Nomenclature

λ,      Buoyancy effect due to the temperature difference;

λ^*,      Buoyancy effect due to the concentration difference;

k^*,      coefficient of the mean absorption;

β_C,      coefficient of the concentration expansion;

ρ,      density of the fluid;

σ,      electrical conductivity;

M,      Hartmann number;

B,      magnetic field strength;

T_m,      mean fluid temperature;

c_s,      ratio of the thermal diffusion;

Re,      Reynolds number;

r,      stagnation point parameter;

Sr,      Soret number;

k,      thermal conductivity;

A,      unsteadiness parameter;

σ^*,      Boltzmann constant;

k^T,      concentration susceptibility;

Γ,      coefficient of the Williamson fluid;

β_T,      coefficient of thermal expansion;

Du,      Dufour number;

g,      gravitational acceleration;

ν,      kinematic viscosity;

D_B,      mass diffusivity coefficient;

Pr,      Prandtl number;

R,      ratio of λ^* to λ;

c_p,      specific heat;

S,      suction parameter;

Sc,      Schmidt number;

Rd,      thermal radiation;

Λ,      Williamson fluid parameter.

1 Introduction

The empathetic nature of non-Newtonian flows and motion near a stagnation region of a solid body needs more investigations recently due to its vital applications in industry, engineering, and other sciences. The thermal oil recovery, high-speed flows, and thrust bearing are few scientific applications, where the role of the Said mechanism is very frequent. The stagnation area of the solid body could be a moving surface or a fixed area in the liquid. The pioneering effort in this domain is presented by Hiemenz^[1], where the two-dimensional (2D) Navier-Stokes equations were transformed into the nonlinear ordinary differential form, and the exact solutions were obtained. Later on, Homann and Angew^[2] reported the three-dimensional axisymmetric stagnation point flow, and obtained the similarity solution of the modeled problem. After these pioneering investigations, several researchers carried out this specific domain by means of different aspects and mechanisms. Makinde et al.^[3] analyzed the influence of buoyancy on the heat transfer and magnetohydrodynamics (MHD) stagnation point nanofluid flow past a convectively heated stretching/shrinking sheet. They showed that, an increase in the buoyancy force intensity resulted in an increase in the local Nusselt number while a decrease in the skin friction coefficient and the local Sherwood number. Haq et al.^[4] studied the influence of thermal radiation and slip on the MHD stagnation point nanofluid flow over a stretching sheet. They showed that, the velocity profile was enhanced when the ratio of the free stream velocity increased, and the temperature profile dropped when the thermal slip parameter increased. Hayat et al.^[5] numerically examined the radiation and melting heat transfer impacts on the stagnation point flow of the carbon-H₂O nanofluid. Tarakaramu and Narayana^[6] examined the Joule heating and nonlinear thermal radiation effects on the MHD stagnation point flow of a nanofluid over a convectively heated stretching surface. They showed that, when the temperature ratio parameter increased, the rate of heat transfer increased. For more details about such research, one may refer to Refs. [7]-[11] and the references therein.

Recently, nanotechnology and heat transport come to be hot topics of research due to their vigorous roles in scientific and practical applications, e.g., solar water heating, engine cooling, and drilling^[12]. Choi^[13] made a pioneering contribution to improve the heat transfer rate. Buongiorno^[14] and Tiwari and Das^[15] proposed mathematical models to study nanofluids. Hayat et al.^[16] studied the Joule heating and slip impacts in mixed convection peristaltic transport of nanofluids. Reddy and Chamkha^[17] analyzed the MHD convective flow of TiO₂-water and Al₂O₃-water nanofluids past a stretching sheet in porous media. For more details about nanofluids, one may refer to Refs. [18]-[21].

The nature of non-Newtonian fluids is complex as compared with that of Newtonian fluids, and many models have been proposed. Williamsonin presented a non-Newtonian nanofluid model in 1929, where the flow of pseudoplastic materials was analyzed experimentally^[22]. The flow model has been successfully used in fields such as the preparation of emulsions and the extrusion of polymer sheets. Amanulla et al.^[23] numerically explored the effects of the Biot number and the thermal radiation on the MHD Williamson fluid flow over a vertical convective surface. Jain and Parmar^[24] carried out a detailed study on the radiation effects on the MHD Williamson fluid flow over a stretching cylinder through a porous medium with heat source. Recently, Hamid et al.^[25] performed a numerical study on the unsteady MHD Williamson nanofluid flow in a permeable channel, and analyzed the effects of thermal radiation and the heat source/sink. Mustafa et al.^[26] examined the effects of buoyancy on the MHD nanofluid flow past a vertical surface with the activation energy and chemical reaction. They showed that the thermal boundary layer thickness decreased due to the concentration and temperature differences as well as the buoyancy, and the mixed convection parameter was directly proportional to the velocity of the fluid. For more details about natural convection, one may refer to Refs. [7] and [27]-[29].

