Heat transfer analysis of the MHD stagnation-point flow of third-grade fluid over a porous sheet with thermal radiation effect: An algorithmic approach

Amjad Ali; Zainab Bukhari; Muhammad Amjad; Sohail Ahmad; Wasim Jamshed; Sayed M. El Din

doi:10.1515/phys-2022-0227

Article Open Access

Heat transfer analysis of the MHD stagnation-point flow of third-grade fluid over a porous sheet with thermal radiation effect: An algorithmic approach

Amjad Ali , Zainab Bukhari , Muhammad Amjad , Sohail Ahmad , Wasim Jamshed and Sayed M. El Din

Published/Copyright: February 24, 2023

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From the journal Open Physics Volume 21 Issue 1

Abstract

The present article aims to investigate the impacts of the thermal radiation and Lorentz force on the stagnation-point flow of third-grade liquid over a porous stretching sheet with suction. The governing equations are transformed using the similarity transformation. The resulting system of ordinary differential equations is solved using a “so-called” hybrid algorithm based on the finite difference method and the shooting method. The influence of the emerging parameters on the velocity and temperature profiles is analyzed. The results are shown in graphical and tabular forms. For the third-grade liquid flow, the velocity profile shows an inciting trend toward the Hartman number (magnetic parameter). The temperature profile shows a declining trend toward the Prandtl number and suction velocity, whereas an inciting trend towards the radiation parameter.

Keywords: third-grade fluid; heat transfer; thermal radiation; stagnation-point; permeable sheet

Nomenclature

B 0: uniform strength
c p: specific heat
k 1: thermal conductivity
M: magnetic parameter
Nu: Nusselt number
p ∼: pressure
Pr: Prandtl number
Rd: radiation parameter
Re x ∼: local Reynolds number
Re: Reynolds number
Sk: skin friction coefficient
T: temperature
T w: temperature of the sheet
T ∞: temperature away from the sheet
U: characteristic flow velocity
u ∼: velocity in x ∼ -direction
v ∼: velocity in y ∼ -direction
ν: kinematic viscosity
v w: suction velocity
α 2: material constant
ψ: stream function
θ: temperature profile
ρ: density

1 Introduction

Over recent years, the analysis of nonlinear problems of non-Newtonian fluid flow models has gained popularity due to their industrial applications. Materials such as polymer mixtures, drilling fluids, blood, paints, oils, grease, and many other fluids are categorized as non-Newtonian. They are relevant to a variety of prime engineering and industrial applications. The mathematical analysis of such models has been challenging for obtaining precise and closed-form solutions. Researchers have been designing algorithms for the investigation of non-Newtonian fluid flow problems with a variety of conditions and contexts. A classical problem of fluid dynamics of stagnation-point flow at a flat surface was investigated using a pioneering work by Hiemenz [1], which was extensively studied by many researchers. Chamkha [2] examined the stagnant-point magnetohydrodynamic (MHD) flow with heat transfer. Wu [3] studied the stagnation-point flow in a permeable space. Teipel [4], Garg and Rajagopal [5,6], and Pakdemirli and Suhubi [7] made efforts for the numerical study of the behavior of viscoelastic stagnation-point flows. Ariel [8,9] used a numerical method for the analysis of the second-grade fluid flow over a porous sheet. Ariel [10] presented a hybrid algorithm for the computations of viscoelastic fluids. Labropulu and Li [11] assumed the slip condition to investigate the second-grade fluid by applying the quasi-linearization method. Abu-Hamdeh et al. [12] numerically inquired about the heat transfer performance and fluid flow characteristic of a helical micro double-tube heat exchanger using the finite volume method. More recent investigations regarding thermally developed flows at different temperatures can be studied in the reference articles [13,14,15,16].

