Activation energy and cross-diffusion effects on 3D rotating nanofluid flow in a Darcy–Forchheimer porous medium with radiation and convective heating

Nomenclature

a

constant

Bi_c

solute Biot number

Bi_T

thermal Biot number

C

solute concentration

Cf

skin friction

C b

drag co-efficient

c p

specific heat at constant temperature

c _s

concentration susceptibility

Df

Dufour parameter

D _m

solute diffusivity

D _T

thermophoretic diffusion coefficient

E

activation energy parameter

E a

activation energy

f, g

non-dimensional velocities

F

inertia co-efficient

Fr

Forchheimer number

Hg

heat generation parameter

h _T

heat transfer coefficient

K p

porous medium permeability

K r

chemical reaction rate constant

k T

thermal diffusion ratio

k #

mean absorption coefficient

n

fitted rate constant

Nb

Brownian motion parameter

Nt

thermophoresis parameter

Nu

Nusselt number

Pr

Prandtl number

q r

heat flux due to thermal radiation

Q

heat generation

Re _x

Reynolds number

Rd

radiation parameter

Sc

Schmidt number

Sh

Sherwood number

Sr

Soret parameter

T

temperature

T _m

mean fluid temperature

u , v , w

velocities in x , y , z directions

x , y , z

Cartesian coordinates

1 Introduction

Heat energy transfer is crucial to many aspects of our daily life, from small-scale to large-scale applications. Through friction, chemical reactions (CRs), or electric losses, heat energy has been produced. Either all the heat must be removed (cooling) or it must be needed in specific areas as efficiently as feasible [1,2]. Thus, thermal management plays a vital role in an extensive range of technical applications, including bearing cooling in wind power plants, cooling electronics, and transportation (brake and radiator cooling) [3,4]. For optimal heat transfer, the medium needs to be highly conducive to efficient transmission. Because traditional media with low heat conductivity, such as water, synthetic oil, molten salt, and ethylene glycol, were developed, scientists created fluid–solid mixtures known as nanofluids [5,6,7]. In other words, liquids are blended with nano-scale (metallic, non-metallic, and polymeric) particles to improve their thermal characteristics and facilitate heat transfer [8,9,10]. Mandal and Pal [11] investigated the mixed convection flow of a hybrid nanofluid over an exponentially shrinking Riga surface. Their research focused on analyzing entropy generation and examining the stability of the flow.

The impact of CR and activation energy (AE) on the Casson nanoliquid stream flowing past a stretched surface with non-linear radiation was investigated by Gireesha et al. [12]. They investigated the effects of heat sources that are thermally dependent and exponentially space-dependent. Using the method of homotopy analysis, Vinodh et al. [13] identified magneto-hydrodynanic (MHD) Casson hybrid nanoliquid current over a moving needle under the impact of Dufour/Soret effects and thermal radiation. When the Casson parameter is raised by 20%, skin friction rises by up to 45%. In the occurrence of AE and CR, Acharya and Das [14] observed the incompressible MHD rotating hybrid-nanofluid (kerosene oil–Al₂O₃–Cu) flow over a stretchable sheet. They concluded that whereas temperature rises as radiation parameter values rise, the opposite phenomenon is seen for concentration when CR values rise. A radially stretched plate created a time-dependent, axisymmetric, chemically reactive Williamson nanofluid flow in the porous medium with Joule heating, AE, and viscous dissipation, according to Saini et al. [15]. The temperature is thought to rise with increasing Brownian motion and temperature ratio parameters, and to decrease with increasing velocity slip. The magneto-convection of a hybrid (methanol, CuO, and MgO) nanofluid flow was examined by Vijay and Sharma [16] using a spinning disk that was decelerating and included Ohmic heating, Soret, and Dufour consequences. Here, we take into consideration the temperature-dependent viscosity of hybrid nanoliquid. The influence of CR and Joule heating on a hybrid nanoliquid across a stretchy sheet with entropy generation is investigated by Joyce et al. [17]. They discovered that raising the slip parameter causes a hybrid nanofluid's flow rate to drop. Using the differential transform method, Chandrapushpam et al. [18] investigated double diffusive MHD squeezed nanoliquid flow in Darcy porous media in the occurrence of CR. They detected that the squeeze number supports mass transfer rate, and it reduces the skin friction when increasing the Biot number.

