Flow and heat transfer of Oldroyd-B nanofluid with relaxation-retardation viscous dissipation and hyperbolic boundary conditions

Abstract

In the present research article, modeling and computations are presented to introduce the novel concept of relaxation-retardation viscous dissipation and hyperbolic time variation boundary conditions on the magnetohydrodynamic transient flow of Oldroyd-B nanofluid past a vertical stretched plate for the first time. In the present work, firstly we implement Buongiorno’s model to illustrate Brownian motion and thermophoretic diffusion which take vital role in heat and mass transportation process. Nonlinear non-dimensional governing equations are solved by fourth order Runge-Kutta method along with shooting technique. We investigate the behavior of influential variables on the velocity, thermal and solutal fields through graphical illustrations. Our results indicate that relaxation and retardation Deborah numbers exhibit completely reverse trend in the flow field. Especially, augmented relaxation- retardation viscous dissipation invigorates the temperature gradient. The results of the current theoretical study may be instrumental for worthful practical applications.

1. Introduction

Authors pay sincere thanks to the researchers for their precious contributions through fair, honest and expeditious investigations to the important progress achieved in the accelerated research world over the past decades. Out of numerous dedicated contributions one remarkable achievement of Maxwell [1] is the development of Maxwell model.

This is a rate type model used for evaluating the viscoelastic nature of materials. Ignoring the possibility of incompressible nature of liquid is another important feature of this model. It is indeed Oldroyd [2] after Maxwell, developed a constitutive relationship for describing actual behavior of nonlinear liquids. The constitutive relations of Oldroyd-B fluid characterizes the influence of both relaxation and retardation times. The Oldroyd-B model developed by Oldroyd becomes Maxwell model without retardation time. Many noteworthy researchers manoeuvred strenuous endeavors to study the flow and heat transfer of Oldroyd-B fluid under different conditions and configurations. The outcomes of their investigations have been addressed and utilized by many others. Bhatnagar et al. [3] showed in their study how free stream velocity influences the flow of an Oldroyd-B fluid due to a stretching sheet. The effect of decay of a potential vortex in the flow of an Oldroyd-B fluid was investigated by Fetecau andFetecau [4]. The stagnation point flow of an Oldroyd-B fluid past a stretching sheet was studied intensively by Sajid et al. [5]. Zheng et al. [6] established exact solutions for MHD flow of generalized Oldroyd-B fluid due to an infinite accelerating plate. In their study, the fractional calculus approach is implemented to establish the constitutive relationship of the Oldroyd-B fluid. The solutions are developed by means of Fourier sine and Laplace transforms. Hayat et al. [7] examined three-dimensional convective flow of an Oldroyd-B fluid over a stretched surface. Their study addressed the convergence of series solutions by the homotopy analysis method (HAM). Abbasbandy et al. [8] analyzed the numerical and analytical solutions for Falkner-Skan flow of MHD Oldroyd-B fluid. The relaxation and retardation times have opposite effects on the velocity components were the results of their investigation. Interestingly, an unsteady helical flow of Oldroyd-B fluids was studied by Jamil et al. [9]. Oscillating motion of an Oldroyd-B fluid between two infinite circular cylinders was analyzed by Fetecau et al. [10]. Impact of Cattaneo-Christov heat flux in MHD flow of Oldroyd-B fluid with homogeneous-heterogeneous reactions was investigated by Hayat et al. [11]. Hayat et al. [12] applied non-Fourier heat flux theory to study on 2D stratified flow of an Oldroyd-B fluid with chemical reaction. They declared in their study that temperature distribution has opposite behavior for thermal relaxation time and variable thermal conductivity parameter. Simultaneous impacts of mixed convection and nonlinear thermal radiation in stagnation point transient flow of Oldroyd-B fluid was discussed by Hayat et al. [13]. They observed that the behavior of thermal relaxation and retardation times on velocity distribution are opposite. They also observed that temperature ratio and radiation parameters augment the temperature distribution. Zhang et al. [14] examined the flow and heat transfer of an Oldroyd-B nanofluid thin film over an unsteady stretching sheet. The influences of various relevant parameters such as unsteady parameter, volume fraction of Cu/Ag and Prandtl number on the flow field were explored in their study.