The researchers extended the literature to discuss the third-grade fluid flow models, which better express the shear thinning and thickening of the viscoelastic fluids than the second-grade fluid flow models. Sajid and Hayat [17] used the homotopy analysis method (HAM) to describe the flow of third-order fluid over an extending surface. Sajid [18] also utilized the HAM to analyze axisymmetric flows for a third-grade liquid. Sahoo [19,20] numerically studied the heat transportation phenomenon in the slip flow of third-grade fluid past a stretchable flat sheet. Hayat et al. [21] analyzed mass and heat transport in a third-grade liquid over a stretchable surface with the chemical reaction. Itishree et al. [22] examined the unsteady convective flow of third-grade fluid flow through a porous medium. Baoku [23] examined the influence of thermal radiation and suction on the heat transfer and flow characteristics in a third-grade liquid over a vertical porous plate. Hayat et al. [24] investigated the transfer of heat in MHD third-grade liquid flow across an extending surface with a heat source and sink. Ramzan et al. [25] interpreted the influence of MHD and convective boundary on 2D third-grade liquid flow with the heat flux and chemical reaction. Abbas et al. [26] investigated the mass transfer in stagnation-point flow second-grade liquid with a chemical reaction. Zubair et al. [27] discussed an application of stagnation-point flow of third-grade liquid with varying thicknesses. Sahoo [28] presented experiments describing third-grade liquids. An account of the history of third-grade liquid was presented by Ikram [29].

An artificial neural network (ANN) was probed by Mishra and Chaudhuri [30] using the Genetic algorithm (GA). The flow was taken between two parallel plates under the influence of uniform heat flux. The ANN model was simplified by the GA, and the results were accomplished. The nonlinear Rosseland approximation to formulate the thermal radiation was considered by Ghasemi and Hatami [31] to interpret the stagnation-point flow over an elongating sheet. They assumed that solar radiation and MHD influenced the flow. Li et al. [32,33] measured the convective heat transfer of nanofluid, which potentially is a cost-effective nano-suspension with plausible heat transfer characteristics. CuO – water nano-suspension was prepared, stabilized, and used with the view to improve the thermal conductivity of water and thereby increase the heat transfer coefficient. Literature related to the collective influence of thermal radiation and convection flow of nanofluid due to a stretching cylinder in a porous medium along with viscous dissipation and slip boundary conditions (BCs) were examined by Pandey et al. [34,35,36]. Ali et al. [37] investigated the pulsatile flow of viscous incompressible MHD nanofluid in a rectangular channel. Further, several other works investigated the thermal-developed flow of third-grade liquid configured past a permeable stretching/shrinking sheet with additional features at different temperatures [38,39,40,41,42,43].

We have found from open literature that only a few works express the third-grade boundary layer flow. The present work is an effort towards determining the flow and thermal characteristics of third-grade fluid taking into consideration the simultaneous impacts of thermal radiation and Lorentz force. Flow is assumed to be a stagnation point occurring over a permeable extending surface with suction. Such a specific study was lacking in the literature. Firstly, the differential equations are transformed into dimensionless ordinary differential equations by using appropriate transformations. The system is solved using a “so-called” hybrid algorithm that is based on the finite difference method and the shooting method. A pseudo-code of the algorithms is also presented. The influence of the emerging parameters on the temperature and velocity profiles is analyzed. Furthermore, the effects of the third-grade fluid parameter, Hartman number (magnetic parameter), Prandtl number, and radiation parameter on the heat transfer rate (in terms of the local Nusselt number) are also examined. The results are shown in graphical and tabular forms. The current study can serve as a catalyst for further stretching flow modeling, especially in polymeric, paper-making, and food-processing applications. We refer the most recent work for further study [44,45,46,47,48,49,50,51,52,53]. The rest of the article is structured as follows. In Section 2, the flow equations are presented. In Section 3, the numerical method is presented for the flow problem. Section 4 is about the results and discussions. Section 5 comprises the concluding remarks.

2 Problem formulation

It is assumed that the third-grade fluid flow is two-dimensional, steady, and incompressible which occurs near a stagnation point over a porous stretchable sheet at y ∼ = 0, as depicted in Figure 1. The porous space is filled with fluid above the sheet, i.e., at y ∼ > 0 . Due to the porosity of the sheet, the suction velocity v w = − v 0 , ( v 0 > 0 ) is taken into account. The flow is due to two opposite and equal forces along the stretchable sheet with external flow velocity to the boundary layer, i.e., U = a x ∼ , where a > 0 is a constant, with a fixed origin. The fluid slightly sticks to the sheet, and thus the fluid motion expresses the slip condition. We apply a transverse magnetic field with uniform strength B 0 . We define T w and T ∞ as the temperature of the sheet and the temperature of the immediate fluid, respectively.

Figure 1

Depiction of the flow geometry.