It is essential to comprehend AE, as it is a prerequisite for nearly all chemical processes. AE aids in our understanding of the energy required for a CR, which governs both the environment and our behaviour [19]. The AE with CR exists in mass and heat transport and has applications in emulsions of multiple suspensions, chemical technology, geothermal reservoirs, food and material processing, etc. [20,21]. The free convective boundary layer stream in porous media with simultaneous mass and heat transfer was studied by Bestman [22] with an AE and a CR. Huang [23] used the Keller box method to quantitatively assess the impact of Arrhenius AE on free convection around a permeable horizontal cylinder maintained at a constant wall concentration and temperature. The effects of the magnetic field, CR, and AE on nanofluid flow over a flexible disc were investigated by Kotresh et al. [24]. It is shown that higher solid volume fraction values improve the drag coefficient and reduce the heat transfer rate. Yesodha et al. [25] investigated the effects of AE and Dufour/Soret effects on the properties of mass and heat transfer of a chemically reacting viscous fluid in a rotating frame. They discovered that the Dufour and Soret parameters both result in a drop-in concentration. The mixed MHD convective non-Newtonian nanoliquid flow with AE and thermal radiation was studied by Younus and Lakshmi [26]. They discovered that local mass transfer decreases with increasing AE and increases with rising CR rate.

The Dufour effect describes how variations in solute concentration affect energy fluxes. The Soret effect refers to the solute fluxes that are impacted by temperature gradients [27]. The heat and solute transfer are accelerated by the emergence of cross diffusion (Dufour and Soret) effects [28,29,30]. The influence of Dufour and Soret effects on MHD convection stagnation-point stream of chemically reacting fluids with radiation and slip condition was inspected numerically by Niranjan et al. [31]. The problem of unsteady MHD convection of an electrically conducting liquid across an (inclined) plate immersed in a porous material in the existence of the Soret effect, a CR, a Hall current, and an aligned magnetic field was solved exactly analytically by Raghunath and Mohanaramana [32]. The Dufour and Soret effects on magneto-convection nanoliquid stream with slip and thermal radiation effects over a cylinder were studied by Jagan and Sivasankaran [33]. It is concluded that while local heat transmission decreases with increasing Dufour number, local mass transfer increases. Geetha et al. [34] evaluated the effects of Dufour and Soret on an MHD bioconvection stream in a channel with a CR. They discovered that increasing thermal radiation induces a progressive rise in heat transport.

In many technical applications, including power plants, re-entry vehicles, high-speed aircraft, and operations involving high temperatures, the impact of thermal radiation on convective flow is significant [35,36] (See, Sharma et al. [35,36], Yasmin et al. [37]). Numerous studies have extensively utilised of the radiative heat fluxes approximated by the Rosseland approximation. The nanoliquid flow via a vertical Riga plate in the occurrence of suction, slip, thermal radiation, and CR was described by Rawat et al. [38]. It was discovered that as thermal radiation improves, temperature and velocity also rise. The Casson fluid's entropy generation over an exponential stretchy surface with thermal radiation, heterogeneous-homogeneous reaction, and slip effects was evaluated by Das et al. [39]. They concluded that a rise in Brinkman number reduces the overall development of entropy. Sivasankaran et al. [40] explored the impact of thermal radiation on the buoyant convection of Casson fluid in a closed domain with various thermal sources. The effects of (linear and nonlinear) radiation on Maxwell fluid flow over a stretchy sheet with convective heating, AE, and viscous dissipation were investigated by Gangadhar et al. [41]. They concluded that as the Biot number rises, the thermal boundary layer improves. Mandal and Pal [42] investigated how thermal radiation influences the convective flow of a nanofluid when a uniform external magnetic field and slip condition are applied. The effects of radiation and CRs with continuous heat and mass flux of convective flow over a vertically placed plate in a porous medium were analytically explored by Choudhury and Sharma [43]. It is known that temperature and velocity decrease as radiation parameter increases. Additionally, it is discovered that when the radiation parameter values increase, the plate's internal friction decreases. Gangadhar et al. [44] conducted a numerical study on the influence of a magnetic field and thermal radiation on the flow of a second-grade nanofluid over a Riga plate with convective boundary heating. Their results showed that the temperature distribution increases with higher values of the radiation parameter and Biot number.