Here we pronounce that the concept of nanofluid is developed by Choi [15] where the proper dilution and suspensions of the poor heat transfer fluids such as oil, water and ethylene glycol takes place with high thermal conductivity material nanoparticles like Cu, CuO, Al₂O₃ ,TiO₂ ,SiO₂ ,ZrO₂ ,ZnO . Nanofluid due to its unique and exceptional characteristic properties ensures a substantial upgradation of heat transfer of conventional (poor heat transfer) fluids. Consequently, such fluids has tremendously overwhelming the demand of heat removal in cooling applications. The most promising and inevitable diversified applications of nanofluids today include process industries, heat exchangers, cooling towers, transportation, magneto-optical devices, biomedical-therapeutic treatment and developing the best quality lubricants and oils and many others. Keeping the above astounding relevance into mind, many researchers succeeded in finding different models, phenomena and mechanisms theoretically and experimentally that would enhance the heat transfer capability of various nanofluids. Many authors have also studied the behavior of different nanofluids subject to several flow configurations and physical boundary conditions. Xuan and Li [16] studied the flow and convective heat transfer of nanofluids. Further, Buongiorno [17] introduced a two-phase model comprising the role of two slip mechanisms namely Brownian diffusion and thermophoresis which account for the thermal conductivity enhancement of nanofluids. Khan and Pop [18] analyzed the boundary-layer stretched flow of a nanofluid.

Moreover, series solutions for Oldroyd-B fluid motion induced by a deforming sheet having exponential velocity were investigated by Hayat et al. [19]. In their study, they considered the wall temperature as exponentially growing function of the horizontal distance. Shivakumara et al. [20] analyzed thermal convective instability in an Oldroyd-B nanofluid embedded in porous layer. In their analysis, the authors implemented the famous Buongiorno model and assumed zero nanoparticle flux at the boundaries. Nayak [21] examined the impact of thermal radiation and viscous dissipation on three dimensional magnetohydrodynamic flow nanofluids by shrinking surface using Homotopy analysis method (HAM). It seen that there is an increase in temperature due to an increase in thermal radiation parameter leads to lower heat transfer rate from the surface of the sheet. Also it is noticed from his study that the local Nusselt number gets reduced indicating lowering heat transfer rate from the surface with increasing Eckert number. Nayak et al. [22] explored the effects of velocity slip and non-linear thermal radiation on three dimensional magnetohydrodynamic convective flows of nanofluids through porous media. Increasing values of slip parameter slow down the axial and transverse fluid velocities and that of temperature parameter yielding greater non-linearity in heat transfer rate from the surface are important outcomes of their investigation. Nayak et al. [23] studied the effect of thermal radiation and natural convection of three dimensional magnetohydrodynamic flows of nanofluids over permeable linear stretching sheet. They declared in their study that rising the buoyancy and radiation parameter uplift the velocity and temperature profiles respectively. Ghadikolaei et al. [24] showed in his study how the porous matrix and thermal radiation influences significantly on flow and heat transfer of Fe3O4–(CH2OH)2 nanofluids. Hayat et al. [25] revealed that the heat generation parameter upgrades the fluid temperature and the related thermal boundary layer in the stretched flow of magnetohydrodynamic Oldroyd-B nanofluid. The flow of magnetized Oldroyd-B fluid over a rotating disk influenced by non- linear radiation and activation energy was analyzed by Khan et al. [15]. In their study, they developed numerical solution and conveyed that temperature ratio parameter augments the temperature distributions in the boundary layer region. Mahanthesh et al. [27] discussed the nonlinear three-dimensional stretched flow of an Oldroyd-B fluid with convective condition. Kumar et al. [28] investigated the effect of Joule heating and viscous dissipation on three- dimensional flow of Oldroyd B nanofluid with thermal radiation where they confirmed that rise in Eckert number led to the escalation of temperature fields in the entire flow domain.

Viscous dissipation finds convenience to take place in stronger gravitational fields, larger planets, heavier gases in space and geological operations. Viscous dissipation is a partial irreversible process that generates an additional heat in the flow process due to fluid friction. Nayak [29] explored the nature of dissipative flow of nanofluids under the influence of transverse magnetic field in association with thermal radiation embedded in a porous medium. It is seen in his study that an increase in Eckert number increases the fluid temperature and the associated thermal boundary layer thickness. This leads to reduction of the rate of heat transfer from the stretched surface. This reduction is further decreased due to presence of porous medium. Therefore, the presence of porous medium acts as an insulator to the surface of the sheet.

Going through the aforementioned literature survey, it is well understood that until now there is only one publication about the aspect of relaxation-retardation viscous dissipation. Only Zhang et al. [30] studied the behavior of relaxation-retardation viscous dissipation of Oldroyd-B fluid thin film. However, the impact of relaxation-retardation viscous dissipation on the flow of Oldroyd-B nanofluid past a stretched vertical plate associated with hyperbolic time varying boundary conditions has not yet studied. The aim of this study is to bridge such gap. The prime objective of our research is to model the Oldroyd-B nanofluid flow subject to relaxation-retardation viscous dissipation past a stretched vertical plate associated with hyperbolic time varying boundary conditions. Heat and mass transfer process are explored due to Buongiorno’s model featured by aspects of Brownian motion and thermophoretic diffusion. The fourth order R-K method along with shooting technique is adopted as a numerical tool to solve the resulting nonlinear governing expressions. The characteristics of velocity, thermal and solutal distributions, surface drag force, heat and mass transfer rates well displayed and discussed.