The governing model in the Cartesian coordinate system is given by

(1) ∂ u ∂ x ∼ + ∂ v ∂ y ∼ = 0 ,

(2) u ∂ u ∂ x ∼ + v ∂ u ∂ y ∼ = − 1 ρ ∂ p ˆ ∂ x ∼ + ν ∂ 2 u ∂ y ∼ 2 + α 1 ρ v ∂ 3 u ∂ y ∼ 3 + ∂ u ∂ x ∼ ∂ 2 u ∂ y ∼ 2 + u ∂ 3 u ∂ x ∼ ∂ y ∼ 2 + ∂ u ∂ y ∼ ∂ 2 v ∂ y ∼ 2 − σ B 0 2 ρ u + 6 ( β 3 ) ρ ∂ u ∂ y ∼ 2 ∂ 2 u ∂ y ∼ 2 ,

(3) ∂ p ˆ ∂ y ∼ = 0 ,

(4) ρ c p u ∂ T ∂ x ∼ + v ∂ T ∂ y ∼ = k 1 ( 1 + Rd ) ∂ 2 T ∂ y ∼ 2 ,

where velocity component u is in the x ∼ -direction while v is in the y ∼ -direction, ν shows the kinematic viscosity, ρ represents the density of a fluid, c p represents the specific heat, T gives the temperature, k 1 represents the thermal conductivity, and α 1 ≥ 0, β 3 ≥ 0 represents the material constants of fluid, Rd represents the thermal radiation effect over the sheet, and p ˆ in Eq. (3) represents the modified pressure, which is defined as follows:

(5) p ˆ = p − ( 2 α 1 + α 2 ) ∂ u ∂ y ∼ 2 ,

where α 2 represents material constant. The BCs for this nonlinear system are given as follows:

(6) u = 0 , v w = − v 0 at y ∼ = 0 u → U ( x ∼ ) as y ∼ → ∞ .

The relevant temperature conditions are as follows:

(7) T ( x ∼ , 0 ) = T w , T ( x ∼ , ∞ ) = T ∞ .

For simplification of the flow model, the following variables are defined:

(8) ω = U ν x ∼ y ∼ , Ψ = U ν x ∼ g ( ω ) , θ ( ω ) = T − T ∞ T w − T ∞ .

Further non-dimensional physical quantities which represent the skin friction coefficient and Nusselt number are given below:

(9) C f = τ w 1 2 ρ ( ax ) 2 , Nu = x q w k ( T w − T ∞ ) ,

where τ w and q w are defined as follows:

τ w = μ ∂ u ∂ y ∼ + α 1 2 ∂ u ∂ x ∼ ∂ u ∂ y ∼ + u ∂ 2 u ∂ x ∼ ∂ y ∼ + 2 β ∂ u ∂ y ∼ 3 y ∼ = 0 , q w = − k ( 1 + Rd ) ∂ T ∂ y ∼ y ∼ = 0 .

Here Ψ represents the stream function, which satisfies

(10) u = ∂ Ψ ∂ y ∼ , v = − ∂ Ψ ∂ x ∼ .

α = a α 1 μ and β = a 2 β 3 μ represents the non-dimensional fluid parameters. Pr = μ c p k 1 represents the Prandtl number, Re x ∼ = a x ∼ 2 ν represents the local Reynolds number, Rd = 16 σ T ∞ 3 3 k k * represents the radiation parameter, g w = v 0 a ν (with v 0 > 0 ) represents the suction velocity, and M 2 = σ B 0 2 ρ a defines the Hartman number M . Note that the parameters α , β , Pr , and n are all positive.

Thus, Eqs. (2) and (4) are transformed into a problem of a system of equations as follows:

(11) g ′ ″ + g g ″ − g ′ 2 + 1 + M 2 ( 1 − g ′ ) + α ( 2 g ′ g ′ ″ − g g ( iv ) − g ″ 2 ) + 6 β Re x ∼ g ″ 2 g ′ ″ = 0 ,

(12) ( 1 + Rd ) θ ″ + Pr g θ ′ = 0 .

Now, the BCs take the following form:

(13) g ( 0 ) = g w , g ′ ( 0 ) = 0 g ′ ( ∞ ) = 1 , g ″ ( ∞ ) = 0 ,

(14) θ ( 0 ) = 1 , θ ( ∞ ) = 0 .

The primes in superscript represent the derivatives with respect to ω . Note that g ′ ( ω ) represents the dimensionless fluid velocity . The dimensionless skin friction coefficient C f , and Nusselt number Nu take the following forms:

(15) C f = 2 Re x [ g ″ + α ( g ′ g ″ − g g ″ ) + 2 β Re x g ″ 3 ] , where α = α 1 a μ ,

(16) Nu = − Re x ( 1 + Rd ) θ ′ ( 0 ) .

