Radiation effects on 3D rotating flow of Cu-water nanoliquid with viscous heating and prescribed heat flux using modified Buongiorno model

In this article, the three-dimensional (3D) flow and heat transport of viscous dissipating Cu-H2O nanoliquid over an elongated plate in a rotating frame of reference is studied by considering the modified Buongiorno model. The mechanisms of haphazard motion and thermo-migration of nanoparticles along with effective nanoliquid properties are comprised in the modified Buongiorno model (MBM). The Rosseland radiative heat flux and prescribed heat flux at the boundary are accounted. The governing nonlinear problem subjected to Prandtl’s boundary layer approximation is solved numerically. The consequence of dimensionless parameters on the velocities, temperature, and nanoparticles volume fraction profiles is analyzed via graphical representations. The temperature of the base liquid is improved significantly owing to the existence of copper nanoparticles in it. The phenomenon of rotation improves the structure of the thermal boundary layer, while, the momentum layer thickness gets reduced. The thermal layer structure gets enhanced due to the Brownian movement and thermo-migration of nanoparticles. Moreover, it is shown that temperature enhances owing to the presence of thermal radiation. In addition, it is revealed that the haphazard motion of nanoparticles decays the nanoparticle volume fraction layer thickness. Also, the skin friction coefficients found to have a similar trend for larger values of rotation parameter. Furthermore, the results of the single-phase nanoliquid model are limiting the case of this study.

model. Homotopy solution for the 3D flow of nanofluid using BM model subjected to the Newton condition and magnetism is proposed by Hayat et al. 9 . The BM model and Oldroyd-B fluid model are utilized by Hayat et al. 10 to study the 3D stretched flow of nanofluid subjected to magnetism. For a more detailed review of the 3D flow of nanofluid using BM model studies reader can refer to references [11][12][13][14][15] .
The studies in Refs. [4][5][6][7][8][9][10][11][12][13][14][15] are more or less similar to the heat and mass transfer problem with the thermodiffusion aspect. Also, the effectual thermophysical properties of nanoliquid affect the flow distributions significantly, hence they can't be ignored and are better to include to obtain accurate results. In this direction, Yang et al. 16 used the Buongiorno nanoliquid model along with effectual thermo-physical properties of nanoliquids to study the convective thermal transport of nanofluids. Later, this model is well known as the Modified Buongiorno Model (MBM). Malvandi et al. 17 used the MBM to examine the completely developed flow of nanofluids in an annular pipe. They concluded that the single-phase nanoliquid model results can be recovered from MBM. Malvandi and Ganji 18 examined the dynamics of nanofluids in a microchannel by using the MBM model. However, the studies related to nanofluid flow using Modified Buongiorno Model are very limited.
The significance of thermal radiation in many industrial and engineering processes like electric power, nondestructive testing, solar cell panels, and many others is vital. Therefore, it is very crucial to comprehend the aspect of thermal radiation to attain the desired quality of products in industrial processes. Sheikholeslami and Rokni 19 studied the importance of Rosseland thermal radiation and Coulomb force on the dynamics of nanofluids in an enclosure saturated by porous space. They found that the thermal radiation process supports improving the thermal layer thickness. Mahmoud and Megahed 20 studied the significance of thermal radiation on the dynamics of non-Newtonian fluid with mixed convection and thermal diffusion. The impact of Rosseland radiative heat is examined by Das et al. 21 on the dynamics of nanoliquid in a micro-channel. Shehzad et al. 22 investigated to study the thermal radiative heat transfer and 3D flow of nano Jeffrey fluid with magnetism.
