Currently, we will explain the physical problem under consideration by assuming that the moving fluid is a Williamson fluid with a time constant *Γ*. Williamson fluid embody the Cauchy stress tensor which fully governed by the extra stress tensor [18]. In addition, the motion for the fluid is yielded from the nonlinearly stretching sheet accompanying with the presence of the radiation and viscous dissipation phenomena. Also, the *x*- axis is selected along the sheet, while the *y*-axis is chosen in the orthogonal direction for the sheet (Fig 1). Due to the stretching process, this may be results in creating the velocity *U*_{w}=*c**x*^{m} for the fluid, where *c* is a constant and *m* is an exponent. Herein, we suppose that both the fluid thermal conductivity *κ* and the fluid viscosity *μ* are altering with the temperature; however, the remaining fluid properties are constant.

After this former explanation and performing the approximations for the problem which we will study, we now have to present the mathematical equations which reflect what has been prescribed. These equations can be introduced in the following form [18]:

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

(1)

$$ u\frac{\partial u}{\partial x}+ v\frac{\partial u}{\partial y}=\frac{1}{\rho_{\infty}}\frac{\partial}{\partial y} \left(\mu(T)\frac{\partial u}{\partial y}+\mu(T)\frac{\Gamma}{\sqrt{2}}\left(\frac{\partial u}{\partial y}\right)^{2}\right), $$

(2)

$$ u\frac{\partial T}{\partial x}+ v\frac{\partial T}{\partial y}=\frac{1}{\rho_{\infty} c_{p}}\frac{\partial}{\partial y}\left(\kappa(T) \frac{\partial T}{\partial y} \right)+\frac{\mu(T)}{\rho_{\infty} c_{p}}\left(1+\frac{\Gamma}{\sqrt{2}}\frac{\partial u}{\partial y}\right)\left(\frac{\partial u}{\partial y}\right)^{2}-\frac{1}{\rho_{\infty} c_{p}}\frac{\partial q_{r}}{\partial y}, $$

(3)

we must refer that, the first equation represents the continuity equation while the second equation expresses the momentum, but the final equation reflects the energy equation. Always as usual, the characters *u* and *v* represent the components of the velocity vector in the *x* and *y* directions, respectively. Also, the constant *ρ*_{∞} is called the fluid density at the ambient, the symbol *T* refers to the temperature for the Williamson fluid, and the symbol *q*_{r} describes the heat flux term which is yielded due to the radiation, while the constant property *c*_{p} is the specific heat at constant pressure.As reported previously, the Rosseland approximation [23], helped us to put *q*_{r} as a function of temperature in the following form:

$$ q_{r}=-\frac{4 \sigma^{*}}{3 k^{*}}\frac{\partial T^{4}}{\partial y}, $$

(4)

where the constant *σ*^{∗} is the Stefan-Boltzmann and *k*^{∗} is the absorption coefficient. In this work, the coefficient *k*^{∗} must be coincide with the Rosseland approximation. As we observed from Eq. (4), the highly nonlinearity of the term *T*^{4} can be simplified by using Taylor expansion about the constant value *T*_{∞} as \(T^{4}\cong 4T_{\infty }^{3}T-3T_{\infty }^{4}\), after ignoring all higher-order terms [24].

In the same context, the nonlinear stretching process for the impermeable sheet and its temperature at the surface and away from it can be reflected in the following conditions:

$$ u=cx^{m},\quad v=0,\quad T_{w}(x)=T_{\infty}+Ax^{r} \quad at \quad y=0 $$

(5)

$$ u \rightarrow 0,\quad T \rightarrow T_{\infty}, \quad as\quad y\rightarrow \infty, $$

