Axisymmetric stagnation-point flow of non-Newtonian nanomaterial and heat transport over a lubricated surface: Hybrid homotopy analysis method simulations

Manzoor Ahmad; Vediyappan Govindan; Sami Ullah Khan; Haewon Byeon; Muhammad Taj; Nadia Batool; Dilsora Abduvalieva; Fuad A. Awwad; Emad A. A. Ismail

doi:10.1515/phys-2023-0148

Article Open Access

Axisymmetric stagnation-point flow of non-Newtonian nanomaterial and heat transport over a lubricated surface: Hybrid homotopy analysis method simulations

Manzoor Ahmad , Vediyappan Govindan , Sami Ullah Khan , Haewon Byeon , Muhammad Taj , Nadia Batool , Dilsora Abduvalieva , Fuad A. Awwad and Emad A. A. Ismail

Published/Copyright: December 6, 2023

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From the journal Open Physics Volume 21 Issue 1

Abstract

The heat transfer phenomenon associated with the lubricated surfaces offers applications in the manufacturing processes, thermal systems, industrial systems, and engineering phenomenon. It is a well-established fact that improvement in heat transfer is recently successfully claimed with the interaction of nanoparticles. Following such motivation in mind, the prime objective of current continuation is to perform the prediction of heat transfer in second-grade material subject to the lubricated surface. The lubricants are filled with non-Newtonian power law material. The varying thickness of the thin lubricating layer permits an imperfect slip surface. The second-grade fluid interfaces with the boundary condition. The modified semi-analytical tool termed as hybrid homotopy scheme is used to perform the simulations. Shooting and homotopy methods are combined in this new approach. Relevant effects of parameters on physical phenomenon are explained. The importance of influencing parameters in relation to the velocity field, temperature, and concentration profiles is investigated graphically. It is claimed that analytical computations existed for shear thinning case. It is observed that there is a noticeable drop in concentration and thermal profiles due to the variation of viscoelastic parameter. The control of free stream velocity is claimed due to the interaction of slip parameters.

Keywords: lubricant surface; slip conditions; homotopy method of analysis; Runge–Kutta method with shooting technique; second-level fluid; nanoparticles; axially symmetric flow

Nomenclature

( u , v ): velocity component ( m s − 1 )
T: nanoparticle temperature ( K )
T ∞: atmospheric temperature ( K )
C: nanoparticle concentration ( kg m − 3 )
C ∞: atmospheric concentration ( kg m − 3 )
ρ f: nanoparticle density ( kg m − 3 )
λ: slip parameter
K: viscoelastic parameter
n: Power law index
θ: temperature
ϕ: concentration
f ': velocity
δ: thickness of the lubrication layer
D T: thermophoretic diffusion coefficient
N t: thermophoresis parameter
Le: Lewis number
D B: Brownian motion constant,
a: constant
Nb: Brownian motion
Pr: Prandtl number
Nt: thermophoresis parameter
ζ: similarity variable
Q: flow rate

1 Introduction

Due to their uses in industry and technology, viscoelastic fluids are much more suitable when related to Newtonian fluids, such as petroleum drilling, food manufacture, paper making, wire drawing, scaffolds for tissue engineering, and other industrial uses of viscoelastic fluids. The latest researchers’ efforts to explore viscoelastic fluids through many aspects have increased steadily. Based on the nonlinear fundamentals between deformation rate and shearing force, diverse kinds of expressions are defined for non-Newtonian materials. The constitutive equation of second-grade liquid is probably most important among the numerous viscoelastic fluids. The dependence of stresses on strain rate for a second-grade fluid raises the rank of the governing differential equation. This leads to the need for additional boundary conditions in order to solve those equations. Rajagopal [1] gave a quick explanation of the issue raised by these additional boundary conditions. For mathematicians, physicists, numerical assimilationists, and engineers, analysis of viscoelastic fluid flows has historically been a challenging research topic. The stagnation analysis in flow system is the fundamental challenge in fluid mechanics. Hiemenz [2] started working on the viscous liquid flow altered by the stagnation point phenomenon. Homann [3] expanded his research on axisymmetric flow based on the investigation of Hiemenz [2]. He attained a similarity solution. Yeckel and Strong [4] were the first to apply the Homann [3] flow to provide the boundary condition on the thin lubricating layer in the stagnation region of a jet to make an impact over a flat surface. Wang [5] conducted research on stagnation-point flow related to boundary slip. Blyth and Pozrikidis [6] defined the stagnation pattern in Newtonian material owing to 3D flow with axisymmetric surface. Andersson and Rousselet [7] arrived at the conclusion of the slip condition over a lubricated spinning disk for flow. Using an axisymmetric flow across determined with lubricated regime, a theoretical investigation was directed by Santra et al. [8]. In the latest research, Sajid et al. [9] generalized the lubrication problem addressed in the study by Santra et al. [8] using a modified slip boundary condition introduced by Thompson and Troian [10]. The constitutive equation of Newtonian fluid is used in the preceding investigations. Sajid et al. [11] prescribed the Walter B distinguished features near the lubricated regime following the stagnation-point impact. Using a hybrid numerical technique introduced by Ariel [12], they offered a solution to the nonlinear differential equation with a nonlinear boundary condition across the surface.