2. Formulation of the problem

In In this problem we deal with the time-varying flow of Oldroyd B nanofluid past a stretched vertical plate. Two familiar phenomena such as thermophoretic motion and Brownian diffusion have been taken into account. Relaxation-retardation viscous dissipation is introduced. Hyperbolic time varying boundary conditions are implemented. Assume that u and v are respectively the velocity components in x- and y- directions where x-axis is along the plate and y-axis is at right angle to it as represented in Figure 1.

The underlying boundary layer equations with the very idea of Boussinesq approximation and above declared assumptions are [3], [18], [26,27,28], [30]:

The required hyperbolic time variation boundary conditions are:

Here, u and v are the fluid velocity components along x and y directions, u_w(x) is the plate velocity, a > 0 is a constant.

The appropriate transformations used are

where F_η(η) is the differentiation with respect to η.

Using eqs (6) and (7), eqs (2-5) take the form

The local skin friction coefficient is written as

The non dimensional local skin friction coefficient is

The local Nusselt and Sherwood numbers are respectively

The non-dimensional local Nusselt and Sherwood numbers can be developed as

3. Solution method and validation

In the present study, the nonlinear and coupled Eqs (8), (9), and (10) with boundary conditions (11) are solved numerically using Runge-Kutta-Fehlberg method with shooting technique for different values of parameters.

Table 1. Comparison of F_ηη(0) for different values of Ω₁ for Ω₂ = ϵ = Z = δ = 0.

Ω1	Abel et al. [33]	Megahed [34]	Waqas [35]	Present Work
0.0	1.000000	0.999978	1.000000	1.00000000
0.2	1.051948	1.051945	1.051889	1.05188988
0.4	1.101850	1.101848	1.101903	1.10185163
0.6	1.150163	1.150160	1.150137	1.15016093
0.8	1.196692	1.196690	1.196711	1.19669083
1.2	1.285257	1.285253	1.285363	1.28525740
1.6	1.368641	1.368641	1.368758	1.36878678
2.0	1.447617	1.447616	1.447651	1.44761469

conveys the comparison of present results with the previously declared results of Abel et al. [33], Megahed [34] and Waqas et al. [35] where it is visualized that there is well and good agreement between them. The good agreement is related to the visualization that augmentation of Deborah number Ω₁ associated with relaxation time upsurges the skin friction significantly.

4. Results and Discussion

This section presents a bold and visionary analysis providing the physical interpretation due to the interaction of interesting assorted variables including Deborah numbers ( Ω₁ and Ω₂ ), Hartmann number M , Prandtl number Pr , modified Eckert number Ec_m , Schmidt number Sc with flow, thermal and solutal fields. The detailed information regarding behaviors of velocity, temperature and concentration fields, viscous drag, Heat transfer rate and rate of mass transfer associated with the time-varying flow of Oldroyd-B nanofluid past a stretched vertical plate influenced by relaxation-retardation viscous dissipation have been well addressed.

4.1. Velocity distribution

The observation of Figure 2 indicates that controlled fluid motion is accomplished due to enhancement of Deborah number Ω₁ associated with relaxation time. However, the adverse nature of fluid motion in response to Deborah number Ω₂ associated with retardation time is visible in Figure 3. From Figure 2 and Figure 3 it is inferred that behavior of Deborah number Ω₂ is quite opposite to that of Ω₁ . In fact, for higher relaxation time the Oldroyd nanofluid boosts up thereby responsible for the decline of the motion of the Oldroyd nanofluid. On the other hand, Ω₂ corresponds to retardation time Г₂ . Physically, an augmentation in retardation time upsurges the flow. It is well agreed with the observation in Mustafa et al. [31]. Figure 4 has rightly stated that increase in Hartmann number M contributes decelerated motion and the related shrinking velocity boundary layer. Please bear in mind that the flow over the plate is restrained by Lorentz force which is generated due to the interaction between applied magnetic field and the conducting fluid.

4.2. Temperature distribution

The outcomes of the present analysis reveal many interesting aspects of the heat transfer in a flow of Oldroyd-B nanofluid under the impact of various pertinent thermal parameters. We start with Figure 5 describes that fluid with high Prandtl number accounts for abatement of fluid temperature θ(η) and upgradation of heat transfer from the plate. The basic point is that Prandtl number has a reverse association with thermal diffusivity. So it is not surprised that for high Prandtl fluid (having larger value of Pr ) momentum diffusivity dominates over thermal diffusivity which responds to decaying of temperature. This is an excellent agreement with Hayat et al. [32].

