Optimal homotopic strategy for thermal and mass transport in Williamson model flow PDEs under heat generation/absorption and Joule heating

Abstract

In current paper, we considered the 3-D occurrence of magneto Williamson fluid subjected to linear penetrating expanding sheet. The Williamson liquid is taken into account under the essence of bio convection bordering with Joule heating, heat production. Moreover, Brownian movement combined with thermophoresis diffusion was also analyzed in current study. Central PDEs are modified into system of ODEs on the basis of appropriate similarity transformation utilization. For solution purpose, an optimal approach namely HAM was opted. The critical review based on all involved multifarious constraints across the non- dimensional velocity contour $(along x - axes and also y - axes),$ energy and concentration distribution. Moreover, across the motile organism distribution was also conducted graphically and depicts total covenant with the former published research work. The drag force effect, Nusselt number, Sherwood number, and motile organism density in tabular form is also proposed in this study. Moreover, Motile organism profile graphically demonstrates decreased demeanor for considered bio convection Lewis number, Peclet number, and also temperature difference constraint whereas results of motile organism density augmented in tabular version for the considered bio convection Lewis number, Peclet number, and also temperature difference constraint.

Keywords

Williamson Nano fluid MHD Joule heating Brownian movement and also thermophoresis diffusion Joule heating bio-convection

Introduction

On account of extensive functional relevance in multitudinous Industrial as well as Engineering procedures multiple researchers have kept in view on the exploration of convective heat transportation analysis and steady occurrence under consideration of incompressible fluids. The standard relevance to conclusive product is heavily based on the rate of thermal transportation and also skin friction at the surface. Investigation of Williamson liquids with bio convection applications gives a multidisciplinary scheme bridging biology, fluid mechanics, and applied mathematics. It provides both practical applications as well as fundamental understanding in the fields ranging from biotechnology to environmental science. Moreover, mathematical modeling with the imposition of bio convection phenomena along with Williamson liquid helps in understanding and predicting complex liquid microorganism interaction. These models considered pivotal in optimizing procedures and also predicting outcomes in numerous engineering and biological applications. Bio convection within the Williamson liquid can be affected by environmental factors such as chemical gradients or temperature gradients. Furthermore, in biotechnological procedures involving microorganisms, understanding of their demeanor toward the non-Newtonian fluids is most significant. Applications owing to the microbial procedures efficiency dependence on fluid rheology, could range from bioreactors to the environmental remediation.

Shah et al.¹ examined the radial magnetized Williamson liquid bounded by unsteady penetrating expanding sheet. An optimal computation in accordance to 3-D Darcy Forchheimer Williamson Nano fluidic occurrence past over an expanding surface by invoking convective conditions was performed by Dawar et al.² Gul et al.³ explored the thermal transportation in reference to flow demeanor of thin film specifically of Williamson fluid subjected to oscillatory and also inclined spinning plate. Rasool et al.⁴ aims to investigate the chemically reactive magneto-Williamson Nano-liquid flow demeanor across the non-linear permeable surface. Shafiq et al.⁵ implemented the novel numerically computation solver namely artificial neural networks (ANN) for the examination of electrically conducting and unsteady Williamson liquid across a radial surface. Prasad et al.⁶ analyzed the mixed convected 2-D hydro-magnetic Williamson Nano liquid occurrence exposed to spinning disk with the imposition of zero mass flux condition. Thermal transportation concerning to radial Williamson liquid occurrence exposed to expandable curvy geometry was studied by Raza et al.⁷ Entropy optimization was conducted by Khan et al.⁸ toward the Williamson hybrid Nano liquid occurrence influenced by Joule heating together with viscous dissipation across the thin vertical shape needle. Moreover, the problem appertaining to Williamson liquid was addressed under the impacts of thermal source/sink and investigated the mass and also thermal transportation in consideration of convective conditions bounded by oscillatory extending geometry. Song et al.⁹ focused on the investigation of magneto-Williamson liquid comprising of motile organisms across the bilateral non-linearly expanding geometry. Computational aspects was explored by Khan et al.¹⁰ in accordance to variable viscosity toward the Williamson Nano fluidic occurrence owing to solutal and also thermal stratification across the non-linear expanding geometry.

Thermal transmission together with mass diffusion analyzed by Khan et al.¹¹ based on magnetized Casson fluid exposed to variable viscosity under the assumption of convective conditions. Shah et al.¹² study based on the examination of second grade liquid flow demeanor in the dispersion of thermophoretic particles influenced by variable concentration diffusivity, varying thermal conductivity and also variable viscosity with the imposition of convective conditions. Thermal efficiency appertaining to mixed convected couple stress Darcy-Forchheimer specified by ternary hybrid Nano liquid occurrence bounded by non-linear extending surface was reported by Gul and Saeed.¹³ Mebarek-Oudina et al.¹⁴ discussed the magnetized thermal naturally convected stability across the molten metal filled inclined cylindrical annulus. Thermal transfer toward the chemically reactive convected Casson liquid occurrence subjected to Darcy-Forchheimer media was studied by Bilal et al.¹⁵ Unsteady 3-D bioconvected occurrence based on chemically reactive Maxwell Nano liquid past over exponentially extending surface exposed to varying thermal conductivity was analyzed by Ahmad et al.¹⁶ Flow and also thermal transfer has been investigated by Khan et al.¹⁷ in accordance to occurrence of bio-convected hybrid Nano liquid influenced by triple stratification impacts. Mixed convected occurrence based on micropolar Hybrid Nano liquid bounded by Riga surface subjected to permeable medium analyzed mathematically by Ahmad et al.¹⁸ Wakif et al.¹⁹ numerically investigated the thermal convection produced by internal heating exposed to spinning medium in accordance to radial nano liquid.

Magnetic field also show vast impactness on the fluid and flow in case of both natural and industrial processes. Earth magnetic field working as a shield in front of the fatal radiations. Solar magnetic field considered to be source of production for sunspots and sun flares. We entitled all above phenomenon with single word as Magnetohydrodynamics (MHD). Mass as well as thermal transportation across the magnetized thin film specifically of second grade possessing variable features bounded by extending surface exposed to thermophoresis impacts together with thermal radiation was addressed by Khan et al.²⁰ 3-D Casson Nano fluidic thin film occurrence across the inclined spinning disk under consideration of heat production/consumption and also thermal radiation impacts was addressed by Saeed et al.²¹

The connective motion owing to density gradient specifically at microscopic level is termed as Bioconvection initially in correspondence to motile organisms. These self-driven motile organisms in specified direction intensified the density of fluid and hence instigating bioconvection. Magnetized bioconvection Nano fluidic occurrence under the combine impacts of Joule heating along with viscous dissipation impacts was covered by Jawad et al.²² Moreover, Jawad et al.²³ also optimized the entropy approach within the magneto bioconvection 3-D Darcy-Forchheimer occurrence specifically of Casson Nano liquid configured by spinning disk. Saeed and Gul²⁴ aims to explore the Darcy Forchheimer across the 3-D bioconvection radial Casson Nano fluidic flow demeanor in terms of spinning disk influenced by Arrhenius activation energy. Similarly, Bioconvectional Nano fluidic occurrence was scrutinized by Gul et al.²⁵ owing to the gyrotactic microorganisms and also thermophoresis impacts amongst the cone and disk gap. Shah et al.²⁶ inspected the chemically reactive mixed bioconvection occurrence concerning to magnetized third-grade fluid under dispersion of Nano-size particles and also gyrotactic microorganisms subjected to expanding sheet influenced by Arrhenius activation energy together with viscous dissipation impacts. Hayat et al.²⁷ considered the combined impacts of Joule heating along with thermal radiation based on third grade liquid occurrence toward the radiative surface. Rasool et al.²⁸ numerically computed the magnetized water based hybrid Nano fluidic occurrence under consideration of viscous dissipation along with Joule heating impacts subjected to a penetrating sheet. Khan et al.²⁹ analyzed the Joule heating as well as chemical response toward the magnetized Carreau-Yasuda Nano liquid flow demeanor bounded by non-linear extending sheet with the imposition of slip and also convective conditions. Entropy approach across the dissipative occurrence appertains to hybrid Nano liquid in the existence of Joule heating together with thermal radiation impacts was investigated by Xia et al.³⁰ Malik et al.³¹ numerically examined the Williamson fluid occurrence induced by expanding cylinder and thermal transmission exposed to varying thermal conductivity and also heat production/absorption. An extensive analysis was performed by Awais et al.³² across the unsteady Williamson liquid configured by melting wedge in the existence of heat absorption/production impacts. 3-D hydromagnetic convected flow demeanor under the chemical response together with non-uniform heat production/absorption impacts within the Williamson fluid was explored by Kumar et al.³³ Magnetized Williamson fluid and also heat transmission bounded by non-linear extending sheet embedded in a permeable medium influenced by viscous dissipation together with heat generation effects was discussed by Abbas et al.³⁴ Mishra et al.³⁵ investigated the magnetized Williamson micropolar liquid occurrence past over a non-linearly expanding sheet with the imposition of heat production/absorption impacts. Thermophoresis is the force generated on account of temperature gradient amongst the hot gas and cold wall influence the particular movement toward the cold wall. In 1827, Robert Brown examines the arbitrary movement of particles of pollen grains in water. The random movement of suspended particles in liquid results from the collection with the fast-moving molecules within the liquid. Owing to collision, Brownian movement is the arbitrary movement of particles which have an origin from Greek word “Leaping.” Naveed et al.³⁶ has given a unique idea by exploring the impacts of Thermophoresis and also Brownian movement in consideration of Blasius flow demeanor appertains to Nano fluid over a curved geometry. Obalulu et al.³⁷ work based on analysis of Brownian movement along with thermophoresis diffusion within the reactive Casson-Williamson Nano-size liquid configured by vertically moving cylinder. Asjad et al.³⁸ investigated the influence of activation energy toward the magnetized Williamson liquid occurrence in the existence of bio convection. Abdal et al.³⁹ explored the combined features of bio convection along with thermal radiation exposed to Williamson Nano size liquid transportation due to cone rotation. Priyadharshini et al.⁴⁰ studied the bio convection together with the magneto hydrodynamic features across the symmetrically shrinking sheet.