3 Numerical method with algorithm

For solving the system in Eqs. (11) and (12) subjected to BCs (13) and (14), we obtain a finite domain [ 0 , 1 ] from the infinite domain ( 0 , ∞ ) using the transformation

(17) ψ = e − c ω ,

where c is not a fixed but a changeable parameter. It mainly describes the thickness of the boundary layer for distinct parametric values used in the equation. Further, we transform the dependent variables as follows:

(18) G ( ω ) = ω − g ( ω ) , θ = ϕ ,

which raises the following boundary value problem

(19) G ′ ″ + ( ω − G ) ( G ″ ) − 2 G ′ + G ′ 2 − M 2 G ′ − α [ − 2 ( 1 − G ′ ) ( G ′ ″ ) + ( ω − G ) G ( iv ) − G ″ 2 ] + 6 β Re x ∼ G ″ 2 G ′ ″ = 0 ,

(20) ( 1 + Rd ) ϕ ″ + Pr ( ω − G ) ϕ ′ = 0 ,

with BCs

(21) G ( 1 ) = − g w , G ′ ( 1 ) = 1 G ′ ( 0 ) = 0 , G ″ ( 0 ) = 0 ,

(22) ϕ ( 1 ) = 1 , ϕ ( 0 ) = 0 .

Next we perform the transformation as follows:

(23) y ∼ 1 = G , y ∼ 2 = G ′ , y ∼ 3 = G ″ , z ∼ = ϕ ,

and through transformation (17), the system of Eqs. (19) and (20) take the form as given below:

(24) y ∼ 2 = − c ψ d y ∼ 1 d ψ ,

(25) y ∼ 3 = − c ψ d y ∼ 2 d ψ ,

(26) c ψ d y ∼ 3 d ψ + ( c − 1 ln ψ + y ∼ 1 ) y ∼ 3 + 2 y ∼ 2 − y ∼ 2 2 + M 2 y ∼ 2 + α 2 ( 1 − y ∼ 2 ) c ψ d y ∼ 3 d ψ − ( c − 1 ln ψ + y ∼ 1 ) c 2 ψ d d ψ ψ d y ∼ 3 d ψ − y ∼ 3 2 + 6 β Re x ∼ y ∼ 3 2 c ψ d y ∼ 3 d ψ = 0 ,

(27) ( 1 + Rd ) c 2 ψ d d ψ ψ d z ∼ d ψ + Pr ( c − 1 ln ψ + y ∼ 1 ) c ψ d z ∼ d ψ = 0 .

Similarly, the BCs (24) and (25) assume the forms

(28) y ∼ 2 ( 0 ) = 0 , y ∼ 3 ( 0 ) = 0 , y ∼ 1 ( 1 ) = − g w , y ∼ 2 ( 1 ) = 1 ,

(29) z ( 0 ) = 0 , z ( 1 ) = 1 .

The Eqs. (24) and (25) are identically satisfied by the conditions at ψ = 0 . The mesh on the ψ -axis is formed as follows:

(30) ψ = ih where i = 0 , 1 , 2 , ⋯ , N ,

where N is an integer. The derivatives in the model are discretized using the central differences as follows:

(31) ψ d y ∼ 3 d ψ i = ψ i y ∼ 3 i + 1 − y ∼ 3 i − 1 2 h ,

(32) ψ d d ψ ψ d y ∼ 3 d ψ i = ψ i h ψ d y ∼ 3 d ψ i + 1 / 2 − ψ d² y ∼ 3 d ψ 2 i − 1 / 2 ,

(33) = ψ i ψ i + 1 / 2 ( y ∼ 3 i + 1 − y ∼ 3 i ) − ψ i − 1 / 2 ( y ∼ 3 i − y ∼ 3 i − 1 ) h 2 ,

(34) d y ∼ 1 d ψ i = ψ i + 1 / 2 y ∼ 1 i + 1 + y ∼ 1 i h ,

Thus, Eqs. ( 24 ) – ( 27 ) assumes the form as follows:

