1 Introduction

Boundary layer flow via stretching surface has mesmerized many scientists and researchers due to its practical applications in polymer industries and engineering processes, e.g., cooling of continuous strips or filaments, wire drawing, glass blowing, glass fiber, wire drawing, artificial fibers, spinning of fibers, hot rolling, continuous stretching of plastic films, and paper production. Pioneering work on flow due to stretching sheet was done by Crane^[1]. Later on, various researchers explored different aspects of stretching sheet^[2-4]. In the literature, the flow by flat stretching sheet has been extensively discussed, while the flow due to curved stretching surface has seldom been studied. Sajid et al.^[5] explained the flow by curved stretching sheet. Naveed et al.^[6] elucidated the fluid motion due to curved stretching surface in presence of magnetohydrodynamics (MHD). Sajid et al.^[7] studied the motion of a micropolar fluid by curved stretching sheet. Rosca and Pop^[8] elucidated the flow due to curved shrinking/stretching surface. Naveed et al.^[9] presented the radiative flow of a micropolar fluid with curved stretching surface. Abbas et al.^[10] studied the radiative slip flow of a nanofluid by curved surface with heat generation. Okechi et al.^[11] studied the exponential stretching of curved surface. Some recent research regarding curved stretching surface can be seen in Refs. [12]-[15].

Non-Newtonian fluids are very useful in industrial and engineering sectors. However, they have much more complicated mathematical expressions as compared with viscous fluids. Many common substances exhibit non-Newtonian fluid flow, e.g., soap solutions, cosmetics, toothpaste, ketchup, jam, cheese, blood, saliva, semen, mucus, and synovial fluid. Non-Newtonian fluids are classified into three categories, i.e., differential, integral, and rate. The Jeffrey fluid model for the rate-type fluids is to show the characteristics of the retardation time and the relaxation time. Shahzad et al.^[16] described the Jeffrey nanofluid flow with nonlinear thermal radiation. Gao and Jian^[17] examined the Jeffrey fluid flow in a circular micropolar with the MHD effects. Some recent studies about non-Newtonian fluids can be found in Refs. [18]-[24].

Heat transfer with dynamic fluids is a very useful topic because of its engineering and industrial applications, e.g., nuclear reactor for cooling, medical applications, e.g., drug targeting magnetically and heat conduction in tissues, etc. The Fourier law of heat conduction^[25] is pioneer in studying the heat transfer mechanism. However, it has some drawback. The initial disturbance is sensed by the medium immediately. This is impossible in reality. Therefore, in the literature, this drawback is called "paradox of heat conduction". Cattaneo^[26] introduced the thermal relaxation time in the Fourier law for heat flux to resolve this enigma. Christov^[27] worked on Cattaneo's model, and replaced the ordinary derivatives with Oldroyld's upper convective derivatives. Tibullo and Zampoli^[28] studied the unique results for the non-Fourier heat flux model in incompressible fluids. Straughan^[29] described the non-Fourier heat flux model by considering thermal convection. Ciarletta and Straughan^[30] provided the structural stability and uniqueness for non-Fourier heat flux. Mustafa et al.^[31] explored the Maxwell fluid flow with non-Fourier heat flux. Hayat et al.^[32] investigated the effects of chemical reactions and non-Fourier heat flux on the Jeffrey fluid flow. Han et al.^[33] studied the viscoelastic fluid flow, and analyzed the heat transfer with the non-Fourier heat flux. Hayat et al.^[34] analyzed the heat transfer in a chemically reactive fluid with the non-Fourier heat flux theory. Some other researchers^[35-36] also analyzed heat transfer with non-Fourier heat flux model.

However, the motion of the Jeffrey fluid by curved stretching surface with the Cattaneo-Christov heat flux has not been analyzed yet. In this paper, similarity variables are used to form the system of equations, the homotopy analysis method (HAM)^[37-40] is used to get the convergent series solutions, and the behaviors of different embedded parameters on the physical quantities are analyzed.

2 Constitutive equations

The two-dimensional (2D) motion of the Jeffrey fluid due to the curved stretching sheet coiled in a circle with the radius R is considered. The sheet is stretched in the r-direction with the velocity u=U_w=as. The s-direction is perpendicular to the r-direction, and the origin is fixed (see Fig. 1). The heat transfer of the flow is characterized by the Cattaneo-Christov heat flux theory. The temperature of the sheet is kept constant as T_w. T_∞ is the uniform temperature of the ambient fluid, and T_w≥ T_∞.

With the above assumptions, we write the equations for velocity and temperature as follows:

(1)

(2)

(3)

(4)

where v and u are the velocities along the r- and s-directions, respectively, p is the pressure, V is the velocity vector, ρ is the density, c_p is the specific heat, q is the heat flux, τ_rr, τ_rs, and τ_ss are the shear stresses, and

(5)

(6)

(7)

(8)

In the above equations, μ is the dynamic viscosity, λ₁ is the ratio of the relaxation time to the retardation time, λ₂ is the retardation time, and q is the heat flux satisfying^{[27, 31-32]}

(9)

where k is the thermal conductivity, and λ₃ is the thermal relaxation time. Omitting q from Eqs. (4) and (9), we have

(10)

where

(11)

Substituting Eqs. (5)-(8) and (11) in Eqs. (2), (3), and (10), we have

(12)

(13)

(14)

After applying the boundary layer approximations in Eqs. (12)-(14), we have

(15)

(16)

(17)

(18)

with the boundary conditions

(19)

We use the following transformations:

