Sage Journals: Discover world-class research

Abstract

The aim of the present analysis was to the best part of the effect of a magnetohydrodynamic micropolar fluid over the convective surface boundary condition into cross-diffusion and linear radiative heats were assumed. The impact on chemical reaction and viscous dissipation is considered. This governing equation about flow fields was transformed with a non-dimensional design to apply relevant correlation variables. That ODE was defined with the bvp4c method. This result is entrusted into plots and tables about the consequence of diverse flow variables in the flow fields. They observed into appreciative temperature in the increase in the Dufour number, although the unfavorable force was recognized in the Richardson number. It was noticed that the higher velocity was placed on the Newtonian liquid on difference into that micropolar liquid. Concentration was the reducing function on chemical reaction parameters and Schmidt number.

Keywords

Micropolar liquid bvp4c radiation convective boundary condition stagnation point

Introduction

Motivating the use of micropolar fluids in engineering and industrial applications position to analyze heat change was highly paramount. A short time ago, investigating the circumstance of heat change on the fluid into microscopic designs had magnetized consideration. That had been establishing the existence of wall-such polymer or particles extra circulated on the fluid could importantly impact the heat transfer process. This mass and energy transfer on the micropolar fluid had bio-medical applications and numerous industrial, e.g. treatment of cancer, glass engendered, and polymer processing and blood filtration. As the application on micropolar fluids, investigators analyzed for thermal and flow investigation on micropolar fluids.

Eringen¹ announced the concept of thermomicropolar fluid. Siddiqui and Turkylmazoglu² considered the thermal and mass flow of micropolar fluid on the medium employing absorbent and affecting bars crosswise into the flow direction. Rauf et al.³ marked the axisymmetric flow of micropolar nanofluid flowing among both rotating disks. Saraswathy et al.⁴ had supposed the micropolar fluid flow on the channel into changeable activation energy and thermal conductivity. Pasha et al.⁵ examined that micropolar heat transfer and fluid flow over both parallel plates. They conclude that heat transfer rises to develop of the Peclet number. Mahmood et al.⁶ investigated the recent boundary condition regarding the error in peristaltic transport of the micropolar fluid within the asymmetric medium. They examined that the magnitude of shear stress reduces in improving the extent of lubrication.

The stagnation point flow had been utilized into the high-rapid flow bodies and performed by the drag contraction. This type of stagnation field that match to greatest pressure, mass deposition, and heat transfer was significant in the industrial, engineering, and technological fields. Xie and Wang⁷ considered the heat transfer on power-law fluid and stagnation-point flow passed over the stretching surface into a heat generation effect. Zainal et al.⁸ considered the fluid flow and heat transfer in unsteady stagnation point flow toward the porous shrinking or stretching Riga plate with thermal radiation.

Rana et al.⁹ considered the time-dependent improvement of the heat transport and nonlinear thermal buoyancy-driven flow on the MWCNT-MgO nanofluid to that stagnation point by the rotating sphere. This magnetized mixed convection unsteady Ag/MgO-water nanofluid through the revolving sphere nearby the stagnation point was analyzed by Acharya et al.¹⁰ Zainal et al.¹¹ discussed unsteady mixed convection to the nanofluid nearby the stagnation point previously in the vertical plate. They examined the heat transfer rate was decreased on the buoyancy on the opposing and assisting flow in presence of the thermal slip effect.

The free convective transport of heat was the important division of fluid dynamics owed into its applications such as oil reservoirs, geothermal engineering, paramedical sciences, geo, astrophysics, etc. This idea of thermal radiation was the technique in which internal energy was converted over electromagnetic waves. The utilization of nonlinear thermal radiation, thermal radiation applied into different fields such as space technology, hypersonic fights, composing to glass and paper, space vehicles, and gas turbines. Waini et al.¹² explored the magnetohydrodynamic (MHD) impact on the flow and radiation against the stagnation point on the aggressive shrinking sheet on the hybrid nanofluid. They examined that radiation takes the greater heat transfer rate. The disposed MHD micropolar fluid flow when the stretching or shrinking sheet on that existence of mass transportation and thermal radiation is examined by Mahabaleshwar et al.¹³ Gireesha et al.¹⁴ presented that thermal performance in the wet stretching or shrinking longitudinal fin on aggressive profile into internal heat generation. Ibrahim et al.¹⁵ induced the time-dependent viscous fluid on the stretch plate into radiation through a porous medium. Mahmood and Khan¹⁶ have performed the nonlinear radiation for aggregation effects of the unsteady nanofluid flow of a stagnation point region.

Permeable media that was assembled and constructed as high thermal conductive materials like metal foams could again enhance the heat transfer. The primary sense of improved thermal performance on permeable media is the development of fluid–solid contact when the variable motion on fluid among openings of the mixing of fluid and permeable materials. Wang et al.¹⁷ inspected the heat and fluid flow on the near-cell permeable medium. Moosavi et al.¹⁸ analyzed numerically the thermal performance calculation in the microchannel into various permeable media including composition. Luo et al.¹⁹ have presented heat transfer characteristics on permeable materials in view of the dual models of radiation and conduction. This mass transfer about solute transport over remaining unsaturated permeable media is examined by Dou et al..²⁰ Wang et al.²¹ discussed effective thermal conductivity on irregular permeable media and the sphere-packed. Pati et al.²² researched the critical analysis of forced convective heat transfer on permeable media. The impact on permeable media and swirling flow to triple coaxial divisions increase diffusion flame in the temperature is marked by Kotb et al.²³ Alsedais et al.²⁴ have inspected mixed convection on the nanofluid composed of a permeable cavity with the help of heat generation and radiative.

