Flow and heat transfer of dusty hyperbolic tangent fluid over a stretching sheet in the presence of thermal radiation and magnetic field
 K. Ganesh Kumar^{1}Email author,
 B. J. Gireesha^{1} and
 R. S. R. Gorla^{2}
https://doi.org/10.1186/s4071201800888
© The Author(s). 2018
Received: 25 August 2017
Accepted: 2 January 2018
Published: 22 January 2018
Abstract
Background
This paper explores the impact of thermal radiation on boundary layer flow of dusty hyperbolic tangent fluid over a stretching sheet in the presence of magnetic field. The flow is generated by the action of two equal and opposite. A uniform magnetic field is imposed along the yaxis and the sheet being stretched with the velocity along the xaxis. The number density is assumed to be constant and volume fraction of dust particles is neglected. The fluid and dust particles motions are coupled only through drag and heat transfer between them.
Methods
The method of solution involves similarity transformation which reduces the partial differential equations into a nonlinear ordinary differential equation. These nonlinear ordinary differential equations have been solved by applying RungeKuttaFehlberg forthfifth order method (RKF45 Method) with help of shooting technique.
Results
The velocity and temperature profile for each fluid and dust phase are aforethought to research the influence of assorted flow dominant parameters. The numerical values for skin friction coefficient and Nusselt number are maintained in Tables 3 and 4. The numerical results of a present investigation are compared with previous published results and located to be sensible agreement as shown in Tables 1 and 2.
Conclusion
It is scrutinized that, the temperature profile and corresponding boundary layer thickness was depressed by uplifting the Prandtl number. Further, an increase in the thermal boundary layer thickness and decrease in momentum boundary layer thickness was observed for the increasing values of the magnetic parameter.
Keywords
Review
The main objective of the present inspection is to study the MHD flow and radiative heat transfer of hyperbolic tangent fluid over a stretching sheet with fluidparticle suspension. The method of solution involves similarity transformation which reduces the partial differential equations into a nonlinear ordinary differential equation. These nonlinear ordinary differential equations have been solved by applying RungeKuttaFehlberg forthfifth order method (RKF45 Method) with help of Shooting technique. The temperature and velocity profiles for various values of flow parameters are presented in the figures. The numerical results of present investigation were compared with existing results and are found to be in good agreement.
Introduction
Boundary layer flow and heat transfer under the influence of stretching surface has received considerable attention in recent years. The problem has scientific and chemical engineering applications such as aerodynamic extrusion of plastic sheets and fibers, tinning of copper wire, drawing, crystal growing and glass blowing, annealing and paper production, metallurgical process, polymer extrusion process, continuous stretching, drawing, annealing and tinning of copper wires. Sakiadis (1961) was the pioneer of studying the boundary layer flow over a stretched surface moving under a constant velocity. Rajagopal and Gupta (1987) examined the boundary layer flow over a stretching sheet for a various class of nonNewtonian fluids. Rauta and Mishra (2014) investigated the heat transfer characteristics of twodimensional flow in a porous medium over a stretching sheet with internal heat generation. Thereafter, exhaustive amount of researches were made related to boundary layer flow and heat transfer (see Ishak (2010), Ali (2006) and Yao et al. (2011)).
In recent years, many researchers are concentrated on the area of dusty fluid due to its wide range of applications such as fluidization, centrifugal separation of matter from fluid, purification of crude oil, dust collection, petroleum industry, powder technology, nuclear reactor cooling, performance of solid fuel rocket nozzles, and paint spraying. Saffman (1962) analyzed the flow of a dusty gas in which the fluid suspension particles are uniformly distributed. Further, Mohan Krishna et al. (2013) have discussed the flow of a dusty viscous fluid in the presence of transverse magnetic field with first order chemical reaction. Datta and Mishra (1982) presented the boundary layer flow of a dusty fluid over a semiinfinite flat plate along with the drag force. Due to these applications, a number of theoretical and experimental studies have been carried out by numerous researchers on dusty fluid (see (Palani and Ganesan 2007; Ramesh et al. 2017; Kumar et al. 2017a, 2017d, 2017f; Makinde et al. 2017)).