The motivation of the current work is to analyze the effects of buoyancy assisting and opposing on the stagnation point flow of the Williamson nanofluid in the presence of thermal radiation. Appropriate similarity techniques transform the governing flow equations into a nonlinear system of ordinary differential equations. Numerical simulations are performed by means of the Chebyshev wavelet method. The significant effects of the emerging parameters are presented graphically. It is concluded that the modification in traditional Chebyshev wavelets brings more accuracy and it is a compatible technique to tackle nonlinear flow problems.

2 Mathematical and geometrical analysis

Consider the 2D unsteady viscous stagnation point flow of an incompressible Williamson nanofluid past a permeable stretched surface (see Fig. 1). The velocity is

where c and ϵ are the stretching coefficient and the unsteadiness of the problem, respectively,

and ϵ>0. The free stream velocity is

where a>0 is the stagnation flow strength. Further, we assume that the temperature and the concentration at the sheet are, respectively, as follows:

where b>0. The ambient temperature and concentration are defined as T_∞ and C_∞, respectively. Moreover, in the y-axis direction (normal to the sheet), a magnetic field having the strength B is applied. The effects of the Soret and Dufour numbers are considered. The above restrictions provide a modeled mathematical problem as follows:

(1)

(2)

(3)

(4)

(5)

where u and v are the velocity components along the x- and y-axes, respectively. σ is the electrical conductivity. ν is the kinematic viscosity. Γ is the coefficient of the Williamson fluid. β_T is the coefficient of thermal expansion. g is the gravitational acceleration. β_C is the coefficient of concentration expansion. B is the magnetic field strength. ρ is the density. k is the thermal conductivity. c_s is the ratio of thermal diffusion. σ^* is the Boltzmann constant. c_p is the specific heat. k^* is the coefficient of mean absorption. D_B is the mass diffusivity coefficient. k_T is the concentration susceptibility. T_m is the mean fluid temperature.

The following similarity transformations are used to nondimensionalize the nonlinear flow equations (1)-(5)^[10]:

(6)

The partial differential equation (PDE) system (2)-(5) takes the following form after incorporating the similarity variables in Eq. (6):

(7)

(8)

(9)

The boundary conditions are

(10)

In Eqs. (7)-(10), ± characterize the assisting flow and the opposing flow, respectively, and R>0. Λ is the Williamson fluid parameter. A is the unsteadiness parameter. M is the Hartmann number. λ represents the buoyancy effect due to the temperature difference. λ^* represents the buoyancy effect due to the concentration difference. R is the ratio of λ^* to λ. r is the stagnation point parameter. Rd is the thermal radiation. Du is the Dufour number. Pr is the Prandtl number. Sr is the Soret number. S>0 is the suction parameter. They are defined as follows:

The skin friction coefficient (C_f), the local Nusselt number (Nu), and the Sherwood number (Sh) are the physical quantities of interest, and are given as follows:

By means of Eq. (6), the nondimensional form of the skin friction coefficient C_f, the local Nusselt number Nu, and the Sherwood number Sh are

(11)

where Re=xu_e/ν is the Reynolds number.

3 Chebyshev polynomials and properties

The Chebyshev polynomials and their properties are discussed in this section briefly. The Chebyshev polynomials T_m of the mth-order must satisfy

The Chebyshev polynomials can be attained via the recurrence relation stated as follows:

The Chebyshev polynomials of the third kind are given as follows^{[8, 30-32]}:

where cos θ=η. The Chebyshev polynomials of the third kind are orthogonal in [-1, 1] w.r.t. the weight function i.e.,

The third kind of Cheyshev polyomials can be obtained by means of the following relation^{[8, 30-32]}:

(12)

The properties of the third kind Chebyshev polynomials are as follows^{[8, 30-32]}:

(ⅰ) C_m(1)=1, C_m (-1)=(-1)^m (2m+1).

(ⅱ)

3.1 Wavelets and Chebyshev wavelets

Many numerical algorithms have been proposed to study non-Newton fluids. The wavelet method is one of the most widely used one among all these methods. It has been successfully used in the engineering field, the physical field, and biological sciences^{[8, 30-32]}. The Chebyshev wavelets of the third kind are defined as follows^[30]:

(13)

where

C_q(η) is the qth-order Chebyshev polynomials of the third kind. A function g(η) from the -space defined in [0, 1) can be expanded in terms of the third kind Chebyshev wavelets as follows:

(14)

where

Equation (14) can be truncated as follows:

(15)

where and ψ(η) are 2^k-1 M×1 orders, which are given in Refs. [8] and [30]-[32].

