Nomenclature
C,	concentration;	T1eiωt,	temperature at the lower plate;
C1eiωt,	concentration at the lower plate;	T2eiωt,	temperature at the upper plate;
C2eiωt,	concentration at the upper plate;	C*,	dimensionless concentration,
T*,	dimensionless temperature,		mass transfer rate;
υ,	kinematic viscosity;	Rd,	radiation parameter,
h,	distance between parallel plates;	Nb,	Brownian motion parameter,
Ec,	Eckert number,	Nt,	thermophoresis parameter,
p,	fluid pressure;	Br,	Brinkman number, Pr· Ec;
Da,	Darcy parameter;	Sc,	Schmidt number,
k1,	permeability parameter;	V1eiωt,	injection velocity at the lower plate;
Re,	Reynolds number,	V2eiωt,	suction velocity at the upper plate;
Ha,	Hartmann number;	a,	suction-injection ratio;
R,	ideal gas constant;	T,	temperature;
D,	molecular diffusion coefficient;	t,	time;
Sh,	Sherwood number,	i, j,	unit vectors along X- and Y-directions, respectively;
Pr,	Prandtl number,	u,	X-direction velocity component;
Grs	solutal Grashof number,	v,	Y-direction velocity component.
Grt,	thermal Grashof number,
Greek Letters
β1,	second grade fluid parameter,	α1,	thermal diffusivity,
λ,	nondimensional coordinate,	λ1,	diffusive constant parameter,
Ω,	nondimensional temperature difference parameter,	ζ,	dimensionless axial variable,
α,	nondimensional concentration difference parameter,	ψ,	nondimensional frequency parameter, ωt.

