**4. Zeroth Order Deformation**

The zeroth order deformation problem is defined as follows:

$$((1-p)L\_f[\tilde{f}(\eta;p) - f\_0(\eta)] = p\hbar\_f N\_f[\tilde{f}(\eta;p), \tilde{g}(\eta;p)] \tag{28}$$

$$(1 - p)L\_{\mathfrak{J}}[\tilde{\mathcal{g}}(\eta; p) - \mathcal{g}\_0(\eta)] = p\hbar\_{\mathfrak{J}}N\_{\mathfrak{J}}[\tilde{f}(\eta; p), \tilde{\mathcal{g}}(\eta; p)]\tag{29}$$

$$\mathcal{L}(1-p)L\_0[\tilde{\partial}(\eta;p) - \partial\_0(\eta)] = p\hbar\_0\mathcal{N}\_0[\tilde{f}(\eta;p), \tilde{g}(\eta;p), \tilde{\partial}(\eta;p), \tilde{\phi}(\eta;p)] \tag{30}$$

$$\tilde{\Phi}\_{\Phi}(1-p)L\_{\phi}[\tilde{\phi}(\eta;p)-\phi\_{0}(\eta)]=p\hbar\_{\phi}N\_{\phi}[\tilde{f}(\eta;p),\tilde{g}(\eta;p),\tilde{\theta}(\eta;p),\tilde{\phi}(\eta;p)]\tag{31}$$

$$\begin{array}{llll} \stackrel{\sim}{f}(0;p)=0 & \stackrel{\sim}{f}(0;p)=1+\gamma\_{1}\stackrel{\sim}{f}^{\prime\prime\prime}(0;p)+\gamma\_{2}\stackrel{\sim}{f}^{\prime\prime\prime}(0;p) & \stackrel{\sim}{f}(\infty;p)=0\\ \stackrel{\sim}{g}(0;p)=0 & \stackrel{\sim}{g}(0;p)=\emptyset+\gamma\_{3}\stackrel{\sim}{g}^{\prime\prime}(0;p)+\gamma\_{2}\stackrel{\sim}{g}^{\prime\prime\prime}(0;p) & \stackrel{\sim}{g}(\infty;p)=0\\ \stackrel{\sim}{f}(0;p)=1-\mathcal{S}\_{1} & \stackrel{\sim}{f}(\infty;p)=0 & \stackrel{\sim}{\phi}(0;p)=1-\mathcal{S}\_{2} & \stackrel{\sim}{\phi}(\infty;p)=0 \end{array} \tag{32}$$

$$\begin{split} N\_{f}[\tilde{f}(\eta;p),\tilde{g}(\eta;p)] &= \frac{\partial^{3}\tilde{f}(\eta;p)}{\partial\eta^{3}} - \left(\frac{\tilde{\vartheta}(\eta;p)}{\partial\eta}\right)^{2} + (\tilde{f}+\tilde{g})\frac{\partial^{2}\tilde{f}(\eta;p)}{\partial\eta^{2}} + \\\ \mathrm{We} &\frac{\partial^{2}\tilde{f}(\eta;p)}{\partial\eta^{2}}\frac{\partial^{3}\tilde{f}(\eta;p)}{\partial\eta^{3}} + \lambda\left(1+\beta\_{2}\tilde{\Theta}\right)\tilde{\Theta} + \lambda\lambda\mathrm{\mathcal{T}}(1+\beta\_{3}\tilde{\phi})\tilde{\phi} - \mathsf{M}\frac{\partial\tilde{f}(\eta;p)}{\partial\eta} \\\ N\_{g}[\tilde{f}(\eta;p),\tilde{g}(\eta;p)] &= \frac{\partial^{2}\tilde{g}(\eta;p)}{\partial\eta^{3}} - \left(\frac{\partial\tilde{g}(\eta;p)}{\partial\eta}\right)^{2} + (\tilde{f}+\tilde{g})\frac{\partial^{2}\tilde{g}(\eta;p)}{\partial\eta^{2}} + \\\ \mathrm{\mathcal{W}c}\frac{\partial^{2}\tilde{g}(\eta;p)}{\partial\eta^{2}}\frac{\partial^{3}\tilde{g}(\eta;p)}{\partial\eta^{3}} - \mathsf{M}\frac{\partial\tilde{g}(\eta;p)}{\partial\eta} \end{split} \tag{33}$$

