Third grade (3rd) fluid is a subcategory of differential types that can explain the impacts of both shear thickening and thinning phenomena. The stability of third grade (3rd) liquid is explored by Fosdick and Rajagopal [1]. Viscoelastic mixed convective flow is developed by Mastroberardino [1] caused by stretchable surfaces. The study of thermodynamics for the third grade (3rd) liquid with heat generation has been done by Adesanya and Makinde [2]. Ellahi and Riaz [3] systematically thought about third grade (3rd) fluid flow with variable viscosity. Third grade squeezing flow is securitized by Hayat et al. [4]. Sajid et al. [5] investigated the coating process of a non-Newtonian material. Hayat et al. [6] evaluate Soret Dufour's effects on third-grade fluid.

Nomenclature

Nano fluids are produced by dispersing solid particles of a nano-meter size into various conventional liquids such as water, gasoline and ethylene glycol respectively. Nano-liquids have numerous applications in manufacturing and automotive cooling, sensing, production of new types of fuels, microelectronic cooling, hybrid-powered engine efficiency and home appliance heating/cooling etc. Choi [7] discovered the term nano fluid. The infusion of metallic nano particles hooked on conventional liquids would greatly improve the thermal efficiencies of those liquids as explained by him. The Nano fluid convective transportation model with the Brownian and thermophoresis effects was analysed by Buongiorno [8]. Non-effective Prandtl numbers by considering nanoparticles and entropy analysis are researched by Hayat et al. [9]. Entropy optimization with different fluid models is studied by Ahmad et al. [10].

Fourier heat conduction expression [11] give much information about heat transfer through the flux. The paradox of heat conduction is leads by this expression. The Cattaneo [12] has revised the relation of thermal relaxation time. Suggestions about the Oldroyed upper convected derivatives should be considered in Ref. [13, 14] instead of material differentiations are given by Christov. Thermal variability in a porous medium by the retentive Cattaneo-Christov (CC) model is analysed by Haddad [15]. The impact of Cattaneo-Christov model considering different fluid models is discussed by some researchers [16, 17, 18, 19].

Here, third grade (3rd) nano liquid flow towards a stretchable sheet is discussed. Bio-convection, radiative heat, dissipative impacts and heat generation are also discussed. By using the optimal method [20, 21, 22, 23] obtained the analytical solution of the governing equations. The results for various physical variables are discussed through graphs.

The simultaneous study of generalized (non-Fourier, non-Fick) heat and mass flux conditions inside a mixed convective flow regime and nonlinear third-grade nanofluid behavior is what makes this work innovative. The integration of third-grade fluids and nanofluid dynamics under nonlinear rheological behavior has not been extensively studied, particularly when generalized boundary conditions that take into consideration thermal and solutal relaxation effects are present. Furthermore, by considering both buoyancy-driven and forced convective effects both of which are crucial in real-world applications including industrial heat exchangers, biomedical flows, and electronic device cooling—this work integrates a realistic description of mixed convection.

In this study, a two-dimensional (x,y) incompressible non-Newtonian fluid flow over a stretching sheet is considered. Both mixed convection and bioconvective effects are taken into account. The sheet is stretched along the x-axis with a linear stretching velocity Uw=a⁢x. The surface temperature and concentration are denoted by Tw and Cw, respectively, while T∞ and C∞ represent the ambient temperature and concentration far from the surface. The free stream velocity outside the boundary layer is represented by Ue. The schematic diagram of the flow configuration is illustrated in Figure 1.

Figure 1.

Flow geometry.

In this work, a mixed convective, laminar, two-dimensional flow over a moving surface is investigated. The analysis incorporates a third-grade non-Newtonian fluid model in the presence of gyrotactic micro-organisms. Heat and mass transport are examined using the Cattaneo–Christov (CC) flux model, which accounts for relaxation effects beyond the classical laws. Furthermore, the influences of Brownian motion and thermophoresis are considered to capture nanoscale transport phenomena. Thermal radiation and viscous dissipation are included as additional heat sources in the energy equation. The governing equations relevant to this study are presented below.