In the aforementioned cited work, it is clearly noted that some studies associated with lubrication flow of viscous fluid are available. However, the decomposition of non-Newtonian materials with nanofluid associated with the lubricated surface is more fascinating and applicable in industrial and engineering systems. The purpose behind this research is to focus toward the lubricated regime flow due to viscoelastic second-grade fluid with stagnation-point constraints. The motivations for selecting the viscoelastic fluid are associated with the novel memory effects. Moreover, this model also offers a shear thinning and thickening impact. The resultant model is governed by complex boundary constraints, which are treated using combining homotopy analysis, which is seen in previously published studies [13–18]. Recently, Li et al. [19], Sun et al. [20], and Zhou et al. [21], respectively, studied the surface pressure pulsation characteristics, heat flow transfer characteristics, and multi-stage thermoelectric cooler combined with microchannel heat sink. Recent developments in shooting methods are indicated in previously published studies [22,25,26,27,28,29,30,31,32,33,34]. Yang et al. [35], Sun et al. [36], and Kuang et al. [37] worked on the deformation mechanism of an amorphous wrapped nanolamellar heterostructure, hear-thickening fluids, and thermodynamic extremal principle to diffusion-controlled phase transformations, respectively.

2 Problem formulation

2.1 Flow equations of second-grade fluid

Assume the flow of second-grade nanofluid at its axisymmetric stagnation point across a disk coated in a thin layer of lubricant with a power law. The z-axis of mathematical modeling goes outward from the disk and is represented in cylindrical coordinates ( r , θ , z ) . The illustration of problem is shown in Figure 1. No effect of θ on velocity is detected. The azimuthal velocity component vanishes due to axisymmetric flow.

Figure 1

Physical configuration.

Assume that the velocity regime for fluid and lubricant is, respectively, expressed as [ u ( r , z ) , 0 , w ( r , z ) ] and [ U ( r , z ) , 0 , W ( r , z ) ]. A thin coating of lubricant with varying thickness h ( r ) forms over a disk out of a point source at the origin. The lubricant that flows at a steady rate is expressed as:

(1) Q = ∫ h ( r ) 0 U ( r , z ) 2 π r d z .

For the second-grade fluid, the governing equations [24] are as follows:

(2) ∂ u ∂ r + u r + ∂ w ∂ z = 0 ,

(3) ∂ u ∂ r + w ∂ u ∂ z = − 1 ρ ∂ p ∂ r + ν ∂ 2 u ∂ r 2 + ∂ ∂ r u r + ∂ 2 u ∂ z 2 u + α 1 ρ 2 u ∂ 3 u ∂ r 3 + 2 ∂ u ∂ r ∂ 2 u ∂ r 2 + ∂ w ∂ r ∂ 2 u ∂ r ∂ z + 2 w ∂ 3 u ∂ r 2 ∂ z + 2 ∂ w ∂ r ∂ 2 w ∂ r 2 + u ∂ 3 u ∂ z 2 ∂ r + ∂ u ∂ z ∂ 2 w ∂ r 2 + u ∂ 3 w ∂ r 2 ∂ z + w ∂ 3 u ∂ z 3 + 2 ∂ w ∂ z ∂ 2 w ∂ r ∂ z + w ∂ 3 u ∂ z 2 ∂ r + ∂ u ∂ r ∂ 2 u ∂ z 2 − ∂ u ∂ r ∂ 2 w ∂ r ∂ z + ∂ w ∂ r ∂ 2 w ∂ z 2 − ∂ u ∂ z ∂ 2 w ∂ z 2 + 2 u r ∂ 2 u ∂ r 2 + 2 w r ∂ 2 u ∂ r ∂ z + 1 r ∂ w ∂ r 2 − 1 r ∂ u ∂ z 2 − 2 u r 2 ∂ u ∂ r − 2 w r 2 ∂ u ∂ z + 2 u 2 r 3 ,