(35) y ∼ 1 i + 1 = y ∼ 1 i − y ∼ 2 i + 1 + y ∼ 2 i c ( 2 i + 1 ) ,

(36) y ∼ 3 i + 1 = ci 2 − α − ci ( 1 − y ∼ 2 i ) + ci i + 1 2 ( ln ψ i + c y ∼ 1 i ) + 3 ci β Re x ∼ ( y ∼ 3 i ) 2 − 1 × ci 2 y ∼ 3 i − 1 − ( c − 1 ln ψ i + y ∼ 1 i ) y ∼ 3 i + ( y ∼ 2 i ) 2 − ( 2 + M 2 ) y ∼ 2 i + 3 ci β Re x ∼ ( y ∼ 3 i ) 2 y ∼ 3 i − 1 + α ci ( 1 − y ∼ 2 i ) y ∼ 3 i − 1 + ( y ∼ 3 i ) 2 + ci ( ln ψ i + c y ∼ 1 i ) − 2 i y ∼ 3 i + i − 1 2 y ∼ 3 i − 1 ,

(37) z ∼ i + 1 = ( 1 + Rd ) c 2 i i + 1 2 + Pr ( ln ψ + c y ∼ 1 i ) i 2 − 1 × Pr ( ln ψ + c y ∼ 1 i ) i 2 z ∼ i − 1 + ( 1 + Rd ) c 2 i 2 i z ∼ i − i − 1 2 z ∼ i − 1 ,

with respect to BCs:

(38) y ∼ 2 0 = 0 , y ∼ 3 0 = 0 , y ∼ 1 N = − g w , y ∼ 2 N = 1 .

(39) z ∼ N = 1 , z ∼ 1 = 0 .

Thus, the procedure of the hybrid method as adapted from (Ariel [10], Abbas et al. [22], and Hayat et al. [24]) can be described in the form of pseudocode as below:

INPUTS : N , count of mesh nodes y ∼ 2 1 = 0 y ∼ 1 N + 1 = − g w c , adjustable constant y ∼ 3 1 = 0 y ∼ 2 N + 1 = 1 α z ∼ 1 = 0 z ∼ N + 1 = 1 β M Re Pr Rd

OUTPUT : Y ∼ 1 = [ y ∼ 1 1 , y ∼ 1 2 , ⋯ , y ∼ 1 N , y ∼ 1 N + 1 ] T Y ∼ 2 = [ y ∼ 2 1 , y ∼ 2 2 , ⋯ , y ∼ 2 N , y ∼ 2 N + 1 ] T Y ∼ 3 = [ y ∼ 3 1 , y ∼ 3 2 , ⋯ , y ∼ 3 N , y ∼ 3 N + 1 ] T Z ∼ = [ z 1 , z 2 , ⋯ , z N , z N + 1 ] T

Step 1: Set h = 1 N and ψ i + 1 = ih , for i = 0 , 1 , 2 , ⋯ , N

Step 2: Construct the values of y ∼ 3 2 and y ∼ 1 1 such that y ∼ 2 N + 1 = 1 and y ∼ 1 N + 1 = − g w

Step 3: Construct the values of z ∼ 2 such that z ∼ N + 1 = 1

Step 4: Set y ∼ 2 2 = y ∼ 2 1 − y ∼ 3 2 + y ∼ 3 1 3 c Set y ∼ 1 2 = y ∼ 1 1 − y ∼ 2 2 + y ∼ 2 1 3 c

Step 5: for i = 2 , 3 , 4 , ⋯ , N perform steps 6–9

Step 6: Compute y ∼ 3 i + 1 as described in Eq. (37)

Step 7: Compute y ∼ 2 i + 1 as described in Eq. (36)

Step 8: Compute y ∼ 1 i + 1 as described in Eq. (35)

Step 9: Compute z ∼ i + 1 as described in Eq. (38)

Step 10: Print the outputs.

STOP.

4 Results and discussion

The problem as described in Eqs. (30)–(35) is solved using the algorithm as described in Section 3. The local Reynolds number Re x ∼ is considered as 1 for the study. Some comparisons between the present results and the results in the literature are presented to illustrate the validity and accuracy of the solution [25]. The numerical results are presented in Figures 2–9 graphically, which shows the development in the velocity and temperature profiles at various values of the control parameters.

Figure 2

Effects of the suction velocity on the dimensionless fluid velocity.

$Figure 3 Effects of α \alpha on g ′ ( ω ) {g}^{^{\prime} }\left(\omega ) with g w = 10 {g}_{w}=10 .$

Figure 3

Effects of α on g ′ ( ω ) with g w = 10 .

