parameter (β m ) .Skin friction co-efficient increased due to the unsteadiness parameter's amplification.Every time the ferrohydrodynamic interaction parameter increases, the surface drag decreases.Figure 21 explains how the unsteadiness parameter behaves in relation to the thermal buoyancy parameter ( T ) .Surface drag is shown to decrease as the unsteadiness parameter (δ * ) approaches its maximum value.The skin friction coefficient increases with the thermal buoyancy parameter.Figure 22 illustrates how the radiation parameter (Rd) responds to the heat dissipation factor ( m ) on Nusselt number Nu S Re − 1 2 .The heat transfer rate marginally decreases when the sheet is exposed to intense radiation.The rate is also slightly increased by magnifying ( m ) .The higher the heat dissipation factor number, the greater the rate of heat transmission.Figure 23 illustrates the characteristics of the unsteadiness parameter (δ * ) for different heat dissipation factors ( m ) .Increases in the unsteadiness parameter (δ * ) result in the lowest heat transfer rate.
In a 2D contour plot (streamlines) Fig. 24, the trajectory followed by the Casson-Williamson fluid particles is described for magnetic parameter at M * = 0.1 and M * = 3 .In Fig. 25, the streamlines for the unsteadiness parameters δ * = −0.5 and δ * = 0.2 are also explained.Figure 26 shows a plot with contours (isotherms) show- ing similar temperatures at places over a stretched area for Ec = 0.01 and Ec = 0.9.