Although the transportation phenomena (heat, mass) arise together on mass fluxes, liquid movement, then enthalpy, and driving potentials relations were much more difficult. That is important to confess to heat flux was neither singly produced as a result of the temperature gradient, but again it was when the concentration gradient also. Soret impact analyzes the mass flux derivative by thermal gradient because Dufour impact embellished the heat transport diffusion over the solutal gradient. In different cases, like form was unused for we were under magnitude that correlated into marked as Fick–Fourier relations. Exceptional work on Dufour and Soret phenomena on hydrology and petrology consist of doubled alloys solidification, isotope separation, chemical reactors, pollutant migration of groundwater, and multi-component melts. Aly and Raizah²⁵ scrutinized the MHD free convection flow on the nanofluid by the permeable cavity containing two circular cylinders and rotating hexagonal below the effects on Dufour and Soret numbers. Lee et al.²⁶ have considered thermophoretic microfluidic cells for studying the Soret coefficient to colloidal particles. Hayat et al.²⁷ have studied the bio-convective thixotropic nanomaterial flow when the stretchable surface. Their results showed that the greater Dufour number outcomes on temperature reduction during the greater Soret number decreased concentration. Some other relevant studies are due to articles.^28–36 Saleh et al.³⁷ explored the chemical modification of polyvinyl chloride (PVC) using captopril as a modifying agent. Ahmed et al.^38,39 described the application of thermal stability of PVC films.

The uniqueness of this work was to introduce the effect of permeable medium over the heat and mass transport on mixed convective radiative micropolar liquid flow previous on a vertical surface with the convective condition. They studied the flow configuration into a Cartesian coordinate system. Utilizing suitable dimensionless quantities and comparison transformations, these complex differential equations might be converted into the interpreted nonlinear system of ODEs. These converted flow equations are numerically implemented using the bvp4c method. The graphical outcomes were obtained about concentration, velocity, and sundry parameters on the temperature and microrotation.

Mathematical modeling

The substantial, two-dimensional, incompressible viscous electrically conducting micropolar liquid nearby the stagnation point over the perpendicular to the hot sheet was studied. This condition on that no-slip boundary was simulated about the liquid medium. That heated plate was assisted to concentration $C_{w} (x) (> C_{\infty})$ and temperature $T_{f} (x) (> T_{\infty})$ . The migration field was conditional to the identical uniform magnetic field B₀ as the concerned approved on that sheet. Every magnetic Re on that governing liquid was simulated into lesser on the Hall effort and convinced magnetic field might be abandoned. This physical geometry in this present analysis was conferred in Figure 1.

Figure 1.

Physical model and correlative structure.

The expected boundary layer assumptions and aforementioned, this governing momentum, concentration, and energy equations about the steady micropolar liquid flow against the stagnation point were taken as.^35–36

\frac{\partial \hat{v}}{\partial \hat{y}} + \frac{\partial \hat{u}}{\partial \hat{x}} = 0,

(1)

ρ (\hat{v} \frac{\partial \hat{u}}{\partial \hat{y}} + \hat{u} \frac{\partial \hat{u}}{\partial \hat{x}}) = \hat{U} ρ \frac{d \hat{U}}{d \hat{x}} + (k + μ) \frac{\partial^{2} \hat{u}}{\partial {\hat{y}}^{2}} + k \frac{\partial \hat{N}}{\partial \hat{y}} + (\frac{σ B_{0}^{2}}{1 + m^{2}} + \frac{ρ μ}{K_{p}^{*}}) (\hat{U} - \hat{u}) + ρ g β_{c} (\hat{C} - {\hat{C}}_{\infty}) + ρ g β_{T} (\hat{T} - {\hat{T}}_{\infty}),

(2)

ρ j (\hat{v} \frac{\partial \hat{N}}{\partial \hat{y}} + \hat{u} \frac{\partial \hat{N}}{\partial \hat{x}}) = (\frac{k}{2} + μ) j \frac{\partial^{2} \hat{N}}{\partial {\hat{y}}^{2}} - k (2 \hat{N} + \frac{\partial \hat{u}}{\partial \hat{y}}),

(3)

ρ c_{p} (\hat{v} \frac{\partial \hat{T}}{\partial \hat{y}} + \hat{u} \frac{\partial \hat{T}}{\partial \hat{x}}) = α \frac{\partial^{2} \hat{T}}{\partial {\hat{y}}^{2}} + \frac{D K_{T}}{c_{s}} \frac{\partial^{2} \hat{C}}{\partial {\hat{y}}^{2}} - \frac{\partial q_{r}}{\partial \hat{y}} + (μ + k) {(\frac{\partial \hat{u}}{\partial \hat{y}})}^{2} + 2 k \hat{N} (\frac{\partial \hat{u}}{\partial \hat{y}}) + (μ + \frac{k}{2}) j (\frac{\partial \hat{N}}{\partial \hat{y}}) + 2 k {\hat{N}}^{2},