$Figure 4 Effect of β \beta on g' ( ω ) {g\text{'}}\left(\omega ) , when g w = 1 {g}_{w}=1 , β = 1 \beta =1 , and α = 2 \alpha =2 .$

Figure 4

Effect of β on g' ( ω ) , when g w = 1 , β = 1 , and α = 2 .

$Figure 5 Effect of M M on g' ( ω ) {g\text{'}}\left(\omega ) , when g w = 1 {g}_{w}=1 , β = 1 \beta =1 , and α = 2 \alpha =2 .$

Figure 5

Effect of M on g' ( ω ) , when g w = 1 , β = 1 , and α = 2 .

$Figure 6 Effect of Pr {\rm{\Pr }} on ϕ ( ω ) \phi \left(\omega ) , when β = 1 \beta =1 , α = 2 \alpha =2 , M = 0.5 M=0.5 and g w = 10 {g}_{w}=10 .$

Figure 6

Effect of Pr on ϕ ( ω ) , when β = 1 , α = 2 , M = 0.5 and g w = 10 .

$Figure 7 Effect of Rd {\rm{Rd}} on ϕ ( ω ) \phi \left(\omega ) , when β = 1 \beta =1 , α = 2 \alpha =2 , g w = 10 {g}_{w}=10 , M = 0.5 M=0.5 and Pr = 0.2 {\rm{\Pr }}=0.2 .$

Figure 7

Effect of Rd on ϕ ( ω ) , when β = 1 , α = 2 , g w = 10 , M = 0.5 and Pr = 0.2 .

$Figure 8 Effect of g w {g}_{w} at ϕ ( ω ) \phi \left(\omega ) , when β = 1 \beta =1 , α = 2 \alpha =2 , M = 0.5 M=0.5 and Pr = 0.2 {\rm{\Pr }}=0.2 .$

Figure 8

Effect of g w at ϕ ( ω ) , when β = 1 , α = 2 , M = 0.5 and Pr = 0.2 .

$Figure 9 Effect of β \beta on Nu {\rm{Nu}} with (a) Pr = 0 . 05 , 0 . 1 , 0 . 2 , and 0 . 3 {\rm{\Pr }}=0.05,0.1,0.2,{\rm{and}}0.3 and (b) Rd = 0 . 1 , 0 . 3 , 0 . 5 , and 0 . 8 {\rm{Rd}}=0.1,0.3,0.5,{\rm{and}}0.8 , when α = 2 \alpha =2 , g w = 10 {g}_{w}=10 , and M = 0 . 5 M=0.5 .$

Figure 9

Effect of β on Nu with (a) Pr = 0 . 05 , 0 . 1 , 0 . 2 , and 0 . 3 and (b) Rd = 0 . 1 , 0 . 3 , 0 . 5 , and 0 . 8 , when α = 2 , g w = 10 , and M = 0 . 5 .

Figure 2 shows the impact of change in g w , for the second- and third-grade fluids, on g ′ ( ω ) . For the case of the second-grade fluid (with α = 0.2 and β = 0 ), g ′ ( ω ) exhibits slight increments as g w is increased from 0 to 10. This is because the fluid is of second-grade type (with α = 0.2 ), due to which the suction boundary layer does not appear even for large values of g w . The plots for the case of the second-grade fluid are found in full agreement with those of previous studies [8] and [22]. For the case of the third-grade fluid (with α = 0.2 and β = 1.0 ), g ′ ( ω ) exhibits relatively larger increments as g w is increased from 0 to 10.

Figure 3 shows the effects of α on g ′ ( ω ) , taking g w = 10 and β = 0 . In the absence of the viscoelasticity (i.e., α = 0 ), the momentum boundary layer effect is quite apparent because of the suction velocity. For the increasing values of α , the values of g ′ ( ω ) decrease, although the momentum boundary thickness grows with α . The velocity profiles tend to be flattened as α is increased. For higher values of the viscoelastic parameter, no significant variation is found in g' ( ω ) . The graphs in Figure 3 are in full agreement with the corresponding plots in previous studies [8] and [22].

Figure 4 illustrates the effect of the third-grade fluid parameter β on g ′ ( ω ) , when α = 2 and g w = 1 . g ′ ( ω ) decreases when β increases. It is because of the shear thickening effect that grows with β . Also, the third-grade parameter causes an increase in the momentum boundary layer thickness.

