Melting heat transfer of hyperbolic tangent fluid over a stretching sheet with fluid particle suspension and thermal radiation

The current investigation deals with melting heat transport in hyperbolic tangent fluid past a stretching sheet with fluid particle suspension and thermal radiation. Numerical solution is obtained via shooting technique after the applying of similarity variables to the governing partial equations. The effects of key parameters like magnetic parameter, fluid particle interaction parameter, Eckert number, Prandtl number, Weissenberg number and melting parameter are analyzed and discussed in detail with help of graphs and tables. Finally, the numerical values of the friction co-efficient and local Nusselt number are tabulated. Checked our results with existing work found excellent agreement. Results reveals that in the presence of melting parameter, the higher values of We and n depreciates the rate of heat transfer.


Introduction
The study of boundary layer flow over a stretching surface plays a pivotal role in industrial and engineering processes, such as aerodynamics and polymers, artificial fibers, crystal growing, cable coating etc. Sakiadis [1] initiated the boundary layer flow over a moving surface.Tsou et al [2] experimental studied the flow on a moving surface.The boundary layer flow induced by stretched surface has been carried out by Magyari et al [3].Ishak et al [4] have initiated the mixed convection flow of a viscous fluid over a stretching vertical sheet.Meanwhile, the process of heat transfer has been examined in several studies mentioned in [5]- [11] which involve various flow of fluid over stretching sheet.Study of heat transfer phenomina during the melting process plays on important role in industrial manufacturing.Roberts [12] studied the melting of a body of ice which presents a plane surface transverse to a stream of hot air.Tien and Yen [13] analyzed convective heat transfer of a melting body.This study also spreads out the recent work of [14]- [16] to examine the marked effects of melting heat transfer on the flow system.
On the other hand, non-Newtonian fluids are found in many industrial and engineering processes, such as food mixing, flow of blood, plasma, mercury amalgams and lubrications with heavy oils and greases.One of the imperative branches of non-Newtonian fluid is the hyperbolic tangent fluid, which is capable of describing the shear thinning effects.Safia and Sohail [17] obtained perturbation and numerical solutions to study peristaltic flow of a hyperbolic tangent fluid model in a vertical channel.Nadeem and Maraj [18] considered curved channel to study peristaltic flow of hyperbolic tangent fluid.Malik et al [19] studied the flow of tangent hyperbolic fluid over a stretching cylinder.In recent years many researchers are concentrated on the area of dusty fluid due to its numerous applications such as fluidization, dust collection, petroleum industry, powder technology, nuclear reactor cooling and performance of solid fuel rocket nozzles.Saffman [20] analyzed the flow of a dusty gas in which the fluid suspension particles are uniformly distributed.The effect of pressure gradient on heat transfer in a boundary layer flow of dusty fluid has been carried out by Agranat [21].Recent studies related to the topic are mentioned in [22]- [25].
The current study is to analyze the two-dimensional boundary layer flow and heat transfer of non-Newtonian hyperbolic tangent dusty fluid over a stretching surface.In addition, the effects of melting heat transfer and transverse magnetic field are included.The governing equations are solved in section 2-4 by making use of RKF-45 method and the graphical presentation of results are shown in section 5 while the final conclusion was done in last section.

Mathematical Formulation
Here we extended laminar flow of tangent hyperbolic fluid by stretching surface with suspended particles.The x axis is along the axis of the sheet and y axis normal to it.Assumed that sheet velocity u w (x), melting surface temperature (T m ), magnetic filed B 0 is exposed on y direction.The size and shape of the dust particles is uniform and spherical.With all these assumption the governing equations are [17], [22], here (u, u p ) and (v, v p ) are denotes components velocity of the fluid and dust phase respectively, ρ and ρ p = Nm * for density of the fluid and dust phase respectively, m * and N are the mass of the particles and number density of the dust particles,ν for kinematic viscosity, σ for electrical conductivity, µ for coefficient of viscosity, K = 6π µr for Stokes drag coefficient, r for radius of dust particle, B 0 for magnetic field and n for power law index.
The appropriate boundary conditions applicable to the present problem are: here u w (x) = bx is a stretching sheet velocity with b(> 0) is the stretching rate.
Following similarity transformations are used to convert the coupled PDE's to coupled ODE's, continuity equations are (2.1) and (2.3) are satisfied and momentum equations (2.2) and (2.4) are reduced to; ) Appropriate boundary conditions becomes; where l = Nm * ρ the mass concentration parameter of dust particles, is the Weissenberg number and n is the power law index parameter.