(4)

\hat{v} \frac{\partial \hat{C}}{\partial \hat{y}} + \hat{u} \frac{\partial \hat{C}}{\partial \hat{x}} = D \frac{\partial^{2} \hat{C}}{\partial {\hat{y}}^{2}} + \frac{D K_{T}}{T_{m}} \frac{\partial^{2} \hat{T}}{\partial {\hat{y}}^{2}} - K_{c}^{*} (\hat{C} - {\hat{C}}_{\infty}),

(5)

This applicable boundary conditions can be bring by:

\hat{u} = 0, \hat{v} = 0, \hat{N} = - n \frac{\partial \hat{u}}{\partial \hat{y}}, - k_{f} \frac{\partial \hat{T}}{\partial \hat{y}} = h_{f} ({\hat{T}}_{f} - \hat{T}), \hat{C} = {\hat{C}}_{w} at \hat{y} = 0,

(6)

\hat{u} \to \hat{U} (\hat{x}), \hat{N} \to 0, \hat{T} \to {\hat{T}}_{\infty}, \hat{C} \to {\hat{C}}_{\infty} as \hat{y} \to \infty .

(7)

Notable to g as the gravity acceleration,

\hat{u}, \hat{v}

are velocities on

\hat{x}

and

\hat{y}

directions,

\hat{T}

as the fluid temperature,

ν, μ

and k are the kinematic, vortex and dynamic viscosities,

\hat{C}

indicates the concentration to the liquid on the boundary layer, appropriately,

B_{0}

as the magnetic fields strength,

β_{c}

as the concentration expansion coefficient, j as the microinertia density,

β_{T}

as the thermal expansion coefficient,

ρ

as the fluid density,

α

as the thermal diffusivity, D as the mass diffusivity coefficient,

K_{p}^{*}

as the permeability parameter,

K_{c}^{*}

as the reaction rate,

K_{T}

as the thermal diffusive ratio,

c_{p}

indicates the specific heat at constant pressure, m as parameter Hall current,

σ

as the electrical conductivity,

c_{s}

as the concentration susceptibility,

\hat{N}

as the microrotation vector and n as the constant like as

0 \leq n \leq 1

. This fact

= 0

, was known as strong concentration, that represents

\hat{N} = 0

nearby the wall, indicates concentrated particle flows on that the microelements nearby that wall exterior was not able to rotate. This fact

n = 1 / 2

represents the disappearance on anti-symmetric factor to the stress tensor and indicates anemic concentrations. The fact

n = 1

was applied about the imitation on turbulent boundary layer flows.

Furthermore, the radiative heat flux $q_{r}$ was interpreted over Rosseland similarity such as:

q_{r} = - \frac{\partial {\hat{T}}^{4}}{\partial \hat{y}} \frac{4 σ_{1}}{3 k^{*}},

(8)

Whichever

σ_{1}

and

k^{*}

the Stefan–Boltzmann constant and as the mean absorption coefficient, appropriately. Every liquid phase temperature variation interior the flow was assumed that the lesser; so,

{\hat{T}}^{4}

could be established to the temperature linear function. That was finished over increasing

{\hat{T}}^{4}

on the Taylor series of exclusion higher-order terms in free-stream and arrival temperature

{\hat{T}}_{\infty}

{\hat{T}}^{4} = 4 {\hat{T}}_{\infty}^{3} \hat{T} - 3 {\hat{T}}_{\infty}^{4} .

(9)

They recommend the equations on Cauchy–Riemann by:

\hat{u} = \frac{\partial Ψ}{\partial \hat{y}}, \hat{v} = - \frac{\partial Ψ}{\partial \hat{x}} .

(10)

Where

Ψ

as stream function. Equation (10) naturally contented continuity equation (1)

We consider the transformations for converting the above expressions (PDEs) into ODEs

\begin{aligned} η = {(\frac{a}{ν})}^{1 / 2} \hat{y}, ω (η) = \frac{Ψ}{{(a ν)}^{1 / 2} \hat{x}}, f (η) = \frac{N}{a {(\frac{a}{ν})}^{1 / 2} \hat{x}}, \\ θ (η) = \frac{\hat{T} - {\hat{T}}_{\infty}}{{\hat{T}}_{f} - {\hat{T}}_{\infty}}, ϕ (η) = \frac{\hat{C} - {\hat{C}}_{\infty}}{{\hat{C}}_{w} - {\hat{C}}_{\infty}} . \end{aligned}

(11)

Condition for incompressibility is verified while other expressions become

(1 + Γ) ω^{″'} + 1 - ω^{' 2} + ω ω^{″} + Γ ω^{'} + (\frac{M}{1 + m^{2}} + K_{p}) (1 - ω^{'}) + θ λ + ϕ δ = 0,

(12)

(1 + \frac{Γ}{2}) {f^{'}}^{'} + ω f^{'} - ω^{'} f - Γ (2 f + {ω^{'}}^{'}) = 0,

(13)

\frac{1}{Pr} (1 + \frac{4}{3} R d) {θ^{'}}^{'} + ω θ^{'} - ω^{'} θ + D u {ϕ^{'}}^{'} + E c [(1 + Γ) {ω^{'}}^{'}^{2} + 2 Γ f^{2} + 2 Γ f {ω^{'}}^{'} + (1 + \frac{Γ}{2}) {f^{'}}^{2}] = 0,

(14)

\frac{1}{S c} {ϕ^{'}}^{'} + ω ϕ^{'} - ω^{'} ϕ - γ ϕ + S r {θ^{'}}^{'} = 0.