(4) u ∂ w ∂ r + w ∂ w ∂ z = − 1 ρ ∂ p ∂ z + ν ∂ 2 w ∂ r 2 + 1 r w r + ∂ 2 w ∂ z 2 + α 1 ρ 2 w ∂ 3 w ∂ z 3 + 2 ∂ w ∂ z ∂ 2 w ∂ z 2 + ∂ u ∂ z ∂ 2 w ∂ r ∂ z + 2 u ∂ 3 w ∂ z 2 ∂ r + 2 ∂ u ∂ z ∂ 2 u ∂ z 2 + w ∂ 3 u ∂ z 2 ∂ r + ∂ w ∂ r ∂ 2 u ∂ z 2 + w ∂ 3 w ∂ r 2 ∂ z + u ∂ 3 w ∂ r 3 + u ∂ 3 u ∂ r 2 ∂ z + 2 ∂ u ∂ r ∂ 2 u ∂ r ∂ z + ∂ w ∂ z ∂ 2 w ∂ r 2 + ∂ u ∂ z ∂ 2 u ∂ r 2 − ∂ w ∂ z ∂ 2 u ∂ r ∂ z − ∂ w ∂ r ∂ 2 u ∂ r 2 + u r ∂ 2 w ∂ r 2 + w r ∂ 2 w ∂ r ∂ z + u r ∂ 2 u ∂ z ∂ r + 1 r ∂ u ∂ r ∂ u ∂ z + 1 r ∂ w ∂ r ∂ w ∂ z − 1 r ∂ u ∂ r ∂ w ∂ r + w r ∂ 2 u ∂ z 2 − 1 r ∂ u ∂ z ∂ w ∂ z ,

(5) u ∂ T ∂ r + w ∂ T ∂ z = α ∂ 2 T ∂ z 2 + τ D T T ∞ ∂ T ∂ z 2 + D B ∂ c ∂ z ∂ T ∂ z ,

(6) u ∂ C ∂ r + w ∂ C ∂ z = D T T ∞ ∂ 2 T ∂ z 2 + D B ∂ 2 C ∂ z 2 ,

where ( u , v , w ) are the velocity components, ρ is the density, T is the fluid temperature, T ∞ is the free stream temperature, D B is the Brownian diffusion, C is the concentration of material, p is the pressure, D T is the diffusion thermophoresis, ν is the kinematic viscosity, α 1 is the material coefficient, and τ is the fluid heat capacity ratio of nanoparticles and base.

2.2 Conditions at interfaces and boundaries

The space h ( r ) < z < 1 got filled by the second-grade fluids and 0 < z < h ( r ) is the space occupied by power law lubricants. At the wall, the usual no-slip condition is provided by:

(7) U ( r , z ) , W ( r , z ) = 0 , T ( 0 ) = T ∞ , C ( 0 ) = C w .

Let us assume that inside the lubricated layer, where there is no axial velocity, we obtain

(8) W ( r , z ) = 0 for z ∈ [ 0 , h ( r ) ] .

When the second-grade fluid and power law lubricants engage in the interfacial region, the velocities and shear stresses of both fluids should be consistent. Continuity of shear stresses present at z = h ( r ) produces the following:

(9) μ ∂ w ∂ r + ∂ u ∂ z + α 1 u ∂ 2 w ∂ r 2 + u ∂ 2 u ∂ r ∂ z + w ∂ 2 u ∂ z 2 + w ∂ 2 w ∂ r ∂ z + ∂ u ∂ r ∂ u ∂ z + ∂ w ∂ r ∂ w ∂ z − ∂ u ∂ z ∂ w ∂ z − ∂ u ∂ r ∂ w ∂ r = k ∂ U ∂ z n

where k denotes the consistency index, n denotes the power law index, and μ denotes the dynamic viscosity μ . Following the previous studies [8,26], the power law lubricant’s radial component of velocity U shows the variation linearly across:

(10) U ( r , z ) = U ˆ ( r ) z h ( r ) ,

where U ˆ ( r ) denotes the radial velocity interaction component.

