Heat Transfer Analysis of Unsteady Natural Convection Flow of Oldroyd-B Model in the Presence of Newtonian Heating and Radiation Heat flux

The main focus of this theoretical inspection is to explore the control of Newtonian heating on heat transfer for an unsteady natural convection flow of Oldroyd-B fluid confined to an infinitely long, vertically static plate. Partial differential equations are constructed effectively to describe the fluid flow and heat transfer. Some appropriate dimensionless quantities and Laplace transformation are employed as basic tools to evaluate the solutions of these differential equations. However, due to the complex nature of velocity field, solution is approximated by using Durbin’s numerical Laplace inverse algorithm. This solution is further validated by obtaining the velocity solution through algorithms proposed by Stehfest and Zakian. The temperature and velocity gradient are also determined to anticipate the heat transfer rate and skin friction at wall. Some well known results in literature are also deduced from the considered model. Conclusively, to have a deep understanding of the physical mechanism of considered model, and influence of implanted parameters, some outcomes are elucidated with the assistance of tables and graphs. As a result, it is found that under the effect of Newtonian heating, freely convective viscous fluid has greater velocity than Oldroyd-B fluid, Maxwell fluid and second grade fluid.

In numerous present day technologies, non-Newtonian fluids are attaining invaluable attraction because of their indispensable implications. A few subdivisions of non-Newtonian fluid are plastics, greases, toothpaste and foodstuff. There exist many mathematical models to efficiently anticipate the key features of different non-Newtonian fluids. However, shear stress and shear rate share a non-linear relation for such fluids, which leads to form higher order equations. Theses succeeding equations are more intricate than Navier-stokes equation however, a controlled handling of this additional non-linear term results in precise anticipation of behavior of fluid. A wide range of fluids reveal a combination of viscous and elastic behavior such as toothpaste, polymer solutions and melts, clay, and oil. A capable and simple model, which records the flow history and provides an adequate approximation of viscoelastic nature of fluids is recognized as Oldroyd-B model. Since James G. Oldroyd provided this model to forecast the elasticity and memory effects, therefore it is remembered as Oldroyd-B model [1]. The current model is capable of retaining rheological effects in case of unidirectional flows, while it contains a non-physical singularity when extensional flows are under consideration. Furthermore, in shear flow, Oldroyd-B fluid provides an excellent approximation of viscoelatic fluids. The model can also be presented split into viscoelastic part separately from the solvent part. Extensively, if solvent has zero viscosity, this model reduces to Upper Convected Maxwell model. In this regard, Upper Convected model can be viewed as a special case of considered model and moreover conventional Maxwell and viscous fluids can be deduced from it by making simple substitutions [2]. According to authors's knowledge, first exact solutions of Oldroyd-B model were evaluated by Tanner [3]. Later, Waters and King [4], [5] employed Laplace transformation to calculate the analytic results of these viscoelastic fluids. Fetecau et al. [6] provided a note on flow of Oldroyd-B model induced as a result of an accelerating surface. A systematic study was conducted by Fetecau et al. [7] to investigate the time-dependent motion of Oldroyd-B fluid due to spontaneous movement of a surface inside a channel. Khan et al. [8] analytically probed the role of porosity in magnetohydrodynamic (MHD) motion of Oldroyd-B model. Shakeel et al. [9] studied the role of slip condition in flow of Oldroyd-B model and drawn a comparison with zero-slip condition at boundary. Khan et al. [10] examined the porosity effects on Oldroyd-B model by calculating the exact solutions. Gul et al. [11] analyzed transient MHD thin motion of Oldroyd-B model near an inclined oscillating surface. Riaz et al. [12] explored the fractional flow of Oldroyd-B model in a circular duct by deriving the analytic results. Khan et al. [13] discussed the hydromagnetic rotatory motion of Oldroyd-B model in a permeable material. Recently, some new global results for incompressible Oldroyd-B fluid were discussed by Wan [14]. Application of Oldroyd-B model to hemodynamics and its numerical simulation was presented by Elhanafy et al. [15]. Hullender [16] studied pretransient turbulent motion in circular lines by employing Oldroyd-B transient model. Tiwana et al. [17] evaluated the influence of ramped temperature and ramped boundary motion on transient MHD convection flow of Oldroyd-B model. Keeping in mind the above literature, it is found that effect of Newtonian heating on Oldroyd-B fluid has not been investigated yet. This study is an attempt to fulfill this gap by analyzing the role of Newtonian heating and nonlinear heat flux in unsteady naturally convective motion of Oldroyd-B fluid past a vertically static plate.
