Influence of viscous dissipation and double stratification on MHD OldroydB fluid over a stretching sheet with uniform heat source
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Abstract
The present report explores the discussion on 3D flow of dual stratified OldroydB fluid in the direction of thermal radiation, viscous dissipation and heat source/sink. Mathematical modeling is formulated with an applied magnetic field through a stretching surface. The present flow governing system has been transformed as nonlinear ODE via suitable transformations and then concluded by using bvp4c. The graphs are described and illustrated for various nondimensional parameters. Thermal stratification and Prandtl number parameters reduce the temperature, whereas thermal radiation and heat source parameters show reverse behavior. Numerical results are used to obtain the values of frictional drag, rate of heat and mass transfers. Finally, numerical results are compared with previous established work in a limiting case.
Keywords
3D OldroydB fluid Dual stratification MHD Thermal radiation Heat source/sink1 Introduction
In present days, the review of nonNewtonian fluids is an interesting concept to the novel investigators in an aspect of their utilizations in the production and technology. Foodstuffs, material industries, shampoos, form oil and bioengineering are some examples of nonNewtonian fluids. Fluids those does not pursue the Newton’s law are called nonNewtonian fluids. Maxwell [1] developed the onedimensional Maxwell fluid model, which includes a subclass of ratetype fluids, and it is found to have only relaxation time but not retardation time. Oldroyd [2] introduced the OldroydB fluid; it is the branch of nonNewtonian fluid. It shows both relaxation and retardation time properties of a nonNewtonian fluid. Fetecau [3] investigated the Maxwell fluid is highly applicable for large relaxation time polymer extrusion process, drawing of metals. Temperature and concentration effects on nonNewtonian fluid were investigated by ChuoJeng Huang [4]. Abdul Gaffar et al. [5] were analyzed by a nonNewtonian fluid across a nonisothermal wedge, utilizing implicit finite difference method for obtaining graphical representations. Soret and Dufour effects on Casson and Maxwell fluids across a stretching sheet were investigated by Ramana Reddy et al. [6] by implementing RK Felhberg technique for obtaining solutions and presenting graphical representations to investigate the consequences for both the nonNewtonian fluids. Raju et al. [7] analyzed the impacts of nonNewtonian fluid over homogeneous/heterogeneous reactions, exhibiting graphical representations for both stretching and shrinking surfaces. Consequently, many researchers were involved in examining the different physical conditions of OldroydB fluid in various regions (see [8, 9, 10, 11, 12, 13, 14, 15, 16, 17]).
Magnetohydrodynamic flow across a stretching/shrinking disk was analyzed by Khuzaimah Soid et al. [18], investigating stability of the multiple solutions. Hydromagnetic flow over a stretchable curved space was studied by Hayat et al. [19] and presented graphical representations for friction factor, rate of heat and mass transfers to estimate curvature. MHD flow of a nanofluid through a saturating permeability porous was analyzed by Siva Kumar Reddy et al. [20] by using HAM and carried out the study of numerical representations. HAM solutions on MHD 3D effects were scrutinized by Hayat et al. [21] and concluded that temperature and concentration fields are enhancing for escalating values of M. Influence of heat transfer effects on MHD flow of thermal radiation and viscous dissipation was investigated by Nayak [22]. Ganesh Kumar et al. [23] distinguish the Joule heating and viscous dissipation on 3D OldroydB nanofluid flow along thermal radiation. Impacts of Joule and viscous heating effects on 3D hydromagnetic flow including mass and heat fluxes were examined by Muhammad et al. [24]. Jaber [25] investigated the power law heat flux with Joule heating and viscous dissipation on 3D hydromagnetic through a porous stretching sheet.
Heat and mass transfer flows across a stretching sheet with double stratification have a main role in polymer production, chemical industries, wire drawing, paper production etc. Heat and mass transfer has several applications like glass fiber, extrusion process and damage of crops due to freezing. Impact of hydromagnetic tangent—hyperbolic nanofluid—was explored by Sajid et al. [26] and analytically solved by using HAM. Impact of heat and mass transfer effects across a stretching sheet was considered by Mishra and Bhatti [27] determined numerically with the aid of successive linearization method and Chebyshev spectral collocation method. 2D heat and mass transfer effects of Casson fluid in the presence of wall mass transfer was explained by Ali et al. [28]. 2D boundary layer flow of double stratification of Eyring–Powell fluid was probed by Khalil et al. [29] and plotted graphical representations for friction factor, rate of heat and mass transfers for both the cylinder and plate. Hydromagnetic nonNewtonian fluid across a stretching cylinder with double stratification was examined by Khalil et al. [30] and concluded that \(\varepsilon_{1}\) enhances the rate of heat transfer decays.
In the raised references, the influence of viscous dissipation is not addressed. The objective of the current paper carried out the study of 3D flow of OldroydB dual stratified fluid through a stretching sheet in direction of temperaturedependent heat source/sink, chemical reaction and viscous dissipation in the magnetic field.
2 Model problem
3 Method of solution
\(M\left( { = \frac{{\sigma B_{0}^{2} }}{\rho m}} \right)\) the magnetic field parameter, \(c \left( { = \frac{n}{m}} \right)\) the stretching ratio parameter, \(\beta_{1} \left( { = \lambda_{1} m} \right)\) and \(\beta_{2} \left( { = \lambda_{2} n} \right))\) are Deborah numbers, \(Pr\left( { = \frac{{\rho c_{p} \nu }}{k}} \right)\) the Prandtl number, \(Nr( = \frac{{4\sigma^{*} T_{\infty }^{3} }}{{kk^{*} }})\) the thermal radiation parameter, \(Ec_{x} = \frac{{u_{w}^{2} }}{{c_{p} \left( {T_{w}  T_{0} } \right)}}\), \(Ec_{y} = \frac{{v_{w}^{2} }}{{c_{p} \left( {T_{w}  T_{\infty } } \right)}}\) the Eckert number along x and y directions, \(Q\left( { = \frac{{Q_{0} }}{{\rho c_{p} m}}} \right)\) the heat source/sink parameter, \(\varepsilon_{1} \left( { = \frac{{m_{2} }}{{m_{1} }}} \right)\) the thermal stratification parameter, \(\varepsilon_{2} \left( { = \frac{{n_{2} }}{{n_{1} }}} \right)\) the solutal stratification parameter, \(Sc\left( { = \frac{\nu }{D}} \right)\) the Schmidt number, \(\gamma \left( { = k_{0} /m} \right)\) the chemical reaction parameter.
 1.
\(\beta_{2} = 0\), then the present problem performs Maxwell fluid model.
 2.
\(\beta_{1} = \beta_{2} = 0\), then the present problem performs classical Newtonian fluid.
c  Hayat et al. [36]  Rudraswamy et al. [37]  Ganesh et al. [38]  Present results  