3.2 Modified Chebyshev wavelet method (MCWM) and solution procedure

This section is devoted to a comprehensive study of the MCWM^[8] and its applications to solve the flow problem (7)-(10). The solution procedure for the MCWM is explained as follows:

Step 1   Consider the problem

(16)

(17)

(18)

Step 2   Introduce the following trial solutions suggested by the traditional Chebyshev wavelet method to solve Eqs. (16)-(18):

(19)

(20)

(21)

where and ψ(η) are presented in Ref. [8]. The trial solutions (19)-(21) can be rewritten as follows:

(22)

where χ(η) and γ_i (i=1, 2, 3) are presented in Ref. [8]. Now, we introduce a new set of unknowns {ζ₀ⁱ, ζ₁ⁱ, ζ₂ⁱ, …} for i=1, 2, 3 such that

(23)

(24)

The trial solutions given in Ref. [8] takes the following forms after incorporating Eqs. (23) and (24):

(25)

After incorporating the boundary conditions in Eq. (10), we can rewrite the trial solutions in Eq. (25) as follows:

(26)

(27)

(28)

Step 3   The residuals for F(η), θ(η), and ϕ(η) can be obtained after inserting the trial solutions (26)-(28) into Eqs. (16)-(18) as follows:

(29)

(30)

(31)

Step 4   In Step 3, the given problem is reduced to an equivalent number of algebraic equations having 3M-7 unknowns. Therefore, to find the unknowns ζ₀ⁱ, ζ₁ⁱ, ζ₂ⁱ, … (i=1, 2, 3), we need 3M-7 equations. Here, we use the collocation points for Eqs. (29)-(31) to generate the equations as and respectively.

Step 5   After solving the system of equations achieved in the last step, we can obtain the values of the unknowns ζ's. The values of ξ's can be obtained by using the relations (23) and (24). Finally, we can obtain the approximate solutions of the problem (7)-(10) after substituting the values of ξ's into Eqs. (19)-(21).

4 Results and discussion

Herein, the thermal radiation effects and the buoyancy assisting and opposing effects on the stagnation point flow of the Williamson nanofluid are briefly discussed. The flow is 2D unsteady, viscous, and incompressible over a porous stretching surface. The combined effects of the thermal radiation and the conduction or convection in the presence of buoyancy forces appear in various areas of science including spacecrafts, high temperature heat exchangers, combustion chambers, thermal insulations, industrial furnaces, etc. In account of these practical applications, a model study is conducted. A wavelet algorithm is adopted to analyze the problem theoretically. The emerging parameters, the physical quantities, and their effects on the velocity the temperature θ(η), and the concentration ϕ(η) of the fluid are plotted graphically and described on the basis of the already reported values of parameters^{[10, 23, 33-35]}. Various suitable ranges for the dimensionless velocity, the temperature, and the concentration are deliberated, and particular exertions are made on the effects of these parameters on the fluid flow and heat transport. The selection of the values for the aforementioned parameters is based on the physical characteristics of the proposed model, and is related to the stability of the solution. A tabular form of comparison with the existing literature is enclosed in the discussion part. Table 1 shows the credibility of our technique. The comparison analysis with the existing literature^{[10, 33-35]} is endorsing that wavelet algorithms are efficient and compatible tools to tackle nonlinear flow problems.

4.1 Behavior of the velocity profiles

The effects of various emerging parameters, including the Hartmann number M, the ratio parameter λ, the stagnation parameter r, the unsteadiness parameter A, the Williamson fluid parameter Λ, and the Suction parameter S, on the velocity profiles F'(η) are shown in Figs. 2-7 according to Eqs. (2)-(7).

Figure 2 is plotted to show the behavior of a magnetic parameter for the assisting and opposing flow regions. It is noted that, when M increases, F'(η) is enhanced for the opposing case while decreases for the assisting case. Figure 3 shows the effects of the ratio parameter λ. It can be clearly seen that, when λ increases the velocity increases for the assisting case while decreases for the opposing case. The variation is significant for higher values of the ratio parameter for both the assisting and opposing cases.

The variations in the stagnation point parameter r are presented in Fig. 4. It can be seen that an increase in the values of r results in an increase in the velocity profile F'(η) for both the assisting and opposing cases. It is perceived that the assisting effects are more significant than the opposing ones. As shown in Fig. 5, when the unsteadiness parameter A increases, the velocity of the fluid increases for the assisting case while decreases for the opposing case.

Figure 6 shows the effects of the Williamson parameter Λ on the velocity profiles. From the figure, we can see that when Λ increases, the velocity decreases for both the assisting and opposing cases. Moreover, Williamson parameter Λ has negligible effects on the velocity profile.