(15)

The dimensionless boundary conditions are

\begin{aligned} ω (0) & = 0, ω^{'} (0) = 0, \\ f (0) & = - n {ω^{'}}^{'} (0), θ^{'} (0) + B i (1 - θ (0)) = 0, \\ ϕ (0) & = 1 at η = 0, \end{aligned}

(16)

\begin{aligned} ω^{'} (η) & \to 1, f (η) \to 0, θ (η) \to 0, \\ ϕ (η) & \to 0 as η \to \infty . \end{aligned}

(17)

Where

\begin{aligned} M = & \frac{σ B_{0}^{2}}{ρ a}, G r = \frac{g β_{T} (T_{f} - T_{\infty}) x^{3}}{ν^{2}}, λ = \frac{G r}{{Re}_{x}^{2}}, δ = = \frac{G c}{{Re}_{x}^{2}}, \\ G c = & \frac{g β_{c} (C_{w} - C_{\infty}) x^{3}}{ν^{2}}, {Re}_{x} = \frac{U x}{ν}, R d = \frac{4 σ_{1} T_{\infty}^{3}}{k^{*} k_{f}}, \\ K_{p} = & \frac{ν}{a K_{p}^{*}}, Pr = \frac{ν}{α}, Γ = \frac{k}{μ}, D u = \frac{D K_{T} (C_{w} - C_{\infty})}{c_{s} c_{p} μ (T_{w} - T_{\infty})}, \\ E c = & \frac{U^{2}}{c_{p} (T_{w} - T_{\infty})}, B i = \frac{h_{f}}{k_{f}} \sqrt{\frac{ν}{a}}, γ = \frac{K_{c}^{*}}{a}, \\ S r = & \frac{D K_{T} (T_{w} - T_{\infty})}{T_{m} ν (C_{w} - C_{\infty})} . \end{aligned}

(18)

In equation (12),

M, λ, δ, R d, K_{p}, P r, Γ, D u, S r, E c, B i

and

γ

are the magnetic parameter, mixed convection parameter, solutal buoyancy parameter, radiation parameter, permeability parameter, Soret number, Prandtl number, Dufour number, Biot number, material parameter, Eckert number, and chemical reaction parameter.

Nusselt number $N u$ , Sherwood number $S h$ and skin friction coefficient $C_{f}$ were the physical capacity on the engineering importance, and this was

\begin{aligned} C_{f} & = \frac{τ_{w}}{ρ U^{2} / 2}, N u = \frac{\hat{x} q_{w}}{k_{f} ({\hat{T}}_{f} - {\hat{T}}_{\infty})}, \\ S h & = \frac{\hat{x} M_{n}}{D ({\hat{C}}_{w} - {\hat{C}}_{\infty})} . \end{aligned}

(19)

The wall shear stress

τ_{w}

the heat flux

q_{w}

, and mass flux

M_{n}

\begin{aligned} τ_{w} & = {((μ + k) \frac{\partial \hat{u}}{\partial \hat{y}} + k \hat{N})}_{\hat{y} = 0}, \\ q_{w} & = - k_{f} {(\frac{\partial \hat{T}}{\partial \hat{y}})}_{\hat{y} = 0}, M_{n} = - D {(\frac{\partial \hat{C}}{\partial \hat{y}})}_{\hat{y} = 0}, \end{aligned}

(20)

And the use on corelation transformations, they take

\frac{1}{2} C_{f} R e_{x}^{1 / 2} = (1 + \frac{Γ}{2}) {ω^{'}}^{'} (0) + Γ ω (0), \frac{N u}{R e_{x}^{1 / 2}} = - θ^{'} (0), \frac{S h}{R e_{x}^{1 / 2}} = - ϕ^{'} (0) .

(21)

Numerical solution procedure

The portion contributes to the numerical solution process about dimensionless nonlinear temperature and momentum equations. Equations (12)–(17) are achieved as having ordinary differential equations solver technique; bvp4c is the selection method that works Lobatto IIIA formula. To calculate results into the pattern, that is becoming the basic assumption toward appeasing the boundary conditions. Later, utilizing the appropriate basic conditions another process, especially the finite difference method was worked that change the basic assumption about advance repetition. To eliminate and execute this pattern, it was fundamental to have the basic value problems in the system (12)–(17). They did that, the ensuing recent variables were introduced by

\begin{aligned} ω = & Ψ_{1}, ω^{'} = Ψ_{2}, {ω^{'}}^{'} = Ψ_{3}, {ω^{'}}^{'}^{'} = Ψ Ψ_{1}, f = Ψ_{4}, \\ f^{'} = & Ψ_{5}, {f^{'}}^{'} = Ψ Ψ_{2}, θ = Ψ_{6}, θ^{'} = Ψ_{7}, {θ^{'}}^{'} = Ψ Ψ_{3}, \\ ϕ = & Ψ_{8}, ϕ^{'} = Ψ_{9}, {ϕ^{'}}^{'} = Ψ Ψ_{4}, \end{aligned}