In general, the novelty lies within the combination of bio convection phenomena with the non-Newtonian rheology specified by Williamson fluid, provides understanding of how fluid characteristics influenced the biological procedures at the microscale. This interdisciplinary scheme bridges rheology, fluid dynamics, and biological sciences offering advanced avenues for both practical applications and also theoretical exploration.

Mathematical modeling

Here, our analysis based on unsteady 3-D incompressible Williamson liquid occurrence configured by linear expanding sheet. Moreover, sheet is supposed to be stretched in the direction of $xy - axes .$ Fluid motion causes the expansion in sheet whilst space occupies by the fluid is in $z > 0 .$ The expanding sheet considered to be penetrating (Figure 1).

Figure 1.

Geometry problem.

Under consideration of boundary layer approximation, the continuity and also momentum equations for Williamson liquid are as follows:

\nabla . \vec{V} = 0,

(1)

ρ \frac{d \vec{V}}{dt} = div \vec{S} + ρ g .

(2)

Here, $ρ$ represents density, $S$ is the stress tensor, $V, g$ , and $\frac{d}{dt}$ demonstrates velocity vector, body force, and material time derivative respectively.

Mathematically, the expression for Williamson liquid are represented by^41–44:

\vec{S} = p \vec{I} + τ,

(3)

τ = [\frac{μ_{o} - μ_{\infty}}{1 - Γ γ^{*}}] \vec{A_{1}} .

(4)

Where, $A_{1}$ defined as first Rivlin-Erickson tensor. Moreover, $A_{1}$ and $γ^{*}$ defined as:

\vec{A_{1}} = \nabla ν + {(\nabla ν)}^{T},

γ^{*} = \sqrt{\frac{1}{2}} π .

(5)

Where, $π$ is the second invariant tensor given by:

π = trace {(\vec{A_{1}})}^{2},

(6)

γ^{*} = {[{(\frac{\partial u}{\partial x})}^{2} + \frac{1}{2} {(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x})}^{2} + {(\frac{\partial v}{\partial y})}^{2}]}^{\frac{1}{2}} .

(7)

Assume $μ_{\infty} = 0$ and $Γ γ^{*} < 1$ therefore, equation (4) in modified form is:

τ = [μ_{o} {(1 - Γ γ^{*})}^{- 1}] \vec{A_{1}} .

(8)

Where, components of extra stress tensor are as follows:

τ_{xx} = 2 μ_{o} [1 + Γ γ^{*}] (\frac{\partial u}{\partial x}), τ_{xy} = τ_{yx} = 2 μ_{o} [1 + Γ γ^{*}] (\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y}),

τ_{yy} = 2 μ_{o} [1 + Γ γ^{*}] (\frac{\partial u}{\partial y}), τ_{xz} = τ_{yz} = τ_{zx} = τ_{zy} = τ_{zz} = 0 .

For considered assumptions, the mathematical modeling expressed as follows:

\frac{\partial u^{o}}{\partial x} + \frac{\partial v^{o}}{\partial y} + \frac{\partial w^{o}}{\partial z} = 0,

(9)

u^{o} \frac{\partial u^{o}}{\partial x} + v^{o} \frac{\partial u^{o}}{\partial y} + w^{o} \frac{\partial u^{o}}{\partial z} = ν \frac{\partial^{2} u^{o}}{\partial z^{2}} + \sqrt{2 ν} Γ \frac{\partial u^{o}}{\partial z} \frac{\partial^{2} u^{o}}{\partial z^{2}} - \frac{σ B_{o}^{2}}{ρ} u^{o} - \frac{ν}{k} u^{o},

(10)

u^{o} \frac{\partial v^{o}}{\partial x} + v^{o} \frac{\partial v^{o}}{\partial y} + w^{o} \frac{\partial v^{o}}{\partial z} = ν \frac{\partial^{2} v^{o}}{\partial z^{2}} + \sqrt{2 ν} Γ \frac{\partial v^{o}}{\partial z} \frac{\partial^{2} v^{o}}{\partial z^{2}} - \frac{σ B_{o}^{2}}{ρ} v^{o} - \frac{ν}{k} v^{o},

(11)

\begin{matrix} u^{o} \frac{\partial T}{\partial x} + v^{o} \frac{\partial T}{\partial y} + w^{o} \frac{\partial T}{\partial z} = \frac{k}{ρ C_{p}} \frac{\partial^{2} T}{\partial z^{2}} - \frac{1}{ρ C_{p}} \frac{\partial q_{r}}{\partial z} \\ + τ_{1} [D_{B} \frac{\partial C}{\partial z} \frac{\partial T}{\partial z} + \frac{D_{T}}{T_{\infty}} {(\frac{\partial T}{\partial z})}^{2}] + \frac{Q_{o}}{ρ C_{p}} (T - T_{\infty}) \\ + \frac{σ B_{o}^{2}}{ρ} ({u^{o}}^{2} + {v^{o}}^{2}) + \frac{μ_{o}}{ρ C_{p}} (u^{o}'^{2} + v^{o}'^{2}), \end{matrix}

(12)

u^{o} \frac{\partial C}{\partial x} + v^{o} \frac{\partial C}{\partial y} + w^{o} \frac{\partial C}{\partial z} = D_{B} \frac{\partial^{2} C}{\partial z^{2}} + \frac{D_{T}}{T_{\infty}} {(\frac{\partial T}{\partial z})}^{2},

(13)

u^{o} \frac{\partial n}{\partial x} + v^{o} \frac{\partial n}{\partial y} + w^{o} \frac{\partial n}{\partial z} + \frac{c_{b} W_{c}}{(C_{w} - C_{\infty})} [\frac{\partial}{\partial z} (n \frac{\partial C}{\partial z})] = D_{m} (\frac{\partial^{2} n}{\partial z^{2}}) .

(14)

Here, $u^{o}$ represents velocity component in the direction $x - axes$ , $v^{o}$ represents velocity component in the direction $y - axes$ , and $w^{o}$ represents velocity component in the direction $z - axes . ν, Γ$ indicates kinematic viscosity and Williamson liquid respectively. Moreover, $k$ , $C_{p}, T$ , $D_{B}$ , $D_{T} and τ_{1}$ represents thermal conductivity, specific heat, temperature, Brownian movement coefficient, thermophoretic coefficient, and heat capacity ratio respectively.

Moreover, the radiative heat flux expressed as follows:

q_{r} = - \frac{4 σ^{*}}{3 k^{*}} \frac{\partial T^{4}}{\partial z} .

(15)

Here, $T^{4} = 4 T_{\infty}^{3} T - 3 T_{\infty}^{4},$ furthermore, symbols $k^{*}, σ^{*}$ represents Stefan Boltzmann constant and absorption coefficient. By utilizing equation (15) in equation (12), it reduces as:

\begin{matrix} u^{o} \frac{\partial T}{\partial x} + v^{o} \frac{\partial T}{\partial y} + w^{o} \frac{\partial T}{\partial z} = \frac{k}{ρ C_{p}} (1 + \frac{4 σ^{*} T_{\infty}^{3}}{k^{*}}) \frac{\partial^{2} T}{\partial z^{2}} \\ + τ_{1} [D_{B} \frac{\partial C}{\partial z} \frac{\partial T}{\partial z} + \frac{D_{T}}{T_{\infty}} {(\frac{\partial T}{\partial z})}^{2}] + \frac{Q_{o}}{ρ C_{p}} (T - T_{\infty}) \\ + \frac{σ B_{o}^{2}}{ρ} ({u^{o}}^{2} + {v^{o}}^{2}) + \frac{μ_{o}}{ρ C_{p}} (u^{o}'^{2} + v^{o}'^{2}) . \end{matrix}

(16)

The equivalent prior conditions for the presumed problem are demonstrated as follows:

\begin{matrix} u^{o} = U_{w} = ax, v^{o} = V_{w} = by, w^{o} = 0, \\ - k \frac{\partial T}{\partial z} = h_{f} (T_{f} - T), - D_{B} \frac{\partial C}{\partial z} h_{s} (C_{f} - C), n = n_{f} at η = 0, \\ u^{o} \to 0, v^{o} \to 0, w^{o} \to 0, T \to T_{\infty}, C \to C_{\infty}, n \to n_{\infty} as η \to \infty . \end{matrix}

(17)

Here, $h_{f}$ represents convective thermal transmission whereas, $h_{s}$ represents mass transportation coefficients. $U_{w} and V_{w}$ defined as expanding velocities. Moreover, $a, b$ are positive constants and also $c = \frac{a}{b} . T_{f}$ indicates convective liquid temperature and $C$ demonstrates concentration under the moving sheet.