Substituting Eq. (8) into Eq. (1), the layer thickness is known as:

(11) h ( r ) = Q π r U ˆ ( r ) .

As a result, the boundary condition in Eq. (7) changes as follows:

(12) μ ∂ w ∂ r + ∂ u ∂ z + α 1 u ∂ 2 u ∂ r ∂ z + u ∂ 2 w ∂ r 2 + w ∂ 2 u ∂ z 2 + w ∂ 2 w ∂ r ∂ z + ∂ u ∂ r ∂ u ∂ z + ∂ w ∂ r ∂ w ∂ z − ∂ u ∂ z ∂ w ∂ z − ∂ u ∂ r ∂ w ∂ r = k π Q n r n u 2 n .

Another benefit of the continuous axial velocity at the fluid–fluid contact is that it offers the following:

(13) w ( r , h ( r ) ) = W ( r , h ( r ) ) ,

which further gives

(14) w ( r , h ( r ) ) = 0 .

The pressure at the interface is as follows [8]:

(15) p ( r , h ( r ) ) = − ρ A 2 r 2 2 .

The shape of the fluid’s velocity, temperature, and concentration when it is away from the plate seem to be:

(16) u = A r , w = − A z , T ( ∞ ) = T ∞ , C ( ∞ ) = C ∞ ,

in which A can be any constant that is positive.

2.3 Dimensional analysis

Introducing the dimensionless variables [8,26]:

(17) η = z c ν , u = c r f ' ( η ) , w = − 2 c ν f ( η ) , p = c μ p ¯ ( η ) − ρ c r 2 2 , T = T ∞ + ( T w − T ∞ ) θ ( η ) , C = C ∞ + ( C w − C ∞ ) ϕ ( η ) .

Eq. (2) seems to be fulfilled identically, and Eqs. (3)–(6), (12), (14), and (16) are constructed as follows:

(18) f ″ ′ + 2 f f ″ − f ′ 2 + 1 + K ( f ″ 2 + 2 f ′ f ″ ′ − 2 f f ″ ″ ) = 0 ,

(19) p ¯ ′ = − 4 f f ′ − 2 f ″ + 2 K ( 2 f f ‴ + 8 f ′ f ″ + δ f ″ f ‴ ) ,

(20) θ ″ + Pr ( 2 f θ ′ + N t θ ′ 2 + N b θ ′ ϕ ′ ) = 0 ,

(21) ϕ ″ + Nt Nb θ ″ + 2 Le f ϕ ′ = 0 ,

(22) f ( 0 ) = 0 , f ″ ( 0 ) = λ { f ′ ( 0 ) } 2 n 1 + 4 K f ′ ( 0 ) f ′ ( ∞ ) = 1 and p ¯ ( 0 ) = 0 , θ ( 0 ) = 1 , ϕ ( 0 ) = 1 , θ ( ∞ ) = 0 , ϕ ( ∞ ) = 0 .

where

(23) λ = k ν μ π Q n ν r 3 n − 1 c 2 n − 3 2 , K = α 1 C μ , and δ = c r 2 ν .

When n = 1 / 3 , it is obvious from Eq. (23) that λ is independent on r , and just in this condition, a similarity solution can be found.

There are also provided non-similar solutions for other values of the parameter n . Eq. (18) is solved while being taken into consideration the boundary conditions in Eq. (23). The following part will examine the solution of problem using a hybrid homotopy analysis technique. It is simply used to determine the pressure by integrating Eq. (19) after the solutions to Eq. (18) have been found (19).

3 Hybrid homotopy analysis method

We initially apply the shooting method [22] referring to attaining the first-order system of governing problem defined via Eqs. (18)–(22) as:

(24) f ′ ( 0 ) = s 1 ,

(25) θ ′ ( 0 ) = s 2 ,

(26) ϕ ′ ( 0 ) = s 3 .

Differentiating Eqs. (19) and (21)–(24) with respect to s 1 , s 2 , and s 3 , respectively, we obtain the following:

(27) g ‴ + 2 g f ″ + 2 f g ″ − 2 f ′ g ′ + K ( 2 f ″ g ″ + 2 f ′ g ‴ + 2 f ‴ g ′ − 2 f g ″ ″ − 2 f ″ ″ g ) = 0,

(28) p ″ + Pr ( 2 f p ′ + 2 Nt p ′ + Nb p ″ ϕ ′ ) = 0 ,

(29) u ″ + Nt Nb θ ″ + 2 Le f u ′ = 0,

subject to the conditions:

(30) p ( 0 ) = 0 , u ( 0 ) = 0 , z ( 0 ) = 0 , G ( 0 ) = λ 2 n s 1 2 n − 1 + 4 k s 1 2 n ( 2 n − 1 ) ( 1 + 4 k s 1 ) 2 , u ' ( 0 ) = s 3 , p ' ( 0 ) = s 2 .