Heat Transfer Analysis
The energy equations for both fluid phase and dust phase are given by; here T and T p are fluid and dust particle temperature respectively, c p and c m are the specific heat of fluid and dust particles respectively, τ t for thermal equilibrium time of fluid, k for thermal conductivity and q r for radiative heat flux.For optically thick layer, approximation of Rosseland expression has the following form; where σ * is the Stefan-Boltzmann constant and k * is the mean absorption coefficient.Here T 4 is a linear function of T , using Taylor series expansion it can be written as, neglect the higher order terms beyond the first degree in (T − T ∞ ), one can get; (3.13) using (3.13) in equation (3.12) we have, Making use of (3.14), the transport equation (3.10) as, Applicable boundary conditions for the temperature as; where E is the latent heat of the fluid and C s is the heat capacity of the solid surface, The dimensional fluid phase temperature θ (η) and dust phase temperature θ p (η) are are expressed as: Applying (3.17) into (3.11) and (3.15), the equations reduces to: with, where Pr = bτ T for fluid-particle interaction parameter for temperature, and λ .The physical quantities of interest are the skin friction coefficient C f and the local Nusselt number Nu x , which are defined as; where τ w and q w are the surface shear stress and the surface heat flux, which are given by Using the non-dimensional variables, one can get, where Re x = u 2 w bν is the local Reynolds number.

Mathematical Formulation
The system of ordinary differential equations (2.7)-(2.8)and (3.18)-(3.19)representing the defined flow problem with boundary conditions (2.9) and (3.20) are highly nonlinear.Thus, the equations are solved numerically by Runge-Kutta-Fehlberg fourth-fifth order method along with shooting technique.The equations are first reduced into system of seven first order simultaneous equations having seven unknowns as follows: with boundary conditions as In order to integrate the above equations as an initial value problem, one requires seven initial conditions.Out of required seven initial conditions, four are known and no such values are given at the boundary for p(0), s(0), q(0) and r(0).Remaining initial conditions are accessed with the help of Runge-Kutta-Fehlberg 45 scheme.The RKF-45 algorithm is given by; ) , ) ,    In this part, the impact of thermo physical parameter on velocity and temperature profile for both fluid and particle phases are discussed through figures 1-16. Figure 1-4 depicts the influence of power law index (n) and Weissenberg number (W e ) on velocity and temperature profile.From this figures we observed that, the momentum and thermal boundary layer scale back with an increase in n, where as opposite trend is observed in case of W e .Figure 5-6 shows that, on melting surface, both velocity and temperature profiles decreases with increasing in M.This is because of a resistive force that acts in the direction opposite to that of the flow caused by applied magnetic field.Figures 7 and 8 explains the effect of β 1 and β 2 on velocity and temperature profiles, respectively.Increase of β 1 will decrease the fluid phase velocity and increase the dust phase velocity, while as expected, increase of β 2 will increase the fluid phase temperature and decrease the dust phase temperature.This is because an increase in β 1 is a result of decrease of τ v and it is obvious that the time required by a dust particle to adjust its velocity relative to the fluid also decreases with decrease of τ v .Figure 9-10 illustrates the influence of melting parameter (m) on velocity and temperature distributions.An increase in the melting parameter enhances the momentum and thermal profiles of both fluid and dust phases.The variation of velocity and temperature profiles for both the phases are illustrated for different values of mass concentration parameter (l) and are shown in figure 11 and 12 respectively.Here the velocity profile for both the phase decreases by increasing the values of mass concentration parameter and the opposite effect with temperature profiles, as shown in Figure 12. Figure 13 and 14 shows variation of temperature profiles for various values of radiation parameter (R) and specific heat ratio respectively.It's clear that the increasing values of radiation parameter will decrease the temperature profile and corresponding boundary layer thickness and is shown in figure 13.From the plot 14, is observed that an uplifting value of γ decreases the temperature profile of fluid and dust particles, which result in degreases of boundary layer thickness.
The analysis of Figure 15 reveals that the effect of increasing values of Eckert number (Ec) is to increase temperature distribution of both fluid and dust phase in the flow region.Figure 16 describes the impact of Prandtl number (Pr) over the temperature profile.One can infer from this figure in the presence of melting parameter, an increase in Pr increases the temperature of both fluid and dust phases.Table 3 is prepared to show the impact of melting parameter (m) on Nusselt number for both present and absence case with different physical parameters.From table 3 it is noted that, the rate of heat transfer is higher in the presence of melting parameter (m = 0.5) than in absence.The numerical values of skin friction coefficient and Nusselt number for different physical parameters are recorded in table 2 and 4.These tables shows that, the increasing values of Ec, Pr, β 2 and l is to enhance the Nusselt number at the surface, but a reduction occurred in both skin friction coefficient and Nusselt number when the values of m, β 1 and n enhance.