In accordance to above mentioned equations, similarity variables expressed as follows:

\begin{matrix} u^{o} = axf' (η), v^{o} = byg' (η), w^{o} = - (av)^{\frac{1}{2}} (f (η) + cg (η)), \\ θ = \frac{T - T_{\infty}}{T_{f} - T_{\infty}}, ϕ = \frac{C - C_{\infty}}{C_{f} - C_{\infty}}, χ = \frac{n - n_{\infty}}{n_{f} - n_{\infty}}, η = z \sqrt{\frac{a}{v}} . \end{matrix}

(18)

Keeping in view equation (18), equation (9) satisfies whereas, equations (10), (11), (16), (13), (14), and (17) are given as:

f ″' - (f')^{2} + (f + cg) f ″ + We f ″ f ″' - γ f' - M^{2} f' = 0,

(19)

g ″' - (g')^{2} + (f + cg) g ″ + We g ″ g ″' - γ g' - M^{2} g' = 0,

(20)

\begin{matrix} \frac{1}{\Pr} (1 + \frac{4}{3} Rd) θ ″ + (f + cg) θ' + Nb θ' ϕ' + Nt (θ')^{2} \\ + Q_{o} θ + MEc ({(f')}^{2} + {(g')}^{2}) + (E c_{x} {(f ″)}^{2} + E c_{y} {(g ″)}^{2}) = 0, \end{matrix}

(21)

ϕ ″ + \frac{Nt}{Nb} θ ″ + Le (f + cg) ϕ' = 0,

(22)

χ ″ + Lb (f + cg) χ' - Pe [(χ + δ_{1}) ϕ ″ + χ' ϕ'] = 0 .

(23)

With prior conditions:

\begin{matrix} f (0) & = g (0) = 0, χ (0) = 1, f' (0) = 1, g' (0) = 1, \\ f' (\infty) & = g' (\infty) = 0, θ' (0) = - B i_{1} (1 - θ (0)), \\ ϕ' (0) & = - B i_{2} (1 - ϕ (0)), θ (\infty) = ϕ (\infty) = χ (\infty) = 0 . \end{matrix}

(24)

Here, $We = Γ x \sqrt{\frac{2 a^{3}}{v}}, γ = \frac{v}{ak}, \Pr = \frac{ρ v c_{p}}{k}, Ec = \frac{{(U_{w}^{o})}^{2}}{{(C_{p})}_{f} (T_{w} - T_{\infty})} Rd = \frac{4 σ^{*} T_{\infty}^{3}}{kk *}, Nt = \frac{τ_{1} D_{T} (T_{f} - T_{\infty})}{v T_{\infty}}$ and $Nb = \frac{τ_{1} D_{T} (C_{f} - C_{\infty})}{v C_{\infty}}$ represents Williamson constraint, porosity constraint, Prandtl number, Eckert number, thermal radiation constraint, thermophoresis, and Brownian movement constraint respectively. Moreover, $B i_{1} = \frac{\frac{h_{f}}{k}}{\sqrt{\frac{v}{a}}}, B i_{2} = \frac{\frac{h_{s}}{D_{B}}}{\sqrt{\frac{v}{a}}}, Lb = \frac{α}{D_{m}}, δ_{1} = \frac{(T_{f} - T_{\infty})}{T_{\infty}} and Pe = \frac{c_{b} W_{c}}{v_{o}} α$ are Biot numbers, bio convection Lewis number, distinct temperature difference constraint, and Peclet number respectively.

The drag force in mathematical form:

C_{fx} = \frac{τ_{wx}}{ρ U_{w}^{2}},

C_{fy} = \frac{τ_{wy}}{ρ U_{w}^{2}} .

(25)

Where, after usage of boundary layer approximation, equation (24) represents shear stress rate and also skin friction coefficient whereas the skin friction in non-dimensional form is given as follows:

\frac{1}{\sqrt{2}} C_{fx} {Re}_{x}^{1 / 2} = [f ″ (0) + We {(f ″ (0))}^{2}],

\frac{1}{\sqrt{2}} C_{fy} {Re}_{x}^{1 / 2} = [g ″ (0) + We {(g ″ (0))}^{2}] .

(26)

Heat transfer efficiency in mathematical form defined as:

N u_{x} = \frac{x q_{w}}{k (T_{f} - T_{\infty})} .

(27)

Where, the external heat transportation is:

q_{w} = - \frac{\partial T}{\partial z} - \frac{16 σ^{*} T_{\infty}^{3}}{3 k^{*}} \frac{\partial T}{\partial z} at z = 0 .

With the utilization of equation (18), the preceding solution in reduced form is as follows:

N u_{x} {Re}_{x}^{1 / 2} = - (1 - Rd) θ' (0) .

(28)

Sherwood number defined as follows:

S h_{x} = \frac{x q_{m}}{k (C_{f} - C_{\infty})} .

(29)

Here, surface mass flow defined as:

q_{m} = - D_{B} \frac{\partial C}{\partial z} at z = 0 .

With the utilization of equation (18), above equation in dimensionless version is:

S h_{x} {Re}_{x}^{1 / 2} = - ϕ' (0) .

(30)

Moreover, microorganism density is defined as follows:

N n_{x} = \frac{x q_{n}}{k (n_{f} - n_{\infty})} .

(31)

Here, $q_{n}$ demonstrates the motile organism flux and denoted as:

q_{n} = - D_{n} \frac{\partial n}{\partial z} at z = 0 .

With the utilization of equation (18), above equation in dimensionless version is:

N n_{x} {Re}_{x}^{1 / 2} = - χ' (0) .

(32)

Where, local Reynolds number owing to stretching velocity $U_{w} (x),$ termed as $R e_{x}$ = $\frac{U_{x} (x)}{x}$ .

Methodology

An analytical method for resolving nonlinear differential equations is the Optimal Homotopy Analysis Method (OHAM).^45–48 Modifying convergence control settings minimizes approximation error and produces more precise and adaptable solutions by combining homotopy analysis with optimization techniques. The OHAM (Optimal Homotopy Analysis Method) is utilized in order to rectify the modeled system of streaming simulations (ODEs) by invoking BCs. Boundary physical constraints have inefficient paired ODE structure. By invoking OHAM technique to our obtained homotopy solutions, which converges.^45–48

The following auxiliary operators and initial deformations are appropriate for OHAM solutions:

f_{0} (η) = 1 - \frac{1}{e^{η}}, g_{0} (η) = 1 - \frac{1}{e^{η}}, θ_{0} (η) = \frac{B i_{1}}{(1 + B i_{1}) e^{η}},

ϕ_{0} (η) = \frac{B i_{2}}{(1 + B i_{2}) e^{η}}, χ_{0} (η) = \frac{1}{e^{η}},

(33)

{\overset{⌣}{ℒ}}_{f} = f^{‴} - f^{'}, {\overset{⌣}{ℒ}}_{g} = g^{‴} - g^{'}, {\overset{⌣}{ℒ}}_{θ} = θ^{″} - θ,

{\overset{⌣}{ℒ}}_{ϕ} = ϕ^{″} - ϕ, {\overset{⌣}{ℒ}}_{χ} = χ^{″} - χ .

(34)

The preceding set of linear operators is followed:

{\begin{matrix} {\overset{⌣}{ℒ}}_{f} [{\overset{ˇ}{ℬ}}_{1} + {\overset{ˇ}{ℬ}}_{2} e^{η} + {\overset{ˇ}{ℬ}}_{3} \frac{1}{e^{η}}] = 0, \\ {\overset{⌣}{ℒ}}_{g} [{\overset{ˇ}{ℬ}}_{4} + {\overset{ˇ}{ℬ}}_{5} e^{η} + {\overset{ˇ}{ℬ}}_{6} \frac{1}{e^{η}}] = 0, \\ {\overset{⌣}{ℒ}}_{θ} [{\overset{ˇ}{ℬ}}_{7} e^{η} + {\overset{ˇ}{ℬ}}_{8} \frac{1}{e^{η}}] = 0, \\ {\overset{⌣}{ℒ}}_{ϕ} [{\overset{ˇ}{ℬ}}_{9} e^{η} + {\overset{ˇ}{ℬ}}_{10} \frac{1}{e^{η}}] = 0, \\ {\overset{⌣}{ℒ}}_{χ} [{\overset{ˇ}{ℬ}}_{11} e^{η} + {\overset{ˇ}{ℬ}}_{12} \frac{1}{e^{η}}] = 0 . \end{matrix}}

(35)

Where ${\overset{ˇ}{ℬ}}_{h} (\overset{ˇ}{h} = 1 - 12)$ displays the random constants. The zeroth-order deformation problems are given by:

(1 - \overset{\cdot\cdot}{Q}) {\overset{⌣}{ℒ}}_{f} [\overset{⌣}{f} (η, \overset{\cdot\cdot}{Q}) - f_{0} (η)] = \overset{\cdot\cdot}{Q} ℏ_{f} {\overset{⌣}{N}}_{f} [\overset{⌣}{f} (η, \overset{\cdot\cdot}{Q}), \overset{⌣}{g} (η, \overset{\cdot\cdot}{Q})],

(36)

