The incompressible Casson type micropolar nanoliquid is considered for the analysis of energy and mass transportation over a stretching surface. Thermal and solutal stratification impacts are considered for this study. The stretching velocity is \({u}_{w}=ax\), which is taken along x-axis, while y-axis is taken normal to the x-axis. A magnetic field is taken along y-axis with strength \({B}_{0}\).

The flow equations in view of boundary layer approximation are developed for current problem, for reference see [23–25]

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

1

,

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\nu \left(1+\frac{1}{\beta }+K\right)\frac{{\partial }^{2}u}{\partial {y}^{2}}+\left(\frac{{K}_{1}^{*}}{\rho }\right)\frac{\partial {N}^{*}}{\partial y}+g\left[{\beta }_{t}\left(T-{T}_{\infty }\right)+{\beta }_{c}\left(C-{C}_{\infty }\right)\right]-\left(\frac{\sigma {{B}_{0}}^{2}}{\rho }\right)u$$

2

,

$$u\frac{\partial {N}^{*}}{\partial x}+v\frac{\partial {N}^{*}}{\partial y}=\left(\frac{{\gamma }^{*}}{{j}^{*}\rho }\right)\frac{{\partial }^{2}{N}^{*}}{\partial {y}^{2}}-\left(\frac{{{K}_{1}}^{*}}{{j}^{*}\rho }\right)\left(2{N}^{*}+\frac{\partial u}{\partial y}\right)$$

3

,

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}= \alpha \frac{{\partial }^{2}T}{\partial {y}^{2}}+\tau \left[{D}_{B}\frac{\partial C}{\partial y}\frac{\partial T}{\partial y}+\frac{{D}_{T}}{{T}_{\infty }}{\left(\frac{\partial T}{\partial y}\right)}^{2}\right]$$

4

,

$$u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}={D}_{B}\frac{{\partial }^{2}C}{\partial {y}^{2}}+\frac{{D}_{T}}{{T}_{\infty }}\frac{{\partial }^{2}T}{\partial {y}^{2}}$$

5

.

Where \(u\)and \(v\), are taken along\(x\)and \(y\) directions, thermal diffusivity parameter is denoted by \(\alpha\) and the ratio between the effective heat capacity of the nanoparticle and heat capacity of the liquid is represented by \(\tau =\frac{{\left(\rho c\right)}_{p}}{{\left(\rho c\right)}_{f}}\).

The boundary settings for above flow models are\(u={u}_{w}\left(x\right)=ax, v=0 , T={T}_{w} , { N}^{*}={-m}_{0}\frac{\partial u}{\partial y} , C={C}_{w} at y=0,\)

$$u\to {u}_{\infty }=0, v\to 0 , T\to {T}_{\infty } , {N}^{*}\to 0, C\to {C}_{\infty } at y\to \infty .$$

6

Where\({T}_{\infty }\left(x\right)={T}_{\infty ,0}+{A}_{1}{x}^{2} , {C}_{\infty }\left(x\right)={C}_{\infty ,0}+{B}_{1}{x}^{2},\)

The set temperature and concentration on the stretchable surface are understood to be of the form\({T}_{w}\left(x\right)={T}_{\infty ,0}+{ M}_{1}{x}^{2} , {C}_{w}\left(x\right)={C}_{\infty ,0}+{N}_{1}{x}^{2} ,\)

The velocity components in view of stream function are

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

7

Similarity variables for this flow problem are presented as

$$u=ax{f}^{{\prime }}\left(\eta \right), v=-\sqrt{av}f\left(\eta \right), \eta =y\sqrt{\frac{a}{v}}, {N}^{*}=ax\left(\sqrt{\frac{a}{v}}\right)h\left(\eta \right),$$

,

$$\theta \left(\eta \right)= \frac{T-{T}_{\infty ,0}}{\varDelta T}-\frac{{A}_{1}{x}^{2}}{\varDelta T}, \varphi \left(\eta \right)= \frac{C-{C}_{\infty ,0}}{\varDelta C}-\frac{{B}_{1}{x}^{2}}{\varDelta C},$$

8

Where,

$$\varDelta T={T}_{w}\left(x\right)-{T}_{\infty ,0}={N}_{1}{x}^{2} , \varDelta C={C}_{w}\left(x\right)-{C}_{\infty ,0}={M}_{1}{x}^{2},$$