∂ u ∂ x + ∂ v ∂ y = 0 ,

u ⁢ ∂ u ∂ x + v ⁢ ∂ u ∂ y = U e ⁢ ∂ U e ∂ x + α 1 ρ ⁢ ( ∂ u ∂ x ⁢ ∂ 2 u ∂ y 2 + u ⁢ ∂ 3 u ∂ x ⁢ ∂ y 2 + 3 ⁢ ∂ u ∂ y ⁢ ∂ 2 u ∂ x ⁢ ∂ y + v ⁢ ∂ 3 u ∂ y 3 ) + α 2 ρ ⁢ ( 2 ⁢ ∂ u ∂ y ⁢ ∂ 2 u ∂ x ⁢ ∂ y ) + 6 ⁢ α 3 ρ ⁢ ( ( ∂ u ∂ y ) 2 ⁢ ∂ 2 u ∂ y 2 ) + g ⁢ β t ⁢ ( T − T ∞ ) + g ⁢ β t ⁢ ( C − C ∞ ) σ ⁢ Q o ⁢ Q o ρ ⁢ a ⁢ u + ν ⁢ ∂ 2 u ∂ y 2 } ,

The relevant periphery conditions are:

u = U w = a ⁢ x , ⁢ v = 0 ⁢ a ⁢ t ⁢ y = 0 , u = U e = b ⁢ x ⁢ when ⁢ y → ∞ } .

Drag force is:

C f ⁢ x = ( τ w ρ f ⁢ ( U w ) 2 ) ,

here:

τ w = ( μ ⁢ ∂ u ∂ y ) y = 0 + [ α 1 ρ ⁢ ( u ⁢ ∂ 2 u ∂ x ⁢ ∂ y + 2 ⁢ ∂ u ∂ x ⁢ ∂ u ∂ y + v ⁢ ∂ 2 u ∂ y 2 ) + 2 ⁢ α 3 ρ ⁢ ( ∂ u ∂ y ) 3 ] y = 0 .

The energy and concentration expression for Cattaneo-Christove (CC) model are:

u ⁢ ∂ T ∂ x + v ⁢ ∂ T ∂ y + δ E ⁢ Ω E = α ⁢ ∂ 2 T ∂ y 2 + τ ⁢ [ D B ⁢ ∂ C ∂ y ⁢ ∂ T ∂ y + D T T ∞ ⁢ ( ∂ T ∂ y ) 2 ] + Q ρ ⁢ C p ⁢ ( T − T ∞ ) − 16 ⁢ σ ∗ ⁢ T 3 3 ⁢ ρ ⁢ c p ⁢ k 1 ⁢ ∂ 2 T ∂ y 2 + μ ρ ⁢ C p ⁢ ( ∂ u ∂ y ) 2 + α 1 ρ ⁢ C p ⁢ [ ( u ⁢ ∂ u ∂ y ⁢ ∂ 2 u ∂ x ⁢ ∂ y + v ⁢ ∂ u ∂ y ⁢ ∂ 2 u ∂ y 2 ) ] + 2 ⁢ α 3 ρ ⁢ C p ⁢ ( ∂ u ∂ y ) 4 } ,

here ΩE is given by:

Ω E = u ⁢ ∂ u ∂ x ⁢ ∂ T ∂ x + v ⁢ ∂ u ∂ y ⁢ ∂ T ∂ y + u ⁢ ∂ v ∂ x ⁢ ∂ T ∂ y + v ⁢ ∂ u ∂ y ⁢ ∂ T ∂ x + 2 ⁢ u ⁢ v ⁢ ∂ 2 T ∂ y ⁢ ∂ x + u 2 ⁢ ∂ 2 T ∂ x 2 + v 2 ⁢ ∂ 2 T ∂ y 2 − Q ρ ⁢ C p ⁢ ( u ⁢ ∂ T ∂ x + v ⁢ ∂ T ∂ y ) − μ ρ ⁢ C p ⁢ ( 2 ⁢ u ⁢ ∂ u ∂ y ⁢ ∂ 2 u ∂ x ⁢ ∂ y + 2 ⁢ v ⁢ ∂ u ∂ y ⁢ ∂ 2 u ∂ y 2 ) + α 1 ρ ⁢ C p ⁢ [ u 2 ⁢ ( ∂ 2 u ∂ x ⁢ ∂ y ⁢ ∂ 2 u ∂ x ⁢ ∂ y + ∂ u ∂ y ⁢ ∂ 3 u ∂ x 2 ⁢ ∂ y ) + 2 ⁢ u ⁢ v ⁢ ( ∂ u ∂ y ⁢ ∂ 2 u ∂ x ⁢ ∂ y + ∂ u ∂ y ⁢ ∂ 3 u ∂ x ⁢ ∂ y 2 ) + v 2 ⁢ ( ∂ 2 u ∂ y 2 ⁢ ∂ 2 u ∂ y 2 + ∂ u ∂ y ⁢ ∂ 3 u ∂ y 3 ) ] + α 3 ρ ⁢ C p ⁢ ( 4 ⁢ u ⁢ ∂ u ∂ y ⁢ ∂ 2 u ∂ x ⁢ ∂ y + u ⁢ v ⁢ ∂ u ∂ y ⁢ ∂ 2 u ∂ y 2 ) − τ ⁢ D B ⁢ ( v ⁢ ∂ T ∂ y ⁢ ∂ 2 C ∂ y 2 + v ⁢ ∂ C ∂ y ⁢ ∂ 2 T ∂ y 2 + u ⁢ ∂ 2 C ∂ x ⁢ ∂ y ⁢ ∂ T ∂ y + u ⁢ ∂ C ∂ y ⁢ ∂ 2 T ∂ x ⁢ ∂ y ) + 2 ⁢ τ ⁢ D T T ∞ ⁢ ( v ⁢ ∂ T ∂ y ⁢ ∂ 2 T ∂ y 2 + u ⁢ ∂ T ∂ y ⁢ ∂ 2 T ∂ x ⁢ ∂ y ) + R d ⁢ ( u ⁢ ∂ 2 T ∂ x ⁢ ∂ y + v ⁢ ∂ 2 T ∂ y 2 ) } .

u ∂ C ∂ x + v ∂ C ∂ y + δ F Ω F = D B ∂ 2 C ∂ y 2 + D T T ∞ ∂ 2 T ∂ y 2 } ,

here ΩF is given as:

Ω F = u 2 ⁢ ∂ 2 C ∂ x 2 + u ⁢ ∂ u ∂ x ⁢ ∂ C ∂ x + u ⁢ ∂ v ∂ x ⁢ ∂ C ∂ y + 2 ⁢ u ⁢ v ⁢ ∂ 2 C ∂ x ⁢ ∂ y ⁢ ∂ T ∂ x + v ⁢ ∂ u ∂ y ⁢ ∂ C ∂ x + v 2 ⁢ ∂ 2 C ∂ y 2 + v ⁢ ∂ v ∂ y ⁢ ∂ C ∂ y − D B ⁢ ( u ⁢ ∂ 3 C ∂ x ⁢ ∂ y 2 + v ⁢ ∂ 3 C ∂ y 3 ) − D T T ∞ ⁢ ( v ⁢ ∂ 3 T ∂ y 3 + u ⁢ ∂ 3 T ∂ x ⁢ ∂ y 2 ) } .

The relevant conditions are:

T → T w ⁢ at ⁢ y = 0 , T → T ∞ ⁢ when ⁢ y → ∞ , } .

C → C w ⁢ at ⁢ y = 0 , C → C ∞ ⁢ at ⁢ y → ∞ , } .

In above equation DB,DT denotes Brownian and thermosphere quantity, (Q>0) be the heat generation rate, the Boltzmann constant is σ∗,k1 denote the coefficient of mean absorption and the specific heat is the Cp.