(1 - \overset{\cdot\cdot}{Q}) {\overset{⌣}{ℒ}}_{g} [\overset{⌣}{g} (η, \overset{\cdot\cdot}{Q}) - g_{0} (η)] = \overset{\cdot\cdot}{Q} ℏ_{g} {\overset{⌣}{N}}_{g} [\overset{⌣}{f} (η, \overset{\cdot\cdot}{Q}), \overset{⌣}{g} (η, \overset{\cdot\cdot}{Q})],

(37)

\begin{matrix} (1 - \overset{\cdot\cdot}{Q}) {\overset{⌣}{L}}_{θ} [\overset{⌣}{θ} (η, \overset{\cdot\cdot}{Q}) - θ_{0} (η)] \\ = \overset{\cdot\cdot}{Q} ℏ_{θ} {\overset{⌣}{N}}_{θ} [\overset{⌣}{f} (η, \overset{\cdot\cdot}{Q}), \overset{⌣}{g} (η, \overset{\cdot\cdot}{Q}), \overset{⌣}{θ} (η, \overset{\cdot\cdot}{Q})], \end{matrix}

(38)

\begin{matrix} (1 - \overset{\cdot\cdot}{Q}) {\overset{⌣}{L}}_{ϕ} [\overset{⌣}{ϕ} (η, \overset{\cdot\cdot}{Q}) - ϕ_{0} (η)] \\ = \overset{\cdot\cdot}{Q} ℏ_{ϕ} {\overset{⌣}{N}}_{ϕ} [\overset{⌣}{f} (η, \overset{\cdot\cdot}{Q}), \overset{⌣}{g} (η, \overset{\cdot\cdot}{Q}), \overset{⌣}{ϕ} (η, \overset{\cdot\cdot}{Q})], \end{matrix}

(39)

\begin{matrix} (1 - \overset{\cdot\cdot}{Q}) {\overset{⌣}{L}}_{χ} [\overset{⌣}{χ} (η, \overset{\cdot\cdot}{Q}) - χ_{0} (η)] \\ = \overset{\cdot\cdot}{Q} ℏ_{χ} {\overset{⌣}{N}}_{χ} [\overset{⌣}{f} (η, \overset{\cdot\cdot}{Q}), \overset{⌣}{g} (η, \overset{\cdot\cdot}{Q}), \overset{⌣}{χ} (η, \overset{\cdot\cdot}{Q})], \end{matrix}

(40)

\begin{matrix} \overset{⌣}{f} (0, \overset{\cdot\cdot}{Q}) & = 0, \overset{⌣}{f}' (0, \overset{\cdot\cdot}{Q}) = 1, \overset{⌣}{f}' (\infty, \overset{\cdot\cdot}{Q}) = 0, \\ \overset{⌣}{g} (0, \overset{\cdot\cdot}{Q}) & = 0, \overset{⌣}{g}' (0, \overset{\cdot\cdot}{Q}) = 1, g' (\infty, \overset{\cdot\cdot}{Q}) = 0, \\ \overset{⌣}{θ}' (0, \overset{\cdot\cdot}{Q}) & = - B i_{1} (1 - \overset{⌣}{θ} (0, \overset{\cdot\cdot}{Q})), \overset{⌣}{θ} (\infty, \overset{\cdot\cdot}{Q}) = 0, \\ \overset{⌣}{ϕ}' (0, \overset{\cdot\cdot}{Q}) & = - B i_{2} (1 - \overset{⌣}{ϕ} (0, \overset{\cdot\cdot}{Q})), \overset{⌣}{ϕ} (\infty, \overset{\cdot\cdot}{Q}) = 0, \\ \overset{⌣}{χ} (0, \overset{\cdot\cdot}{Q}) & = 1, \overset{⌣}{χ} (\infty, \overset{\cdot\cdot}{Q}) = 0, \end{matrix}

(41)

\begin{matrix} {\overset{⌣}{N}}_{f} [\overset{⌣}{f} (η; \overset{\cdot\cdot}{Q}), \overset{⌣}{g} (η; \overset{\cdot\cdot}{Q})] = \frac{\partial^{3} \overset{⌣}{f}}{\partial η^{3}} - {(\frac{\partial^{2} \overset{⌣}{f}}{\partial η^{2}})}^{2} + (f + cg) \frac{\partial^{2} \overset{⌣}{f}}{\partial η^{2}} \\ + We \frac{\partial^{2} \overset{⌣}{f}}{\partial η^{2}} \frac{\partial^{3} \overset{⌣}{f}}{\partial η^{3}} - γ \frac{\partial \overset{⌣}{f}}{\partial η} - M^{2} \frac{\partial \overset{⌣}{f}}{\partial η}, \end{matrix}

(42)

\begin{matrix} {\overset{⌣}{N}}_{g} [\overset{⌣}{f} (η; \overset{\cdot\cdot}{Q}), \overset{⌣}{g} (η; \overset{\cdot\cdot}{Q})] = \frac{\partial^{3} \overset{⌣}{g}}{\partial η^{3}} - {(\frac{\partial^{2} \overset{⌣}{g}}{\partial η^{2}})}^{2} + (f + cg) \frac{\partial^{2} \overset{⌣}{g}}{\partial η^{2}} \\ + We \frac{\partial^{2} \overset{⌣}{g}}{\partial η^{2}} \frac{\partial^{3} \overset{⌣}{g}}{\partial η^{3}} - γ \frac{\partial \overset{⌣}{g}}{\partial η} - M^{2} \frac{\partial \overset{⌣}{g}}{\partial η}, \end{matrix}

(43)

\begin{matrix} {\overset{⌣}{N}}_{θ} [\overset{⌣}{f} (η; \overset{\cdot\cdot}{Q}), \overset{⌣}{g} (η; \overset{\cdot\cdot}{Q}), \overset{⌣}{θ} (η; \overset{\cdot\cdot}{Q})] = \frac{1}{\Pr} (1 + \begin{matrix} 4 \\ 3 \end{matrix} Rd) \frac{\partial^{2} \overset{⌣}{θ}}{\partial η^{2}} \\ + (f + cg) \frac{\partial \overset{⌣}{θ}}{\partial η} + Nb \frac{\partial \overset{⌣}{θ}}{\partial η} \frac{\partial \overset{⌣}{ϕ}}{\partial η} + Nt {(\frac{\partial \overset{⌣}{θ}}{\partial η})}^{2} + Q_{0} \overset{⌣}{θ} + MEc \\ ({(\frac{\partial \overset{⌣}{f}}{\partial η})}^{2} + {(\frac{\partial \overset{⌣}{g}}{\partial η})}^{2}) + (E c_{x} {(\frac{\partial^{2} \overset{⌣}{f}}{\partial η^{2}})}^{2} + E c_{y} {(\frac{\partial^{2} \overset{⌣}{g}}{\partial η^{2}})}^{2}), \end{matrix}

(44)

\begin{matrix} {\overset{⌣}{N}}_{ϕ} [\overset{⌣}{f} (η; \overset{\cdot\cdot}{Q}), \overset{⌣}{g} (η; \overset{\cdot\cdot}{Q}), \overset{⌣}{ϕ} (η; \overset{\cdot\cdot}{Q})] & = \frac{\partial^{2} \overset{⌣}{ϕ}}{\partial η^{2}} + \frac{Nt}{Nb} \frac{\partial^{2} \overset{⌣}{θ}}{\partial η^{2}} \\ + Le (f + cg) \frac{\partial \overset{⌣}{ϕ}}{\partial η}, \end{matrix}

(45)

\begin{matrix} {\overset{⌣}{N}}_{χ} [\overset{⌣}{f} (η; \overset{\cdot\cdot}{Q}), \overset{⌣}{g} (η; \overset{\cdot\cdot}{Q}), \overset{⌣}{χ} (η; \overset{\cdot\cdot}{Q})] & = \frac{\partial^{2} \overset{⌣}{χ}}{\partial η^{2}} + Lb (f + cg) \frac{\partial \overset{⌣}{χ}}{\partial η} \\ - Pe [(\overset{⌣}{χ} + δ_{1}) \frac{\partial^{2} \overset{⌣}{ϕ}}{\partial η^{2}} + \frac{\partial \overset{⌣}{χ}}{\partial η} \frac{\partial \overset{⌣}{ϕ}}{\partial η}] . \end{matrix}

(46)

In the axioms overhead, $\overset{\cdot\cdot}{Q} \in [0, 1]$ represents the embedding parameter, $ℏ_{f}, ℏ_{g}, ℏ_{θ}, ℏ_{ϕ}$ , and $ℏ_{χ}$ the non-zero auxiliary parameters, and ${\overset{⌣}{N}}_{f}$ , ${\overset{⌣}{N}}_{g}$ , ${\overset{⌣}{N}}_{θ}$ , ${\overset{⌣}{N}}_{ϕ}$ , and ${\overset{⌣}{N}}_{χ}$ , the nonlinear operators.

\overset{⌣}{f} (η; 0) = f_{0} (η), \overset{⌣}{f} (η; 1) = f (η),

(47)

\overset{⌣}{g} (η; 0) = g_{0} (η), \overset{⌣}{g} (η; 1) = g (η),

(48)

\overset{⌣}{θ} (η; 0) = θ_{0} (η), \overset{⌣}{θ} (η; 1) = θ (η),

(49)

\overset{⌣}{ϕ} (η; 0) = ϕ_{0} (η), \overset{⌣}{ϕ} (η; 1) = ϕ (η),

(50)

\overset{⌣}{χ} (η; 0) = χ_{0} (η), \overset{⌣}{χ} (η; 1) = χ (η) .

