For an incompressible fluid, the equations of motion and continuity are as follows: ∇ . V = 0 , (1) d i v T = − J × B + ρ D V D t − ρ b . (2)

Here, ρ, V, T, b, J, and B are density of the fluid, velocity vector, Cauchy stress tensor, body force, electric current, and external magnetic field, respectively. Following Rajagopal and Fosdick [31], the third-grade fluid’s stress tensor is as follows: T + p I = S , (3) S = μ B 1 + α 1 ̂ B 2 + α 2 ̂ B 1 2 + β 1 ̂ B 3 , (4) + β 2 ̂ ( B 1 B 2 + B 2 B 1 ) + β 3 ̂ B 1 t r c B 1 2 . Here, p, I, T, and S are the pressure, identity tensor, Cauchy stress tensor, and the extra stress tensor, respectively. Furthermore, α k ̂ ( k = 1,2 ) , β j ̂ ( j = 1,2,3 ) are metallic constants and B _i (i = 1, 2, 3) is the kinematic tensor defines as: B 1 = L T + L , B n = D B n−1 D t + B n−1 L + L T B n−1 , n = 2,3 where , L = ∇ V . Here, D D t represents the substantial derivative, and it is stated as: D () D t ≡ V . ∇ () + ∂ () ∂ t .

In the case of third-grade fluid, the material moduli satisfy Clausius–Duhem inequality stated as: β 1 ̂ = β 2 ̂ = 0 , α 1 ̂ ≥ 0 , μ ≥ 0 , β 3 ̂ ≥ 0 , (5) ∣ α 2 ̂ + α 1 ̂ ∣ ≤ 24 μ β 3 ̂ . (6)

So, the stress tensor takes the following form. T = μ B 1 + α 1 ̂ B 2 + α 2 ̂ B 1 2 + β 3 ̂ B 1 t r c B 1 2 . (7)

The field equations for the micropolar fluid following Papautsky et al. [32] are stated as follows: ∇ . ρ V + ∂ ρ ∂ t = 0 , (8) λ ∗ + κ + 2 μ ∇ ∇ . V + κ ∇ × G − κ + μ ∇ × ∇ × V + ρ f = ρ V ′ + ∇ p , (9) α ̄ + γ ∗ + β ̄ ∇ ∇ . G + κ ∇ × V − − 2 κ G − γ ∗ ∇ × ∇ × G = − ρ l + ρ j G ′ . (10)

Eqs. 8–10, respectively, denote conservation of mass, linear momentum, and angular momentum. Furthermore, ρ represents the micropolar fluid density, G is the (gyration) micro-rotation vector, V depicts velocity vector, j represents micro-inertia, p is the pressure, l denotes body couple per unit mass vector , λ∗ reflects Eringen second-degree viscosity parameter, f depicts the body strength per unit mass vector, κ represents coefficient of vortex viscosity, μ denotes dynamic viscosity, and α ̄ , γ∗, and β ̄ denote the swirl gradient viscosity coefficient. Also, (′) denotes the time derivative. For α ̄ = κ = β ̄ = γ ∗ = 0 and in the absence of l and f, the micro-rotation vector G approaches zero, and Eq. 9 is simplified to the Navier–Stokes equations. For κ = 0, the gyration vector G and velocity vector V become detached, and the micro-movements have no influence on the entire motion of the fluid.

In this section, the steady, laminar, two-dimensional, and incompressible third-grade viscoelastic micropolar fluid stream along with thermal radiation caused by an exponentially stretched surface is reported. The x-axis is marked parallel to the sheet, while the y-axis is directed orthogonal to the sheet. The uniform magnetic field is supplied to the surface in a normal direction, and the fluid is believed to be electrically conductive. The sheet and the third-grade micropolar liquid are both originally maintained at the identical temperature. The temperature is elevated to T _f , and T _f > T _∞ where T _∞ is the ambient temperature, which remains constant as (see Figure 1). The concentration at the sheet is taken as C _f , while the ambient concentration is C _∞ (C _f > C _∞ ). This is supposed that the flow’s magnetic Reynolds number is not low in magnitude, and thus the produced magnetic field is not negligible. The evoked magnetic field H ₂ is also assumed to be supplied along the x-axis. The parallel component H ₁ of the induced magnetic field tends to H _e (x) in the free stream flow, and the slope of the magnetic field approaches zero along y-axis.