We can infer from the literature that no research has been conducted to examine how AE, radiation, Dufour, and Soret effects influence convective mass and heat transport. Therefore, these effects are investigated numerically in the current work on three-dimensional doubly diffusive rotating nanoliquid flow in Darcy–Forchheimer porous media in the presence of AE, radiation, Dufour, and Soret effects in a rotating surface.

2 Mathematical formulation

We consider a three-dimensional laminar, steady, incompressible flow over a rotating frame inserted into a porous medium as shown in Figure 1. The frame is rotating at an angular velocity “Ω.” The heat and solute transfer is included in the flow to examine the factors. The internal heat generation, thermo-diffusion (Soret effect), and diffusion-thermo (Dufour effect) effects (that is, the cross diffusion consequence) are included here. The thermal radiation is taken into account based on Rosseland approximation and the medium is optically thick. An AE-driven CR is taking place. The non-Darcy–Forchheimer porous model is taken and it supports the nonlinear flow behaviour with high velocities. The isotropic porous matrix is thermally equilibrium with local liquid. The surface of the plane z = 0 is stretched at a stretching rate a ( > 0 ) . The velocity components u , v , and w are located along the x, y, and z axes, respectively. Convective heating and salting are applied on the stretchy surface. The following assumptions are taken into the study.

• The flow is incompressible, steady, and laminar.

• The surface is stretched with uniform velocity U 0 .

• The porous medium is isotropic and thermally equilibrium with local fluid.

• The medium is optically thick and thermal radiation is considered.

• The AE-driven CR is taking place.

Figure 1

Physical model.

The governing models are [45]:

The solute diffusivity ( D m ), porous medium permeability ( K p ), thermal diffusivity ( α m ), kinematic viscosity ( ν ), density ( ρ ), specific heat ( c p ), drag co-efficient ( C b ), and non-uniform inertia co-efficient F = C b K 1 / 2 are physical quantities in the above-mentioned model. The term K r 2 T T ∞ n e − E a κ T ( C − C ∞ ) denotes the Arrhenius function with AE ( E a ), CR rate constant ( K r 2 ), Boltzmann constant ( κ = 8.61 × 10 − 5 eV / K ), and fitted rate constant ( n ).

The heat flux due to thermal radiation, based on the Rosseland approximation, is defined as

where k # & σ * stand for the mean absorption coefficient and the Stefan–Boltzmann constant, respectively. It is acknowledged that the temperature differential within the stream is too tiny to allow T 4 to be expressed by Taylor’s series. Extending T 4 about T ∞ and ignoring higher level terms, we can then find T 4 ≅ 4 T T ∞ 3 − 3 T ∞ 4 . Then, substitute this into Eq. (5) to obtain the energy equation.

The values at the boundary are

Here, T ∞ and C ∞ are ambient (free stream) temperature and concentration, respectively.

The subsequent variables are presented to transform the governing model.

The continuity Eq. (1) is usually satisfied. The model (2–6) becomes

The valid constrains at surface are:

where E = E a κ T ∞ is the activation energy, Hg = Q a ρ c p is the heat generation parameter, F r = C b K 1 / 2 is inertia coefficient, Pr = ν α is the Prandtl number, Rd = 4 σ * T ∞ 3 k k # is the parameter for radiation, β = Ω a is the rotational parameter, λ = ν K p a represents the permeability parameter, Sc = ν D m is the Schmidt number, Df = D m k T ( C w − C ∞ ) C s C p ( T w − T ∞ ) υ is the Dufour number, Sr = D m k T ( T w − T ∞ ) T m υ ( C w − C ∞ ) is the Soret number, δ = T w − T ∞ T ∞ is a parameter for temperature difference, and σ = K r 2 α is the dimensionless reaction rate.