(6)

where *T*_{∞} is the constant ambient temperature, *A*,*r* are constants. Now, we will introduce the non-dimensional functions *f* and *θ* in which they are function of the variable *η* as follows:

$$ \eta=\left(\frac{cx^{m-1}}{\nu_{\infty}}\right)^{\frac{1}{2}}y, \quad \quad \psi(x,y)= \left(cx^{m+1}\nu_{\infty}\right)^{\frac{1}{2}}f(\eta),\quad \quad \theta(\eta)= \frac{T-T_{\infty}}{T_{w}-T_{\infty}}, $$

(7)

where *ν*_{∞} is the kinematic viscosity at the ambient and *ψ*(*x*,*y*) is the stream function which fulfill Eq. (1) according to:

$$ u=\frac{\partial \psi}{\partial y}, \quad \quad \quad v=-\frac{\partial \psi}{\partial x}. $$

(8)

On the other hand, some of the important assumptions in this research is that the viscosity is changing exponentially with the temperature, while the thermal conductivity is altering linearly with temperature according to these equations [25]:

$$ \mu=\mu_{\infty}e^{-\alpha\theta},\quad \quad \kappa=\kappa_{\infty}\left(1+\varepsilon\theta\right). $$

(9)

The final relations present the coefficient *μ*_{∞} and *κ*_{∞} which represent the viscosity and the thermal conductivity at the ambient, respectively. Also, from the same relations, the parameter *α* is the viscosity but the parameter *ε* is the thermal conductivity.

Employing Eq. (7) in the principal Eqs. (1)–(3), we trusted that Eq. (1) is exactly satisfied, while the other equations reduce to:

$$ e^{-\alpha\theta}\left(\left(1+ \delta f^{\prime\prime}\right)f^{\prime\prime\prime}-\alpha\theta'f^{\prime\prime}\left(1 +\frac{\delta}{2}f^{\prime\prime}\right) \right)+\left(\frac{m+1}{2}\right)ff^{\prime\prime}-mf'^{2}=0, $$

(10)

$$ \frac{1}{Pr}\left(\varepsilon\theta'^{2}+(1+R+\varepsilon\theta){\theta}^{\prime\prime}\right)+\left(\frac{m+1}{2}\right)f\theta'-rf'\theta+Ec \left(1 +\frac{\delta}{2}{f}^{\prime\prime}\right){f}^{\prime\prime{2}} e^{-\alpha\theta}=0, $$

(11)

and the reduced boundary conditions are

$$ f(0)=0,\quad\quad f'(0)=1,\quad\quad \theta(0)=1, \quad\quad $$

(12)

$$ f'\rightarrow 0, \quad\quad \theta\rightarrow0,\quad\quad \text{at}\quad\quad \eta\rightarrow \infty, $$

(13)

where \(\delta =\left (\frac {\sqrt {2}c^{\frac {3}{2}}x^{\frac {3m-1}{2}}}{\sqrt {\nu _{\infty }}}\right)\Gamma \) is the local Williamson fluid parameter, \(\text {Pr}=\frac {\mu _{\infty } c_{p}}{\kappa _{\infty }}\) is the Prandtl number, \(R=\frac {16\sigma ^{*}T_{\infty }^{3}}{3\kappa _{\infty }k^{*}}\) is the radiation parameter, and \(\text {Ec}=\frac {{U^{2}_{w}} }{c_{p}(T_{w}-T_{\infty })}=\frac {c^{2}x^{2m-r}}{\text {Ac}_{p}}\) is the local Eckert number. After finishing this analyzing, we observe that both the *δ* parameter and the Ec number are dependent of *x*. To overcome this dilemma which results in a non-similar solution for our problem, we should take \(r=2m=\frac {2}{3}\). So, these parameters take the form, \(\delta =\left (\frac {\sqrt {2}c^{\frac {3}{2}}}{\sqrt {\nu _{\infty }}}\right)\Gamma \) is the Williamson fluid parameter and \(\text {Ec}=\frac {{U^{2}_{w}} }{c_{p}(T_{w}-T_{\infty })}=\frac {c^{2}}{\text {Ac}_{p}}\) is the Eckert number.