1 Introduction

In the last few decades, many researchers have been attracted to study the problems of non-Newtonian fluid flows due to their wide range of applications in industry and engineering. These may include polymeric liquids, soaps, shampoos, tomato paste, paints, paper production, glass fiber, slurry transport, petroleum industry etc. The present second grade fluid is a subclass of differential type non-Newtonian fluids, which was first introduced by Rivlin and Ericksen^[1]. Rajagopal^[2] discussed the creeping flow of homogeneous second grade fluids with various properties. Massoudi and Phuoc^[3] considered the second grade fluid flows down a heated plane where the viscosity is assumed to be dependent on temperature. Donald-Ariel^[4] obtained analytical solutions of two different problems of viscoelastic second grade fluids, one being two parallel flat walls, and the other an annulus. Emin-Erdogǎn and Erdem-Imrak^[5] considered an unsteady laminar unidirectional second grade fluid flow and discussed the various flow characteristics in six cases. Hayat et al.^[6] discussed the second grade fluid flow induced by a stretching surface and used the homotopy analysis method for solutions. Shrestha^[7] examined magnetohydrodynamics (MHD) Newtonian fluid flows through a porous channel. Raftari et al.^[8] analyzed an MHD viscoelastic second grade fluid flow and heat transfer confined between two parallel porous flat plates and used the homotopy analysis method for a solution. Ramzan and Bilal^[9] examined MHD viscoelastic second grade nanofluid flows over a porous stretching sheet under the influence of radiation and mixed convection. Labropulu et al.^[10] considered the stagnation point flows of second grade fluids over an infinite flat plate. Hayat et al.^[11] discussed the squeezing flow of the second grade fluid flow between two parallel plates. Cheng^[12] studied the fully developed steady laminar natural convective micro polar fluids between two vertical plates. Abdulaziz and Hashim^[13] described the steady free convective flow, heat and mass transfer of a micro polar fluid between porous vertical plates and used the homotopy analysis method for solutions. Chamkha et al.^[14] examined an unsteady MHD free convection flow of a micro polar fluid in a heated vertical porous plate under the influence of chemical reaction, Joule-heating and thermal radiation and obtained a solution by using an implicit finite-difference approach. Singh and Gorla^[15] provided a steady laminar natural convective fluid flow past a vertical porous plate under the influence of joule heating, Hall current and thermal diffusion. Hayat et al.^[16] analyzed the effects of thermophoresis and Brownian motion on three-dimensional steady incompressible MHD radiative couple stress nanofluids. Hayat et al.^[17] addressed two-dimensional MHD third grade fluid flows by an exponentially stretching sheet under the influence of thermal radiation and chemical reaction. Ahmed et al.^[18] observed the viscous fluid flow over an asymmetric channel with the non-linear thermal radiation effect, obtained numerical solutions by Galerkin's method and the Runge-Kutta Fehlberg method, and found good agreement between the two methods. Sudarsana-Reddy et al.^[19] reported mixed convective MHD flows, heat and mass transfer of a radiative nanofluid over a vertical plate saturated by a porous medium with thermophoresis and Brownian motion and obtained a solution with the finite element method. Dogonchi et al.^[20] studied the two-dimensional laminar radiative MHD flow of a viscous nanofluid between parallel plates, and the reduced system of ordinary differential equations (ODEs) is solved by the Duan-Rach approach. Eegunjobi et al.^[21] carried out MHD chemically reacting mixed convective viscous fluid flows past a stretching surface in a porous medium with thermal radiation and used the fourth-fifth order Runge-Kutta-Fehlberg method with the shooting technique for solutions. Bhatti et al.^[22] investigated MHD stagnation point flows over a permeable shrinking sheet, and the reduced governing equations are solved by a combination of successive linearization and Chebyshev spectral collocation methods. Zhu et al.^[23] considered heat transfer of MHD nanofluid flows with the chemical reaction and thermal radiation. Abolbashari et al.^[24] considered the second law analysis in MHD nanofluids over a porous accelerating stretching surface, and the solution is obtained by the homotopy analysis method. Rashidi et al.^[25] analyzed the second law analysis of nanofluid flows in a rotating porous disk with the magnetic field and used the fourth-order Runge-Kutta method for solutions. Mahmoodi and Kandelousi^[26] presented the second law analysis of steady kerosene-alumina nanofluids between two horizontal parallel plates in the presence of thermal radiation, Brownian motion and thermophoresis and found out an analytical solution with the differential transform method (DTM). Srinivas et al.^[27] described the second law analysis of two immiscible radiative couple stress fluids between two parallel horizontal plates. Makinde^[28] studied the second law analysis of MHD viscous fluid flows over a flat plate. Noghrehabadi et al.^[29] examined the entropy generation of a nanofluid in the presence of Brownian motion and thermophoresis over a stretching sheet with heat generation/absorption. Shit et al.^[30] discussed the second law analysis of MHD nanofluid flows over a stretching surface with thermal radiation, Brownian motion and thermophoresis. Abolbashari et al.^[31] investigated the second law analysis of Casson nanofluid flows over a stretching surface under the influence of Brownian motion and thermophoresis with convective boundary conditions and the velocity slip. Jbara et al.^[32] presented the entropy generation analysis of natural convective viscous fluid flows inside a porous enclosure with thermal radiation. Das et al.^[33] studied the second law analysis of MHD nanofluid flows past a stretching sheet. More works on the entropy generation analysis for various fluids with different properties can be viewed in the available references^[34-46]. Presently, research is going on heat and mass transform of nanofluids, because they have a wide range of applications in industrial and technology such as textile, chemical, cooling and heating systems in petrochemical, food and other processing plants. The following authors found out the effects of thermophoresis and Brownian motion on nanofluids. Choi and Estman^[47] observed thermal conductivities of nanofluids. Buongiorno^[48] reported the nanofluid model by considering the Brownian motion and thermophoresis effects. Sheikholeslami et al.^[49] considered the steady incompressible laminar MHD nanofluid flow, heat and mass transfer between parallel plates in the presence of Brownian motion and thermophoresis and obtained the solution with the DTM. Hayat et al.^[50] provided the influence of Brownian motion and thermophoresis on unsteady incompressible MHD squeezing flows of viscous nanofluids over a porous stretching sheet. Ramana-Reddy et al.^[51] noticed that an unsteady electrically conducting flow, heat and mass transfer of nanofluid over a slandering stretching sheet with slip effects. Guha and Samanta^[52] analyzed a steady MHD free convective viscous fluid flow past a semi-infinite plate under the influence of thermophoresis and Brownian motion. Awad et al.^[53] discussed the effects of thermophoresis and Brownian motion on an unsteady laminar Oldroyd-B nanofluid flow over a stretching sheet, and the reduced governing flow field equations are solved with the spectral relaxation method. Qayyum et al.^[54] worked on nonlinear convective MHD Jeffrey nanofluid flows over a stretching surface with thermal radiation and chemical reaction. To the best of our knowledge, no such attempt has been made in studying the entropy generation of a second grade nanofluid with thermophoresis and Brownian motion between parallel plates. With the motivation from the above analysis in mind, the aim of the present study is to investigate the second law analysis in free convective viscoelastic second grade nanofluid flows confined between parallel plates through a porous medium in the presence of thermal radiation, thermophoresis and Brownian motion. The flow is generated by periodic suction and injection. The reduced flow field equations are solved by the shooting method with the fourth-order Runge-Kutta scheme. The nondimensional primary, secondary velocity components, temperature distribution, concentration, entropy, Bejan number, skin friction, heat transfer rate, and mass transfer rate corresponding to distinct fluid and geometric parameters are discussed and displayed in the form of graphs and tables.