(51)

When $\overset{\cdot\cdot}{Q}$ changes as of 0–1 then $\overset{⌣}{f} (η; \overset{\cdot\cdot}{Q}), \overset{⌣}{g} (η; \overset{\cdot\cdot}{Q}), \overset{⌣}{θ} (η; \overset{\cdot\cdot}{Q}), \overset{⌣}{ϕ} (η; \overset{\cdot\cdot}{Q}) and \overset{⌣}{χ} (η; \overset{\cdot\cdot}{Q})$ fluctuate from the initial guesses $f_{0} (η), g_{0} (η), θ_{0} (η), ϕ_{0} (η)$ and $χ_{0} (η)$ to the consequences $f (η), g (η), θ (η), ϕ (η) and χ_{0} (η),$ correspondingly. The Taylor series expansion’s employment provides:

\overset{⌣}{f} (η; \overset{\cdot\cdot}{Q}) = f_{0} (η) + \frac{{\sum^{⌣}}_{\overset{⌣}{m} = 1}^{\infty} f_{\overset{⌣}{m}} (η) {\overset{\cdot\cdot}{Q}}^{\overset{⌣}{m}}, f_{\overset{⌣}{m}} (η) = \frac{1}{\overset{⌣}{m}!} \partial^{\overset{⌣}{m}} f (η, \overset{\cdot\cdot}{Q})}{\partial {\overset{\cdot\cdot}{Q}}^{\overset{⌣}{m}}} |_{\overset{\cdot\cdot}{Q} = 0},

(52)

\overset{⌣}{g} (η; \overset{\cdot\cdot}{Q}) = g_{0} (η) + {\sum^{⌣}}_{\overset{⌣}{m} = 1}^{\infty} g (η) {\overset{\cdot\cdot}{Q}}^{\overset{⌣}{m}}, g_{\overset{⌣}{m}} (η) = \frac{1}{\overset{⌣}{m}!} \frac{\partial^{\overset{⌣}{m}} \overset{⌣}{g} (η, \overset{\cdot\cdot}{Q})}{\partial {\overset{\cdot\cdot}{Q}}^{\overset{⌣}{m}}} |_{\overset{\cdot\cdot}{Q} = 0},

(53)

\overset{⌣}{θ} (η; \overset{\cdot\cdot}{Q}) = θ_{0} (η) + \frac{{\sum^{⌣}}_{\overset{⌣}{m} = 1}^{\infty} θ_{\overset{⌣}{m}} (η) {\overset{\cdot\cdot}{Q}}^{\overset{⌣}{m}}, θ_{\overset{⌣}{m}} (η) = \frac{1}{\overset{⌣}{m}!} \partial^{\overset{⌣}{m}} θ (η, \overset{\cdot\cdot}{Q})}{\partial {\overset{\cdot\cdot}{Q}}^{\overset{⌣}{m}}} |_{\overset{\cdot\cdot}{Q} = 0},

(54)

\overset{⌣}{ϕ} (η; \overset{\cdot\cdot}{Q}) = ϕ_{0} (η) + \frac{{\sum^{⌣}}_{\overset{⌣}{m} = 1}^{\infty} ϕ_{\overset{⌣}{m}} (η) {\overset{\cdot\cdot}{Q}}^{\overset{⌣}{m}}, ϕ_{\overset{⌣}{m}} (η) = \frac{1}{\overset{⌣}{m}!} \partial^{\overset{⌣}{m}} ϕ (η, \overset{\cdot\cdot}{Q})}{\partial {\overset{\cdot\cdot}{Q}}^{\overset{⌣}{m}}} |_{\overset{\cdot\cdot}{Q} = 0} .

(55)

\overset{⌣}{χ} (η; \overset{\cdot\cdot}{Q}) = χ_{0} (η) + \frac{{\sum^{⌣}}_{\overset{⌣}{m} = 1}^{\infty} χ_{\overset{⌣}{m}} (η) {\overset{\cdot\cdot}{Q}}^{\overset{⌣}{m}}, χ_{\overset{⌣}{m}} (η) = \frac{1}{\overset{⌣}{m}!} \partial^{\overset{⌣}{m}} χ (η, \overset{\cdot\cdot}{Q})}{\partial {\overset{\cdot\cdot}{Q}}^{\overset{⌣}{m}}} |_{\overset{\cdot\cdot}{Q} = 0} .

(56)

The convergence of equations (52)–(56) powerfully is dependent on production the right conclusions of $ℏ_{f}, ℏ_{g}, ℏ_{θ}, ℏ_{ϕ} and ℏ_{χ}$ . Considering that $ℏ_{f}, ℏ_{g}, ℏ_{θ}, ℏ_{ϕ} and ℏ_{χ}$ are selected in a way that equations (52)–(56) converge at $\overset{\cdot\cdot}{Q} = 1$ then

f (η) = f_{0} (η) + \sum_{\overset{⌣}{m} = 1}^{\infty} f_{\overset{⌣}{m}} (η),

(57)

g (η) = g_{0} (η) + \sum_{\overset{⌣}{m} = 1}^{\infty} g_{\overset{⌣}{m}} (η),

(58)

θ (η) = θ_{0} (η) + \sum_{\overset{⌣}{m} = 1}^{\infty} θ_{\overset{⌣}{m}} (η),

(59)

ϕ (η) = ϕ_{0} (η) + \sum_{\overset{⌣}{m} = 1}^{\infty} ϕ_{\overset{⌣}{m}} (η),

(60)

χ (η) = χ_{0} (η) + \sum_{\overset{⌣}{m} = 1}^{\infty} χ_{\overset{⌣}{m}} (η) .

(61)

The $\overset{⌣}{m}$ th-order deformation issues are described as follows:

{\overset{⌣}{ℒ}}_{f} [f_{\overset{⌣}{m}} (η) {\overset{⌣}{χ}}_{\overset{⌣}{m}} f_{\overset{⌣}{m} - 1} (η) = ℏ_{f} R_{f}^{\overset{⌣}{m}} (η)],

(62)

{\overset{⌣}{ℒ}}_{g} [g_{\overset{⌣}{m}} (η) {\overset{⌣}{χ}}_{\overset{⌣}{m}} g_{\overset{⌣}{m} - 1} (η) = ℏ_{g} R_{g}^{\overset{⌣}{m}} (η)],

(63)

{\overset{⌣}{ℒ}}_{θ} [θ_{\overset{⌣}{m}} (η) {\overset{⌣}{χ}}_{\overset{⌣}{m}} θ_{\overset{⌣}{m} - 1} (η) = ℏ_{θ} R_{θ}^{\overset{⌣}{m}} (η)],

(64)

{\overset{⌣}{ℒ}}_{ϕ} [ϕ_{\overset{⌣}{m}} (η) {\overset{⌣}{χ}}_{\overset{⌣}{m}} ϕ_{\overset{⌣}{m} - 1} (η) = ℏ_{ϕ} R_{ϕ}^{\overset{⌣}{m}} (η)],

(65)

{\overset{⌣}{ℒ}}_{χ} [χ_{\overset{⌣}{m}} (η) {\overset{⌣}{χ}}_{\overset{⌣}{m}} χ_{\overset{⌣}{m} - 1} (η) = ℏ_{χ} R_{χ}^{\overset{⌣}{m}} (η)],

(66)

\begin{matrix} f_{\overset{⌣}{m}} (0) = 0, f_{\overset{⌣}{m}}' (0) = 1, f_{\overset{⌣}{m}}' (\infty) = 0, g_{\overset{⌣}{m}} (0) = 0, g_{\overset{⌣}{m}}' (0) \\ = 1, g_{\overset{⌣}{m}}' (\infty) = 0, θ_{\overset{⌣}{m}} (0) = - B i_{1} (1 - θ (0)), θ_{\overset{⌣}{m}} (\infty) \\ = 0, ϕ_{\overset{⌣}{m}}' (0) = - B i_{2} (1 - ϕ (0)), ϕ_{\overset{⌣}{m}} (\infty) = 0, χ_{\overset{⌣}{m}} (0) \\ = 1, χ_{\overset{⌣}{m}} (\infty) = 0 . \end{matrix}

(67)

Convergence analysis

ε_{\overset{⌣}{m}}^{f} = \frac{1}{k + 1} \sum_{j = 0}^{k} {\overset{⌣}{N}}_{f} {[{(\sum_{i = 0}^{\overset{⌣}{m}} \overset{⌣}{f} (η), \sum_{i = 0}^{\overset{⌣}{m}} \overset{⌣}{g} (η),)}_{η = j δ η}]}^{2},

(68)

ε_{\overset{⌣}{m}}^{g} = \frac{1}{k + 1} \sum_{j = 0}^{k} {[{\overset{⌣}{N}}_{g} {(\sum_{i = 0}^{\overset{⌣}{m}} \overset{⌣}{f} (η), \sum_{i = 0}^{\overset{⌣}{m}} \overset{⌣}{g} (η),)}_{η = j δ η}]}^{2},

(69)

\begin{matrix} ε_{\overset{⌣}{m}}^{θ} = \frac{1}{k + 1} \sum_{j = 0}^{k} {[{\overset{⌣}{N}}_{θ} {(\sum_{i = 0}^{\overset{⌣}{m}} \overset{⌣}{f} (η), \sum_{i = 0}^{\overset{⌣}{m}} \overset{⌣}{g} (η), \sum_{i = 0}^{\overset{⌣}{m}} \overset{⌣}{θ} (η),)}_{η = j δ η}]}^{2} \end{matrix},

(70)

\begin{matrix} ε_{\overset{⌣}{m}}^{ϕ} = \frac{1}{k + 1} \sum_{j = 0}^{k} {[{\overset{⌣}{N}}_{ϕ} {(\sum_{i = 0}^{\overset{⌣}{m}} \overset{⌣}{f} (η), \sum_{i = 0}^{\overset{⌣}{m}} \overset{⌣}{g} (η), \sum_{i = 0}^{\overset{⌣}{m}} \overset{⌣}{ϕ} (η),)}_{η = j δ η}]}^{2} \end{matrix},

(71)

\begin{matrix} ε_{\overset{⌣}{m}}^{χ} = \frac{1}{k + 1} \sum_{j = 0}^{k} {[{\overset{⌣}{N}}_{χ} {(\sum_{i = 0}^{\overset{⌣}{m}} \overset{⌣}{f} (η), \sum_{i = 0}^{\overset{⌣}{m}} \overset{⌣}{g} (η), \sum_{i = 0}^{\overset{⌣}{m}} \overset{⌣}{χ} (η),)}_{η = j δ η}]}^{2} \end{matrix} .

