Effects of Radiative Electro-Magnetohydrodynamics Diminishing Internal Energy of Pressure-Driven Flow of Titanium Dioxide-Water Nanofluid due to Entropy Generation

The internal average energy loss caused by entropy generation for steady mixed convective Poiseuille flow of a nanofluid, suspended with titanium dioxide (TiO2) particles in water, and passed through a wavy channel, was investigated. The models of thermal conductivity and viscosity of titanium dioxide of 21 nm size particles with a volume concentration of temperature ranging from 15 °C to 35 °C were utilized. The characteristics of the working fluid were dependent on electro-magnetohydrodynamics (EMHD) and thermal radiation. The governing equations were first modified by taking long wavelength approximations, which were then solved by a homotopy technique, whereas for numerical computation, the software package BVPh 2.0 was utilized. The results for the leading parameters, such as the electric field, the volume fraction of nanoparticles and radiation parameters for three different temperatures scenarios were examined graphically. The minimum energy loss at the center of the wavy channel due to the increase in the electric field parameter was noted. However, a rise in entropy was observed due to the change in the pressure gradient from low to high.


Introduction
In recent ages, heat transfer enhancement has gained much attention in the field of technological and industrial applications-like thermal devices-owing to the cooling rate, which highly affects the manufactured product with the desired topographies. Moreover, over the years many methodologies and techniques have been used to investigate heat exchange in fluids. An electrically conducted Poiseuille fluid flow passing through straight walls with porosity was probed by Chauhan and Rastogi [1]. The electro-magnetohydrodynamic heat transfer characteristics for an incompressible fluid by virtue of ohmic and thermal dissipations was numerically testified by Pal and Mondal [2]. Authors examined that the velocity distribution increased with the increase of the electric field parameter, while the temperature decreased with the electric field parameter. Furthermore, heat transfer can be improved by presenting nanoparticles with high heat transfer characteristics in a low volume fraction within the nanofluids. Nanofluid is a new idea of nanotechnology, which is used to enhance the property of the thermal conductivity in fluids [3]. An experimental study on titanium dioxide ρC p n f (V.∇)T = k n f ∇ 2 T + Φ + 1 Equations (1)

Formulation
The nanofluid flow model with mixed convection under gravitational force [33][34][35] could be framed as: Equations (1-3) hold: Flow vector V , current density J , gravitational acceleration vector g , dimensional temperature T , dimensional pressure p , viscous dissipation  , radiative heat flux External forces (5) ρC p n f u ∂T ∂x Heat Conductivity The allied boundary conditions are: Entropy 2019, 21, 236 4 of 23 The interrelated viscosity and thermal conductivity models [36] are: The most imperative nanofluid models for density ρ n f [37], heat capacity C p n f [38], thermal coefficient β n f [39] and electrical conductivity σ n f [40] with a nanoparticle volume fraction φ are referred for detailed study of the readers.
The Rosseland approximation for radiative heat flux q r [41] is: By using the following transformation: Equations (4)-(6) in a dimensionless form is acquired as: ∂u ∂x Where: The key parameters contain wavelength λ, non-dimensional wave number δ, non-dimensional velocity components (u,v), lower wall temperature T 1 , upper wall temperature T 2 and dimensionless temperature θ. When a fluid is moving with a constant pressure gradient with: U m = − a 2 2µ f ∂p ∂x then ∂p ∂x = P as given by Reference [42].
Under the long wavelength approximation, Equations (12)- (14) along with linked boundary in a dimensionless form are renewed as: The significant characteristics of nanoparticles with a base fluid are specified in Table 1.

Drag Force (Skin Friction)
The drag force C f [45] is well-defined by: By using Equations (11) and (15), and neglecting dimensionless flow properties, Equation (19) is reformed as:

Heat Transfer Ratio (Nusselt Number)
The Heat transfer ratio Nu, [46] may be laid out as: , with q w (wall heat flux) = k n f ∂T ∂y y=H 1 and H 2 With the same analogy, in view of Equations (11) and (15), by neglecting the flow properties in a dimensionless form of Equation (21), one has:

Analysis of Energy Loss
The local entropy generation E G in a nanofluid with effective influences of electro-magnetohydrodynamics (EMHD) and thermal radiative heat flux is described in the subsequent relation as:  The entropy generation rate E G 0 is determined by: The entropy generation number N G is signified in b.
Such that: Hence, the total entropy generation is: Br Where: The Bejan number Be can be made as: Br Ω ∂u ∂y Br From Equations (29) and (30), it follows that: Br Average entropy generation is calculated through: Here: Or: ,

Code Validation and Convergence Analysis
The velocity and temperature results in Equation (41) contain the auxiliary parameters u and θ , respectively. As pointed out by the originator of homotopy analysis method, a faster convergence can be achieved by the optimum selection of the involved auxiliary parameters [48]. Figure 2 portrays the -curves at thirtieth-order approximations for velocity and temperature, to estimate the suitable interval of convergence, that visibly predicts admissible ranges for u and θ to lie between −2.0 to 0.5 and −1.5 to 0.5.
For the optimum values of u and θ , the residual errors were computed up to thirtieth-order approximations over an embedding parameter ξ ∈ [0, 1] of velocity E u and temperature distributions E θ , by the succeeding formulas: Eventually, Figure 3 and Figure 4 bear witness that the best optimum values of the  -curves for velocity and temperature, within admissible ranges, are respectively. The residual errors for the convergence of analytical solutions are further elaborated in Table 2.
Eventually, Figure 3 and Figure 4 bear witness that the best optimum values of the  -curves for velocity and temperature, within admissible ranges, are respectively. The residual errors for the convergence of analytical solutions are further elaborated in Table 2.