(22)

\begin{aligned} Ψ Ψ_{1} = - \frac{1}{1 + Γ} & (1 - Ψ_{2}^{2} + Ψ_{1} Ψ_{3} + Γ Ψ_{5} + (\frac{M}{1 + m^{2}} + K_{p}) \\ (1 - Ψ_{2}) + λ Ψ_{6} + δ Ψ_{8}), \end{aligned}

(23)

Ψ Ψ_{2} = - \frac{2}{2 + Γ} (Ψ_{1} Ψ_{5} - Ψ_{2} Ψ_{4} - Γ (2 Ψ_{4} + Ψ_{3})),

(24)

Ψ Ψ_{3} = - \frac{3 P r}{3 + 4 R d} (Ψ_{1} Ψ_{7} - Ψ_{2} Ψ_{6} + D u Ψ Ψ_{4} + E c Ψ_{3}^{2}),

(25)

Ψ_{1} Ψ_{1} = - S c (Ψ_{1} Ψ_{9} - Ψ_{2} Ψ_{8} - γ Ψ_{8} + S r Ψ Ψ_{3}),

(26)

With conditions

\begin{aligned} Ψ_{1} (0) = 0, Ψ_{2} (0) = 0, Ψ_{4} (0) + n Ψ_{3} (0) = 0, \\ Ψ_{7} (0) + B i (1 - Ψ_{6} (0)) = 0, Ψ_{8} (0) = 1, \end{aligned}

(27)

Ψ_{2} (\infty) = 1, Ψ_{4} (\infty) = 0, Ψ_{6} (\infty) = 0, Ψ_{8} (\infty) = 0.

(28)

Code of verification

To find out this design's efficacy, they had excerpted the decreased Nusselt number, skin frictional and Sherwood values about different values of $P r, δ$ and $S c$ , appropriately. That numerical solutions was recorded on Tables 1 –3 and related from Baag et al.,³⁵ Ishak et al.³⁶ It displays superior deal into the earlier researches.

Table 1.

Comparison on values of $(1 / 2) C_{f} R e_{x}^{1 / 2}$ for different values of $P r, δ$ and $S c$ when $Γ = R d = E c = M = K_{p} = 0$ and $λ = 1$ .

$P r$	$δ$	$S c$	Present	Baag et al.³⁵	Ishak et al.³⁶
0.71	0	0	1.705094	1.70328	1.7063
1	0	0	1.675437	1.67490	1.6755
7	0	0	1.517913	1.51800	1.5179
0.71	1	0	2.277162	2.27761	–
0.71	1	0.22	2.240780	2.24014	–

Table 2.

Correlation on values of $N u R e_{x}^{- 1 / 2}$ about different values of $P r$ and $δ$ when $Γ = R d = E c = S c = M = K_{p} = 0$ and $λ = 1$ .

$P r$	$δ$	Present	Baag et al.³⁵	Ishak et al.³⁶
0.71	0	0.768081	0.76636	0.7641
0.71	1	0.840434	0.84009	–
1	0	0.870779	0.87139	0.8708
1	1	0.955206	0.95488	–
7	0	1.722382	1.72246	1.7225

Table 3.

Correlation on values of $S h R e_{x}^{- 1 / 2}$ about different values of $γ$ and $δ$ when $Γ = R d = E c = S c = M = K_{p} = 0, P r = 0.71$ and $λ = 1$ .

$γ$	$δ$	Present	Baag et al.³⁵
0	0	0.718789	0.71785
0	1	0.761781	0.76031
1	1	1.044748	1.04230
−1	1	0.397230	0.39318

Results and discussion

On that division, every graphical and numerical outcomes on the physical parameters, Hartmann number M, Dufour number $D u$ , Schmidt number $S c$ , Prandtl number $P r$ , Soret number $S r$ , micropolar fluid parameter $Γ$ , Eckert number $E c$ , chemical reaction parameter $γ$ , radiation parameter $R d$ and Biot number $B i$ . The parametric values are taken as $Γ = 5.0, M = 0.1, m = 0.5, K_{p} = 0.2, P r = 0.71, λ = 0.2, δ = 0.5, R d = 6.0, D u = 0.1, E c = 0.2, S c = 0.24, S r = 0.1, γ = 0.1, B i = 2.0, n = 0.1$ unless otherwise stated.

The impact of material parameter for micropolar fluid Γ, mixed convection parameter λ, and permeability parameter Kp on velocity distribution $ω^{'} (η)$ is displayed in Figure 2. In that Richardson dispute, where the boundary layer velocity improves. It was substantially compelling as a result of it was the ratio of Gr as the equal on the Re, Grashof number (Gr) exists on temperature difference, hence to that density reduces for the raises on temperature difference and details the fluid increase. From Figure 2 the velocity decreases into increasing values on the micropolar fluid parameter. Likewise, every greater velocity was noticed on the Newtonian fluid model into esteemed the micropolar fluid model. Furthermore, the greater velocities are determined with augmentation in the porous permeability parameter.

Figure 2.

Impact on $Γ, λ$ and $K_{p}$ of velocity.