Idea of Newtonian heating was initiated by Merkin [18], as he noted that by applying Newtonian heating from surface, convective flows can be set up. Such kind of flows are called conjugate convective flows. He provided a numerical solution comprised of analytic solution near the leading edge and numerical solution far downstream. The mechanism of Newtonian heating occurs in numerous engineering processes such as heat exchanger, petroleum industry, solar radiation, and conjugate heat transfer about fins. Salleh et al. [19] studied the impact of Newtonian heating on energy and fluid motion near a stretching surface. Haq et al. [20] explored the influence of convective boundary conditions and magnetic field on two dimensional flow of Casson nanofluid near a stretching surface. Nadeem et al. [21] further extended this study to numerically probe the same flow in three dimensions. Consequences of imposed Newtonian heating, MHD, and slip condition on Casson fluid flow over a nonlinearly stretching surface embedded in porous material were explored by Ullah et al. [22]. Imran et al. [23] examined control of Newtonian heating and slip effect on MHD boundary layer flow of generalized Maxwell model. Hayat et al. [24] analyzed power law nanofluid's steady motion near stretching sheet with Newtonian heating. Ramzan [25] reported the influence of Joule heating, Newtonian heating, and viscous dissipation on three dimensional MHD flow of nanofluid.
The three major modes of convection are known as natural, mixed and forced convection [26]. From these three mechanisms, natural convection is considered in this study. In natural convection, temperature gradient in turn causes the heat transfer and buoyancy force induces the flow. The flows based on natural convection are particularly significant in electric machinery, electronic power supplies, designing of spaceships, drying of porous substances in textile industries, and nuclear reactors. Furthermore applications of natural convection in engineering and sciences include solar ponds, oceanic and atmospheric circulation, and formation of micro-structure [27], [28]. Initially, Siegel [29] provided the foundation to study natural convection by investigating the time dependent natural convective flow over a semi infinite vertical plate. Ahmed [30] analyzed the effects of heat source, Hall current, MHD, thermal diffusion, and porosity of medium on transient natural convective motion over an upright plate. Exact analysis of unsteady naturally convective motion of a radiative gas under imposed magnetic field past an infinite inclined plate was reported by Narahari [31]. Impacts of ramped temperature and slip condition at wall for transient free convective flow of viscous fluid were examined by Haq et al. [32]. Shen et al. [33] studied unsteady natural convection flow of second grade nanofluid with a new definition of time-space fractional applied to heat conduction.
Thermal radiation has a vital role when it comes to space technology and many other environmental process which take place at high temperature. For instance, hypersonic flights, rocket combustion chambers, gas-cooled nuclear reactors, missile reentry, and power plants for interplanetary flights. All these practical applications have urged the researchers to put their special attention on this significant energy transferring mechanism and interpret the outcomes in an improved and understandable fashion. Das et al. [34] examined the influence of radiative flux and Newtonian heating on transient fee convection motion over a vertical wall. Closed form solutions for freely convective nanofluid motion near a moving plate influenced by heat radiation and magnetic field were reported by Das and Jana [35]. Izadi et al. [36] scrutinized thermogravitational flow of a micro-polar nanoliquid to evaluate the impression of radiative heat flux and MHD in porous channel.
In the light of above investigations, this study is an attempt to analyze the profiles of unsteady free convective flow of Oldroyd-B fluid over a static vertical plate and radiative heat transfer with Newtonian heating at boundary. Semi analytic solutions of flow and heat governing equations are obtained by applying Laplace transformation. Durbin's numerical algorithm [37] is executed to approximate the velocity solution in real time domain and this approximation is further validated by using Stehfest's [38] and Zakian's algorithm [39]. The temperature gradient and Skin friction are precisely evaluated at wall in pursuance to their significant applications in mechanics and engineering. Finally, current velocity, temperature and velocity of some limiting fluids' cases are graphically portrayed to get a clear insight of physical features of considered model.

