## 2 Mathematical Methods
Next, the similarity solution is applied to the mathematical model as shown:  
\[\psi=\sqrt{2v_{f}U_{0}L}e^{x/2L}f(\eta),\eta=\sqrt{\frac{U_{0}}{2v_{f}L}}e^{x/ 2L}y,\theta(\eta)=\frac{T-T_{\infty}}{T_{w}-T_{\infty}} \tag{9}\]  
where \(\psi\) as a stream function is defined as \(u=\frac{\partial\psi}{\partial y}\) and \(v=-\frac{\partial\psi}{\partial x}\). Thus, \(u\) and \(v\) are obtained as:  
\[ \begin{array}{l}u=\frac{\partial\psi}{\partial y}=U_{0}e^{x/L}f^{\prime}( \eta),\\ v=-\frac{\partial\psi}{\partial x}=-\sqrt{\frac{U_{0}v_{f}}{2L}}e^{x/2L}[\eta f ^{\prime}(\eta)+f(\eta)]\end{array}  \tag{10}\]  
where (\({}^{\prime}\)) is differentiation with respect to \(\eta\). It found that:  
\[v_{w}(x)=-\sqrt{\frac{U_{0}v_{f}}{2L}}e^{x/2L}S \tag{11}\]  
where \(v_{w}(x)\) is the mass flux surface, \(S\) is the mass flux parameter, \(U_{0}\) is constant velocity, \(v_{f}\) is kinematic viscosity, and \(L\) is the characteristic length surface. Additionally, it also considers the effective properties of hybrid nanofluid by performing a numerical analysis to identify the behavior of momentum and the thermal boundary layer on the permeable exponentially shrinking surface when nanoparticles are mixed with base fluid. Meanwhile, the effective properties of hybrid nanofluid, as described in the mathematical model, are shown in Table 1. As is well known, the effective thermal conductivity model, \(k_{eff}/k_{b}\), and effective electrical conductivity, \(\sigma_{eff}\)\(\sigma_{b}\), are based on the Hamilton and Crosser model, which is then modified to the Maxwell model when \(n=3/m=3\) and where \(n\) is an empirical shape factor. The Maxwell model is needed because this model considers ideally spherical particles. The Maxwell theory is not considered a space distribution effect of particles on thermal conductivity and electrical conductivity. Table 2 shows the properties of the nanoparticle, hybrid nanoparticle, and base fluid of different types. The values of fixed parameters in Table 2 were chosen based on the standard values used by many researchers [3; 10; 21]. Equation (9) is substituted into Equations (1), (2), (7), and (8) using similarity transformation, hence:  
\[\frac{1}{\left(1-\varphi_{1}\right)^{2.5}\left(1-\varphi_{2}\right)^{2.5}}f^{ \prime\prime\prime}\left(\eta\right)-\frac{\rho_{lnf}}{\rho_{f}}\Big{(}2f^{ \prime 2}(\eta)-f(\eta)f^{\prime\prime}\left(\eta\right)\Big{)}-\frac{\sigma_{ lnf}}{\sigma_{f}}Mf^{\prime}(\eta)+Gr\theta=0 \tag{12}\]  
\[\frac{1}{\mathrm{Pr}}\left(\frac{k_{lnf}}{k_{f}}+\frac{4}{3}Rd\right)\theta^{ \prime\prime}\left(\eta\right)-\frac{\left(\rho C_{p}\right)_{lnf}}{\left( \rho C_{p}\right)_{f}}\big{(}f^{\prime}(\eta)\theta(\eta)-f(\eta)\theta^{ \prime}(\eta)\big{)}+e\theta(\eta)=0 \tag{13}\]  
with a boundary condition as follows:  
\[ \begin{array}{l}f^{\prime}(\eta)=\lambda,f(\eta)=S,\theta(\eta)=1\;\text{ as}\;\eta=0\\ f^{\prime}(\eta)=0,\theta(\eta)=0\;\text{as}\;\eta\rightarrow\infty\end{array}  \tag{14}\]  
The governing equation can be defined as follows:  
\[ \begin{array}{l}\mathrm{Pr}=\frac{v_{f}}{\sigma_{f}},\quad M=\frac{2\sigma_ {f}B_{0}^{2}L}{\rho_{f}L_{0}},\quad\varepsilon=\frac{2Q_{0}L_{0}}{U_{0}\left( \rho C_{p}\right)_{f}}\;,\\ S=-\frac{v_{w}(x)}{\sqrt{\frac{U_{0}v_{f}}{2L}}}e^{-x/2L},\quad Rd=\frac{4 \sigma^{\prime}r_{0}^{3}}{k^{2}k_{f}}\;,\;Gr=g\frac{\left(T_{w}-T_{w}\right) }{a^{2}x}\end{array}