Figure 3 is organized to see the performance of material parameter Γ, Hartmann number M, and mixed convection parameter λ on microorganism distribution $f (η)$ . It was displayed to the microrotation frequently decrease on magnitude over η and convert zero expend taken away the plate. As a consequence, the microrotation ambiance was extra dominant close to the wall. The equal direction was followed on total the plots way to assure the length of the act of the microorganism; later the crucial moment, the reverse act was noticed. Developing values of magnetic strength, the microorganism reduces into crucial points later that position was various. The case once that, the presence of the magnetic field in the electrically conducting liquid affected the Lorentz force, it dealt with the flow. Further Figure 3 displays that the decrease in the microrotation into a rise to the Richardson number; after, the opposite direction was followed. The Micropolar fluid parameter growing of microrotation about the fixed length across direction; after, that position was opposite.

Figure 3.

Impact on $Γ, M$ and $λ$ of microrotation.

The act on diverse parameters such that Richardson number $λ$ , Dufour number Du and Prandtl number Pr via the thermal profile $θ (η)$ is exposed in Figure 4. We noticed from the graph that the escalation in temperature distribution for the development in Dufour number. Actually, the ratio between composition gradient and energy flux was called as Dufour number. The greater temperature was seen about lesser values of Pr and the trend was opposite about larger Pr. Prandtl number Pr had the reverse corporation into thermal diffusivity. The development of Prandtl number Pr prompts low thermal diffusivity that details collapse on temperature and layer of thermal. Further Figure 4 shows the temperature distribution decreases with the mixed convection parameter λ.

Figure 4.

Impacts on $λ, D u$ and $P r$ of temperature.

Figure 5 indicates how radiation parameters Rd, Eckert number Ec, and Biot number Bi impact on thermal distribution $θ (η)$ . They are notable for the growth in values of Rd outcomes on the improvement in the thermal profile. Actually, the growth on Rd increases the moving temperature of the micropolar fluid and reduces to mean absorption coefficient. So, the thermal distribution rises to improve the temperature field. Further, it can visualize to developing the values of the Ec point to improve the thermal profile over the boundary layer. This Eckert number was the ratio of kinetic energy for enthalpy. Hence, the effect on the Ec of the thermal profile was observed for the developing direction of viscous dissipation. Finally, variations on the Biot number arrive to feature an extra conspicuous effect of the fluid temperature. Again, Figure 5 displays to dimensionless fluid temperatures were lesser to below values on the Biot number. That was incoming familiar, to the below Biot number indicating great thermal conductivity on the permeable medium, which was the important factor in the improvement of heat transfer on permeable media.³³

Figure 5.

Impacts on $R d, E c$ and $B i$ of temperature.

Figure 6 displays the concentration profiles for variations in Soret number Sr, mass buoyancy parameter δ, and Schmidt number Sc. It is detected from Figure 6 that there is an obligation on concentration about the aggregation of Soret number. Commonly, mass fluxes activated as temperature gradient is known as Soret number. The mass buoyancy parameter reduces the absorption. The ratio among the thermal diffusivity and mass diffusivity was called as Schmidt number. Larger values of the Schmidt number coincide with lesser mass diffusivity that details the deceleration on the concentration profile.

Figure 6.

Impacts on $S r, δ$ and $S c$ of concentration.

Figure 7 indicates the impact on chemical reaction parameters about $γ < 0$ constructive reaction and destructive reaction $γ > 0$ . The reaction parameter displays the retarding impact of concentration distribution to the reaction returns against effective to the detrimental cases. It was noted to the concentration of the fluid reduces in destructive reactions and the opposite for constructive reactions. The Local Nusselt number was increased in Supplementary Figure S8 for both the Prandtl number and radiation parameter. That linear reversion slope in Supplementary Figure S8 depicts to Prandtl number production heat transport increase to the rate 0.16494 and 0.020923 about radiation. From Supplementary Figure S9, the heat transport aggravates about greater Biot number. Hence cooling procedure convert impressive in that fact.

Figure 7.

Effects of $γ$ on concentration.

Slope analysis on Supplementary Figure S9 determines the increment to the rate 0.056406 when they reduce to the rate −0.00011 for the Eckert number. The local Nusselt number is decreased for the Dufour number and increased for the Richardson number, which is displayed in Supplementary Figure S10. This linear reversion slope reveals to increase of the Richardson number to the rate 0.011005, while it decreases at the rate −0.04868 for Dufour number. Thus, we can observe higher heat transport is produced in Richardson number. To check the variations in $C_{f}, N u$ and $S h$ over different physical parameters on activity, Table 1 –3 was processed. Including that difference, the correlation analysis was again formed by the extant article by Baag et al.³⁵ and Ishak et al..³⁶ It was noted to the conferred citation was on best compliance over the past printed uses to certain cases about particular values on the parameters.

Table 1 presents the variation in the skin friction coefficient. That was noted to the skin friction reduces into developing values of Pr in the absence of chemical reaction and buoyancy effect. The Buoyancy forces enhance to skin friction coefficient. The skin friction decreases over increasing values of Sc. Table 2 was presented into saw the differences in local Nusselt numbers. That was noticed by Nu as the developing function on buoyancy effect and Prandtl number. Table 3 displays to the local Sherwood number increase over developing on γ and buoyancy forces. As a result, it was carried out to lighter diffusive spices kindness constructive reaction was concerned with reducing mass transport rate to the boundary field.