Using Eq. 7 in Eq. of motion (Eq. 2) along with the boundary layer estimations [33–36] in case of third-grade fluid, notably, inside the boundary layer ∂ p ∂ x , ∂ 2 u ∂ x , ∂ u ∂ x , and u are O(1), v and y are O(δ), α j ̂ ρ ( j = 1,2 ) and ν be O(δ ²), and β k ̂ ρ ( k = 1,2,3 ) being O(δ ⁴), and the components of O(δ) are ignored (δ is boundary layer width). The equations for conservation of mass, magnetic field, and momentum are given as Eqs. 11–12. Equations 13–14 rise from the conservation of energy and concentration. Eq. 15 rises from micropolar boundary layer approximation by Papautsky et al. [32], and Eq. 16 resulted from the conservation of induced magnetic field. Hence, the mathematical model overseeing the considered flow problem is as follows: ∂ v ∂ y + ∂ u ∂ x = 0 , ∂ H 1 ∂ x + ∂ H 2 ∂ y = 0 , (11) v ∂ u ∂ y + u ∂ u ∂ x = α 1 ̂ ρ ∂ u ∂ x ∂ 2 u ∂ y 2 + 3 ∂ u ∂ y ∂ 2 v ∂ y 2 + v ∂ 3 u ∂ y 3 + u ∂ 3 u ∂ x ∂ y 2 + 2 α 2 ̂ ρ ∂ u ∂ y ∂ 2 v ∂ y 2 (12) + 6 β 3 ̂ ρ ∂ 2 u ∂ y 2 ∂ u ∂ y 2 + k ρ ∂ N ∂ y + k + μ ρ ∂ 2 u ∂ y 2 − σ B 0 2 ρ u , v ∂ T ∂ y + u ∂ T ∂ x = 1 ρ C p ∂ ∂ y ∂ T ∂ y k T + k + μ ρ C p ∂ u ∂ y 2 − 1 ρ C p ∂ q r ∂ y (13) + τ ∗ D B ∂ C ∂ y ∂ T ∂ y + ∂ T ∂ y 2 D T T ∞ v ∂ C ∂ y + u ∂ C ∂ x = D B ∂ 2 C ∂ y 2 + D T T ∞ ∂ 2 T ∂ y 2 , (14) v ∂ N ∂ y + u ∂ N ∂ x = γ ∗ ρ j ∂ 2 N ∂ y 2 − k j ρ ∂ u ∂ y + 2 N , (15) v ∂ H 1 ∂ y + u ∂ H 1 ∂ x = H 2 ∂ u ∂ y + H 1 ∂ u ∂ x + η o ∂ 2 H 1 ∂ y 2 . (16)

The concerned boundary conditions are as follows: N = − m ∂ u ∂ y , u = U w x = U 0 e x p x / l , T = k ∂ T ∂ y + T f , H 2 = 0 , D B ∂ C ∂ y + D T T ∞ ∂ T ∂ y = 0 , v = 0 , ∂ H 1 ∂ y = 0 , for y → 0 , (17) C → C ∞ , H 1 → H e x T → T ∞ , N → 0 , u → 0 , ∂ u ∂ y → 0 for y → ∞ . (18)

Here, u, v are the velocity coefficients in x and y directions, ρ depicts the liquid’s density , k denotes the vertex viscosity, μ represents dynamic viscosity, N denotes the micropolar fluid’s angular velocity, T represents the temperature, U ₀ depicts the reference velocity, C depicts the concentration, B ₀ denotes the uniform magnetic field, η ₀ denotes the magnetic diffusivity, σ denotes the electrical conductance, C _p represents specific heat, and τ* reveals the quotient of the latent heat of the nanoparticles to the latent heat of the base liquid. Furthermore, D _B reflects the coefficient of Brownian motion, D _T denotes the coefficient of thermophoresis diffusion, and the micro-inertial density is depicted by j. The viscosity of the spin gradient γ* is given as follows: γ ∗ = j μ + k 2 = μ j 1 + K 2 , and j = 2 l μ ρ U 0 exp − x / l , (19) where K = k μ is the micropolar parameter. Temperature-based thermal conductivity is defined as follows: k T = 1 + ϵ T ∞ − T T ∞ − T f k . (20)