(20)

where a is the stretching rate, ν is the kinematic viscosity, and P is the pressure constant. Then, we can rewrite Eqs. (16)-(19) as follows:

(21)

(22)

(23)

(24)

where K is the curvature parameter, β is the Deborah number, Pr is the Prandtl number, γ is the relaxation time, and

(25)

We omit the pressure constant P by using Eqs. (21) and (22) to make the problem simpler. We take the derivative of Eq. (22) with respect to ζ, and substitute it into Eq. (21). Then, we have

(26)

The pressure constant can be obtained from Eq. (22) as follows:

(27)

The skin friction coefficient for the present flow is

(28)

where

(29)

The skin friction coefficient in non-dimensionalize form is

(30)

where

3 Homotopic solutions 3.1 Zeroth-order deformation equations

The initial approximations, auxiliary linear operators, and auxiliary functions for the velocity and temperature profiles are taken as follows:

(31)

(32)

(33)

with

(34)

where A_i (i=1, 2, ..., 6) are constants.

Let the embedding parameter be q∈ [0, 1] and the non-zero auxiliary parameters be ħ_f and ħ_θ. Then, the zeroth-order deformation problems are

(35)

(36)

(37)

(38)

where

(39)

(40)

3.2 mth-order deformation equations

The mth-order deformation equations are

(41)

(42)

(43)

where

(44)

(45)

(46)

The solutions (f_m(ζ), θ_m(ζ)) comprising the special solutions (f_m^* (ζ), θ_m^*(ζ)) are

(47)

(48)

where

(49)

4 Convergence analysis

By setting the appropriate values of ħ_f and ħ_θ, we can control the convergence region in the HAM. To get the convergence regions of f''(0) and θ'(0), we sketch the curves at the 8th iteration (see Figs. 2 and 3). The ranges of the auxiliary parameters are noted as 1.5≤ ħ_f≤ -0.7, and -0.95≤ ħ_θ ≤-0.1.

The convergence of the obtained series solutions is displayed in Table 1. It depicts that the 11th- and 12th-orders of approximations are suitable for the convergence of momentum and temperature equations.

5 Results and discussion

In this section, the behaviors of the velocity, the temperature, the pressure, and the skin friction are noticed via the involved parameters.

5.1 Velocity distribution

Figure 4 portrays the behavior of the ratio of the relaxation time to the retardation time λ₁ on f'(ζ). It shows that, when λ₁ increases, the magnitude of the velocity profile f'(ζ) decreases. Physically, higher λ₁ corresponds to longer relaxation time. Therefore, particles need much time to reach the equilibrium system from the perturbed system.

Figure 5 depicts the effect of the Deborah number β on f'(ζ). When β increases, the magnitude of f'(ζ) increases, and the retardation time increases, which will further enhance the elasticity rate. Since elasticity and viscosity are opposite to each other, when the viscosity decreases, f'(ζ) increases.

Figure 6 depicts the behavior of K on f'(ζ). From the figure, we can see that f'(ζ) increases when K increases. It is due to the reason that the surface radius increases when K increases, which consequently makes the velocity increase.

5.2 Temperature distribution

Figures 7-11 are portrayed to depict the effects of the Prandtl number Pr, the ratio of the relaxation time to the retardation time λ₁, the Deborah number β, the curvature parameter K, and the thermal relaxation time γ on the temperature distribution θ (ζ).

Figure 7 presents the behavior of θ (ζ) for Pr. It can be seen that, when Pr increases, the fluid temperature and thermal boundary layer thickness decay. Since the thermal diffusivity decreases when Pr increases, the fluid temperature decreases.

Figure 8 examines the effects of the thermal relaxation time γ on θ (ζ). It can be seen that, when γ increases, θ (ζ) decreases. Due to the increase in the thermal relaxation time, the particles transfer heat slowly in comparison with the heat transferred by its neighboring particles, showing a non-conducting behavior.

Figure 9 presents the effects of θ (ζ) on β. It can be seen that, when β increases, the fluid temperature decreases. Figure 10 depicts the effects of λ₁ on θ (ζ). It can be seen that, the fluid temperature is an increasing function of λ₁. Figure 11 shows the behavior of the temperature profile θ (ζ) on the curvature parameter K. It can be seen that the temperature profile enhances for larger K.

5.3 Pressure distribution

Figures 12-14 show the effects of the ratio of the relaxation time to the retardation time λ₁, the Deborah number β, and the curvature parameter K on the pressure profile P(ζ). Figure 12 shows the behavior of the pressure profile for the Deborah number β. It can be seen that, the fluid pressure enhances when β increases. Figure 13 depicts the effects of λ₁ on P(ζ). It can be seen that, the fluid pressure decays when λ₁ increases. Figure 14 shows the effects of the curvature parameter K on P(ζ). It can be seen that the magnitude of P(ζ) decreases when the curvature parameter enhances.

5.4 Skin friction coefficient

Figures 15 and 16 depict the behaviors of the curvature parameter K and the Deborah number β on the skin friction coefficient against the ratio of the relaxation time to the retardation time λ₁. It can be seen that the magnitude of the surface drag force enhances when K and β increase.

6 Conclusions

The main conclusions are as follows:

(ⅰ) The ratio of the relaxation time to the retardation time has a decreasing effect on the fluid velocity.

(ⅱ) The temperature and velocity profiles are increasing functions of the curvature parameter.

(ⅲ) The fluid temperature decreases when the Prandtl number and the thermal relaxation time increase.

(ⅳ) The pressure profile decreases when the curvature parameter increases.

(ⅴ) The surface drag force shows a decreasing behavior when the curvature parameter increases.