*N*θ[ ∼ *f*(η; *p*), ∼ *g*(η; *p*), ∼ θ(η; *p*), ∼ φ(η; *p*)] = [(1 + ε ∼ θ) ∂2 ∼ θ(η;*p*) ∂η<sup>2</sup> + <sup>ε</sup> ∂ ∼ θ(η;*p*) ∂η <sup>2</sup> + Pr*Nb* ∂ ∼ θ(η;*p*) ∂η ∂ ∼ φ(η;*p*) ∂η + Pr*Nt* ∂ ∼ θ(η;*p*) ∂η <sup>2</sup> + Pr( ∼ *f* + <sup>∼</sup> *g*) ∂ ∼ θ(η;*p*) ∂η − δ*t*Pr ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ( ∼ *f* + <sup>∼</sup> *g*) <sup>2</sup> <sup>∂</sup><sup>2</sup> ∼ θ(η;*p*) ∂η<sup>2</sup> <sup>−</sup> <sup>2</sup> <sup>∂</sup> ∼ *f*(η;*p*) ∂η ∂ ∼ θ(η;*p*) ∂η ( ∼ *f* + <sup>∼</sup> *g*)+ (*c*) <sup>2</sup> <sup>−</sup> <sup>∂</sup><sup>2</sup> ∼ *f*(η;*p*) ∂η<sup>2</sup> ( ∼ *f* + <sup>∼</sup> *g*) (*S*<sup>1</sup> <sup>+</sup> <sup>θ</sup>)+(<sup>∼</sup> *f* + <sup>∼</sup> *g*) ∂ ∼ *f*(η;*p*) ∂η ∂ ∼ *g*(η;*p*) ∂η ∂ ∼ θ(η;*p*) ∂η ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (34) *N*φ[ ∼ *f*(η; *p*), ∼ *g*(η; *p*), ∼ θ(η; *p*), ∼ <sup>φ</sup>(η; *<sup>p</sup>*)] = <sup>∂</sup><sup>2</sup> ∼ φ(η;*p*) ∂η<sup>2</sup> <sup>+</sup> *Nt Nb* ∂2 ∼ θ(η;*p*) ∂η<sup>2</sup> − PrLe <sup>∂</sup> ∼ *f*(η;*p*) ∂η (*S*<sup>2</sup> + θ) + PrLe( ∼ *f* + <sup>∼</sup> *g*) ∂ ∼ φ(η;*p*) ∂η − PrLeδ*<sup>c</sup>* ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ( ∼ *f* + <sup>∼</sup> *g*) <sup>2</sup> <sup>∂</sup><sup>2</sup> ∼ φ(η;*p*) ∂η<sup>2</sup> <sup>−</sup> <sup>2</sup> <sup>∂</sup> ∼ *f*(η;*p*) ∂η ( ∼ *f* + <sup>∼</sup> *g*) ∂ ∼ φ(η;*p*) ∂η + ⎛ ⎜⎜⎜⎜⎝ ( ∂ ∼ *f*(η;*p*) ∂η ) 2 − ∂2 ∼ *f*(η;*p*) ∂η<sup>2</sup> ( ∼ *f* + <sup>∼</sup> *g*) ⎞ ⎟⎟⎟⎟⎠ (*S*<sup>2</sup> + φ)+ ( ∼ *f* + <sup>∼</sup> *g*) ∂ ∼ *f*(η;*p*) ∂η ∂ ∼ *g*(η;*p*) ∂η ∂ ∼ φ(η;*p*) ∂η ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (35)

Here *<sup>p</sup>* <sup>∈</sup> [0, 1] is embedding parameter and *<sup>f</sup>* , *g*, <sup>θ</sup> and <sup>φ</sup> are the non-zero auxiliary parameters.