Conclusion
An analysis has been developed to investigate the effect of melting on flow and heat transfer of an electrically conducting tangent hyperbolic dusty fluid over a stretching sheet in the presence of a magnetic field and thermal radiation.The impacts of thermo physical parameter on velocity and temperature profile are studied graphically and are summarized as follows: • On melting surface, the higher values of W e and n depreciates the rate of heat transfer.
• Momentum boundary layer thickness reduces due to the influence of Lorenz force.
• Velocity and temperature profiles decreases for fluid phase and increases for dust phase with increase in β 1 and β 2 .
• For uplifting values of melting parameter, the momentum boundary layer thickness increases and thermal boundary layer thickness decreases.
21) and(4.22)are fourth and fifth order Runge-Kutta respectively.Throughout our computation, the step size is taken as ∆η = 0.001 with the convergence criteria 10 −6 and the CPU time is 1.855 seconds.To assess the accuracy of aforementioned numerical method, comparison of skin friction coefficient and local Nusselt number values between the present results and existing results for various values presented in the

Figure 1 :Figure 2 :
Figure 1: Velocity profile for different values of n.

Figure 3 :Figure 4 :Figure 5 :Figure 6 :
Figure 3: Velocity profile for different values of W e .

Figure 14 :
Figure 14: temperature profile for different values of γ.

Figure 15 :Figure 16 :
Figure 15: temperature profile for different values of Ec.

Table 2 :
Numerical values of skin friction coefficient and local Nusselt number for different values M,W e , l, n, and β 1 .

Table 3 :
Numerical values of Nusselt number with the different parameters in the presence and absent of melting parameter.

Table 4 :
Numerical values of local Nusselt number for different values Ec, Pr, γ, R and β 2 .

•
By increasing the values of Ec and Pr, the thermal boundary layer thickness get increases due to influence of melting effect.Stefan number for the liquid phase Nu x local Nusselt number c s (T m −T 0 ) λ temperature dynamic viscosity (kgm −1 s −1 ) γ specific heat ratio ν kinematic viscosity (m 2 s −1 ) ρ base fluid density (kg/m 3 ) σ electrical conductivity of the fluid ρ p dust particles density (kg/m 3 ) σ c f (T ∞ −T m ) λ * Stefan-Boltzmann constant (W m −2 K −4 )