It is clear from Eqs. (18) and (22) that this is the case: conventional Runge–Kutta algorithm fails to integrate the resulting systems. Moreover, at η = 0 and K → 0, the coefficient of the greatest derivative terms in Eqs. (18) and (22) disappears. In this article, we provide a methodology for attaining simulations for the initially defined system with homotopy analysis method (HAM). After each iteration, the first-order initial value problems are created by converting the initial value problems (18) and (20)–(22) in the following manner:

(31) f ′ = F ,

(32) F ′ = G ,

(33) G ′ = − 2 f G + F 2 − 1 − K ( G 2 + 2 F G ′ − 2 f G ″ ) ,

(34) g ′ = Y ,

(35) Y ′ = Z ,

(36) Z ′ = − 2 g G − 2 f Z + 2 F Y − K ( 2 G Z + 2 F Z ′ + 2 G ′ Y − 2 f Z ″ − 2 G ″ g ) ,

(37) θ ′ = M ,

(38) ϕ ′ = N ,

(39) M ′ = − Pr ( 2 f M + Nb M N + Nt M 2 ) ,

(40) N ′ = 2 Le f N − Nt Nb M ′ ,

(41) p ′ = P ,

(42) u ′ = U ,

(43) P ′ = − Pr ( 2 f P + Nb P N + Nt M N ) ,

(44) U ′ = 2 Le f U − Nt Nb M ′ ,

(45) f ( 0 ) = 0 , F ( 0 ) = s 1 , G ( 0 ) = λ s 1 2 n 1 + 4 K s 1 , g ( 0 ) = 0 , Y ( 0 ) = 1 , Z ( 0 ) = λ 2 n s 1 2 n − 1 + 4 K s 1 2 n ( 2 n − 1 ) ( 1 + 4 K s 1 ) 2 ,

(46) θ ( 0 ) = 1 , M ( 0 ) = s 2 , ϕ ( 0 ) = 1 , N ( 0 ) = s 3 , p ( 0 ) = 0 , P = 1 , u ( 0 ) = 0 , U = 1 .

In numerical calculations, we altered ∞ to the value η ∞ and divided the 0 ≤ η ≤ η ∞ domain into subintervals with lengths H i so that

(47) ∑ H i = η ∞ , i = 1 , 2 , 3 .

In order to perform the calculations for such a problem, a fixed length subinterval indicated by [ ( i − 1 ) H , H ] , is specified. i = 1 , 2 , 3 … . Every subinterval’s initial value problem has the following syntaxes:

(48) d f i d η = F i ,

(49) d F i d η = G i ,

(50) d G i d η = − 2 f i G i − 1 − ( F i ) 2 − K ( G i ) 2 + 2 F i d G i d η − 2 f i d 2 G i d η 2 ,

(51) d g i d η = Y i ,

(52) d Y i d η = Z i ,

(53) d Z i d η = − 2 g i G i − 2 f i Z i + 2 F i Y i − K 2 G i Z i + 2 F i d Z i d η + 2 Y i d G i d η − 2 f i d 2 Z i d η 2 − 2 g i d 2 G i d η 2 ,

(54) d θ i d η = M i ,

(55) d ϕ i d η = N i ,

(56) d M i d η = − Pr ( 2 f i M i + Nb M i N i + Nt M i 2 ) ,

(57) d N i d η = 2 Le f i N i − Nt Nb d M i d η ,

(58) d p i dζ = P i ,

(59) d p i d η = − Pr ( 2 f i P i + Nb P i N i + 2 Nt M i N i ) ,

(60) d u i dζ = U i ,

(61) d U i dζ = 2 Le f i U i − Nt Nb d M i dζ ,

subject to the condition as:

(62) f 1 ( 0 ) = 0 , F 1 ( 0 ) = s 1 , G 1 ( 0 ) = λ s 2 n 1 + 4 k s , g 1 ( 0 ) = 0 , Y 1 ( 0 ) = 1 , z 1 ( 0 ) = λ 2 n s 1 2 n − 1 + 4 k s 1 2 n ( 2 n − 1 ) ( 1 + 4 k s 1 ) 2 , θ 1 ( 0 ) = 1 , M 1 ( 0 ) = s 2 , ϕ 1 ( 0 ) = 1 , N 1 ( 0 ) = s 3 , P 1 ( 0 ) = 0 , P 1 ( 0 ) = 1 , u 1 ( 0 ) = 0 , U 1 ( 0 ) = 1 .