2 Mathematical model

We consider an unsteady incompressible viscoelastic second grade fluid flow through a porous medium confined between two parallel plates. The two plates are placed at y=0 and y=h, and the fluid is being sucked and injected periodically through upper and lower plates in the form of V₂e^iωt and V₁e^iωt and | V₂ |≥ | V₁ |. The temperature and concentrations are periodic types at the boundaries. The X- and Y-axes are chosen along and perpendicular to the primary flow direction, respectively, as shown in Fig. 1.

The governing equations of the viscoelastic second grade fluid flow, heat and mass transfer in the presence of thermal radiation are given by^[10-11]

(1)

(2)

(3)

(4)

where q =ui +vj, ρ is the fluid density, μ is the viscosity of the fluid, k is the thermal conductivity, k₁ is the permeability of the porous medium, T is the temperature of the fluid, D_B is the Brownian diffusion coefficient, β_T and β_C are thermal and solutal expansion coefficients, respectively, and D_T is the thermophoretic diffusion coefficient. , and (ρc)_p is the heat capacity of the nanoparticle, (ρc)_f is the heat capacity of the fluid, C is the concentration of the fluid, L is the velocity gradient tensor, τ is the Cauchy's stress tensor, which is defined as

(5)

where p is the hydrostatic pressure, and a₁ and a₂ are the material moduli usually referred to as the normal stress coefficients. The symbols A₁ and A₂ are the first two Rivlin-Ericksen tensors, which are defined as

(6)

The material constants are considered by the inequalities,

and F_b is the buoyancy force defined as

(7)

The velocity, temperature and concentration components are^[7]

(8)

where , U₀ is the average entrance velocity, a is the suction injection ratio, and .

By using the Rosseland approximation, the radiative heat flux q_r is defined as

(9)

and T⁴ can be expressed by using Taylor's series,

(10)

Neglecting the higher order terms beyond the first degree of (T-T₁e^iωt) in the above equation (10), we getT⁴=T₁⁴e^iωt+4T₁³ (T-T₁e^iωt). Substitute this equation into Eq. (9), and we get

(11)

The boundary conditions of the velocity, temperature, and concentration are

(12)

Substitute Eq. (8) into Eqs. (2)-(4). Then,

(13)

(14)

(15)

(16)

(17)

where f, ϕ₁, ϕ₂, g₁, and g₂ are functions of λ to be determined, and the prime denotes the differentiation with respect to λ.

The dimensionless forms of temperature and concentration from Eq. (8) are

The nondimensional boundary conditions (12) in terms of f, ϕ₁, ϕ₂, g₁, and g₂ are

(18)

In the presence of radiation, the local volumetric entropy generation can be expressed as

(19)

In Eq. (19), ϕ is the viscous dissipation, K is the conductive effect, and N_s is the entropy generation number. The dimensionless form of Eq. (19) can be expressed as

(20)

where the term N_H is the entropy generation due to heat transfer, the term N_F is the entropy generation due to fluid friction, the term N_D is the entropy generation due to mass transfer, and N_P is the entropy generation due to the porous media.

The irreversible ratio is defined as the ratio of the sum of entropy generation due to the fluid friction N_F, the entropy generation due to mass transfer N_D, and the entropy generation due to porous media N_P to entropy generation due to heat transfer N_H. It is denoted by

Thus,

Here, ϕ ≥ 0.

The Bejan number is an alternative irreversible distribution parameter, and it is defined as the ratio of entropy generation due to heat transfer to total entropy generation, i.e.,

(21)

The value of Bejan number lies between 0 and 1.

If Be=0, then the irreversibility due to fluid friction, mass transfer and porous media dominates.

If Be =1, then the irreversibility due to heat transfer dominates.

If Be =0.5, then the irreversibility due to heat transfer is equal to the sum of irreversibility due to the mass transfer, fluid friction and porous media contributions in the entropy generation.

The nondimensional skin friction coefficient for the second grade fluid flow is

(22)

where

The rate of heat transfer at the plates is given by

(23)

The ratio of the mass transfer rate to the diffusion rate is

(24)

3 Solutions

The nonlinear ODEs (13), (14), (15), (16), and (17) are converted as a system of first-order differential equations by the following substitution:

(25a)

(25b)

(25c)

(25d)

(25e)

(25f)

(25g)

(25h)

(25i)

(25j)

(25k)

The initial conditions are

(26)

The above equations (25a)-(25k) are solved by using the fourth-order Runge-Kutta method^[55] subject to the boundary conditions (26), where b₁, b_2, b₃, b₄, b₅, and b₆ are unknowns to be determined at the boundary condition λ =1 with the Newton-Raphson method. For convergence, our calculations are set to at least 10^-6.