Figure 5 shows the effect of Hartman number M on g ′ ( ω ) at different values of ω . The figure illustrates that by decreasing the value of M , the value of velocity field g ′ ( ω ) also decreases. The Lorentz force, which opposes the flow, causes the flow to decelerate more quickly because the Hartmann number directly relates to the Lorentz force. The graphs in Figure 5 are in full agreement with the corresponding plots in the previous study [22] for M = 0 and g w = 1 .

Figure 6 shows the changes in temperature field ϕ ( ω ) due to Prandtl number Pr for β = 1 , α = 2 , M = 0.5 , and g w = 10 . As Pr is a ratio of the thermal diffusivity and momentum diffusivity, hence, temperature field is influenced by the Prandtl number Pr . The values of ϕ ( ω ) decrease significantly as the Prandtl number is increased. Figure 7 shows the variation in ϕ ( ω ) due to the radiation Rd when Pr = 0.2 . The values of ϕ ( ω ) increase with the increase in Rd . Figure 8 shows the effects of g w on ϕ ( ω ) when Pr = 0.2 . The values of ϕ ( ω ) decrease with the increments in g w . The velocity of the fluid increases because of suction, which causes a decrease in the temperature.

Figure 9 shows the impact of β , Pr , and Rd on the important physical quantity, the local rate of heat transfer measured in terms of the local Nusselt number, Nu for α = 2 , M = 0.5 , and g w = 10 . Nu is observed to be increasing with an increment in β and Rd , whereas it declines when Pr is increased. Since Pr is the ratio of dynamic viscosity to thermal diffusivity, increasing Pr causes the fluid molecules to disperse and lose energy.

Table 1 shows the variation in the values of the temperature gradient at the sheet surface, ϕ ′ ( 0 ) , for different values of α , β , Pr , and Rd , when g w = 5 and M = 0.2 . The increments in the values of α , β , and Rd result in the decrement of the temperature gradient (the rate of heat transfer) at the sheet surface, whereas it increases as the value of Pr is increased.

Table 1

Variation in the temperature gradient at the sheet surface for different values of α , β , Pr , and Rd with g w = 5 and M = 0.2

α	β	Pr	Rd	ϕ ′ ( 0 )
0.1	1	1	1	2.80197
0.2				2.64761
0.5				2.58806
1.0				2.57900
2.0				2.57660
5.0				2.57588
1	0.1			2.58550
	0.2			2.58468
	0.5			2.58238
	1.0			2.57900
	2.0			2.57344
	5.0			2.56176
2	1	0.2		0.63473
		0.5		1.36135
		1.0		2.57660
		1.5		3.78857
		2.0		5.00131
		1	0.0	5.00131
			0.5	3.38451
			1.0	2.57660
			2.0	1.76740
			5.0	0.95452
			10.0	0.59256

5 Conclusion

In this research, the two-dimensional stagnation-point flow of third-grade liquid over a stretching sheet with uniform suction velocity and thermal radiation is investigated numerically. The governing system of equations is solved by using a hybrid numerical algorithm described in pseudocode form. The impacts of different emerging parameters M , α , β , Pr , and Rd on the velocity g' ( ω ) and temperature ϕ ( ω ) profiles are visualized in graphical and tabular forms. The summary of the impacts is as follows:

g' ( ω ) shows an inciting trend towards the suction velocity g w . However, g' ( ω ) exhibits somewhat higher increments in case of the third-grade fluid than the case of the second-grade fluid with increasing values of g w .
In the absence of magnetic field (i.e., M = 0 ), g' ( ω ) exhibits a declining trend towards each of the second-grade and the third-grade fluid parameters.
In the case of third-grade fluid, g' ( ω ) shows an inciting trend towards M .
In the presence of the magnetic field, ϕ ( ω ) shows a declining trend towards Pr and g w , whereas an inciting trend towards Rd .
The values of the temperature gradient at the sheet surface, ϕ ′ ( 0 ) , decreases for the flow controlling parameters α , β , and Rd , whereas increases for Pr .
The Nusselt number increases with an increase in β and Rd , while it decreases when Pr is increased.

The shooting approach could be applied to a variety of physical and technical challenges in the future [12,54–65].

Funding information: The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: All data generated or analyzed during this study are included in this published article.

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Received: 2022-08-14

Revised: 2022-12-18

Accepted: 2023-01-12

Published Online: 2023-02-24

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2022-0227

Keywords for this article

third-grade fluid; heat transfer; thermal radiation; stagnation-point; permeable sheet

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