Considering transformations:

u = a ⁢ x ⁢ f ′ ⁢ ( η ) , ⁢ v = − a ⁢ ν ⁢ f ⁢ ( η ) , ⁢ t = T − T ∞ T w − T ∞ , J = C − C ∞ C w − C ∞ , ⁢ η = a ν ⁢ y } ,

the eq. (1) is satisfied while others eqs. (2)-(11) are:

f ′′′ + f ⁢ f ′′ − f 2 ′ + α 1 ∗ ⁢ ( f ′ ⁢ f ′′′ − f ′ ⁢ f ′′′ − f ′′′′ ⁢ f ) + ( α 1 ∗ + α 2 ∗ ) ⁢ f ′′ ⁢ f ′′ + 6 ⁢ α 3 ∗ ⁢ R ⁢ e ⁢ f ′′ ⁢ f ′′ ⁢ f ′′′ + S 2 + M ⁢ f ′ + λ ⁢ t + λ ⁢ λ c ⁢ J = 0 } ,

( 1 + R d ) ⁢ t ′′ + Pr ∗ γ 1 ∗ ( f ⁢ f ′ ⁢ t ′ + f ′ ⁢ f ′ ⁢ t ′′ − δ ⁢ f ⁢ t ′ − 2 ⁢ f ⁢ f ′ ⁢ t ′′ ) + P ⁢ r ⁢ f ⁢ t ′ Pr ∗ γ 1 ∗ E ⁢ c ∗ ( 2 ⁢ f ′ ⁢ f ′ ⁢ f ′′′ − f ⁢ f ′′ ⁢ f ′′′ ) − Pr ∗ γ 1 ∗ N B ∗ ( f ⁢ f ′ ⁢ J ′′ − f ⁢ J ′ ⁢ t ′ ) + Pr ∗ E ⁢ c ∗ α 1 ∗ f ′ ⁢ f ′′ ⁢ f ′′ + Pr ∗ γ 1 ∗ E ⁢ c ∗ α 1 ∗ ⁢ ( f ′ ⁢ f ′ ⁢ f ′′ + f ⁢ f ′′ ⁢ f ′′ − f ⁢ f ′ ⁢ f ′ ⁢ f ′′′ − 2 ⁢ f ⁢ f ⁢ f ′′′ ⁢ f ′′′ + f ⁢ f ′ ⁢ f ′ ⁢ f ′′ ) + Pr ∗ E ⁢ c ∗ f ′′ ⁢ f ′′ Pr ∗ γ 1 ∗ E ⁢ c ∗ α 3 ∗ ∗ ( 4 ⁢ f ⁢ f ′′ ⁢ f ′′ + f ⁢ f ′′ ⁢ f ′′′ ) + Pr ∗ E ⁢ c ∗ α 1 ∗ ∗ f ⁢ f ′′ ⁢ f ′′′ + 2 ⁢ Pr ∗ E ⁢ c ∗ α 3 ∗ ∗ R ⁢ e f ′′ f ′′ f ′′ f ′′ − Pr γ 1 N t J ′′ t ′ − Pr γ 1 R d f t ′′ + N B t ′ J ′ + N t t ′ t ′ + Pr δ t ] = 0 , }

J ′′ + L e f J ′ + N B N t t ′′ − L e γ 2 [ f 2 J ′′ + f f ′ J ′ ] − γ 2 N B N t t ′′ = 0 } ,

{ f ⁢ ( 0 ) = 0 , ⁢ t ⁢ ( 0 ) = 1 , f ′ ⁢ ( 0 ) = 1 , J ⁢ ( 0 ) = 1 f ′ ⁢ ( ∞ ) = S , t ⁢ ( ∞ ) = 0 , J ⁢ ( ∞ ) = 0 , } .