II. MATHEMATICAL MODELING OF PHYSICAL PROCESS
The principal equations to express the laminar, unsteady freely convective flow of Oldroyd-B fluid together with applied Boussinesq's approximation for buoyancy force are described as [40]- [42] where g is standard gravitational pull, t is time, β is coefficient of thermal volume expansion, ρ is density, T is temperature, T ∞ is ambient temperature and ∇ is gradient operator. Moreover, Cauchy tensor of stress T and one dimensional velocity of laminar flow V are respectively described as where y is space variable, u is velocity component in x-direction and −pI is the indeterminate stress tensor. Furthermore, relation for extra stress tensor S is given as where µ is dynamic viscosity, λ is relaxation time and λ r is retardation time. Additionally, Rivlin-Ericksen tensor A and material time derivative represented by superposed dot are given as Operating equations (3) − (7) in momentum equation (2), we acquire The considered model is geometrically presented in Fig. 1. Application of Rosseland approximation [43], and assumptions of small temperature difference and small thermal Reynolds number formulate the following equations for velocity, shear stress, and energy of Oldroyd-B fluid where ν is kinematic viscosity, S is nontrivial shear stress, k is thermal conductivity, c p is specific heat at invariant pressure, σ 1 is Stefan-Boltzmann coefficient and K 1 is Rosseland absorption coefficient. The appropriate initial and boundary conditions corresponding to governing equations are The purpose of reducing the number of variables in current model is achieved by introducing following dimensionless terms Using above quantities in equations (9)-(11), we get where non-dimensional numbers are given as The initial and boundary conditions turn out as

III. SOLUTION OF PROBLEM
Laplace transformation [44] is an efficient technique to derive the analytical solutions of current problem. Its formulation in integral form is provided as For considered problem, M ∈ {θ, u 1 , F} and the condition Re(s) > a 0 ensures the convergence of above integral. Here s = a +ib with a, b and a 0 stand for some real constants andi is standard imaginary unit. The inverse Laplace transformation to obtain solutions in τ domain is accomplished by employing following relation  (18) and using initial condition (20) whereθ satisfies the following boundary conditions The solution of equation (25) under conditions in equation (26) is evaluated as Implementation of inverse Laplace transform (24) provides following result To anticipate the rate of heat transfer from plate to fluid, Nusselt number is evaluated by plugging equation (28) into the following expression

B. VELOCITY DISTRIBUTION
Applying Laplace transform on equations (16), and using initial condition (20) 1 emits whereū 1 follows the conditions mentioned below u 1 (0, s) = 0 andū 1 (η, s) → 0 when η → ∞. (31) Introducing equation (27) into equation (30), we obtain The solution of above equation is derived as Sinceū 1 (η, s) is a complex function containing multivalued combinations of Laplace parameter ''s'', therefore numerical inverse Laplace transform is applied to approximate solution in real time domain. Particularly, we operated Durbin's method, which is based on Fourier series expansion such as In order to validate our numerical Laplace inversion, we obtained approximation ofū 1 (η, s) with the help of Stehfest's algorithm and Zakian's algorithm respectively in following manner.
where n is a positive integer and α i and K i are fixed complex values.
The shear stress on the wall F w is approximated by introducing the derivative of equation (33) into following relation