Figure 7 shows the effects of the suction parameter S on the velocity profiles. It can be seen that, when S increases, the velocity decreases for the opposing case while increases for the assisting case.

4.2 Behaviors of the temperature profiles

The buoyancy (assisting and opposing) effects with various physical parameters on the temperature profiles are shown in Figs. 8-11.

Figure 8 shows the effects of the unsteadiness parameter A on the temperature profiles. It can be seen that, when A increases, the temperature increases for both the assisting and opposing cases, while the effects of the opposing flow are a little more dominant than those of the assisting flow.

Figure 9 shows the effects of the radiation parameter Rd on the temperature profiles. It can be seen that, when Rd increases, the temperature increases for both the assisting and opposing cases, while the opposing effects are more significant than the assisting ones.

Figure 10 shows the effects of the Dufour parameter Du on the temperature profiles. It can be seen that, when Du increases, the temperature increases for both the assisting and opposing cases, while the opposing effects are more prevailing than those of the assisting ones.

Figure 11 shows the effects of the Prandtl number Pr on the temperature profiles. It can be seen that, when Pr increases, the temperature decreases for both the assisting and opposing cases, while the assisting effects are more significant than those of the opposing ones.

4.3 Behaviors of the concentration profiles

Figures 12 and 13 are plotted to describe the behaviors of the concentration profiles ϕ(η). From these figures, we can see that, when the Soret number Sr increases, the concentration ϕ(η) increases for both the assisting and opposing cases, and the opposing effects are a slightly dominant as compared with the assisting ones (see Fig. 12). However, when the Schmidt number Sc increases, the concentration ϕ(η) decreases for both the assisting and opposing cases, and the assisting effects are more significant than the opposing ones.

4.4 Behaviors of the physical quantities

Figures 14-18 show the effects of buoyancy and the emerging parameters on the physical quantities briefly.

Figure 14 shows the effects of Λ, M, and S on the skin friction coefficient. The behaviors of the skin friction coefficient for the buoyancy effects and various values of the unsteadiness parameter A, the stagnation parameter r, and the ration parameter Λ are also presented. From Fig. 14, we can see that, when A or S increases, the skin friction increases for the assisting case while decreases for the opposing case. Moreover, The effects of λ provide higher values of the skin friction coefficient for the assisting case while lower values for the opposing case. When the magnetic parameter M increases, the skin friction coefficient increases for both the assisting and opposing cases, while the assisting effects are more significant than the opposing ones.

Figure 15 is designed to show the effects of the Dufour number Du for the assisting and opposing cases. It can be seen that, when Du increases, the local Nusselt number Nu decreases for both the assisting and opposing cases. Here, it is important to mention that the opposing case provides a more significant decrease than the assisting case.

Figure 16 shows the effects of the Prandtl number Pr for the assisting and opposing cases. It can be seen that, when Pr increases, the local Nusselt number Nu increases for both the assisting and opposing cases, and the assisting effects are more dominant than the opposing ones.

Figures 17 and 18 show the effects of the Soret number Sr and the Prandtl number Pr for the assisting and opposing cases. It can be seen that, when Sr or Pr increases, the Sherwood number decreases, and the opposing effects are more dominant than the assisting ones.

5 Concluding remarks

A wavelet analysis is presented for the effects of buoyancy forces on the stagnation point flow of the Williamson nanofluid with thermal radiation. The set of flow equations are reduced to a nonlinear ordinary differential equation system with the appropriate similarity method, and are numerically tackled via a new wavelet algorithm based on Chebyshev polynomials. The key findings are as follows:

(ⅰ) When A, S, or λ increases, F'(η) increases for the assisting case while decreases for the opposing case. When the magnetic parameter increases, the velocity increases for the opposing case while decreases for the assisting case.

(ⅱ) When Du, Rd, or A increases, the temperature increases for both the assisting and opposing cases. When Pr increases, the temperature decreases. Moreover, the opposing effects are slightly more dominant than the assisting ones.

(ⅲ) The concentration decreases when Sr increases while increases when Sc increases. It is perceived that the assisting force causes more decrease than the opposing one.

(ⅳ) When r increases, the skin friction coefficient increases for both the assisting and opposing cases. When λ or A increases, the skin friction coefficient increases in the assisting region while decreases in the opposing region.

(ⅴ) When Pr increases, the local Nusselt number increases while the Sherwood number decreases. When Sr or Sc increases, the Sherwood and Nusselt numbers decrease.

(ⅵ) A worthy comparison of outcomes obtained via the MCWM is presented. It is evident that the MCWM is very accurate, and could be extended to other complex natural nonlinear flow problems.

Acknowledgements  The authors are grateful to the reviewers and editor for suggesting suitable changes in the original manuscript. The first author is also grateful to the China Scholarship Council (CSC) for the financial assistance.