The Rosseland radiative heat flow (q _r ) along y-axis is defined as: q r = − 4 σ ∗ 3 k ∗ ∂ T 4 ∂ y , and T 4 ≈ 4 T ∞ 3 T − 3 T ∞ 4 . (21)

Here, Stefan–Boltzman constant and average absorption coefficient are expressed by σ∗ and k∗, respectively. ∂ q r ∂ y = − 16 T ∞ 3 σ ∗ 3 k ∗ ∂ 2 T ∂ y 2 . (22)

To simplify the analysis, the following appropriate similarity transformations are implemented. ψ x , y = 2 ν l U 0 f η e x p x 2 l , ψ 1 x , y = H 0 ν U 0 g η e x p x 2 l , ϕ η = C − C ∞ C f − C ∞ , N η = U 0 U 0 2 ν l e x p 3 x 2 l h η , θ η = T − T ∞ T f − T ∞ , η = y U 0 2 ν l e x p x 2 l . (23)

Here, ψ and ψ ₁ are stream functions and can be represented as: u = ∂ ψ ∂ y = U 0 f ′ η e x p x / l , v = − ∂ ψ ∂ x = − ν U 0 2 l f + η f ′ e x p x / 2 l , H 1 = ∂ ψ 1 ∂ y = H 0 2 l e x p x / l g ′ η , H 2 = − ∂ ψ 1 ∂ x = − H 0 ν 4 l 2 U 0 e x p x / 2 l g + η g ′ . (24) where η denotes the dimensionless variable, and f′(η), θ(η), ϕ(η), h(η), and g(η) are the dimensionless velocity curve, temperature curve, concentration curve, micropolar curve, and induced magnetic field curve, respectively. Implying Eqs. 23, 24, the continuity equation and magnetic flux equation holds true, and the Eqs. 12–16 are modified into the non-dimensional ODEs as stated: 1 + K f ‴ + 3 β f ‴ f ″ 2 + K h ′ + f f ″ + α 1 − 2 η f ″ f ‴ − f f i v − 9 f ″ 2 + 6 f ′ f ‴ − 2 f ′ 2 − 2 M f ′ − α 2 η f ″ f ‴ + 3 f ″ 2 = 0 , (25) 1 P r θ ″ + ϵ θ θ ″ + ϵ θ ′ 2 + f θ ′ + E c 1 + K f ″ 2 + 4 3 R d θ ″ + N t θ ′ 2 + N b θ ′ ϕ ′ = 0 , (26) ϕ ″ + N t N b θ ″ + L e f ϕ ′ = 0 , (27) 1 + K 2 h ″ + f h ′ − 3 h f ′ − 2 K B f ″ + 2 h = 0 , (28) g ‴ + P r m f g ″ − g f ″ = 0 . (29)

The associated non-dimensional boundary constraints are given as: N b ϕ ′ 0 + N t θ ′ 0 = 0 , g ″ 0 = 0 , f 0 = 0 , h 0 = − m f ″ 0 , h ∞ = 0 , ϕ ∞ = 0 , θ 0 = δ θ ′ 0 + 1 , θ ∞ = 0 , f ′ ∞ = 0 , f ′ 0 = 1 , f ″ ∞ = 0 , g ′ ∞ = 1 , g 0 = 0 . (30)

The non-dimensional parameters are defined as: α 1 = α 1 ̂ U 0 2 ρ ν l e x p x / l , α 2 = α 2 ̂ U 0 ρ ν l e x p x / l , β = β 3 U 0 3 ρ ν 2 l e x p 3 x / l , P r = μ C p k , E c = U 0 2 e x p 2 x / l C p T f − T ∞ , R d = 4 σ ∗ T ∞ 3 μ C p k ∗ , N t = D T T f − T ∞ τ ∗ ν T ∞ , N b = D B τ ∗ ν C f − C ∞ , L e = ν D B , B = ν l U 0 j e x p − x / l , P r m = ν η 0 , δ = k U 0 2 ν l e x p x / 2 l M = σ B 0 2 l ρ U 0 e x p − x / l , K = k μ .