Results and Discussion
The sketches of the key factors, such as the electric field, nanoparticle volume fraction, radiation and group parameter are presented for three different temperatures (15 C, 25 C, 35 C).  whereas the combined effects of the electro-magnetohydrodynamics (EMHD) produced Lorentz forces to resist the fluid velocity. Also, the thickness of the boundary layer increased with the rise of 1 E . However, in Figure 6, the opposite behavior for the fluid temperature was noted, which was due to the applied electric field. The effects of the nanoparticle volume fraction  on the fluid flow are shown in Figure 7. It could easily be examined that when the volume fraction of the nanoparticle upsurges in the base fluid, the base fluid's density increased. Subsequently, the fluid became denser, so the suspension of the particles in the fluid resulted in a reduction in nanofluid velocity. In Figure  8, the temperature distribution of the nanofluid against the radiation parameter d R is displayed.
The boundary layer thickness increased with increasing values of the radiation factor. The temperature of nanofluid could also be controlled with the radiation factor, because the fluid

Results and Discussion
The sketches of the key factors, such as the electric field, nanoparticle volume fraction, radiation and group parameter are presented for three different temperatures (15 • Figure 5, identified that the velocity gradually increased by an upturn of E 1 , whereas the combined effects of the electro-magnetohydrodynamics (EMHD) produced Lorentz forces to resist the fluid velocity. Also, the thickness of the boundary layer increased with the rise of E 1 . However, in Figure 6, the opposite behavior for the fluid temperature was noted, which was due to the applied electric field. The effects of the nanoparticle volume fraction φ on the fluid flow are shown in Figure 7. It could easily be examined that when the volume fraction of the nanoparticle upsurges in the base fluid, the base fluid's density increased. Subsequently, the fluid became denser, so the suspension of the particles in the fluid resulted in a reduction in nanofluid velocity. In Figure 8, the temperature distribution of the nanofluid against the radiation parameter Rd is displayed. The boundary layer thickness increased with increasing values of the radiation factor. The temperature of nanofluid could also be controlled with the radiation factor, because the fluid temperature was very sensitive to Rd, which meant that the heat flux of channel walls would be as large as perceived. Figures 9-14 portray the effects of E 1 , Br/Ω and Rd on N G and Be. Figures 9 and 10 show the behaviors of the electric field parameter E 1 on N G and Be. The entropy generation rate near the walls increased with the increase of the electric field parameter, as shown in Figure 9, while at the left wall, the entropy loss was greater as compared to the right wall. It is further noted that near the center of the channel, energy loss was at a minimum, between y = −0.3 and y = 0.2. This was due to the combined effects of the electro-magnetohydrodynamics, which produced Lorentz forces to resist the fluid flow. In Figure 10, The Bejan number near to the center of the channel with a large electric parameter value gradually accelerated and approached to 1, but near to the walls, a reduction in the Bejan number against large values of the electric field parameter was detected. The impacts of group parameters Br/Ω on N G and Be are shown in Figures 11 and 12. The entropy generation rate escalated with increasing values of the group parameter, as shown in Figure 11. One also noticed that the entropy generation rate at the left wall as compared to the right wall was high due to the increase in buoyancy forces in the system. The upshot of Br/Ω was visible in Figure 12. Here Be attained an extreme value, almost at y = −0.1, because of the escalation of the heat transfer irreversibility for Br/Ω = 0.2, but gradually decreased with the increase of the group parameter values. The effects of the radiation parameter Rd on the entropy generation rate are displayed in Figure 13. Here, the entropy generation was characterized by the nice concave shape and almost symmetrical profiles for all values of Rd. A small change in Rd caused a large variation of N G , as seen in Figure 13. It could also be noted that the energy loss entropy generation rate round the center of the channel was approximately zero, but as one proceeded towards the channel walls, entropy occurred. Figure 14 shows the same increasing results for the radiation parameter Rd on the Bejan number Be, as shown in the case of entropy generation. The Bejan number near the center of the channel was about to attain its extreme position for low radiation evolvement, but near the vicinity of the walls, the Bejan number increased with the growing radiation factor. The increasing results suggested that heat transfer irreversibility plays a dominant role in energy loss.          