(72)

Succeeding Liao⁴⁹:

ε_{\overset{⌣}{m}}^{t} = ε_{\overset{⌣}{m}}^{f} + ε_{\overset{⌣}{m}}^{g} + ε_{\overset{⌣}{m}}^{θ} + ε_{\overset{⌣}{m}}^{ϕ} + ε_{\overset{⌣}{m}}^{χ} .

(73)

Working procedure of OHAM

Optimal Homotopy analysis method (OHAM) is most powerful analytical approach utilized to compute non-linear differential equations that may not provide direct analytical solutions. Below is the general outline of working process of the OHAM

Firstly, consider the non-linear differential equation in order to get the solution of that equation. For instance, consider an equation expressed in the version of $L (u) = 0,$ Here, $u$ is unknown whereas, $L$ defined as nonlinear operator.

As a second step, we introduce and designed auxiliary linear operator $A (u, λ^{*})$ in such a way that

• Original non-linear equation in the form $A (u, 0) = L (u),$

• $A (u, 1)$ is easy to solve and also in linear form.

Here, $λ^{*}$ is an embedding constraint range from 0 to 1. In general, $A (u, λ^{*})$ is an advanced form of $L (u)$ that simplifies when $λ^{*}$ approaches to 1.

3. As a third step, construct a homotopy $H (u, λ^{*})$ that joins $A (u, λ^{*})$ with the original non-linear equation $L (u) = 0 :$

H (u, λ^{*}) = (1 - λ^{*}) A (u, 0) + λ^{*} A (u, 1)

4. Series solution construction

• In terms of $λ^{*},$ expand $u$ in series form:

u (x, λ^{*}) = \sum_{n = 0}^{\infty} u_{n} (x) {λ^{*}}^{n}

• Utilizing this series into $H (u, λ^{*}) = 0,$ equating coefficients of powers of $λ^{*}$ in order to produce sequence of linear and also non-linear algebraic equation particularly for $u_{n} (x)$ .

5. Solve the obtained system of coupled algebraic equations in order to obtain the coefficients $u_{n} (x)$ .

6. Evaluate the solution

Sum the series solution $u (x, λ^{*})$ in order to obtain an approximation $u (x)$ specifically for the original considered problem.

7. Check and verify:

• Compare the obtained solution with original solution and ensure it satisfies the initial conditions.

• By adjusting the required number of terms in the series solution or utilization of advanced techniques if necessary.

Discussion and results

The modified non-dimensional ODEs (19)–(23) are solved by utilization of OHAM approach. The results are validated graphically and also interpreted in tabular form. The outcomes of current analysis are interpreted graphically through Figures 2 to 25. In this section, Figures 2 to 9 interpret the influence of $c$ constraint and Williamson parameter expressed as $We$ on velocity contour (along x-axes and also y-axes). Weissenberg number is known to be quotient of elastic forces to the viscous forces. Higher estimation in Weissenberg number demonstrates strong elastic forces impact in contrast to the viscous forces. Consequently, enhanced relaxation time and therefore feeble velocity contour (along x-axes and also y-axes) is witness. Moreover, porosity constraint $γ$ and magnetic parameter $M$ on velocity contour (along x-axes and also y-axes) are also portrayed. Lorentz force rise up owing to the presence of magnetic field. This force magnitude is directly proportional to the magnetic constraint magnitude. Therefore, with the rise up in $M,$ Lorentz force strengthens. Consequently, provides higher resistivity to the fluid occurrence and therefore momentum tends to decreased with the enhancement in $M .$ This declination across the fluid velocity flow allows Nano-size particles to conduct heat more and therefore temperature enhancement is observed. Moreover, this segment also comprises of Figures 10 to 15 is sketched to visualize the influence of Magnetic constraint $M,$ radiation constraint $Rd$ , Brownian constraint $Nb$ and thermophoresis constraint $Nt$ , heat generation constraint $Q_{o}$ , and also Eckert number $Ec .$ These figures outlines consequential knowledge under the presence of these constraints, temperature profiles tends to be remarkably accelerated. Higher estimates of the thermophoretic constraint fluid nano size particles beyond from the heated surface. Hence, a higher fluid temperature is seen.

Figures 16 and 17 are constructed to witness the involved parameters such as Prandtl number $\Pr$ and $c$ across the energy distribution and hence deteriorated energy distribution is perceived for these parameters. Certainly, Prandtl number $\Pr$ considered to be significant function of viscosity, specific heat, and also thermal conductivity. Prandtl number $\Pr$ expressed as ratio of kinematic viscosity with thermal diffusivity. Moreover, $\Pr$ also manages the boundary layer and also relative velocity thickness. In view of these, one can easily remark that reduced Prandtl liquids accompanied by increased thermal conductivities provides faster heat diffusion in broader thermal boundary layer configuration in contrast to higher Prandtl liquids across the boundary layer thinner region. The influence of $c$ and thermophoretic constraint $Nt$ is illustrated through the Figures 18 and 19. An accelerated demeanor for the concentration distribution is seen for $c and Nt$ whilst declined behavior for Brownian parameter $Nb$ and also for Lewis number $Le$ is schemed in Figures 20 and 21. Biological convection Lewis number $Lb$ , constraint $c$ , Peclet number $Pe$ , and temperature difference constraint $δ_{1}$ impressions are portrayed through Figures 22 to 25. A diminishing motile organism profile is witnessed across these figures. The microorganism diffusivity vanished for higher estimates of biological convection Lewis number. Therefore, fluid velocity tends to drop. As for as concerning to Peclet number, reduction in microorganism diffusivity is witnessed over here. Consequently, that affects the fluid motile density. The influence of drag force factor $Cfx$ (along x-axes) and also $Cfy$ (along y-axes) demeanor owing to distinct influential constraints such as $c, We, γ and M$ can be witnessed in Table 1. It is noticed via Table 1 that with the upsurge values of considered physical constraints drag force factor is enhanced. Table 2 demonstrates the results of Nusselt number for distinct considered parameters like $M, Rd, Nb, Nt, Q_{0}, Ec, P_{r} and c .$ Nusselt number demonstrates decreasing demeanor for $M, Rd, Nb, Nt, Q_{0}, Ec$ and intensified for $P_{r} and c .$ In Table 3 the influence of Sherwood number under consideration of different constraint like $c, Nt, Nb and Le$ is observed. It can be seen through Table 3 that Sherwood number represents enhanced demeanor for $Nb and Le$ whereas, it declines for rise up values of $c and Nt .$ Moreover, result of $- χ'$ can be seen in Table 4 for physical parameters such as $Lb, c, Pe and δ_{1} .$ Table 4 illustrates that microorganism density is enhanced for the involved influential parameters. A convincible comparative analysis for the Nusselt number amongst the proffered study and previous study is conducted and validated the results in Table 5. Nusselt number improved for both studies under the upshot values of $P_{r} and c$ whilst declined for $Nb$ .

Figure 2.

Variation of $c$ within the non-dimensional velocity distribution along x-axes.

Figure 3.

Variation of Williamson constraint $We$ across the non-dimensional velocity distribution along x-axes.

Figure 4.

Variation of porosity constraint $γ$ across the non-dimensional velocity distribution along x-axes.

Figure 5.

Variation of magnetic constraint $M$ across the non-dimensional velocity distribution along x-axes.

Figure 6.

Variation of $c$ within the non-dimensional velocity distribution along y-axes.

Figure 7.

Variation of Williamson constraint $We$ across the non-dimensional velocity distribution along y-axes.

Figure 8.

Variation of porosity constraint $γ$ across the non-dimensional velocity distribution along y-axes.

Figure 9.

Variation of magnetic constraint $M$ across the non-dimensional velocity distribution along y-axes.

Figure 10.

Variation of magnetic constraint $M$ across the non-dimensional temperature distribution.

Figure 11.

Variation of radiation constraint $Rd$ across the non-dimensional temperature distribution.

Figure 12.

Variation of Brownian constraint $Nb$ across the non-dimensional temperature distribution.

Figure 13.

Variation of thermophoresis constraint $Nt$ across the non-dimensional temperature distribution.