Here, α ₁, α ₂ are the non-dimensional viscoelastic coefficients, β is the third-grade fluid constant, M depicts the magnetic parameter, δ denotes the thermal slip parameter , Ec represents the Eckert no, Rd is the radiation constant, Nt represents the thermophoresis constant, K denotes the micropolar fluid coefficient, Pr stands for Prandtl number, Nb indicates the Brownian diffusivity coefficient, Le denotes the Lewis no, B is the micro-inertia density coefficient, and Pr _m denotes the magnetic Prandtl number. Physical variables of interest, such as local couple stress (M _s ), Sherwood number (Sh _x ), Nusselt number (Nu _x ), and skin friction coefficient (Cf _x ) are defined as follows: S h x = x j w D B C f − C ∞ , C f x = τ w 1 2 ρ U w 2 , N u x = x q w k T f − T ∞ , M s = γ ∗ U w 2 ρ l ∂ N ∂ y y = 0 , (31) where j w = − D B ∂ C ∂ y y = 0 , q w = − 16 T ∞ 3 σ ∗ 3 k ∗ + k ∂ T ∂ y y = 0 , τ w = μ + k ∂ u ∂ y + α 1 ̂ v ∂ 2 u ∂ y 2 + 2 ∂ u ∂ y ∂ u ∂ x + u ∂ 2 u ∂ y ∂ x + 2 β 3 ∂ u ∂ y 3 + k N y = 0 .

Utilizing the similarity transformation (Eq. 22), the aforementioned expressions in non-dimensional aspects are stated as follows: 1 2 R e x 1 / 2 C f x = 1 + K f ″ 0 + K h 0 + β f ″ 0 3 + α 1 − f 0 f ‴ 0 + 7 f ′ 0 f ″ 0 , (32) 2 X R e x − 1 / 2 N u x = − 1 + 4 3 R d θ ′ 0 , (33) 2 X R e x − 1 / 2 S h x = − ϕ ′ 0 , (34) R e x M s = 1 + K 2 h ′ 0 , (35) where X = x/l and R e x = l U w ( x ) ν is the local Reynolds number.

Figure 2 depicts the tendency of velocity distribution with the rising values of material constant α ₁. It is clear that velocity and momentum boundary layer thickness increases for the higher values of viscoelastic constant α ₁. Physically, the viscosity of the liquid is inversely proportional to the material constant α ₁ because of that when the stress is applied, it reduces the strain and strengthens the elastic effects between the adjacent layers, and hence the velocity profile enhances. Figure 3 denotes the impact of second material constant α ₂ on the velocity profile. The velocity curve is the decreasing function for larger values of α ₂. This parameter causes shear thickening of the fluid and an increase in resistance that reduces the boundary layer flow, and originates a decrement in the size of the momentum boundary layer width. Figure 4 exhibits the ascending behavior of velocity with the augmentation of the third-grade fluid coefficient β. This coefficient is inversely proportionate to the square of the liquid viscosity. Alternatively, higher β readings indicate superior third-grade material characteristics (higher liquid elasticity) and smaller fluid viscosity. This causes the boundary layer stream to accelerate, resulting in higher f′(η) values. As this value is raised, the fluid needs less stress to flow, encouraging flow acceleration. In Figure 5, the influence of the magnetic field coefficient on velocity profile is examined. It is analyzed that the velocity curve declines monotonically with the augmented values of M ,and the velocity diminishes far away from the sheet. This process assists in controlling the size of the boundary layer. This is because of the fact that the existence of magnetism in an electrically conductive liquid generates a force known as the Lorentz force that operates opposite to the flow direction and forces velocity profile to decline. The velocity diminishes as the retardation to the flow enhances. In Figure 6 the influence of micropolar fluid parameter on the structure of velocity distribution is noticed. The increase in vortex viscosity strongly accelerates the fluid flow. The results show that the momentum transfer layer-by-layer is significantly affected by the rise in viscosity caused by the micro-rotation of the molecules. The micro-elements rotate more strongly, which helps to accelerate the liquid motion in the boundary layer. As a result, linear momentum diffusion is enhanced by micro-polarity, which explains why suspension fluids have thinner boundary layers than regular fluids.