The Nusselt number (Nu) indicates the amount of heat transferred through a fluid as a result of convection. The Sherwood number (Sh) indicates the rate of solute mass transfer across the system. The skin friction (Cf) factor derives the rate of frictional (drag) force along the surface. All the quantities are defined as follows:

The quantity of heat transfer is indicated by the Nusselt number, Nu x = xq w k ( T w − T ∞ ) with q w = − k T ∂ T ∂ z | z = 0 . The solute transfer across the system is indicated by the Sherwood number, Sh x = xj w D ( C w − C ∞ ) with j w = − D ∂ C ∂ z | z = 0 . The rate of frictional (drag) force along the surface is determined by the skin friction factor, C f = VF w ρ U w 2 with VF w = μ ∂ u ∂ z . Finally, we have the dimensionless form of physical quantities:

where R e x = U 0 x v denotes local Reynolds number.

3 Method of solution

The solutions of the governing model are needed to understand the physical phenomena clearly for predicting and controlling numerous engineered systems. The governing model of equations is quite difficult to solve due to the possibility of extremely chaotic behaviour in the solution. The closed form of solutions cannot be established because of the nonlinear nature of the system of equations. As such, we want to find a numerical solution for the governing model. Using the shooting approach, the Runge–Kutta method is implemented to solve the equations. Since a crucial component of the numerical solution is compared with the numerical values, the current computational code is compared to the previous findings, as found by Rashid et al. [45]. It is displayed in Table 1. It is detected that the present results agree well with the existing data, and it gives confidence in our calculations.

4 Results and discussion

The three-dimensional doubly diffusive convective rotating nanoliquid stream in a Darcy–Forchheimer porous medium under the influence of AE, thermal radiation, Dufour and Soret effects is explored numerically under various groupings of appropriate parameters involved herein. The values of the parameter fall into the following range with Pr = 6.7 and Sc = 1 . The rotational parameter ( β ) varies ( 0 ≤ β ≤ 2 ) , radiation parameter ( Rd ) varies ( 0 ≤ β ≤ 2 ) , inertia parameter (Fr) varies ( 0 ≤ Fr ≤ 1.5 ) , Soret number (Sr) varies ( 0 ≤ Sr ≤ 0.8 ) , the Dufour number (Df) varies ( 0 ≤ Df ≤ 0.5 ) , permeability parameter ( λ ) ranges from 0 to 8, AE parameter (E) varies ( 0 ≤ E ≤ 3 ) , thermal Biot number ( Bi T ) and solutal Biot number ( Bi c ) take values from 0 to 1, heat generation/absorption (Hg) takes values ( − 0.5 ≤ Hg ≤ 0.5 ) , Brownian motion parameter ( 0 ≤ Nb ≤ 1.5 ) , thermophoresis parameter ( 0 ≤ Nt ≤ 0.3 ) , ( − 1 ≤ n ≤ + 1 ) , reaction rate takes values ( 0 ≤ σ ≤ 2 ) , and parameter for temperature difference takes values ( 0 ≤ δ ≤ 7 ) .

The effect of the rotational parameter on stream, thermal, and solutal fields is shown in Figure 2(a) and (c). It is evident that the rotational parameter values are strengthened by the stream speed. In other words, the fluid velocity is suppressed by the strength of the rotation. Conversely, the thermal field is supported by the rotational strength, that is, when “β” increases, the temperature also rises within the boundary layer regime. Unlike thermal fields, solutal field is unaffected. Nonetheless, “β” has a very noticeable effect on concentration. As the rotational strength grows, so does the solute concentration, see Figure 2(c). The impact of the inertia parameter on temperature, concentration, and velocity is depicted in Figure 3(a)–(c). Temperature, velocity, and solute distributions have little effect on altering Fr values.

Figure 2

Velocity, temperature and concentration profiles for different values of β with Pr = 6.7, Sc = 1, λ = 0.2, E = 0.5, n = 0.5, Fr = 0.1, σ = 0.5, Bi_T = 0.3, Bi_C = 0.3, Hg = 0.1, δ = 0.5, Nb = 0.3, Nt = 0.1, Sr = 0.2, Df = 0.2 and Rd = 0.5.

Figure 3

Velocity, temperature and concentration profiles for different values of Fr with Pr = 6.7, Sc = 1, λ = 0.2, E = 0.5, n = 0.5, β = 0.5 , σ =0.5, Bi_T = 0.3, Bi_C = 0.3, Hg = 0.1, δ = 0.5, Nb = 0.3, Nt = 0.1, Sr = 0.2, Df = 0.2 and Rd = 0.5.