With the implementation of Eq. (8), equations 2 to 5 are

$${(1+\frac{1}{\beta }+K)f}^{{\prime }{\prime }{\prime }}+f{f}^{{\prime }{\prime }}-{{f}^{{\prime }}}^{2}+K{h}^{{\prime }}+\lambda \left(\theta +N\varphi \right)-Mf{\prime }=0$$

9

,

\(\left(1+\frac{K}{2}\right){h}^{{\prime }{\prime }}+f{h}^{{\prime }}-{f}^{{\prime }}h-K\left(2h+{f}^{{\prime }{\prime }}\right)=0\) , (10) \({\theta }^{{\prime }{\prime }}+\text{Pr}\left(f{\theta }^{{\prime }}-{f}^{{\prime }}\theta +2{f}^{{\prime }}{\epsilon }_{1}\right)+PrNb{\varphi }^{{\prime }}{\theta }^{{\prime }}+PrNt{{\theta }^{{\prime }}}^{2}=0,\) (11)

$${\varphi }^{{\prime }{\prime }}+Le(f{\varphi }^{{\prime }}-{f}^{{\prime }}\varphi +2{f}^{{\prime }}{\epsilon }_{2})+{Nt}_{b}{\theta }^{{\prime }{\prime }}=0$$

12

,

Where,

\(M=\frac{\sigma {{B}_{0}}^{2}}{a\rho }\) , \({N}_{b}=\frac{\tau {D}_{B}\left({C}_{w}-{C}_{\infty }\right)}{\nu }\), \({N}_{t}=\frac{\tau {D}_{t}\left({T}_{w}-{T}_{\infty }\right)}{\nu {T}_{\infty }}\), \(K=\frac{{k}_{1}^{*}}{\mu }\), \({\epsilon }_{1}=\frac{x}{\varDelta T}\frac{d\left({T}_{\infty }\left(x\right)\right)}{dx}\), \(N=\frac{{\beta }_{C}{\Delta }C}{{\beta }_{T}{\Delta }T}\) ,

\({Re}_{x}=\frac{{u}_{w}x}{\nu }\) , \(\lambda =\frac{{Gr}_{x}}{{{Re}_{x}}^{2}}\), \({Nt}_{b}=\frac{{N}_{t}}{{N}_{b}}\), \({\epsilon }_{2}=\frac{x}{\varDelta C}\frac{d\left({C}_{\infty }\left(x\right)\right)}{dx}\), \({Gr}_{x}=\frac{g{\beta }_{T}{\Delta }T{x}^{3}}{{\nu }^{2}}\). (13)

In above factors

\({\epsilon }_{1}\) presents the thermal stratification factor,

\({\epsilon }_{2}\) demonstrates the solutal stratification factor,

\(N\) denotes the buoyancy ratio factor,

\(\lambda\) stands for mixed convection ratio.

The transformed boundary conditions are

$$f\left(\eta \right)=0, { f}^{{\prime }}\left(\eta \right)=1, h\left(\eta \right)=0, \theta \left(\eta \right)=1-{\epsilon }_{1}, \varphi \left(\eta \right)=1-{\epsilon }_{2}, at \eta =0$$

,

$${f}^{{\prime }}\left(\eta \right)\to 0, h\left(\eta \right)\to 0, \theta \left(\eta \right)\to 0, \varphi \left(\eta \right)\to 0 as \eta \to \infty$$

16

.

The involved physical quantities are defined by

\({Nu}_{x}=\frac{x{q}_{w}}{k\left({T}_{w}-{T}_{\infty }\right)}\) , denotes Nusselt number,

\({C}_{f}=\frac{{t}_{w}}{{{{u}_{w}}^{2}\rho }_{f}}\) , presents skin friction,

\({Sh}_{x}=\frac{x{q}_{m}}{{D}_{B}\left({C}_{w}-{C}_{\infty }\right)}\) , stands for Sherwood number,

The associated terms for \({C}_{fx}\left(0\right)={f}^{{\prime }{\prime }}\left(0\right),\) \(-\theta {\prime }\left(0\right)\), and \(-\varphi {\prime }\left(0\right)\) are defined as

$$-\theta {\prime }\left(0\right)=\frac{{Nu}_{x}}{\sqrt{{Re}_{x}}}, {C}_{fx}={C}_{f}\sqrt{{Re}_{x}}, -\varphi {\prime }\left(0\right)=\frac{{Sh}_{x}}{\sqrt{{Re}_{x}}}.$$

17