Figure 7 explains that temperature in the boundary layer increases with progressing micropolar coefficient, that is, higher K values. The regime is greatly heated, and the size of the temperature boundary layer enhances. The vortex’s enhanced viscosity promotes thermal diffusion and serves as a rotator. This boosts up the capacity of thermal diffusion within the fluid’s regime from the micro to the macro level and rapidly carries heat from the sheet boundary into the liquid body with greater intensity. Figure 8 depicts the relationship between the temperature profile and the Prandtl number Pr. The fraction of the momentum diffusion coefficient to heat diffusion coefficient is called the Prandtl number. This is examined that the temperature curve declines with increasing the Prandtl number. The higher values of Prandtl number affect the thermal diffusion. Prandtl number is inversely proportionate to the thermal diffusion; hence, higher Pr values indicate lower thermal diffusion, leading in lower temperature and a weaker temperature boundary layer. When the Prandtl number increases, rate of thermal conductivity gets lower. Consequently, heat is dissipated more quickly, and hence, the temperature boundary layer width and temperature of the liquid both diminish. As a result, the Prandtl number is employed to enhance the cooling tendency of fluids. The temperature-dependent thermal conductivity parameter’s effect over the temperature curve is explored in Figure 9. This is evident that the temperature curve is increasing with the augmentation of ϵ. The consequence of various Eckert number values over the temperature curve is explored in Figure 10. Ec is known as the quotient of the kinetic energy and enthalpy of heat transfer. As the Eckert number (Ec) increases, the liquid’s temperature rises. The fluctuation in the thermal curve θ(η) is examined in Figure 11 in respect to distinct amounts of the radiation parameter Rd. It is identified that a higher thermal radiation constant is related to a greater temperature and a wider temperature boundary layer. As the temperature field grows, the enhanced radiation transfers a significant amount of heat to the fluid. Physically, the involved radiation produces more heat which raises the fluid’s temperature. The influence of thermophoresis coefficient over the thermal distribution is probed in Figure 12. Thermophoresis is a phenomenon that occurs in mixture of sub-micron sized particles, in which the different particles respond differently to the force of a thermal gradient. The particle’s velocity is known as the thermophoretic velocity, and the stress exerted on the dispersed particles because of the temperature difference is known as the thermophoretic force. The thermophoretic force increases as the value of Nt grows, causing the temperature to increase.

The influence of Brownian diffusivity and thermophoresis coefficients over the concentration distribution ϕ(η) is portrayed in Figures 13, 14. According to these graphs, nanofluid constants have opposite impacts on concentration distributions. Particularly, as the thermophoresis constant rises, the width of the concentration boundary layer enhances; nevertheless, as the Brownian motion constant increases, the ϕ(η) values decrease. The thermophoresis constant amplifies the thermophoretic force, resulting in the transfer of nanoparticles from warm to cool locations and an increment in nanoparticle volume. Thermophoresis has numerous uses, including radioactive particle deposition in nuclear reactors, silicon thin film deposition, and aerosol technologies. Furthermore, the progressing amount of the Brownian motion coefficient reduces the micro-mixing of nanoparticles into the fluid’s zone, which diminishes the boundary layer thickness of concentration distribution.

