2. Mathematical Analysis of the problem
Consider two dimensional, steady, MHD boundary layer heat and mass
transfer characteristics of \(Ag/SWCNT\ \)– Water based Maxwell hybrid
nanoliquid over a vertical cone under convective boundary condition with
suction as demonstrated in Fig. 1. The \(Ag/SWCNT\ \)– Water based
Maxwell hybrid nanoliquid flow is chosen as the x-axis over the surface
of the vertical cone. An external magnetic field \(B_{0}\) is applied
along the y-axis. It is assumed that \(T_{w}\) to be determined, as the
result of convective heating process which is characterized by a
temperature \(T_{f}\) and heat transfer coefficient \(h_{f}\) and\(C_{w}\) is the nanoparticle volume fraction at the surface of the cone
and \(T_{\infty}\) and\(\text{\ C}_{\infty}\text{\ \ }\)are the
temperature and nanoparticle volume fraction of the ambient fluid,
respectively. Under the above considerations, the governing equations
describing the momentum, energy and concentration in the presence of
thermal radiation and chemical reaction as follows:
\(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0\)(1)
\(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+\lambda_{1}\left(u^{2}\frac{\partial^{2}u}{\partial x^{2}}+v^{2}\frac{\partial^{2}u}{\partial y^{2}}+2uv\frac{\partial^{2}u}{\partial x\partial y}\right)=\nu_{\text{hnf}}\frac{\partial^{2}u}{\partial y^{2}}+\text{g\ }\left[\beta\left(T-T_{\infty}\right)-\beta^{*}\left(C-C_{\infty}\right)\right]\text{Cosλ}-\frac{\text{σ\ }{B_{0}}^{2}}{\rho_{\text{hnf}}}u\)(2)
\(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+\lambda_{2}\left(u\frac{\partial u}{\partial x}\frac{\partial T}{\partial x}+v\frac{\partial v}{\partial y}\frac{\partial T}{\partial y}+u\frac{\partial v}{\partial x}\frac{\partial T}{\partial y}+v\frac{\partial u}{\partial y}\frac{\partial T}{\partial x}+2uv\frac{\partial^{2}T}{\partial x\partial y}+u^{2}\frac{\partial^{2}T}{\partial x^{2}}+v^{2}\frac{\partial^{2}T}{\partial y^{2}}\right)=\frac{k_{\text{hnf}}}{\left(\rho C_{p}\right)_{\text{hnf}}}\frac{\partial^{2}T}{\partial y^{2}}-\frac{1}{\left(\rho C_{p}\right)_{\text{nhf}}}\frac{\partial q_{r}}{\partial y}\)(3)
\(u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=D_{B}\frac{\partial^{2}C}{\partial y^{2}}-C_{1}(C-C_{\infty})\)(4)
The following physical boundary conditions are
\(u=0,\ \ \ \ v=v_{w},\ \ \ \ -k\frac{\partial T}{\partial y}=\ h_{f}\left(T_{f}-T\right),\ \ C=C_{w}\text{\ a}\)t\(\ y\ =\ 0\)(5)
\(u\rightarrow 0,\ \ \ \ \ \ \ \ \ \ \ \ T\rightarrow T_{\infty}\) ,\(C\rightarrow C_{\infty}\) at \(y\rightarrow\infty\) (6)
The subsequent similarity transformations are presented to streamline
the mathematical study of the problem
\(u=axf^{{}^{\prime}}\left(\eta\right),\ \ \ \ v=-\sqrt{a\upsilon_{f}}\text{\ f}\left(\eta\right),\ \ \ \ \eta=\ \sqrt{\frac{a}{\upsilon_{f}}}y\ ,\ \ \theta\left(\eta\right)=\ \frac{T-T_{\infty}}{T_{f}-T_{\infty}}\),\(S\left(\eta\right)=\frac{\phi\ -\ \phi_{\infty}}{\phi_{w}-\ \phi_{\infty}}\)(7)
By utilizing Rosseland estimation for radiation, the radiative heat flux\(q_{r}\) is demarcated as
\(q_{r}=-\frac{{4\sigma}^{*}}{3K^{*}}\frac{\partial T^{4}}{\partial y}\)(8)
After simplification using Taylor’s series expansion, we get
\(q_{r}=-\frac{{16\sigma}^{*}T^{3}}{3K^{*}}\frac{\partial T}{\partial y}\)(9)
The transformed equations are