On the other hand, nonNewtonian 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. In view of these applications, many studies are focused on nonNewtonian fluid. Govardhan et al. (2015) initiated the magnetohydrodynamics flow of an incompressible micropolar fluid over a stretching sheet for unsteady case. Hayat et al. (2010) have initiated the effect of thermal radiation on a twodimensional stagnationpoint flow and heat transfer of an incompressible magnetohydrodynamic Carreau fluid towards a shrinking surface with convective boundary condition. In modern days, all such interesting applications have motivated scientists and researchers to look for more avenues in the field of nonNewtonian flow (Rudraswamy et al. 2017a, 2017b; Kumar et al. 2017c, 2017e).
One of the imperative branches of nonNewtonian fluid is the tangent hyperbolic fluid capable of describing the shear thinning effects. Recently, Akbar et al. (2013) have discussed the two dimensional tangent hyperbolic fluid flow over a stretching in the presence of magnetic field. Malik et al. (2015) have used Keller box method to study MHD flow of tangent hyperbolic fluid under the influence of stretching cylinder. Nadeem and Akram (2011) analyzed the magnetohydrodynamics (MHD) peristaltic flow of a tangent hyperbolic fluid model in a vertical asymmetric channel (Fig. 1). Recently Naseer et al. (2014) have analyzed the steady boundary layer flow and heat transfer of a tangent hyperbolic fluid flow under the influence of vertical exponentially stretching cylinder. Kumar et al. (2017b) discussed an unsteady squeezed flow of a tangent hyperbolic fluid over a sensor surface in the presence of variable thermal conductivity. Nadeem and Maraj (2013) initiated the mathematical analysis for peristaltic flow of tangent hyperbolic fluid in a curved channel.
The main objective of the present inspection is to study the MHD flow and radiative heat transfer of hyperbolic tangent fluid over a stretching sheet with fluidparticle suspension. The method of solution involves similarity transformation which reduces the partial differential equations into a nonlinear ordinary differential equation. These nonlinear ordinary differential equations have been solved by applying Runge–Kutta–Fehlberg fourthfifth order method (RKF45 Method) with the help of shooting technique. The temperature and velocity profiles for various values of flow parameters are presented in the figures. The numerical results of present investigation were compared with existing results and are found to be in good agreement.
Method
Comparison table of skin friction coefficient (We = n = 0)
Comparison of −θ^{′}(0) for different values of Prandtl number (Pr) when M = 0, We = 0, n = 0, Ec = 0, and R = 0
Mathematical formulation
Let us consider a steady flow of an incompressible hyperbolic tangent fluid over a stretching sheet. The flow is assumed to be confined to region of y > 0. The flow is generated by the action of two equal and opposite forces along the xaxis and yaxis being normal to the flow. A uniform magnetic field B_{0} is imposed along the yaxis and the sheet being stretched with the velocity u_{ w }(x) along the xaxis. The number density is assumed to be constant and volume fraction of dust particles is neglected. The fluid and dust particle motions are coupled only through drag and heat transfer between them. The drag force is modeled using Stokes linear drag theory. Other interaction forces such as the virtual force, the shear lift force, and the spinlift force will be neglected compared to the drag force. The term T_{ w } represents the temperature of fluid at the sheet, whereas T_{∞} denotes the ambient fluid temperature.
Heat transfer analysis
Result and discussion
Numerical values of \( \sqrt{\operatorname{Re}}{C}_f \) and \( \left[\frac{\mathrm{N}{\mathrm{u}}_x}{\mathrm{R}{\mathrm{e}}^{1/2}}\right] \) for different values of W_{e}, n, M, β_{ v }, and l
M  W_{e}  β _{ v }  l  n  \( \sqrt{\operatorname{Re}}{C}_f \)  \( \left[\frac{\mathrm{N}{\mathrm{u}}_x}{\mathrm{R}{\mathrm{e}}^{1/2}}\right] \) 