The behavior of ϕ(η) for the numerous values of Lewis number (Le) is examined in Figure 15. The Lewis number is stated as the rate of heat-to-mass diffusion coefficient. It is utilized to express the flow of liquid, in which heat and momentum transfer occur simultaneously. The concentration distribution becomes steeper when Lewis number is increased. The higher values of ‘Le’ imply the lesser values of mass diffusivity ‘D _B ,’ which causes a weaker penetration depth for the concentration boundary layer. In Figure 16, the concentration curve for the various amounts of the micropolar coefficient is explored. However, when K increases, the micro-rotation velocity ϕ(η) drops down within a shorter range in the concentration boundary layer.

Figures 17, 18 are portrayed for the different values of micro-inertia density coefficient B and micropolar coefficient K. This is clear that the micropolar (angular) speed of the sub-micron sized particles decreases with the increase in the micro-polarity of the fluid. Similarly, the micro-inertia density coefficient B serves as a restricting force for the micro-rotation profile h(η). So, for the progressing amounts of B, the micropolar boundary layer width declines. This is possible because micropolar fluids provide a high barrier to fluid motion.

The influence of magnetic Prandtl number Pr _m over induced magnetic field curve g(η) is exhibited in Figure 19 , where a rise in Pr _m leads to an improvement in the produced magnetic field distribution. This is because the magnetic diffusivity over the boundary layer’s surface decreased while the fluid viscous dispersion rate enhanced. Magnetic Prandtl no.(Pr _m ) is a non-dimensional quantity in Magnetohydrodynamics that estimates the ratio of momentum diffusion coefficient (ν) to magnetic diffusion coefficient (η).

The influence of all the non-dimensional coefficients used in the considered article on the skin friction parameter ( R e x 1 / 2 C f x ) , Nusselt number. ( R e x − 1 / 2 N u x ) , Sherwood number. ( R e x − 1 / 2 S h x ) , and coupled wall stress (Re _x M _s ) are presented in Tables 3, 4. Table 3 probes the significance of involved coefficients on the skin friction. The values of the coefficients other than used in Table 3 are taken to be fixed Pr = 0.7, ϵ = 0.3, Ec = 0.1, K = 0.2, Rd = 0.3, Nt = 0.3, Nb = 1.0, and Le = 1.0. By increasing the value of material parameters α ₁ and α ₂, the skin friction coefficient reduces. It is because of the reason that the viscosity near the surface of sheet is enhanced with the progression in material constant, which causes the decline in R e x 1 / 2 C f x values. As the rise in shear thickening coefficient β, improves the boundary layer thickness that give rise to the skin friction coefficient. The increase in micropolar parameter K, give rise to the value of R e x 1 / 2 C f x , but the opposite behavior is seen for couple stress Re _x M _s . A reduction is caused by a rise in the magnetic field constant M in both R e x 1 / 2 C f x and Re _x M _s values. By increasing the micro-inertia density B, the values of R e x 1 / 2 C f x get enhanced but reverse behavior is seen for couple stress Re _x M _s .

Table 4 depicts the influence of different parameters on the Nusselt number and Sherwood number. The other parameters are taken to be fixed β = 0.5, α ₁ = 0.1, B = 0.5, α ₂ = 0.5, K = 0.2, and M = 0.3. The increment in the values of Prandtl number causes a decline in the thermal diffusivity, and hence resists the rise in the heat transmission rate at the boundary. Boosting the Prandtl number raises the average Nusselt number at the heated surface, while a reverse effect is examined for the R e x − 1 / 2 S h x . As ϵ is enhanced, the Nusslet number decreases while the Sherwood number increases. The increase in the value of Ec causes the Nusselt number to decrease while the Sherwood number increases. The boosting values of micropolar parameter K enhances the Nusselt number but declines the Sherwood number. Higher values of the Rd parameter improve convective heat and matter transmission, which ultimately increases the average Nusselt and Sherwood numbers values.

The higher values of thermophoresis coefficient Nt causes a decline in both the N u x R e x − 1 / 2 and S h x R e x − 1 / 2 values. The increment in the Nb values results in the enhancement of Sherwood number because the mass transfer rate from higher to lower concentration area get increased. The growth in the values of Lewis number Le improves the thermal diffusion and declines the mass conductivity. Thus, the values of N u x R e x − 1 / 2 decrease and that of S h x R e x − 1 / 2 increase.