IV. LIMITING CASES
This section is comprised of special cases deduced from the current work.

A. MAXWELL FLUID WITH NEWTONIAN HEATING
Making λ 2 → 0 in equation (33), we derive the following solution

B. SECOND GRADE FLUID WITH NEWTONIAN HEATING
Making λ 1 → 0 in equation (33), we get the following result

V. RESULTS AND DISCUSSION
To deeply analyze the practical applicability of considered problem, we have studied the physical significance of Grashof number Gr, time τ , relaxation time λ 1 , radiation parameter Nr, retardation time λ 2 and Prandtl number Pr in momentum and energy equations and attained conclusions are interpreted through graphs. The effect of same parameters on rate of heat transfer and wall shear stress is observed and numerical computations are presented in tabular form. The cases Gr = 0 and Nr = 0 correspond to absence of buoyancy force and thermal radiation respectively. The default values of relevant parameters are mentioned inside the respective figures. Fig. 2 presents the relationship between radiation coefficient Nr and dimensionless temperature of fluid θ (η, τ ). It is VOLUME 8, 2020  found that temperature profile gets elevation with increase in value of Nr. Since increase in Nr at fixed values of T ∞ and k, decreases the value of K 1 , therefore gradient of radiative thermal flux ∂q r ∂y increases which leads to increase the radiative heat tranfer rate and eventually temperature of fluid rises. It means that thickness of energy boundary layer reduces and temperature is distributed more uniformly.
The variation of fluid temperature due to Prandtl number Pr is described in Fig. 3. It is witnessed that temperature of fluid goes through a decay due to increasing variation of Pr. It is justified by the fact that a high Pr value is associated to low thermal conductivity, which reduces both conduction and thermal boundary layer thickness. Ultimately, fluid faces more thermal resistance and temperature of fluid decreases. Fig. 4 accounts the transient nature of flow and reveals that temperature rises with the extension of time duration τ . The fluid temperature is high adjacent to the wall and asymptotically drops down to zero value, as fluid creeps away from the wall. Table 1 exhibits an enhancement in Nusselt number due to increasing variation of Nr, while an inverse profile is observed as a response of higher Pr value. This is supported by the fact that increase in Nr means temperature  gradient has strong dominance which results in higher rate of heat transfer. Contrarily, when fluid gains higher Pr value, its thermal conductivity reduces and therefore, its capacity of heat conduction vanishes. Hence thickness of thermal boundary layer decreases and eventually rate of heat transfer reduces. Additionally, respective table indicates that rate of heat transfer increases as time τ evolves.
In order to verify our velocity approximation calculated by Durbin's method, we have also computed the solution by Stehfest's method and Zakian's method. In Fig. 5, velocity solution approximated by these three approaches is presented at time steps τ = 3.0 and τ = 5.0, and an excellent agreement between all the solutions is noticed at both time steps. This comparison validates the reliability of solution. Fig. 6 demonstrates the velocity distribution for different values of relaxation time λ 1 . It is observed that fluid's velocity 92484 VOLUME 8, 2020  increases with an increase in λ 1 . This is realized by the fact that λ 1 reduces the boundary layer thickness and corresponding to this decrease in thickness, velocity illustrates significant behavior in main stream region and later fluid attains zero velocity. The significance of Grashof number Gr in fluid flow is presented in Fig. 7. It is clearly sighted that fluid gets accelerated when Gr increases. This is due to the fact that Gr value is responsible for relative contribution of buoyancy force and viscous force to the fluid flow. Increase in Gr means, buoyancy force dominates the viscous force which leads to more rapid flow of fluid. Therefore, as value of Gr increases, velocity profile of fluid rises. An interesting observation is made that when Gr has zero value fluid has absolutely no motion. This observation justifies the free convection nature of flow and indicates that fluid flow is purely generated by buoyancy force in this case. Fig. 8 exhibits the impact of radiation parameter Nr on velocity of fluid. It is spotted that thickness of momentum boundary layer increases with higher values of Nr. Physically, rate of energy transfer justifies this increment. As Nr enlarges, rate of energy transfer to the fluid increases which in turn results to loose the bonds between fluid particles. As a result,   these loosely connected particles collectively offer a much weaker viscosity to fluid motion and eventually fluid gets accelerated. The effect of retardation time λ 2 on velocity distribution is revealed in Fig. 9. It is realized that flow is retarded due to higher values of λ 2 . This is supported by the fact that increase in λ 2 reduces the momentum boundary layer VOLUME 8, 2020       Newtonian fluid is drawn in Fig. 12. It is found that under the effect of Newtonian heating, Newtonian fluid attains highest velocity and on the other hand, second grade fluid exhibits slowest motion profile. Table 2 shows the computations of velocity by Durbin's, Stehfest's and Zakian's algorithms at different spatial steps to observe the authenticity of solution upto desired accuracy. Table 3 predicts that shear stress at wall is controlled by gradually raising the value of λ 1 , while an inverse behavior is observed for higher values of Grashof number Gr, time τ and retardation time λ 2 . Approximated shear stress computations are also compared for the purpose of verification. Usually, one is attracted to calculate the skin friction (shear stress) for the technical purposes. Moreover enhanced skin friction is considered to be a limitation in engineering exercises. Table 4 is presented to numerically compare the velocity of various fluids at different spatial steps.

