We consider a steady two-dimensional laminar flow and heat transfer of viscous incompressible MHD Cu/Ag-Water nanofluids past a permeable wedge with temperature-dependent viscosity. The coordinate system is selected in such a way that the *x*-axis is aligned with the flow on the surface of the wedge and the *y*-axis is taken normal to it, as shown in Figure 1. The inclined angle of the wedge is taken as \(\Omega= \beta\pi\). The free stream velocity \(U_{e} ( x ) = U_{0} x^{m}\), where \(U_{0}\) is constant and *m* is a pressure gradient parameter related in the inclined angle *βπ* by \(m = \beta/(2-\beta)\). A variable magnetic field of strength \(B ( x ) = B_{0} x^{(m-1)/2}\) is applied along the *y*-direction, where \(B_{0}\) is constant. It is assumed that the temperature on the wedge surface is a constant \(T_{w}\) and the ambient temperature is \(T_{\infty}\). \(V_{w} \) is the velocity of suction (\(V_{w} <0\)) or injection (\(V_{w} >0\)). Further, the magnetic Reynolds number is assumed to be small so that the induced magnetic field can be neglected in comparison with the applied magnetic field. The base fluid water and the nanoparticles are also assumed to be in thermal equilibrium, and there is no slippage between them.

With the above assumptions, using Boussinesq and boundary layer approximations, the governing equations for the continuity, momentum, and energy can be expressed as follows:

$$\begin{aligned}& \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} =0, \end{aligned}$$

(1)

$$\begin{aligned}& u \frac{\partial u}{\partial x} +v \frac{\partial u}{\partial y} = U _{e} \frac{d U_{e}}{dx} + \frac{1}{\rho_{\mathit {nf}}} \frac{\partial}{\partial y} \biggl( \mu_{\mathit {nf}} \frac{\partial u}{\partial y} \biggr) + \frac{ \sigma B ( x ) ^{2}}{\rho_{\mathit {nf}}} ( U_{e} -u ) , \end{aligned}$$

(2)

$$\begin{aligned}& u \frac{\partial T}{\partial x} +v \frac{\partial T}{\partial y} = \alpha_{\mathit {nf}} \frac{\partial^{2} T}{\partial y^{2}} \end{aligned}$$

(3)

with the boundary conditions

$$\begin{aligned}& u=0, \quad\quad v= V_{w}, \quad\quad T= T_{w} \quad \text{at } y=0, \end{aligned}$$

(4)

$$\begin{aligned}& u= U_{e} ( x ) , \quad\quad T= T_{\infty}\quad \text{at } y\rightarrow \infty. \end{aligned}$$

(5)

Here \((u, v)\) are the velocity components along the *x* and *y* directions, respectively, *T* is the temperature, and *σ* is the electrical conductivity. The effective dynamic viscosity \(\mu_{\mathit {nf}}\), the effective density \(\rho_{\mathit {nf}}\), the thermal diffusivity \(\alpha _{n}\), and the heat capacity \((\rho C_{p} )_{\mathit {nf}}\) of the nanofluids are defined as in [29, 30]

$$\begin{aligned}& \mu_{\mathit {nf}} = \frac{\mu_{f}}{(1-\phi)^{2.5}}, \end{aligned}$$

(6)

$$\begin{aligned}& \rho_{\mathit {nf}} = ( 1-\phi ) \rho_{f} +\phi\rho_{s}, \end{aligned}$$

(7)

$$\begin{aligned}& (\rho C_{p} )_{\mathit {nf}} = ( 1-\phi ) ( \rho C_{p} ) _{f} +\phi(\rho C_{p} )_{s}, \end{aligned}$$

(8)

$$\begin{aligned}& \upsilon_{\mathit {nf}} = \frac{\mu_{\mathit {nf}}}{\rho_{\mathit {nf}}},\quad\quad \alpha_{\mathit {nf}} = \frac{k_{\mathit {nf}}}{( \rho C_{p} )_{\mathit {nf}}}, \end{aligned}$$

(9)

where *ϕ* is the solid volume fraction of nanoparticles. The thermal conductivity of nanofluids restricted to spherical nanoparticles is approximated by the Maxwell-Garnett (MG) model [31]:

$$ \frac{k_{\mathit {nf}}}{k_{f}} = \frac{(k_{s} +2 k_{f} )-2\phi( k_{f} - k_{s} )}{(k _{s} +2 k_{f} )+\phi( k_{f} - k_{s} )}, $$

(10)

in which the subscripts *nf*, *f*, and *s* represent the thermophysical properties of the nanofluid, base fluid, and nano solid particles, respectively.

Note that the viscosity of base fluid \(\mu_{f}\) is not constant, but vary as a function of temperature given by the following [24, 32]:

$$ \frac{1}{\mu_{f}} = \frac{1}{\mu_{\infty}} \bigl[ 1+\gamma ( T- T_{\infty} ) \bigr] =a ( T- T_{r} ) , $$

(11)

where \(a=\gamma/ \mu_{\infty}\) and \(T_{r} = T_{\infty} - \gamma ^{-1}\), \(\mu_{\infty}\), a constant, is the cold free stream viscosity, *a* and \(T_{r} \) are constants related to the reference state, and *γ* is a thermal property of the fluid. For nanofluids, \(a>0\). To solve Eqs. (1), (2), and (3) subjected to the boundary conditions (4) and (5), we introduce the stream function \(\psi(x,y)\) (\(u= {\partial \psi} / \partial y\), \(v=-\partial\psi/\partial x \)) and the similarity variables as

$$ \psi= \sqrt{\frac{2 \upsilon_{\infty} U_{\infty}}{m+1}} x^{\frac{m +1}{2}} f ( \eta ) , \quad\quad \eta=y \sqrt{ \frac{m+1}{2} \frac{U _{\infty}}{\upsilon_{\infty}}} x^{\frac{m-1}{2}},\quad\quad \theta= \frac{T- T_{\infty}}{T_{w} - T_{\infty}}. $$