Conclusions

This present analysis abstracted the mixed transport on mass and heat over stagnation point flow on MHD micropolar liquid that the heated location below the effect on viscous dissipation and convection boundary condition. This formation on the issue contained Dufour and Soret effects. That nonlinear governing equation was decreased in the system of ordinary differential equations by utilizing comparison variables. The bvp4c method is again worked to resolve the doubled, nonlinear ordinary differential equations. This allegation on that analysis could be summarized as follows:

This velocity augments the growth of the permeability parameter and Richardson number but opposite for the micropolar fluid parameter.

The Mixed act was noticed on the microrotation into total parameters on effect.

The temperature was the increasing function of the Dufour number, Eckert number, and Biot number.

Temperature reduces with augmentation of the Prandtl number and Richardson number.

The concentration was decreasing function on $δ$ and Schmidt number and opposite for incremental values of Soret number.

The stagnation point flow on MHD micropolar liquid previous the stretching field over the convective condition, the mass and heat transport was especially important to more practical significance on the industry. These applications comprise polymer sheet discharge against the dye, the metallic plate contraction method on the cooling bath, and aerodynamic discharge on sheets. This present analysis is abstracted in micropolar fluid into viscous dissipation and convective boundary condition. These future directions on the present analysis were: the researchers might study micropolar nanofluid flows into microorganisms. This analysis shall advance to grant consequences on the industrial applications.

Supplemental Material

sj-docx-1-pie-10.1177_09544089221144174 - Supplemental material for Cross-diffusion of the stagnation-point solar radiated micropolar liquid flow through a convected surface

Supplemental material, sj-docx-1-pie-10.1177_09544089221144174 for Cross-diffusion of the stagnation-point solar radiated micropolar liquid flow through a convected surface by Pallamkuppam Vinodh Kumar and Yeddula Pedda Obulesu in Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering

Footnotes

Acknowledgments

The authors are very grateful for the editor and reviewers for their constructive suggestions.

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.

ORCID iD

Yeddula Pedda Obulesu

Supplemental material

Supplemental material for this article is available online.

References

Eriengen

. Theory of thermomicrofluids. J Math Anal Appl 1972; 38: 480.

Siddiqui

Turkyilmzoglu

. Slit flow and thermal analysis of micropolar fluids in a symmetric channel with dynamic and permeable. Int Commun Heat Mass Transf 2022; 132: 105844.

Rauf

Faisal and Mushtaq

. Cattaneo–Christov-based study of Al₂O₃–Cu/Eg Casson hybrid nanofluid flow past a lubricated surface with cross diffusion and thermal radiation. Appl Nanosci 2022; 12: 2059–2076.

Saraswathy

Prakash

Muthtamilsevan

, et al. Arrhenius energy on asymmetric flow and heat transfer of micropolar fluids with variable properties: a sensitivity approach. Alex Eng J 2022; 61: 12329–12352.

Pasha

Mirzaei

Zarinfar

. Application of numerical methods in micropolar fluid flow and heat transfer in permeable plates. Alex Eng J 2022; 61: 2663–2672.

Mahmood

Sajid

Ali

, et al. A new interfacial condition for the peristaltic flow of a micropolar fluid. Ain Shams Eng J 2022; 13: 101744.

Xie

Wang

Y-M

. Stagnation-point flow and heat transfer of power-law MHD fluid over a stretching surface with convective heat transfer boundary condition. Int J Numer Method H 2022; 32: 265–282.

Zainal

Nazar

Naganthran

, et al. Unsteady stagnation point flow past a permeable stretching/shrinking Piga plate in Al₂O₃–Cu/H₂O hybrid nanofluid with thermal radiation. Int J Numer Method H 2022; 32: 2640–2658.

Rana

Gupta

. Unsteady nonlinear thermal convection flow of MWCNT-MgO/EG hybrid nanofluid in the stagnation-point region of a rotating sphere with quadratic thermal radiation: RSM for optimization. Int Commun Heat Mass Transf 2022; 134: 106025.

10.

Acharya

Mabood

Badruddin

. Thermal performance of unsteady mixed convective Ag/MgO nanohybrid flow near the stagnation point domain of a spinning sphere. Int Commun Heat Mass Transf 2022; 134: 106019.

11.

Zainal

Nazar

Naganthran

, et al. Slip effects on unsteady mixed convection of hybrid nanofluid flow near the stagnation point. Appl Math Mech-Engl Ed 2022; 43: 547–556.

12.

Waini

Pop

Abu Bakar

, et al. Stagnation point flow toward an exponentially shrinking sheet in a hybrid nanofluid. Int J Numer Methods H 2022; 32: 1012–1024.

13.

Mahabaleshwar

Vishalashi

Hatami

. MHD micropolar fluid flow over a stretching/shrinking sheet with dissipation of energy and stress work considering mass transpiration and thermal radiation. Int Commun Heat Mass Transf 2022; 133: 105966.

14.

Gireesha

Keerthi

Sowmya

. Effects of stretching/shrinking on the thermal performance of a fully wetted convective-radiative longitudinal fin if exponential profile. Appl Math Mech-Engl Ed 2022; 43: 389–402.

15.

Ibrahim

Saeed

Zeb

. Numerical simulation of time-dependent two-dimensional viscous fluid flow with thermal radiation. Eur Phys J Plus 2022; 137: 609.