VI. CONCLUSION
The purpose of this investigation is to analyze the effect of Newtonian heating on rate of heat transfer for an unsteady free convection flow of Oldroyd-B model over a static infinite vertical plate. Some appropriate dimensionless quantities are introduced in principal flow and heat governing equations, and boundary conditions to obtain the unit-less form of current model. Solutions of this unit-less model are calculated by applying Laplace transform. Graphical and tabular illustrations of solutions are provided to observe the physical importance of emerging parameters. Temperature and velocity gradient at wall are evaluated to analyze the Nusselt number and shear stress respectively. A comparison between velocity profile of different fluid models deduced from current model is also drawn to clearly examine the difference between flows.
The key findings of this analysis are outlined as • Temperature has higher profile for increasing values of Nr and τ , while it has lower profile for higher Pr values.
• Rate of heat transfer (Nusselt number) is decreasing function of Pr and it increases for an increment in Nr.
• Fluid has no motion for Gr = 0, which shows that flow is induced due to buoyancy force (free convection).
• Fluid gets accelerated with an increase in Gr, λ 1 , Nr and τ . However, velocity decreases for larger values of λ 2 and Pr.
• Skin friction reduces as value of relaxation time λ 1 increases and has an inverse behavior for similar variation of retardation time λ 2 .
• In Comparison, Newtonian fluid has highest velocity profile and on the other hand second grade fluid has the slowest flow.
• Our approximated velocity solutions through numerical Laplace inversion methods named as Durbin's algorithm, Stehfest's algorithm and Zakian's algorithm are equal.

CONFLICTS OF INTEREST
The author declares that there is no competing interest.  Cluster). His research targeted fixed-point theory, variational analysis, random operator theory, optimization theory, and approximation theory. Also, fractional differential equations, differential game, entropy and quantum operators, fuzzy soft set, mathematical modeling for fluid dynamics and areas of interest Inverse problems, dynamic games in economics, traffic network equilibria, bandwidth allocation problem, wireless sensor networks, image restoration, signal and image processing, and game theory and cryptology. He has provided and developed many mathematical tools in his fields productively over the past years. He has more than 600 scientific articles and projects either presented or published. Moreover, he is on editorial board journals more than 40 journals and also he delivers many invited talks on different international conferences every year all around the world.
ILYAS KHAN is currently with Majmaah University, Saudi Arabia. He is also a Visiting Professor with the Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, Johar Bahru, Malaysia, and also with the Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh, Vietnam. He has published more than 400 articles in various reputed journals. He has authored several books and book chapters. He is working on both analytical and numerical techniques. His research interests include boundary layer flows, Newtonian and non-Newtonian fluids, heat and mass transfer, renewable energy, and nanofluids.
ASIFA received the B.S. degree (Hons.) from the University of Sargodha, Pakistan, and the M.S. degree in mathematics from COMSATS University Islamabad, Pakistan. She is currently doing the research in heat transfer, mathematical modeling, fractional analysis, nanofluid, MHD, and heat exchangers. She has written a few articles on flow and heat transfer phenomenons of fluids. He has authored 129 scientific articles (including 21 articles in IEEE TRANSACTIONS/Magazines) published in Scopus with citations=3,063 times and h-index=29. His current research interests include power electronics, electric drives, electric vehicles, electrical devices (fuel cells, photovoltaic, wind turbine, batteries, and supercapacitors), nonlinear controls, and observers. VOLUME 8, 2020