4 Results and discussion

The numerical outcomes for nondimensional velocity components, temperature distribution, concentration, the entropy generation number, the Bejan number, skin friction, heat transfer rate, and mass transfer rate with respective distinct fluid and geometric parameters such as the second grade fluid parameter β₁, the suction-injection ratio a, the thermophoresis parameter N_t, the radiation parameter R_d, and the Brownian motion parameter N_b are studied in detail and presented in the form of graphs and tables.

The effects of β₁ on velocity components, temperature, concentration, the entropy generation number, and the Bejan number are displayed in Fig. 2. It is observed that, as β₁ increases, the temperature also increases, whereas the radial velocity, concentration, and Bejan number decrease from λ =0 to λ =1. However, the primary velocity decreases towards λ =0.5 then increases with increasing β₁, and the entropy generation number follows the opposite trend of the primary velocity. The effects of the suction-injection ratio a on the flow, heat mass transfer, entropy generation number, and Bejan number are depicted in Fig. 3. From the figure, it is shown that the radial velocity, concentration, and Bejan number are decreasing with increasing a, whereas the axial velocity, temperature, and entropy generation number increase from λ =0 to λ =1. Figure 4 describes the effect of N_t (thermophoresis parameter) on temperature, concentration, entropy generation number, and Bejan number. It is observed that, as N_t increases, the temperature profiles increase towards λ =1 due to gradual enhancement in nanoparticle percentage with N_t, whereas the concentration profiles decrease from λ =0 to λ =1. However, the entropy generation and Bejan numbers increase towards λ =0.5 then decrease. The effects of radiation parameter R_d on the temperature, concentration, entropy generation number, and Bejan number are shown in Fig. 5. It is examined that, as R_d increases, the concentration, entropy generation number, and Bejan number are also increasing, whereas the temperature decreases from the lower plate to the upper plate. The effects of Brownian motion parameter N_b on the temperature distribution, concentration, entropy generation number, and Bejan number are shown in Fig. 6. It is evident that as N_b increases the temperature, concentrations are also increasing towards the upper plate, whereas the Bejan number decreases. However, the entropy generation number increases towards λ =0.5 of the channel then decreases.

Table 1 presents the numerical values of skin friction at the upper plate for Sc=0.3, Da=50, ξ = 0.632 5, Gr_t=20, Ec=0.8, Sh=0.1, Gr_s=20, R_d=0.75, Sc=0.3, N_t=0.4, Pr=0.6, N_b=0.5 by increasing β₁, a, Re, and ψ. It is analyzed that the skin friction at the upper plate is increased for ψ, whereas it is decreased for β₁, a, and Re.

Table 2 demonstrates the numerical results of heat and mass transfer coefficients at the lower and upper plates for ψ=0.2, Da=50, ξ=0.632 5, Gr_t=20, Ec=0.8, Sh=0.1, Gr_s=20, a=0.3, β₁=0.1, Br =0.48, Ω =0.9, λ =0.5, α =0.8, and L=0.3. It is analyzed that, as R_d, Pr, N_t, N_b, Sc, and Re increase, the heat transfer rate at the upper plate is also increasing with Pr, N_t, N_b, Sc, and Re, whereas decreases with R_d, and at the lower plate the heat transfer rate is increasing with Pr, Sc, and Re, whereas decreases with R_d, N_t, and N_b. The mass transfer rate at the lower plate increases with N_t, Sc, and Re, whereas decreases with increasing R_d, Pr, and N_b. The mass transfer rate at the upper plate increases with R_d and N_b, whereas decreases with the increase in Pr, N_t, Sc, and Re.

Table 3 shows comparisons of the present skin friction values with those in Ref. [7] for the Newtonian fluid case, and good agreement is observed.

5 Conclusions

The second law analysis of free convection flow of a second grade nanofluid between two porous parallel plates under the influence of thermophoresis and Brownian motion is presented. The results are interpreted for the dimensionless velocity components, temperature and concentration profiles, entropy generation number, Bejan number, skin friction, heat and mass transfer rates with distinct fluid and geometric parameters. From the above analysis, we conclude that

(ⅰ) The temperature profile of the fluid is increased with thermophoresis and Brownian motion, whereas it is decreased with the increasing radiation parameter.

(ⅱ) The concentration profile of the fluid is decreased with the increasing second grade fluid parameter, whereas it is increased with the radiation parameter.

(ⅲ) The profile of entropy generation number is increased with the suction injection ratio, whereas it is reversed for the radial velocity.

(ⅳ) The Bejan and entropy generation numbers exhibit the similar trend for the thermophoresis parameter.

(ⅴ) The skin friction at the upper plate is increased with the frequency parameter, and it is reversed for the Reynolds number.

(ⅵ) The heat transfer rate at the lower plate exhibits the opposite trend for Prandtl and radiation parameters.

(ⅶ) The behavior of Sherwood number at the boundary is opposite for thermophoresis and Brownian motion parameters.