16.

Mahmood

Khan

. Nanaoparticles aggregation effects on unsteady stagnation point flow of hydrogen oxide-based nanofluids. Eur Phys J Plus 2022; 137: 750.

17.

Wang

Zheng

Zhang

, et al. The effect of porosity and number of unit cell on applicability of volume average approach in closed cell porous media. Int J Numer Methods H 2022; 32: 2778–2798.

18.

Moosavi

Banihashemi

Lin

C-H

. Thermal performance evaluation of a microchannel with different porous media insert configurations. Int J Numer Methods H 2022; 32: 1488–1516.

19.

Luo

Wang

Zhao

, et al. Characteristics of effective thermal conductivity of porous materials considering thermal radiation: a pore-level analysis. Int J Heat Mass Transf 2022; 188: 122597.

20.

Dou

Zhang

Zhuang

, et al. Saturation dependence of mass transfer for solute transport through residual unsaturated porous media. Int J Heat Mass Transf 2022; 188: 122595.

21.

Wang

Yin

, et al. Temperature field prediction for various porous media considering variable boundary conditions using deep learning method. Int Commun Heat Mass Transf 2022; 132: 105916.

22.

Pati

Borah

Boruah

, et al. Critical review on local thermal equilibrium and local thermal non-equilibrium approaches for the analysis of forced convective flow through porous media. Int Commun Heat Mass Transf 2022; 132: 105889.

23.

Kotb

Kamal

Baghdady

, et al. Case study for swirling flow and porous media on triple coaxial ports inverse diffusion flame. Alex Eng J 2022; 61: 2294–2306.

24.

Alsedais

Aly

Mansour

. Local thermal non-equilibrium condition on mixed convection of a nanofluid-filled undulating cavity containing obstacle and saturated by porous media. Ain Shams Eng J 2022; 13: 101562.

25.

Aly

Raizah

. Double-diffusive convection of a nanofluid in a porous cavity containing rotating hexagon and circular cylinders: ISPH simulations. Int J Numer Methods H 2022; 32: 432–452.

26.

Lee

Mohanakumar

Wiegand

. Thermophoretic microfluidic cells foe evaluating Soret coefficient of colloidal particles. Int J Heat Mass Transf 2022; 194: 123002.

27.

Hayat

Inayatullah Khan

, et al. Significance of melting heat transfer in bio-convective thixotropic nanofluid. Ain Shams Eng J 2022; 13: 101625.

28.

Gangadhar

Manasa Seshakumari

Venkata Subba Rao

, et al. Biconvective transport of magnetized couple stress fluid over a radiative paraboloid of revolution. Proc Inst Mech Eng Part E J Process Mech Eng 2022; 236: 1661–1670.

29.

Elbashbeshy

EMA

Asker

. Fluid flow over a vertical stretching surface within a porous medium filled by a nanofluid containing gyrotactic microorganisms. Eur Phys J Plus 2022; 137: 541.

30.

Lakshmi

Siddeshwar

Ismail

, et al. Linear and weakly non-linear stability analyses of Rayleigh-Benard convection in a water-saturated porous medium with different shapes of copper nanoparticles. Eur Phys J Plus 2022; 137: 698.

31.

Safarzadeh

Heidarinejad

Pasdarshahri

. Accuracy of three different versions of flamelet-generated manifold with/without radiation coupling in simulation of pool fire. J Braz Soc Mech Sci Eng 2022; 44: 210.

32.

Alqsair

Ramesh

, et al. Nonlinear heat source/sink and activation energy assessment in double diffusion flow of micropolar (non-Newtonian) nanofluid with convective conditions. Arab J Sci Eng 2022; 47: 859–866.

33.

Torabi

Zhang

Karimi

, et al. Entropy generation in thermal systems with solid structures—a concise review. Int J Heat Mass Transf 2016; 97: 917–931.

34.

Makinde

Khan

. Stagnation point flow of MHD chemically reacting nanofluid over a stretching convective surface with slip and radiative heat. Proc Inst Mech Eng Part E J Process Mech Eng 2017; 231: 695–703.

35.

Baag

Mishra

Dash

, et al. Numerical investigation on MHD micropolar fluid flow toward a stagnation point on a vertical surface with heat source and chemical reaction. J King Saud Univ-Eng Sci 2017; 29: 75–83.

36.

Ishak

Nazar

Pop

. Magnetohydrodynamic (MHD) flow of a micropolar fluid towards a stagnation point on a vertical surface. Comput Math Appl 2008; 56: 3188–3194.

37.

Saleh

Yousif

Al-Tikrity

, et al. Modification of PVC with captopril and complexation reaction for preparing photostability and thermal stability of PVC. Mater Sci Technol 2022; 5: 311–323.

38.

Ahmed

Kandhom

Hadi

, et al. Tetra Schiff bases as polyvinyl chloride thermal stabilizers. Chemistry 2021; 3: 288–295.

39.

Ahmed

Ibrahim

Bufaroosha

, et al. Polyphosphates as thermal stabilizers for poly (vinyl chloride). Mater Today: Proc 2021; 42: 2680–2685.

Supplementary Material

Please find the following supplemental material available below.

For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.

For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.

0.00 MB

0.11 MB