0  1.0547  1.0506  
0.5  1.2641  1.0115  
1  1.4446  0.9788  
0  1.2247  1.0137  
0.3  1.2641  1.0115  
0.6  1.3057  1.0093  
0.5  1.2641  1.0115  
1  1.2958  0.9956  
1.5  1.3144  0.9834  
0.5  1.2641  1.0115  
1  1.3268  1.1973  
1.5  1.3868  1.3598  
0  1.2440  1.1973  
0.1  1.2909  1.0257  
0.2  1.3268  0.9942 
Numerical values of \( \left[\frac{\mathrm{N}{\mathrm{u}}_x}{\mathrm{R}{\mathrm{e}}^{1/2}}\right] \) for different values of Ec, γ, Pr , R, and β_{ t }
Ec  γ  Pr  R  β _{ t }  \( \left[\frac{\mathrm{N}{\mathrm{u}}_x}{\mathrm{R}{\mathrm{e}}^{1/2}}\right] \) 

0  1.0175  
0.1  0.9581  
0.2  0.8987  
1  0.9633  
2  0.9049  
3  0.8718  
2  0.7834  
3  1.0115  
4  1.2055  
0  1.3770  
0.5  1.0115  
1  0.8188  
0.5  0.9833  
1  1.1007  
1.5  1.1786 
Table 3 presents the numerical values of Nusselt number and skin friction coefficient for various values M, W_{e}, β_{ v }. It is observed that skin friction increase and Nusselt number decreases with increasing M, W_{e}, β_{ v }, l, and n. Table 4 presents the numerical values of local Nusselt number for various values Ec, γ, Pr , R, and β_{ t }. From this table, we observed that, Nusselt number increase with an increasing the values of Pr and β_{ t } whereas an opposite trend is observed for different values Ec, γ, and R.
Conclusions

Increasing values of W_{ e } and n will increase the temperature profile. However, decrease the velocity profile.

An increasing values of Ec, γ, and R parameters increase the thermal boundary layer thickness.

The temperature profile and corresponding boundary layer thickness was depressed by uplifting the Prandtl number.

Velocity and temperature profiles decreases by an increasing values of l.

With a rise of β_{ v } and β_{ t }, velocity and temperature profile decreases for fluid phase, but increases for dust phase.

An increase in the thermal boundary layer thickness and decrease in momentum boundary layer thickness was observed for the increasing values of the magnetic parameter M.
Nomenclature
\( {B}_0^2 \) magnetic field
b stretching rate
c_{ p } fluid phase specific heat coefficient (J/kgK)
c_{ m } dust phase specific heat coefficient (J/kgK)
C_{ f } skin friction coefficient
Ec Eckert number
f dimensionless velocity of the fluid phase
F dimensionless velocity of the dust phase
K Stokes drag constant
k thermal conductivity
k^{∗} mean absorption coefficient (m^{−1})
l mass concentration of dust particles parameter
M magnetic parameter
m mass of dust particles
n power law index
N dust particles number density
Nu_{ x } local Nusselt number
Pr Prandtl number
q_{ w } heat flux at the surface
q_{ r } radiative heat flux (Wm^{−2} )
R radiation parameter
r radius of dust particles
Re_{ x } local Reynolds number
Sh_{ x } Sherwood number
T temperature of the fluid phase
T_{ p } temperature of the dust phase
T_{ w } temperature of the fluid at the sheet
T_{∞} ambient fluid temperature
u_{ w } stretching sheet velocity
u, u_{ p } velocity components of fluid phase
v, v_{ p } velocity components of dust phase
W_{e} Weissenberg number
x coordinate along the plate (m)
y coordinate normal to the plate (m)
Greek symbols
β_{ v } fluidparticle interaction parameter for velocity
β_{ t } fluidparticle interaction parameter for temperature
μ dynamic viscosity (kgm^{−1}s^{−1} )
ν kinematic viscosity
σ electrical conductivity of the fluid
σ^{∗} StefanBoltzmann constant
θ temperature of the fluid phase
θ_{ p } temperature of the dust phase
η similarity variable
τ_{ T } thermal equilibrium time
τ_{ w } surface shear stress
τ_{ v } dust particles relaxation time
γ specific heat ratio
ρ base fluid density (kg/m^{3} )
ρ_{ p } dust particles density
Γ the time constant
Superscript:
^{′} derivative with respect to η
Subscript:
p particle phase
∞ fluid properties at ambient condition.
Declarations
Acknowledgements
Not applicable
Authors’ contributions
GK has involved in conception and design of the problem. RSRG has involved in drafting the manuscript or revising it critically for important intellectual content. BJG has given final approval of the version to be published. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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