Fluid temperature θ(η) and the related layer thickness eventually develop the ascending trend in response to rising modified Eckert number Ec_m (portrayed in Figure 6). In fact, increasing Ec_m contributes stronger viscous dissipation associated with the converted internal energy, which in turn boosts the temperature. Further, increase in Hartmann number M (visualized in Figure 7) causes establishment and development of escalated non-dimensional fluid temperature θ(η) and the related thermal - layer thickness. In fact, greater Lorentz force due to larger M produces restrained fluid flow which in turn builds up more heat in the boundary layer.

4.3. Concentration distribution

Figure 8 envisages that nanoparticles concentration ϕ(η) and the associated layer thickness undermine due to increasing Schmidt number Sc . Higher Sc indicating lower molecular diffusivity contributes lesser nanoparticles concentration in the related boundary layer. At certain Sc (for instance, Sc = 5 ), ϕ(η) is maximum in the flow zone contiguous to the solid boundary and then ϕ(η) gradually diminishes as we proceed towards the ambient fluid.

4.4. Local skin friction, Nusselt number and Sherwood number

Figure 9 reveals the surface drag force F_ηη(0) behavior or the skin friction characteristics in response to Deborah numbers Ω₁ and Ω₂ for different Hartmann number M . In this case Ω₁ and Ω₂ influence the wall shear stress in diagonally opposite manner. In this case, surface drag force slows down with increase in Ω₁ while it escalates with rising Ω₂ (opposite behavior is noticed). The skin friction is a decreasing function of Ω₁ while it is an increasing function of Ω₂ . Here we must note that these two Deborah numbers exhibit exactly opposite nature of fluid velocity. This is well observed in Mustafa et al. [31]. Figure 10 narrates how the Nusselt number varies in response to thermophoretic parameter N_t and Brownian motion parameter N_b against different values of Hartmann number M . A reduction in the absolute value of Nusselt number is accomplished due to elevating both N_t and N_b . This implicates that heat transfer rate gets diminution due to the influence of Brownian motion and thermophoresis mechanism. However, descending trend is more eye-catching due to the influence of Brownian motion. In fact, rise in thermophoresis parameter N_t improves the temperature profiles and the related layer thickness. As indicated in different literature, larger thermophoresis parameter indicates stronger thermophoretic force. Stronger thermophoretic force drags enormous nanoparticles through greater diffusion from the hot plate towards the ambient thereby increases the fluid temperature within the domain. An increase in N_b corresponds to the effective random motion of fluid particles generating more heat within the flow. Therefore temperature increases. Figure 11 delineates how the Sherwood number gets affected from the impact of N_t and N_b against different M . As a first step, by increasing N_t leads to a diminishing trend of mass transfer rate while that of N_b yields a reverse trend. This is because one can observe from diversified literature that increase in thermophoretic parameter N_t witnesses an up gradation of nanoparticles concentration while augmented Brownian motion parameter N_b shows a reverse trend. This is due to the fact that rising of N_t indicating strong thermophoretic force leading to larger diffusion of nanoparticles from the hot plate to the ambient fluid and as a result improves nanoparticles concentration profiles. On the other hand, the Brownian force due to Brownian motion makes the particles to move in opposite direction of the concentration gradient and makes the nanofluid more homogenous. Such force leads to low concentration gradient and more uniform nanoparticle concentration distribution. It is interesting to note that thermophoretic force strength due to N_t = 0.6 and Brownian force strength due to N_b = 0.7 contribute almost same mass transportation in the associated boundary layer.

5. Conclusion

There have been reports regarding the flow and heat transfer of Oldroyd-B nanofluid in view of relaxation-retardation viscous dissipation over a vertical stretched plate. This convenient research grows our insights into innumerable significant practical applications, for example production of plastic sheets and extrusion of polymer in polymer industry.

Enhancement of Deborah numbers Ω₁ and Ω₂ associated with relaxation time accounts for diminishing trend of fluid motion while that of Ω₂ (Deborah number associated with retardation number) exhibit adverse behavior. More Hartmann number M indicating more Lorentz force yields diminutive fluid flow and escalates the fluid temperature. Deborah numbers Ω₁ and Ω₂ influence the wall shear stress in diagonally opposite manner for different Hartmann number M . Increasing thermophoretic force causes a diminution of mass transfer rate while a reverse situation is noticed for that of Brownian motion. Smaller thermophoretic force (small N_t ) and less Brownian motion subject to considerably high Prandtl fluids lead to the enhancement of heat transfer rate from the vertical stretched plate.

At last, it has been suggested that the present model can be adopted for entropy optimized three dimensional rotating flow problems, which will be a focus in our future work.