(12)

Then Eqs. (1)-(3) are reduced to

$$\begin{aligned}& \begin{aligned}[b] & f^{\prime\prime\prime} + \frac{f^{\prime\prime} \theta'}{\theta_{r} ( 1- \frac{\theta}{ \theta_{r}} ) } + \frac{2mA}{m+1} \biggl( 1- \frac{\theta}{\theta _{r}} \biggr) \bigl( 1- f^{\prime\, 2} \bigr) +A \biggl( 1- \frac {\theta}{ \theta_{r}} \biggr) f f^{\prime\prime} \\ &\quad{} + \frac{2MA}{m+1} \biggl( 1- \frac{\theta}{\theta_{r}} \biggr) \bigl( 1- f ' \bigr) =0, \end{aligned} \end{aligned}$$

(13)

$$\begin{aligned}& \theta^{\prime\prime} +B P_{r} f \theta' =0, \end{aligned}$$

(14)

where

$$ {A}= ( 1-\phi ) ^{2.5} \biggl[ ( 1- \phi ) +\phi \frac{\rho_{s}}{\rho_{f}} \biggr] , \quad\quad B = \frac{ [ k_{s} +2 k_{f} +\phi ( k_{f} - k_{s} ) ] }{k_{s} +2 k _{f} -2\phi ( k_{f} - k_{s} ) } \biggl[ ( 1-\phi ) +\phi \frac{ ( \rho C_{p} ) _{s}}{ ( \rho C_{p} ) _{f}} \biggr] , $$

prime denotes differentiation with respect to *η*, \(\upsilon_{ \infty}\) is the cold free stream kinematic viscosity, the magnetic field parameter \(M= \sigma B_{0}^{2} / U_{\infty} \rho_{\mathit {nf}}\), the Prandtl number \(P_{r} = \upsilon_{\infty} (\rho C_{p} )_{f} / k_{f}\), and the variable viscosity parameter \(\theta_{r} =( T_{r} - T_{\infty } )/( T_{w} - T_{\infty} )\).

According to the definition of \(\theta_{r}\), we obtain \(\mu_{f} = \mu_{\infty} /[1-\theta(\eta)/ \theta_{r} ]\). Since the viscosity of liquids decreases with increasing temperature, \(\theta_{r}\) is negative for nanofluids. When \(\theta_{r} \rightarrow-\infty\), \(\mu_{f} \rightarrow\mu_{\infty}\), i.e., the viscosity variation in the boundary layer is negligible.

The boundary conditions (4) and (5) can be converted into

$$\begin{aligned}& f= f_{w}, \quad\quad f ' =0,\quad\quad \theta=1 \quad \text{at } \eta=0, \end{aligned}$$

(15)

$$\begin{aligned}& f ' =1, \quad\quad \theta=0 \quad \text{at } \eta\rightarrow\infty, \end{aligned}$$

(16)

where \(f_{w} =- V_{w} \sqrt{2x/[ ( m+1 ) \upsilon_{ \infty} U_{e} ]}\), \(f_{w} <0 \) for injection and \(f_{w} >0\) for suction, while \(f_{w} =0\) for impermeable wedge surface.

The quantities of physical interest are the skin friction coefficient \(C_{f}\) and the local Nusselt number \(\mathit {Nu}_{x}\), which are defined as [24]

$$\begin{aligned}& C_{f} = \frac{2 \tau_{w}}{\rho_{\infty} U_{e}^{2}}, \quad\quad \tau_{w} = \mu _{\mathit {nf}} \biggl( \frac{\partial u}{\partial y} \biggr) \bigg|_{y=0}, \end{aligned}$$

(17)

$$\begin{aligned}& \mathit {Nu}_{x} = \frac{x q_{w}}{k_{f} ( T_{w} - T_{\infty} )}, \quad\quad q_{w} = - k _{\mathit {nf}} \biggl( \frac{\partial T}{\partial y} \biggr)\bigg|_{y=0}, \end{aligned}$$

(18)

where \(\tau_{w}\) is the wall shear stress on the surface and \(q_{w}\) is the surface heat flux. Using the similarity transformation (12), we obtain

$$\begin{aligned}& \frac{C_{f} ( \mathit {Re}_{x} )^{1/2}}{2} = \frac{\sqrt{\frac{m+1}{2}}}{ ( 1-\phi ) ^{2.5} ( 1- \frac{\theta}{\theta_{r}} ) } f^{\prime\prime} ( 0 ) , \end{aligned}$$

(19)

$$\begin{aligned}& \mathit {Nu}_{x} ( \mathit {Re}_{x} )^{-1/2} =- \frac{k_{\mathit {nf}}}{k_{f}} \sqrt{ \frac{m+1}{2}} \theta' ( 0 ) , \end{aligned}$$

(20)

where \(\mathit {Re}_{x} = U_{e} x/ \upsilon_{\infty}\) is the local Reynolds number. So, \(f^{\prime\prime} ( 0 ) \) represents the skin friction coefficient \(C_{f} \) and \(- \theta' ( 0 ) \) represents the local Nusselt number \(\mathit {Nu}_{x}\).