The HAM algorithm is implemented for incorporating the computations of Eqs. (32)–(46). The numerical value computed at i th subinterval is used as initial approximations for HAM solution obtained at ( i + 1 ) th subinterval.

4 Validation of results

Tables 1 and 2 present a comparison between the existing results and investigation of Santra et al. [8]. Clearly, favorable accuracy of results is noted between both studies. The results are further verified with continuation of Sajid et al. [9] in Table 2. Again, an excellent accuracy is noticed with both investigations. A complete mathematical step involved in HAM technique is presented in Figure 2. Furthermore, to verify the precision and validity of computed solution via the hybrid homotopy analysis method, the residual error curve is shown in Figure 3. This error curve is displayed for a specific set of parameters. The figure clearly demonstrates that the solutions obtained exhibit an accuracy level of 10 − 7 .

Table 1

Comparison of results with analysis of Santra et al. [8] when K = 0 and λ → ∞

ζ	f ′ ( ζ )	f ′ ( ζ )
ζ	Santra et al. [8]	Present results
0.0	0.0	0.0
0.6	0.60870994	0.60870992
1.2	0.89597727	0.89597727
1.8	0.98315816	0.98315813
2.4	0.99845935	0.99845934
3.0	0.99992397	0.99992392

Table 2

Verification of results for f ' ' ( 0 ) with specific values of λ when K = 0

λ	Sajid et al. [9]	Present
λ	f ″ ( 0 )	f ″ ( 0 )
0.01	0.009999	0.009997
0.02	0.019924	0.019925
0.05	0.049246	0.049244
0.10	0.096638	0.096636
0.20	0.186043	0.186044
0.50	0.414732	0.414732
1.00	0.687618	0.687618
2.00	0.979048	0.979047
10.0	1.275870	1.275869

Figure 2

Flowchart for methodology.

Figure 3

Residual error of obtained solutions.

5 Results and discussion

The physical impact for flow parameters on different profiles is graphically entertained. The physical illustration and applications of parameters are presented. Current investigation focuses on the rheology of viscoelastic parameter subject to shear thinning aspects. Figure 4 describes the assessment of f ′ due to slip parameter λ. A fall in velocity is disclosed owing to larger λ. Such outcomes are associated with the slippage of velocity due to lubricant surface. Same consequences are determined in Figure 5 for f . The behavior identified via Figure 6 reports that key sense of viscoelastic constant K on f ′ . The achieving observations convey that viscoelastic parameter causes an increase in velocity and a decrease in boundary thickness. The fluid has reduced viscosity when the parameter K is big, which results in a decreased fluid resistance and an increase in fluid velocity. The velocity f has produced the same findings (Figure 7). Non-identical solutions for the velocity components f and f ′ for various values for power law index n are shown in Figures 8 and 9. According to Figure 8, as m is increased, velocity rises and boundary thickness falls layer by layer. The velocity field of the shear thickening lubricants is greater than that of the Newtonian lubricants, whereas it is lower for the shear thinning lubricants. The parameter n has similar properties on the velocity f , which is shown in Figure 9. The influence of temperature and concentration profiles is shown in Figures 10–17. The temperature field for slip parameter is shown in Figure 10. We have noted that by increasing slip parameter, θ ( η ) decreases. Physically, we noted that by increasing the slip parameter, the fluid speed also increases but the temperature at wall decreases. The same variations can be observed for concentration as well as microorganism profiles ϕ ( η ) , which is shown in Figure 11. By increasing the slip parameters, the species of concentration as well as the density of motile organism also increases. Figures 12 and 13 highlight the significance of the second-grade parameter K on thermal profile and concentration fields. The effect of K on θ ( η ) for different values is described in Figure 12. We have noted the fall in temperature for larger value of K . This decrease in temperature profile for large K becomes more obvious if we lubricate the surface. The same observations are seen for concentration fields ϕ ( η ) in Figure 13 but decrement case is more reflective in concentration species. Figure 14 is sketched for the purpose of observing the effect of Prandtl Pr on the temperature distribution for the partial slip case. It is clearly noted from the figure that larger Pr highly inflicted the temperature in a declining manner. It is noted that thermal diffusivity behaves as inverse reflection of Pr . Figure 15 aims to provide the significance of Nt on θ . For greater values of Nt , an increment is recorded in the temperature and corresponding boundary layer thickness. For checking the behavior of Nb , and Le in contrast with concentration distribution ϕ ( η ) , Figures 16 and 17 are sketched. When λ = 1 , we have noted that concentration distribution usually decreases against Nb and Le , but increases for Le . We have observed that ϕ ( η ) decreases very prominently on the lubricated surface but Nt increases more obviously on the rough surface.