Figure 14.

Variation of heat generation constraint $Q_{o}$ across the non-dimensional temperature distribution.

Figure 15.

Variation of Eckert number $Ec$ across the non-dimensional temperature distribution.

Figure 16.

Compartment in Prandtl number $\Pr$ across the non-dimensional temperature distribution.

Figure 17.

Compartment in $c$ across the non-dimensional temperature distribution.

Figure 18.

Compartment in $c$ across the non-dimensional concentration distribution.

Figure 19.

Variation of thermophoresis constraint $Nt$ across the non-dimensional concentration distribution.

Figure 20.

Variation of Brownian constraint $Nb$ across the non-dimensional concentration distribution.

Figure 21.

Variation of Lewis number $Le$ across the non-dimensional concentration distribution.

Figure 22.

Variation of bio convection Lewis number $Lb$ across the non-dimensional motile microorganism profile.

Figure 23.

Variation of $c$ across the non-dimensional motile microorganism profile.

Figure 24.

Variation of Peclet number $Pe$ across the non-dimensional motile microorganism distribution.

Figure 25.

Variation in distinct temperature difference constraint $δ_{1}$ across the non-dimensional motile microorganism distribution.

Table 1.

Numerically computed values of Drag force against the distinct values of the involved parameters.

$c$	$We$	$γ$	$M$	$Cfx$	$Cfy$
$0.1$				$- 0.588604$	$- 0.595921$
$0.2$				$- 0.582411$	$- 0.591870$
$0.3$				$- 0.575704$	$- 0.587554$
$0.4$				$- 0.568479$	$- 0.582975$
	$0.1$			$- 1.194300$	$- 1.224640$
	$0.2$			$- 1.024590$	$- 1.043260$
	$0.3$			$- 0.827111$	$- 0.828351$
	$0.4$			$- 0.560737$	$- 0.578132$
		$0.1$		$- 0.298742$	$- 0.374270$
		$0.2$		$- 0.231215$	$- 0.335186$
		$0.3$		$- 0.151924$	$- 0.290820$
		$0.4$		$- 0.059917$	$- 0.241172$
			$0.1$	$- 0.039898$	$- 0.230609$
			$0.2$	$- 0.008816$	$- 0.214368$
			$0.3$	$0.045875$	$- 0.186243$
			$0.4$	$0.128766$	$- 0.144650$

Table 2.

Numerically computed values of Nusselt number against the distinct values of the involved parameters.

$M$	$Rd$	$Nb$	$Nt$	$Q_{0}$	$Ec$	$P_{r}$	$c$	$- θ' (0)$
$0.1$								$0.053714$
$0.3$								$0.051491$
$0.5$								$0.049371$
$0.7$								$0.047368$
	$0.1$							$0.069374$
	$0.3$							$0.051491$
	$0.5$							$0.033812$
	$0.7$							$0.017561$
		$0.1$						$0.052622$
		$0.3$						$0.052565$
		$0.5$						$0.052508$
		$0.7$						$0.052452$
			$0.1$					$0.060482$
			$0.3$					$0.060439$
			$0.5$					$0.060396$
			$0.7$					$0.060353$
				$0.1$				$0.060481$
				$0.3$				$0.059570$
				$0.5$				$0.058569$
				$0.7$				$0.057472$
					$0.1$			$0.061842$
					$0.3$			$0.060002$
					$0.5$			$0.058161$
					$0.7$			$0.056320$
						$0.1$		$0.595231$
						$0.3$		$0.060001$
						$0.5$		$0.062857$
						$0.7$		$0.063454$
							$0.1$	$0.061661$
							$0.3$	$0.061992$
							$0.5$	$0.062312$
							$0.7$	$0.062624$

Table 3.

Numerically computed values of Sherwood number against the distinct values of the involved parameters.

$c$	$Nt$	$Nb$	$Le$	$- ϕ' (0)$
$0.1$				$0.128674$
$0.3$				$0.128667$
$0.5$				$0.128649$
$0.7$				$0.128619$
	$0.1$			$0.144781$
	$0.3$			$0.136364$
	$0.5$			$0.128102$
	$0.7$			$0.119995$
		$0.1$		$0.144882$
		$0.3$		$0.147646$
		$0.5$		$0.148220$
		$0.7$		$0.148465$
			$0.1$	$0.146040$
			$0.3$	$0.147814$
			$0.5$	$0.149562$
			$0.7$	$0.151283$

Table 4.

Numerically computed values of motile microorganism against the distinct values of the involved parameters.

$Lb$	$c$	$Pe$	$δ_{1}$	$- χ' (0)$
$0.1$				$0.352985$
$0.3$				$0.400121$
$0.5$				$0.448896$
$0.7$				$0.499267$
	$0.1$			$0.372708$
	$0.3$			$0.379939$
	$0.5$			$0.386996$
	$0.7$			$0.393876$
		$0.1$		$0.352984$
		$0.3$		$0.387332$
		$0.5$		$0.421668$
		$0.7$		$0.455991$
			$0.1$	$0.368125$
			$0.3$	$0.372195$
			$0.5$	$0.376266$
			$0.7$	$0.380336$

Table 5.

Comparative analysis with Shah et al.⁵⁰ for heat transfer rate values expressed numerically for various embedded parameters.

$Nb$	$P_{r}$	$c$	$- θ'$ Previous study	$- θ'$ Present study
$0.1$			$0.155044$	$0.15526220$
$0.3$			$0.154874$	$0.15456597$
$0.5$			$0.154704$	$0.15425089$
	$0.1$		$0.127300$	$0.12723989$
	$0.3$		$0.134579$	$0.13406001$
	$0.5$		$0.140705$	$0.14062857$
		$0.1$	$0.119212$	$0.11061661$
		$0.3$	$0.120401$	$0.12061992$
		$0.5$	$0.121608$	$0.12062312$

Conclusion

In this analysis, unsteady, incompressible three-dimensional Williamson liquid exposed to linear expanding sheet is numerically analyzed by invoking optimal approach namely HAM. The pivotal conclusions of the investigation are as follows:

The velocity distribution $f'$ (along x-axes) and also $g'$ (along y-axes) depicts decreasing demeanor by increasing $c, We, γ and M;$

Energy distribution pointedly enhanced with $M, Rd, Nb, Nt, Q_{o} and Ec$ versus the impact of $P_{r} and c;$

Concentration profile is augmented under the consequences of constraints like $c and Nt$ whilst declines for $Nb and Le;$

Motile organisms profile shows reduced behavior for $Lb, c, Pe and δ_{1}$ .

Drag force $Cfx$ (along x-axes) and also for $Cf y$ (along y-axes) magnified demeanor is witnessed in Table 1 with the increment in values of $c, We, γ and M$ .

The results of Nusselt number in Table 3 represents enhancement with the augmented values of $P_{r} and c$ whereas decreasing demeanor is witnessed for $M, Rd, Nb, Nt, Q_{0}, Ec$ .

The results of Sherwood number is depicted via Table 3 and it illustrates augmented demeanor for $Nb and Le$ whereas, it drops off for larger vales of $c and Nt$ .

Motile organism density in Table 4 depicts enhancement under consideration of involved parameters like $Lb, c, Pe and δ_{1}$ .

Footnotes

Handling Editor: Chenhui Liang

Author contributions

Sana Akbar (S.A): Writing-original draft preparation; writing-editing; methodology; analysis; investigation; revision and writing-reviewing; data curation; original draft preparation. Muhammad Sohail (M.S): Investigation and graphical abstract; supervision; revision and writing-reviewing; project administration; conceptualization; analysis. Syed Tehseen Abbas (S.T.A): Conceptualization; analysis; investigation. Nacer Badi (N.B): Investigation and graphical abstract; funding acquisition.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Availability of data and material

The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request.

ORCID iD

Muhammad Sohail

References

Shah

Bonyah

Islam

, et al. Radiative MHD thin film flow of Williamson fluid over an unsteady permeable stretching sheet. Heliyon 2018; 4: e00825.

Dawar

Shah

Islam

, et al. An optimal analysis for Darcy–Forchheimer three-dimensional Williamson nanofluid flow over a stretching surface with convective conditions. Adv Mech Eng 2019; 11: 1687814019833510.

Gul

Khan

Islam

, et al. Heat transfer investigation of the unsteady thin film flow of Williamson fluid past an inclined and oscillating moving plate. Appl Sci 2017; 7: 369.

Rasool

Zhang

Chamkha

, et al. Entropy generation and consequences of binary chemical reaction on MHD Darcy–Forchheimer Williamson nanofluid flow over non-linearly stretching surface. Entropy 2019; 22: 18.

Shafiq

Çolak

Sindhu

, et al. Estimation of unsteady hydromagnetic Williamson fluid flow in a radiative surface through numerical and artificial neural network modeling. Sci Rep 2021; 11: 14509.

Prasad

Setty

Mebarek-Oudina

, et al. Mixed convective Williamson nanofluid flow over a rotating disk with zero mass flux. Z Angew Math Mech 2022; 102: e202100117.

Raza

Mabood

Naz

, et al. Thermal transport of radiative Williamson fluid over stretchable curved surface. Therm Sci Eng Prog 2021; 23: 100887.

Khan

Ahmad

Wang

, et al. Enhancement in the efficiency of heat recovery in a Williamson hybrid nanofluid over a vertically thin needle with entropy generation. Heliyon 2023; 9: e17665.