The determination of permeability parameter ( λ ) on a stream, thermal, and solutal fields is depicted in Figure 4(a)–(c). Permeability has a very strong impact on fluid flow and thermal and solutal distributions inside the boundary layer regime. As the permeability parameter values increase, the velocity of the fluid particles declines. Conversely, temperature and concentration rises with increasing permeability parameter values. The permeability of the porous matrix decreases, resulting in a slowdown of the stream flow rate. However, the porous structure enhances the thermal exchange and increases the temperatures inside the boundary layer. The concentration exhibits a peculiar effect with permeability parameter. The solute concentration declines near the boundary and it enhances slightly away from boundary, which is clearly seen from Figure 4(c). Figure 5(a) and (b) presents the outcome of thermal Biot number on solute and temperature profiles. The thermal Biot number supports the thermal and solutal fields.

Figure 4

Velocity, temperature and concentration profiles for different values of λ with Pr = 6.7, Sc = 1, Fr = 0.2, E = 0.5, n = 0.5, β = 0.5 , σ = 0.5, Bi_T = 0.3, Bi_C = 0.3, Hg = 0.1, δ = 0.5, Nb = 0.3, Nt = 0.1, Sr = 0.2, Df = 0.2 and Rd = 0.5.

Figure 5

Temperature and concentration profiles for different values of Bi_T with Pr = 6.7, Sc = 1, Fr = 0.2, E = 0.5, n = 0.5, β = 0.5 , σ = 0.5, λ = 0.2, Bi_C = 0.3, Hg = 0.1, δ = 0.5, Nb = 0.3, Nt = 0.1, Sr = 0.2, Df = 0.2 and Rd = 0.5.

Figure 6(a) and (b) shows the influence of solutal Biot number on thermal and solutal distributions with β = E = σ = δ = n = Rd = 0.5 , Bi T = Nb = 0.3 , λ = Fr = 0.2 , Nb = 0.3, Nt = Hg = 0.1 , Sr = Df = 0.2 . Both thermal and solute concentration improve when rising the values of Bi C . Both Biot numbers ( Bi T and Bi C ) provide a remarkable impact on thermal and concentration fields. Figure 7(a) and (b) examines how temperature and concentration are affected by the Brownian motion parameter with Rd = 0.5, Hg = Nt = 0.1 , Bi T = Bi c = 0.3 , Fr = 0.2 , E = n = β = δ = σ = 0.5 , and λ = Sr = Df = 0.2 . The Brownian motion parameter shows a strong impact on the thermal field compared to that on the solute field, which is clearly indicated in Figure 7(a) and (b). It is also found that concentration and temperature exhibit opposite trends when rising the values of Nb. That is, temperature enhances and solute concentration declines on rising the values of Nb. Figure 8(a) and (b) illustrates the influence of thermophoresis parameter (Nt) on solute concentration and temperature fields with n = E = δ = β = σ = 0.5 , Nb = Bi T = Bi c = 0.3 , Hg = 0.1, Fr = λ = Sr = Df = 0.2 , and Rd = 0.5. Comparing both figures, it is understood that thermophoresis parameter has a greater impact on the solute profile at higher values of Nt. There is an overshoot of the solute profile as the values of Nt continue to rise. The solute concentration enriches when the values of Nt are raised. On the contrary, there is no considerable effect on the temperature profile on changing the values of Nt.

Figure 6

Temperature and concentration profiles for different values of Bi_C with Pr = 6.7, Sc = 1, Fr = 0.2, E = 0.5, n = 0.5, β = 0.5 , σ = 0.5, Bi_T = 0.3, λ = 0.2, Hg = 0.1, δ = 0.5, Nb = 0.3, Nt = 0.1, Sr = 0.2, Df = 0.2 and Rd = 0.5.

Figure 7

Temperature and concentration profiles for different values of Nb with Pr = 6.7, Sc = 1, Fr = 0.2, E = 0.5, n = 0.5, β = 0.5 , σ = 0.5, Bi_T = 0.3, Bi_C = 0.3, Hg = 0.1, δ = 0.5, λ = 0.2, Nt = 0.1, Sr = 0.2, Df = 0.2 and Rd = 0.5.