\( f^{{{\prime \prime }}} \left( 0 \right)\)  \( g^{{{\prime \prime }}} \left( 0 \right)\)  \( f^{{{\prime \prime }}} \left( 0 \right)\)  \( g^{{{\prime \prime }}} \left( 0 \right)\)  \( f^{{{\prime \prime }}} \left( 0 \right)\)  \( g^{{{\prime \prime }}} \left( 0 \right)\)  \( f^{{{\prime \prime }}} \left( 0 \right)\)  \( g^{{{\prime \prime }}} \left( 0 \right)\)  
0  1  0  1  0  1  0  1  0 
0.25  1.048810  0.19457  1.04881  0.19457  1.04881  0.19457  1.04882  0.19458 
0.5  1.093095  0.465205  1.09309  0.46522  1.093095  0.46522  1.093095  0.46521 
0.75  1.134500  0.794620  1.13450  0.79462  1.13450  0.79462  1.13449  0.79464 
1.0  1.173721  1.173721  1.17372  1.17372  1.17372  1.17372  1.17373  1.17373 
Numerical values of \( \theta^{{\prime }} \left( 0 \right)\) and \( \phi^{{\prime }} \left( 0 \right)\) for some values of Pr = 1.2, Ec = 0.1, Sc = 0.8, \(\gamma\) = 0.1, Q = 0.2 and Nr = 1.0
\(\beta_{1}\)  \(\beta_{2}\)  M  \(\varepsilon_{1}\)  \(\varepsilon_{2}\)  \( \theta^{{\prime }} \left( 0 \right)\)  \( \phi^{{\prime }} \left( 0 \right)\) 

0.2  0.5  0.5  0.1  0.1  0.602263  1.025792 
0.7  0.499512  0.964644  
1.2  0.413240  0.914331  
1.7  0.341643  0.872434  
0.5  0.5  0.5  0.1  0.1  0.546207  0.987944 
0.7  0.586368  1.013971  
1.0  0.631435  1.043420  
1.5  0.682537  1.077392  
0.5  0.5  1.0  0.1  0.1  0.532954  0.969914 
2.0  0.487148  0.936630  
4.0  0.412381  0.882465  
6.0  0.353748  0.840044  
0.5  0.5  0.5  0.1  0.1  0.546207  0.987944 
0.2  0.513247  0.987944  
0.3  0.480288  0.987944  
0.4  0.447328  0.987944  
0.5  0.5  0.5  0.1  0.0  0.578881  1.056408 
0.1  0.578881  0.990429  
0.2  0.578881  0.924451  
0.3  0.578881  0.858473 
4 Results and discussion
The investigation is focused to explore the consequences of double stratification on 3D flow of an OldroydB fluid past a stretching surface under the effect of a transverse magnetic field, thermal radiation in the direction of temperaturedependent heat source/sink, viscous heating and firstorder chemical reaction. To understand the physics of the problem under investigation, the flow variables, viz. velocity, temperature and species concentration, are computationally evaluated at some fixed values of nondimensional parameters \(\beta_{1} = \beta_{2} = 0.5, \varepsilon_{1} = \varepsilon_{2 } = 0.1\), M = 0.5, Pr = 1.2, \(Ec_{x} = Ec_{y}\) = 0.1, Sc = 0.8, \(\gamma\) = 0.1, Q = 0.2, Nr = 1.0, c = 0.8 except the diverse values as addressed in the respective figures and illustrated through plots. Physical interpretations of the results are discussed in detail.
5 Conclusions

The Deborah number (\(\beta_{1}\)) and magnetic field parameters have a decreasing effect on velocity while temperature and concentration show a reverse behavior.

The parameters \(\varepsilon_{1}\) and Pr are reducing the temperature, whereas thermal radiation and heat source parameters have an increasing effect.

Species concentration is depreciated with regard to Schmidt number, chemical reaction and solutal stratification parameters.

Frictional drag coefficient is decreased with respect to the Deborah number (\(\beta_{1}\)) and magnetic field parameters.

Nusselt number increases with Prandtl number, while it decreases with parameters Nr and \(\varepsilon_{1}\).

Rate of mass transfer increases with solutal stratification parameter (\(\varepsilon_{2}\)).
Notes
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
Human and animals rights
This article does not contain any studies with human participants or animals performed by any of the authors.
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