Song

Khan

, et al. Bidirectional non-linear stretched flow of Williamson nanofluid with swimming of motile gyrotactic microorganisms. Appl Math Comput 2021; 411: 126502.

10.

Khan

Salahuddin

Malik

, et al. Change in viscosity of Williamson nanofluid flow due to thermal and solutal stratification. Int J Heat Mass Transf 2018; 126: 941–948.

11.

Khan

Shahid

Salahuddin

, et al. Heat and mass diffusions for Casson nanofluid flow over a stretching surface with variable viscosity and convective boundary conditions. J Braz Soc Mech Sci Eng 2018; 40: 1–10.

12.

Shah

Yook

Tosin

Analytic simulation of thermophoretic second grade fluid flow past a vertical surface with variable fluid characteristics and convective heating. Sci Rep 2022; 12: 5445.

13.

Gul

Saeed

Nonlinear mixed convection couple stress tri-hybrid nanofluids flow in a Darcy–Forchheimer porous medium over a nonlinear stretching surface. Waves Random Complex Media. Epub ahead of print 23 May 2022. DOI: 10.1080/17455030.2022.2077471.

14.

Mebarek-Oudina

Bessaih

Mahanthesh

, et al. Magneto-thermal-convection stability in an inclined cylindrical annulus filled with a molten metal. Int J Numer Methods Heat Fluid Flow 2021; 31: 1172–1189.

15.

Bilal

Sohail

Naz

Heat transport in the convective Casson fluid flow with homogeneous–heterogeneous reactions in Darcy–Forchheimer medium. Multidiscipline Model Mater Struct 2019; 15: 1170–1189.

16.

Ahmad

Naveed Khan

Nadeem

Unsteady three dimensional bioconvective flow of Maxwell nanofluid over an exponentially stretching sheet with variable thermal conductivity and chemical reaction. Int J Ambient Energy 2022; 43: 6542–6552.

17.

Khan

Ahmad

Nadeem

Flow and heat transfer investigation of bio–convective hybrid nanofluid with triple stratification effects. Phys Scr 2021; 96: 065210.

18.

Ahmad

Nadeem

Rehman

Mathematical analysis of thermal energy distribution in a hybridized mixed convective flow. J Nanofluids 2021; 10: 222–231.

19.

Wakif

Boulahia

Sehaqui

Numerical study of a thermal convection induced by a purely internal heating in a rotating medium saturated by a radiating nanofluid. Int J Comput Appl 2016; 135: 33–42.

20.

Khan

Gul

Islam

, et al. Thermophoresis and thermal radiation with heat and mass transfer in a magnetohydrodynamic thin-film second-grade fluid of variable properties past a stretching sheet. Eur Phys J Plus 2017; 132: 1–20.

21.

Saeed

Shah

Islam

, et al. Three-dimensional Casson nanofluid thin film flow over an inclined rotating disk with the impact of heat generation/consumption and thermal radiation. Coatings 2019; 9: 248.

22.

Jawad

Saeed

Bilal

, et al. The impact of magnetohydrodynamic on bioconvection nanofluid flow with viscous dissipation and joule heating effects. Eng Res Express 2021; 3: 015030.

23.

Jawad

Saeed

Khan

, et al. MHD bioconvection Darcy-Forchheimer flow of Casson nanofluid over a rotating disk with entropy optimization. Heat Transf 2021; 50: 2168–2196.

24.

Saeed

Gul

Bioconvection Casson nanofluid flow together with Darcy-Forchheimer due to a rotating disk with thermal radiation and Arrhenius activation energy. SN Appl Sci 2021; 3: 78.

25.

Gul

Ahmed

Jawad

, et al. Bio-convectional nanofluid flow due to the thermophoresis and gyrotactic microorganism between the gap of a disk and cone. Braz J Phys 2021; 51: 687–697.

26.

Shah

Dawar

Nasir

, et al. Application of Arrhenius chemical process on unsteady mixed bio-convective flows of third-grade fluids having temperature-dependent in thermo-rheological properties. Waves Random Complex Media. Epub ahead of print 16 January 2022. DOI: 10.1080/17455030.2021.2023781.

27.

Hayat

Shafiq

Alsaedi

Effect of Joule heating and thermal radiation in flow of third grade fluid over radiative surface. PLoS One 2014; 9: e83153.

28.

Rasool

Wang

Yashkun

, et al. Numerical treatment of hybrid water based nanofluid flow with effect of dissipation and Joule heating over a shrinking surface: stability analysis. J Magn Magn Mater 2023; 571: 170587.

29.

Khan

Shahid

Malik

, et al. Chemical reaction for Carreau-Yasuda nanofluid flow past a nonlinear stretching sheet considering Joule heating. Results Phys 2018; 8: 1124–1130.

30.

Xia

Hafeez

Khan

, et al. Entropy optimized dissipative flow of hybrid nanofluid in the presence of non-linear thermal radiation and Joule heating. Sci Rep 2021; 11: 16067.

31.

Malik

Bibi

Khan

, et al. Numerical solution of Williamson fluid flow past a stretching cylinder and heat transfer with variable thermal conductivity and heat generation/absorption. AIP Adv 2016; 6: 035101.

32.

Awais

Bilal

Ur Rehman

, et al. Analysis on flow features of unsteady Williamson fluid inaugurated by melted wedge in the presence of heat generation–absorption: an extensive computational study. Can J Phys 2019; 97: 1277–1287.

33.

Kumar

Varma

SVK

Kiran Kumar

RVMSS

, et al. Three-dimensional hydromagnetic convective flow of chemically reactive Williamson fluid with non-uniform heat absorption and generation. Int J Chem Reactor Eng 2018; 17: 20180118.

34.

Abbas

Jeelani

Alnahdi

, et al. MHD Williamson nanofluid fluid flow and heat transfer past a non-linear stretching sheet implanted in a porous medium: effects of heat generation and viscous dissipation. Processes 2022; 10: 1221.

35.

Mishra

Kumar

Reddy

, et al. MHD Williamson micropolar fluid flow pasting a non-linearly stretching sheet under the presence of non linear heat generation/absorption. J Indian Chem Soc 2023; 100: 100845.

36.

Naveed

Abbas

Sajid

Thermophoresis and Brownian effects on the Blasius flow of a nanofluid due to a curved surface with thermal radiation. Eur Phys J Plus 2016; 131: 1–9.

37.

Obalalu

Olayemi

Odetunde

, et al. Significance of thermophoresis and Brownian motion on a reactive Casson-Williamson nanofluid past a vertical moving cylinder. Comput Therm Sci 2023; 15.

38.

Asjad

Zahid

Inc

, et al. Impact of activation energy and MHD on Williamson fluid flow in the presence of bioconvection. Alex Eng J 2022; 61: 8715–8727.

39.

Abdal

Siddique

Eldin

, et al. Significance of thermal radiation and bioconvection for Williamson nanofluid transportation owing to cone rotation. Sci Rep 2022; 12: 22646.

40.

Priyadharshini

Karpagam

Shah

, et al. Bio-convection effects of MHD Williamson fluid flow over a symmetrically stretching sheet: machine learning. Symmetry 2023; 15: 1684.

41.

Williamson

RV.

The flow of pseudoplastic materials. Ind Eng Chem 1929; 21: 1108–1111.

42.

Satya Narayana

Tarakaramu

Moliya Akshit

, et al. MHD flow and heat transfer of an Eyring-Powell fluid over a linear stretching sheet with viscous dissipation-a numerical study. Front Heat Mass Transf 2017; 9: 1–5.

43.

Hayat

Saif

Ellahi

, et al. Simultaneous effects of melting heat and internal heat generation in stagnation point flow of Jeffrey fluid towards a nonlinear stretching surface with variable thickness. Int J Therm Sci 2018; 132: 344–354.

44.

Khan

Irfan

Khan

, et al. Modeling and simulation for 3D magneto Eyring–Powell nanomaterial subject to nonlinear thermal radiation and convective heating. Results Phys 2017; 7: 1899–1906.

45.

Sohail

Rafique

Singh

, et al. Engagement of modified heat and mass fluxes on thermally radiated boundary layer flow past over a stretched sheet via OHAM analysis. Discov Appl Sci 2024; 6: 240.

46.

Ali

Abbas

Sohail

, et al. Study of bioconvection phenomenon in Jefferey model in a Darcy-Forchheimer porous medium. Bionanoscience. Epub ahead of print 27 April 2024. DOI: 10.1007/s12668-024-01412-1.

47.

Rafique

Ilyas

Sohail

Utlization of generalized heat flux model on thermal transport of Powell-Eyring model via oham with heat geneartion aspects: thermal transport of Powell-Eyring model. Babylonian J Math 2024; 2024: 19–33.

48.

Sohail

Ilyas

Rafique

, et al. OHAM analysis on bio-convective flow of partial differential equations of Casson nanofluid under thermal radiation impact past over a stretching sheet. Bionanoscience 2024; 14: 1572–1582.

49.

Liao

An optimal homotopy-analysis approach for strongly nonlinear differential equations. Commun Nonlinear Sci Numer Simul 2010; 15: 2003–2016.

50.

Shah

Dawar

Alzahrani

, et al. Hall effect on couple stress 3D nanofluid flow over an exponentially stretched surface with Cattaneo Christov heat flux model. IEEE Access 2019; 7: 64844–64855.