$Figure 4 Assessment of f ′ f^{\prime} due to λ.$

Figure 4

Assessment of f ′ due to λ.

Figure 5

Assessment of f due to λ.

$Figure 6 Assessment of f ′ f^{\prime} due to K K .$

Figure 6

Assessment of f ′ due to K .

Figure 7

Assessment of f with K .

$Figure 8 Assessment of f ′ f^{\prime} due to n n .$

Figure 8

Assessment of f ′ due to n .

Figure 9

Assessment of f due to n .

$Figure 10 Assessment of θ ( η ) \theta (\eta ) due to λ.$

Figure 10

Assessment of θ ( η ) due to λ.

$Figure 11 Assessment of ϕ ( η ) \phi (\eta ) due to λ.$

Figure 11

Assessment of ϕ ( η ) due to λ.

$Figure 12 Assessment of θ ( η ) \theta (\eta ) due to K K .$

Figure 12

Assessment of θ ( η ) due to K .

$Figure 13 Assessment of ϕ ( η ) \phi (\eta )\hspace{.25em} due to K K .$

Figure 13

Assessment of ϕ ( η ) due to K .

$Figure 14 Assessment of θ ( η ) \theta (\eta ) due to Pr \text{Pr} .$

Figure 14

Assessment of θ ( η ) due to Pr .

$Figure 15 Assessment of θ ( η ) \theta (\eta ) due to Nb \text{Nb} and Nt . \text{Nt}.$

Figure 15

Assessment of θ ( η ) due to Nb and Nt .

$Figure 16 Assessment of ϕ ( η ) \phi (\eta ) due to Nb \text{Nb} .$

Figure 16

Assessment of ϕ ( η ) due to Nb .

$Figure 17 Assessment of ϕ ( η ) \phi (\eta ) due to Le \text{Le} .$

Figure 17

Assessment of ϕ ( η ) due to Le .

6 Concluding remarks

Second-grade nanofluid flow over the lubricated surface is presented in a novel analysis. It has been investigated how important thermophoresis and Brownian motion are in slip impact. With the proper boundary constraints, partial differential equations are used to frame the issue. These equations are then similarly converted into ordinary differential equations (ODEs). The dimensionless ODE system was then numerically and analytically solved by hybrid homotopy analysis method. Key findings of this analysis are as follows:

The slip parameter reduces the free stream velocity profile.
When the slip parameter has high values, the thickness of the boundary layer is decreased and velocity varies and increases.
The temperature within the boundary layer fluctuates for large viscoelastic parameters.
A decreasing effect of power law index n on transport phenomenon has been predicted.
The flow of nanofluid over the lubricated disk is more ideal for improving heat transfer. Problems with thermodynamics and heat transfer benefit more from the current findings.

byeon@inje.ac.kr

Acknowledgments

The authors acknowledge Researchers Supporting Project Number (RSPD2023R1060), King Saud University, Riyadh, Saudi Arabia.

Funding information: This research received funding from King Saud University through Researchers Supporting Project Number (RSPD2023R1060), King Saud University, Riyadh, Saudi Arabia.
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 the data are available from the corresponding author and can be accessed via corresponding email after clearly stating the intention and permission to conduct research that requires our data.

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Received: 2023-07-30

Revised: 2023-10-27

Accepted: 2023-11-09

Published Online: 2023-12-06

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

Articles in the same Issue

https://doi.org/10.1515/phys-2023-0148

Keywords for this article

lubricant surface; slip conditions; homotopy method of analysis; Runge–Kutta method with shooting technique; second-level fluid; nanoparticles; axially symmetric flow

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