Figure 9 and Figure 10 show the behaviors of the electric field parameter 1 E on G N and Be . The entropy generation rate near the walls increased with the increase of the electric field parameter, as shown in Figure 9, while at the left wall, the entropy loss was greater as compared to the right wall. It is further noted that near the center of the channel, energy loss was at a minimum, between 0.3 y   and 0.2 y  . This was due to the combined effects of the electro-magnetohydrodynamics, which produced Lorentz forces to resist the fluid flow. In Figure 10, The Bejan number near to the center of the channel with a large electric parameter value gradually accelerated and approached to 1, but near to the walls, a reduction in the Bejan number against large values of the electric field parameter was detected. The impacts of group parameters Br  on G N and Be are shown in Figure 11 and    Figure 9 and Figure 10 show the behaviors of the electric field parameter 1 E on G N and Be . The entropy generation rate near the walls increased with the increase of the electric field parameter, as shown in Figure 9, while at the left wall, the entropy loss was greater as compared to the right wall. It is further noted that near the center of the channel, energy loss was at a minimum, between 0.3 y   and 0.2 y  . This was due to the combined effects of the electro-magnetohydrodynamics, which produced Lorentz forces to resist the fluid flow. In Figure 10, The Bejan number near to the center of the channel with a large electric parameter value gradually accelerated and approached to 1, but near to the walls, a reduction in the Bejan number against large values of the electric field parameter was detected. The impacts of group parameters Br  on G N and Be are shown in Figure 11 and   Figure 15a, phi diagrams are displayed against the magnetic parameter for different M. In Figure 15b, the phi diagrams show the performance of the electric field for different E 1 . In Figure 15c, the phi drawings deal with the nanoparticle volume fraction for different φ. In Figure 15d, the phi drawings describe the radiation parameter for different values of Rd. The effects of the magnetic parameter for different values of M are given in the phi diagrams, as shown in Figure 16a, whereas Figure 16b, show the phi diagrams against the electric field parameter for different values of E 1 . In Figure 16c, the phi diagrams depict the effects of the nanoparticle volume fraction for different values of φ, while Figure 16d, show the effects of the radiation parameter for different values of Rd via phi diagrams. In all phi diagrams, it was determined that when the pressure gradient increased, the average entropy loss and consequently entropy generation increased in the system. Thus, one can say that the reported results about electro-magnetohydrodynamics (EMHD), thermal radiation and entropy generation on Poiseuille flow with Titanium dioxide nanoparticles are very effective to reduce the energy losses and escalate the heat transfer in wavy surfaces. The said analysis is very informative for food industries, as in the presence of titanium dioxide in the consumer packaging, which helps to preserve food for a considerable time period. the right wall was high due to the increase in buoyancy forces in the system. The upshot of Br  was visible in Figure 12. Here Be attained an extreme value, almost at 0.1 y   , because of the escalation of the heat transfer irreversibility for 0.2 Br   , but gradually decreased with the increase of the group parameter values. The effects of the radiation parameter d R on the entropy generation rate are displayed in Figure 13. Here, the entropy generation was characterized by the nice concave shape and almost symmetrical profiles for all values of d R . A small change in d R caused a large variation of G N , as seen in Figure 13. It could also be noted that the energy loss entropy generation rate round the center of the channel was approximately zero, but as one proceeded towards the channel walls, entropy occurred. Figure 14 shows the same increasing results for the radiation parameter d R on the Bejan number Be , as shown in the case of entropy generation. The Bejan number near the center of the channel was about to attain its extreme position for low radiation evolvement, but near the vicinity of the walls, the Bejan number increased with the growing radiation factor. The increasing results suggested that heat transfer irreversibility plays a dominant role in energy loss.   The numeric features of C f and Nu on both opposite walls-with respect to three different temperature/conditions, as suggested by Duangthongsuk and Wongwises [36]-against different values of the nanoparticle, volume fraction, electric element and magnetic factor, are calculated in Tables 3 and 4, respectively. It could be noted that the skin friction reduced at the lower wall, with increasing values of φ, E 1 and M, while the opposite effects occurred at the wall of the concerned parameters. In heat transfer phenomena, the heat rate increased at the lower wall but decreased at the upper wall, with large values of φ, E 1 and M.         Table 3. Numeric attributes of C f on opposite walls with respect to three different temperature/ conditions against different points of φ, E 1 and M when Gr = 2.0 and Rd = 1.0.

Conclusions
The electro-magnetohydrodynamics (EMHD) and entropy generation on the Poiseuille flow synthesis with nanoparticles through a wavy channel were investigated here. The most vital findings were: 1) The electric field E 1 applied on a tangential direction to the fluid affected both the velocity and temperature distributions, which produced a reduction in the temperature and an increase in the velocity.
2) The suspension of nanoparticles φ in the base fluid caused a slowdown in nanofluid velocity.
3) The thermal boundary layer increased against the growing radiation parameter Rd, which was why an increase in temperature was observed. 4) The entropy generation near the boundary of the channel prolonged, while was very insufficient at the vicinity of the center for the electric field E 1 .

5)
Initially, Be attained a high impact near the middle of channel, but gradually it fell for a large value of the electric field parameter near the walls. 6) The entropy generation for the group parameter BrΩ −1 and the radiation parameter Rd at the intermediate of the channel was approximately zero, while an enhancement was noted near the walls. 7) The average energy loss was due to a rise in the pressure gradient.

Conflicts of Interest:
The authors declare no conflict of interest.