Dimensionless form of skin friction is given as:

R ⁢ e x ⁢ C f ⁢ x = f ′′ ⁢ ( 0 ) + α 1 ∗ ⁢ ( 3 ⁢ f ′ ⁢ ( 0 ) ⁢ f ′′ ⁢ ( 0 ) − f ⁢ ( 0 ) ⁢ f ′′′ ⁢ ( 0 ) ) + α 2 ∗ ⁢ ( f ′′ ⁢ ( 0 ) ) 3 .

where Pr(=μ⁢Cpk) is Prandtl number, Hartman number M(=σ⁢Mo⁢Moρ⁢a), ratio of rates is the S(=ba), thermophoresis parameter Nt(=τ⁢DT⁢(Tw−T∞)ν⁢T∞), the radiation parameter is Rd(=16⁢σ∗⁢T∞33⁢k1⁢k), Schmidt number is S⁢c(=νDB), Brownian motion parameter NB(=τ⁢DB⁢(Cw−C∞)ν), the third grade fluid parameters are (α1∗(=α1⁢aμ),α2∗(=α2⁢aμ),α3∗(=α3⁢aμ)), heat generation parameter δ(=Qρ⁢Cp), λ(=g⁢βt⁢(Tw−T∞)a) mixed convective parameter, Eckert number is E⁢c(=Uw2Cp⁢(Tw−T∞)), thermal relaxation parameter γ(=a⁢δE) and solutal concentration parameter γ1(=a⁢δF).

We determined the series solutions by using homotopy analysis:

ε k ∗ f ⁢ ( h f ) = 1 N ∗ + 1 ⁢ ∑ j ∗ = 0 N ∗ ∗ [ ∑ i = 0 k ∗ ( f i ) η = j ∗ ⁢ Π ⁢ η ] 2 ,

ε k ∗ t ⁢ ( h f , h t , h J ) = 1 N ∗ + 1 ∗ ∑ j ∗ = 0 N ∗ ∗ [ ∑ i = 0 k ∗ ∗ ( f i ) η = j ∗ ⁢ Π ⁢ η , ∑ i = 0 k ∗ ∗ ( t i ) η = j ∗ ⁢ Π ⁢ η , ∑ i = 0 k ∗ ∗ ( J i ) η = j ∗ ⁢ Π ⁢ η ] 2 ,

ε k ∗ J ⁢ ( h f , h t , h J ) = 1 N ∗ + 1 ⁢ ∑ j ∗ = 0 N ∗ ∗ [ ∑ i = 0 k ∗ ∗ ( f i ) η = j ∗ ⁢ Π ⁢ η , ∑ i = 0 k ∗ ∗ ( t i ) η = j ∗ ⁢ Π ⁢ η , ∑ i = 0 k ∗ ∗ ( J i ) η = j ∗ ⁢ Π ⁢ η ] 2 ,

ε k ∗ t ∗ = ε k ∗ f + ε k ∗ t + ε k ⁣ ∗ J ,

where the total square residual error is represented by εk∗t∗. When S=0.1,M=0.2,Pr=1.2,NB=0.5,α1∗=0.1,α2∗=0.2,NT=0.1,α1∗=0.3,γ=γ1=0.2,R⁢e=0.1,E⁢c=1.0,λ=0.1 and D⁢a−1=0.4. Then the total usual squared residual error is abated by employing Mathematica BVPh2.0. The optimal values of convergence control variables are hf=−0.816767,ht=−0.287548, hJ=2.35602, and hϕ=−0.4562,. The total residual error is εk∗t∗=0.79552. The variation of the total residual error with respect to the approximation order is illustrated in Figure 2.

Table 1

Residual errors.

Figure 2.

Total residual error.

In this study, a comprehensive numerical investigation has been carried out to explore the behavior of nonlinear third-grade nanofluids under generalized heat and mass flux conditions in a mixed convective flow regime. The results highlight the significant influence of fluid nonlinearity, nanoparticle concentration, and relaxation parameters on the thermal and concentration boundary layers.

The following are the main results:

M, α1∗,α2∗ and α3∗ are the higher estimations of the fluid velocity while reduces via D⁢a−1.