Figure 8

Temperature and concentration profiles for different values of Nt with Pr = 6.7, Sc = 1, Fr = 0.2, E = 0.5, n = 0.5, β = 0.5 , σ = 0.5, Bi_T = 0.3, Bi_C = 0.3, Hg = 0.1, δ = 0.5, Nb = 0.3, λ = 0.2, Sr = 0.2, Df = 0.2 and Rd = 0.5.

Figure 9(a) and (b) shows the impact of “ δ ” on the profile of θ and ϕ for E = n = β = Rd = σ = 0.5, Bi T = Bi C = 0.3 , Sr = Df = 0.5, λ = 0.2. Both temperature and solute decline as the values of “ δ ” increase. The influence of AE parameter (E) on thermal and solutal distribution is illustrated in Figure 10(a) and (b) for σ = n = β = Rd = δ = 0.5 , r = λ = 0.2, Sr = Df = 0.5. The impact of AE is more pronounced on solutal transport than that on thermal transport because the parameter “E” is directly involved in the solute transport equation. The solute profile enhances with an increase in the AE parameter values. A slight increase in temperature is observed as the value of E rises. The influence of σ on temperature and concentration is clearly seen in Figure 11(a) and (b) with Fr = 0.2, β = n = δ = E = 0.5 , λ = Sr = Df = 0.2 , Bi T = Bi C = 0.3 , Hg = 0.1, Nt = 0.1, Nb = 0.3, and Rd = 0.5. The solute transport shows more variation as the “ σ ” values increase. That is, the solutal distribution declines with increasing “ σ ” values and and a corresponding decrease in temperature is also observed.

Figure 9

Temperature and concentration profiles for different values of δ with Pr = 6.7, Sc = 1, Fr = 0.2, E = 0.5, n = 0.5, β = 0.5 , σ = 0.5, Bi_T = 0.3, Bi_C = 0.3, Hg = 0.1, λ = 0.2, Nb = 0.3, Nt = 0.1, Sr = 0.2, Df = 0.2 and Rd = 0.5.

Figure 10

Temperature and concentration profiles for different values of E with Pr = 6.7, Sc = 1, Fr = 0.2, λ = 0.2, n = 0.5, β = 0.5 , σ =0.5, Bi_T = 0.3, Bi_C = 0.3, Hg = 0.1, δ = 0.5, Nb = 0.3, Nt = 0.1, Sr = 0.2, Df = 0.2 and Rd = 0.5.

Figure 11

Temperature and concentration profiles for different values of σ with Fr = 0.2, E = 0.5, n = 0.5, β = 0.5 , λ = 0.2, Bi_T = 0.3, Bi_C = 0.3, Hg = 0.1, δ = 0.5, Nb = 0.3, Nt = 0.1, Sr = 0.2, Df = 0.2 and Rd = 0.5.

Figure 12(a) and (b) provides the internal heat generation (Hg) effects on the fields of temperature and solutal concentration. It is evidently seen that the internal heat generation boosts up the temperature within the system, resulting in an increase in thermal profiles. The concentration gets less important with Hg. The power of Dufour parameter on thermal and solutal fields is displayed in Figure 13(a) and (b) with Fr = 0.2, β = δ = E = σ = n = 0.5 , σ = 0.5, Bi T = Bi C = 0.3 , Hg = 0.1, Nt = 0.1, Sr = λ = 0.2, Nb = 0.3, and Rd = 0.5. Figure 13(a) shows the Dufour number yields more impression on thermal field. That is, the temperature of the system increases with the increase in the Dufour parameter. On the other side, an inconsiderable impact is found on solutal distribution with Df. Figure 14(a) and (b) provides the outcome of the Soret parameter on θ and ϕ , with λ = Fr = Df = 0.2 , β = n = σ = E = δ = 0.5 , Bi T = Bi C = 0.3 , Hg = 0.1, δ = 0.5, Nb = 0.3, Nt = 0.1, and Rd = 0.5. The solute distribution enhances with Soret number. An overshoot is detected near the boundary for higher values of Sr. No noteworthy fact on temperature is discovered on changing the Sr values. Figure 15(a) and (b) provides the consequences of thermal radiation on θ and ϕ . The radiation parameter strongly influences the thermal field which is clearly seen from the temperature profiles in Figure 15(a). The temperature increases significantly with increasing thermal radiation values. However, no notable impact on ϕ is observed with Rd values.