Recently, Raza et al. 23 and Wakif et al. 24 are investigated the significance and applications of the thermal radiation aspect. Moreover, Alam et al. 25 used the turbulent SST model to capture the heat transfer characteristics of micro-pin-fin. Several researchers numerically solved the derived 3D flow and energy equations of nanofluids or hybrid nanofluid by the Runge-Kutta-Fehlberg-method (RKF). Furthermore, Baslem et al. 26 investigated the thermal behavior of porous fin fully wetted with various nanofluids under natural convection and radiation condition. They found that Cu-water was best to enhance the fin heat transfer amongst the investigated nanofluids Al 2 O 3 -water and TiO 2 -water. In addition. Ganesh Kumar et al. 27 explained heat transfer behavior influenced by both tangent hyperbolic nanofluid and magnetic field over a moving stretched surface. Additionally, Punith Gowda et al. 28 presented a 3D nonlinear model for nanofluid flow over an expansion and contraction of a rotating disk featured with thermophoretic particle deposition. In addition, Ahmadian et al. 29 used homotopy solution and PCM to show the influence of 3D unsteady flow caused by the wavy rotating disk and magnetic field on hybrid nanofluids Ag/MgO-Water. Ahmadian et al. 30 studied the Brownian motion and thermophoresis effect on maxwell nanofluid between two stretchable horizontal rotating disks under a magnetic field. They solved their model by boundary value solver (Bvp4c) and RK4. Finally, Lv et al. 31 used PCM to explore the impact of thermal radiation, magnetic field, and the upshot of Hall current on flow and heat transfer characteristics of carbon nanotubes, and iron ferrite nanofluids flow over a spinning disk.
Inspired by the above-indicated literature and applications, the prime purpose of the current research is to study the 3D rotating flow of nanoliquid over a stretched plate using the modified Buongiorno model (MBM). The modified Buongiorno model comprised haphazard movement and thermo-migration of nanoparticles along with effectual thermophysical properties. The influences of viscous heating, thermal radiation, and prescribed surface heat flux boundary conditions are also scrutinized. The nonlinear partial differential boundary value problem is solved numerically and the results are analyzed. Further, the velocity, temperature, heat transport rate, and mass transport rate are examined for various parameters.

Mathematical formulation
The rotating 3D of water-based Cu nanoliquid over a stretched plate subjected to a rotating frame is considered. The surface heat flux boundary conditions are included. Figure 1 represents a schematic diagram for the problem under investigation.
The transport mechanism for nanoliquid has been addressed by the Modified Buongiorno Model (MBM) that includes the effective nanoliquid properties, haphazard motion of Cu nanoparticles, and thermo-migration www.nature.com/scientificreports/ mechanism. The rectangular coordinate framework is aligned with xy−plane and the fluid region is considered at z ≥ 0 . The Cu nanoliquid rotates unvaryingly about z-axis with an unvarying rate ω . To characterize the effective thermal conductivity and effectual dynamic viscosity of nanoliquid, the Maxwell-Garnetts 2 and Brinkman 3 models are utilized. These models correspond to spherical shape nanoparticles of the volume fraction less than 5-6%. Utilizing the above-mentioned assumptions, and following Lund et al. 32 , the subsequent boundary-layer expressions are (see 3,6,9,16 ); The last term from the right-hand side of Eq. (4) corresponds to the thermal radiation term and it is modeled by using Rosseland's approximation. The q R (radiative heat flux) is given below (see 11 ) Upon simplifying Eq. (6) through low temperature difference yields (see 11 ) Using Eq. (7) in Eq. (4) where u, v and w are velocities along x, y and z-directions, ν = µ ρ is the kinematic viscosity, µ is the dynamic viscosity, ρ is the density, T is the temperature, C is the dimensionless nanoparticle volume fraction, α = k ρC p is the thermal diffusivity, k is the thermal conductivity, ρC p is the specific heat, D B is the coefficient of Brownian diffusion, D T is the coefficient of thermo-migration diffusion, α is the Stefan-Boltzmann constant and β is the mean absorption factor. The boundary conditions are where a > 0 is stretching rate and q w = (T w − T ∞ )k l √ a/ν l is constant heat flux. The effective density and specific heat of water-based Cu nanoliquid are as follows: Brinkman model for dynamic viscosity and Maxwell model for thermal conductivity are used.
(1) where Re x = xu w ν l is local Reynolds number.