\(f^{{}^{\prime\prime\prime}}+\ \frac{A_{2}}{A_{1}}\text{\ f\ }f^{{}^{\prime\prime}}-\frac{A_{2}}{A_{1}}\left(f^{{}^{\prime}}\right)^{2}-\alpha\ A_{1}[f^{2}f^{{}^{\prime\prime\prime}}-2ff^{{}^{\prime}}f^{{}^{\prime\prime}}]-A_{1}\text{M\ }f^{{}^{\prime}}+A_{1}\ [\theta-Nr\ S]Cos\lambda=0\)(10)
\(\left(1+A_{4}R\right)\theta^{{}^{\prime\prime}}-Pr\ \ \frac{A_{3}}{A_{4}}\text{f\ }\theta^{{}^{\prime}}-Pr\ \ \beta\ {A_{3}\ (f}^{2}\theta^{{}^{\prime\prime}}+ff^{{}^{\prime}}\theta^{\prime})=0\)(11)
\(S^{{}^{\prime\prime}}-Sc\ f\ S^{\prime}-C_{r}\ Sc\ S=0\) (12)
The associated converted boundary conditions are
\(\eta=0,\ \ \ \ \ \ f=V0,\ \ f^{{}^{\prime}}=1,\ \ \ \ \theta^{{}^{\prime}}\left(0\right)=-\text{Bi}\left(1-\theta\left(0\right)\right),\ \ \ S=1\)(13)
\(\eta\longrightarrow\infty,\ \ \ \ \ \text{f\ }^{\prime}=0\ ,\ \ \ \ \ \ \ \ \theta=0\ ,\ \ \ \ \ \ \ \ S=0\text{.\ \ \ }\)(14)
The associated non-dimensional parameters are defined as
\(\operatorname{Pr=}\frac{\upsilon_{f}}{\alpha_{f}}\),\(\beta=\lambda_{2}\ a\) ,\(M=\frac{\sigma B_{0}^{2}}{\rho a}\),\(\text{\ \ \ \ \ \ \ \ \ \ }\alpha=\lambda_{1}\ a\),
\(Sc=\ \frac{\upsilon_{f}}{D_{B}}\), \(C_{r}=\frac{C_{1}}{a}\),
R\(=\frac{16T_{\infty}^{3}\sigma^{*}}{3k^{*}k_{f}}\),\(V0=\frac{v_{w}}{\sqrt{a\upsilon_{f}}}\),\(Bi=\frac{h_{f}\ }{\text{k\ }}\left(\frac{\nu_{f}}{a}\right)^{1/2}.\)
\(A_{1}=\frac{1}{\left(1-\phi_{1}\right)^{2.5}\left(1-\phi_{2}\right)^{2.5}}\ \ ,\ \ A_{2}=\left(1-\phi_{2}\right)\left[\left(1-\phi_{1}\right)+\phi_{1}\left(\frac{\rho_{s1}}{\rho_{f}}\right)+\phi_{2}\left(\frac{\rho_{s2}}{\rho_{f}}\right)\right],\ \)
\(A_{3}=\left(1-\phi_{2}\right)\left[\left(1-\phi_{1}\right)+\phi_{1}\left(\frac{{(\rho c_{p})}_{s1}}{{(\rho c_{p})}_{f}}\right)+\phi_{2}\left(\frac{{(\rho c_{p})}_{s2}}{{(\rho c_{p})}_{f}}\right)\right],\ \ A_{4}=\frac{k_{\text{hnf}}}{k_{f}},\)
The density\(\text{\ \ ρ}_{\text{hnf}}\), thermal conductivity\(k_{\text{hnf}}\), dynamic viscosity \(\mu_{\text{hnf}}\), and heat
capacitance \({(\rho c_{p})}_{\text{hnf}}\) of the hybrid nanoliquid are
specified by:
\(\mu_{\text{hnf}}=\frac{\mu_{f}}{\left(1-\phi_{1}\right)^{2.5}\left(1-\phi_{2}\right)^{2.5}}\),\(\rho_{\text{hnf}}=\left(1-\varphi_{2}\right)\left[\left(1-\varphi_{1}\right)\rho_{f}+\varphi_{1}\rho_{s1}\right]+\varphi_{2}\rho_{s2}\),
\({{(\rho c}_{p})}_{\text{hnf}}=\left(1-\varphi_{2}\right)\left[\left(1-\varphi_{1}\right){(\rho c_{p})}_{f}+\varphi_{1}{(\rho c_{p})}_{s1}\right]+\varphi_{2}{(\rho c_{p})}_{s2}\),
\(k_{\text{hnf}}=k_{\text{nf}}*\left(\frac{k_{s2}+2k_{\text{nf}}-2\varphi_{2}\left(k_{\text{nf}}-k_{s2}\right)}{k_{s2}+2k_{\text{nf}}+2\varphi_{2}\left(k_{\text{nf}}-k_{s2}\right)}\right),\ \)where\(k_{\text{nf}}=k_{f}*\left(\frac{k_{s1}+2k_{f}-2\varphi_{1}\left(k_{f}-k_{s1}\right)}{k_{s1}+2k_{f}+2\varphi_{1}\left(k_{f}-k_{s1}\right)}\right).\)
The another object of this problem is to calculate skin – friction
coefficient \(\left(C_{f}\right)\), Nusselt number\(\left(\text{Nu}_{x}\right)\) and Sherwood number \((\text{Sh}_{x})\)and are given as
\(C_{\text{fx}}=\frac{\tau_{w}}{\rho U_{w}^{2}}\),\(Nu_{x}=\frac{xq_{w}}{k_{f}\left(T_{w}-T_{\infty}\right)}\) ,\(\text{\ S}h_{x}=\frac{xq_{m}}{D_{B}\left(C_{w}-C_{\infty}\right)}\)(15)
Where,
\(\tau_{w}=\mu_{\text{hnf}}(1+\alpha)\left(\frac{\partial u}{\partial y}\right)_{y=0}\),\(q_{w}=\left.\ -k_{\text{hnf}}\frac{\partial T}{\partial y}\right|_{y=0}\),\(q_{m}=\left.\ -D_{B}\frac{\partial C}{\partial y}\right|_{y=0}\), (16)