Figure 12

Temperature and concentration profiles for different values of Hg with Fr = 0.2, E = 0.5, n = 0.5, β = 0.5 , σ = 0.5, Bi_T = 0.3, Bi_C = 0.3, λ = 0.2, δ = 0.5, Nb = 0.3, Nt = 0.1, Sr = 0.2, Df = 0.2 and Rd = 0.5.

Figure 13

Temperature and concentration profiles for different values of Df with Fr = 0.2, E = 0.5, n = 0.5, β = 0.5 , σ = 0.5, Bi_T = 0.3, Bi_C = 0.3, Hg = 0.1, δ = 0.5, Nb = 0.3, Nt = 0.1, Sr = 0.2, λ = 0.2, and Rd = 0.5.

Figure 14

Temperature and concentration profiles for different values of Sr with Fr = 0.2, E = 0.5, n = 0.5, β = 0.5 , σ = 0.5, Bi_T = 0.3, Bi_C = 0.3, Hg = 0.1, δ = 0.5, Nb = 0.3, Nt = 0.1, λ = 0.2, Df = 0.2 and Rd = 0.5.

Figure 15

Temperature and concentration profiles for different values of Sr with Pr = 6.7, Sc = 1, Fr = 0.2, E = 0.5, n = 0.5, β = 0.5 , σ = 0.5, Bi_T = 0.3, Bi_C = 0.3, Hg = 0.1, δ = 0.5, Nb = 0.3, Nt = 0.1, λ = 0.2, Df = 0.2 and Sr = 0.2.

Table 2 provides the impact of − f ″ ( 0 ) , − g ″ ( 0 ) , − θ ′ ( 0 ) , − ϕ ′ ( 0 ) values with different combinations of pertinent parameters involved in the study. The skin friction ( − f ″ ( 0 ) ) increases with increasing the values of Fr, λ , β . − g ″ ( 0 ) declines with increasing the values of Fr, λ and it rises on strengthening “ β ” values. The local heat energy transport suppresses on strengthening the values of the inertia parameter (Fr), permeability parameter ( λ ), and rotational parameter ( β ). The rate of local solute transfer diminishes with increasing the values of λ . Tables 3 and 4 demonstrate the consequences of local heat and mass transfer rate under several groupings of appropriate parameters presented in the research. Both Biot numbers have opposite behaviours to local heat/mass transport rates. That is, the heat (mass) transport enhances (declines) with increasing the thermal Biot number. On the contrary, the local heat (solute) transfer suppresses (augments) on strengthening the solutal Biot number. The AE parameter supports local energy transfer and it suppress local mass transport. The Brownian motion of the particle boosted-up the local mass transport within the system. With increasing the value of Df and Sr, the local heat transfer rate decreases. The local heat energy transfer decreases with higher values of Hg, Nb, Nt, n, and Rd. The local mass transfer augments as the values of n, Sr, Rd, Hg increase. However, the value of − ϕ ′ ( 0 ) decreases with increasing the Nt and Df values.

5 Conclusion

The influence of AE, thermal radiation, cross diffusion (Dufour & Soret) effects on three-dimensional doubly diffusive convective rotating nanoliquid stream in a Darcy–Forchheimer porous medium is discovered numerically under several groupings of parameters involved. The following are deduced from the study.

• The strength of the rotation suppresses the fluid velocity while supports the thermal and solutal fields.

• The temperature enhances largely on rising the thermal radiation, Dufour values.

• The AE is more pronounced on solutal transport than that of on thermal transport.

• The skin friction ( − f ″ ) rises on increasing the values of Fr, λ , β . The skin friction ( − g ″ ) deteriorates by rising the values of Fr, λ and it rises on strengthening “ β ” values.

• The local heat transfer drops on raising the values of Hg, Nb, Nt, n and Rd values.

• The both Biot numbers makes opposite behaviour on local heat & mass transport rate.

• The local solute transfer augments when increasing the n, Sr, Rd, & Hg values.

• The rate of heat transport suppresses on strengthening the values of inertia parameter, permeability parameter, and rotational parameter.

• The results may applicable to the thermal systems involving cooling of electronic equipment, food processing, paper and sheet production.