The stimulus of rotation factor ( Ro ) on velocities (f ′ (ζ ), g(ζ )) , temperature (θ(ζ )) , and nanoparticle volume fraction (�(ζ )) is presented in Figs. 2, 3, 4 and 5 respectively. The magnitude of axial velocity f ′ (ζ ) and transverse velocity decreases by increasing values of rotation factor ( Ro ). The rotation factor ( Ro ) is a ratio of angular velocity to stretching rate. Larger values of Ro indicate to lower stretching rate, due to which, the velocities are diminished for larger Ro . Further, in the absence of rotation ( Ro = 0 ) there is no g(ζ ) and the f ′ (ζ ) is found to be higher. An increasing trend is observed for the thermal and nanoparticle volume fraction layer structure   Figures 6 and 7 describe how the haphazard movement of nanoparticles ( Nb ) affects the temperature (θ(ζ )) and nanoparticle volume fraction (�(ζ )) fields. A diminishing trend of nanoparticle volume fraction (�(ζ )) is perceived for advanced values of Nb, while, an opposite trend is observed for temperature (θ(ζ )) profile. The haphazard collision of suspended nanoparticles in water produces supplementary heat in the system and thereby the magnitude of the temperature field increases. The consequence of Nt on temperature (θ(ζ )) and nanoparticle volume fraction (�(ζ )) distributions are delineated in Figs. 8 and 9. The thermodiffusion factor is a quantity that measures the intensity of force produced by the thermal gradient in the liquid system. As Nt increases, the additional internal heat supplied to the liquid system via thermal gradient increases, and thereby both the temperature (θ(ζ )) and nanoparticle volume fraction (�(ζ )) are enhanced. Figures 10 and 11 display the impact of viscous heating ( Ec ) on temperature (θ(ζ )) and nanoparticle volume fraction (�(ζ )) . A significant improvement occurred in both θ(ζ ) and �(ζ ) via higher Ec . Higher Ec corresponds  www.nature.com/scientificreports/ to stronger kinetic energy which causes an enhancement in the magnitude of temperature and volume fraction of Cu nanoparticles. Figures 12 and 13 are drawn to visualize the influence of Rosseland thermal radiation ( Rd ) on temperature (θ(ζ )) and nanoparticle volume fraction (�(ζ )) . Both θ(ζ ) and �(ζ ) are upsurged for increasing numeric values of Rd . Physically, the factor of mean absorption has an inverse relation with Rd . Consequently, larger Rd implies a lower mean absorption factor which improves supplementary heat, and as a result, nanoliquid temperature is enhanced. Figures 14 and 15 are drawn to examine the influence of φ on θ(ζ ) and �(ζ ) . Enhancing tendency of temperature θ(ζ ) and nanoparticle volume fraction �(ζ ) is seen for larger values of φ . The thermal conductivity of base liquid water enhances by conveying Cu nanoparticles. Stronger thermal diffusivity is responsible for the development of thermal and solute layer structures. Figures 16 and 17 are plotted to analyze the behavior of friction factors at the plate along x and y directions ( Re 0.5 x Sf x , Re 0.5 x Sf y ) , Nusselt number Re −0.5 x Nu x and Sherwood number Re −0.5 x Sh x when Nb = Nt = 0.2, Le = 1, Pr = 6.0674, Ro = 0.5, Ec = 0.2, φ = 3%, and Rd = 0.2 except when they are diverse.  x Sf x and Re 0.5 x Sf y . Figure 16 exhibit that Re 0.5 x Sf x decreases for larger Ro and φ . A similar trend is observed for Re 0.5 x Sf y when Ro and φ are gets increased. Figure 18 designate that the Re −0.5 x Nu x is an increasing property of both Nb and Ec . Figure 19 demonstrates the impact of Nb and Ec on Re −0.5 x Sh x . It is seen that Re −0.

Final remarks
The key outcomes of the present study are: • The temperature is found to be higher due to constant heat flux condition.
• The viscous dissipation effect leads to an enhancement of nanoparticle volume fraction and temperature profiles. • The rotation of the plate exhibits decreasing behavior of velocity components.
• The nanoparticle volume fraction revelations increasing behavior of thermal and solute layer thickness.