An unsteady laminar boundary layer flow of an incompressible viscous Maxwell nanofluid over a shrinking surface in two dimensions is considered here. The shrinking sheet velocity is *u*_{w}(*x*,*t*) while the mass transfer velocity is *v*_{w}(*x*,*t*), *t* denotes time and *x* is measured along the sheet. Using the nanofluid model proposed by Buongiorno [22], the conservation equations for mass, momentum, thermal energy and nanoparticles for a Maxwell fluid are:
$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}u}{\mathrm{\partial}x}+\frac{\mathrm{\partial}v}{\mathrm{\partial}y}=0,}\end{array}$$(1)
$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}u}{\mathrm{\partial}t}+u\frac{\mathrm{\partial}u}{\mathrm{\partial}x}+v\frac{\mathrm{\partial}u}{\mathrm{\partial}y}=\nu \frac{{\mathrm{\partial}}^{2}u}{\mathrm{\partial}{y}^{2}}-{k}_{0}\left({u}^{2}\frac{{\mathrm{\partial}}^{2}u}{\mathrm{\partial}{x}^{2}}\right.}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\left.+2uv\frac{{\mathrm{\partial}}^{2}u}{\mathrm{\partial}x\mathrm{\partial}y}+{v}^{2}\frac{{\mathrm{\partial}}^{2}u}{\mathrm{\partial}{y}^{2}}\right)}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}-\frac{\sigma {B}^{2}}{{\rho}_{f}}\left(u-kv\frac{\mathrm{\partial}u}{\mathrm{\partial}y}\right),}\end{array}$$(2)
$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}T}{\mathrm{\partial}t}+v\frac{\mathrm{\partial}T}{\mathrm{\partial}y}+u\frac{\mathrm{\partial}T}{\mathrm{\partial}x}=\alpha \frac{{\mathrm{\partial}}^{2}T}{\mathrm{\partial}{y}^{2}}}\\ {\displaystyle +\frac{1}{{\rho}_{f}{c}_{p}}\frac{\mathrm{\partial}}{\mathrm{\partial}y}\left[\kappa (T)\frac{\mathrm{\partial}T}{\mathrm{\partial}y}\right]}\\ {\displaystyle +\frac{\mu}{{\rho}_{f}{c}_{p}}{\left(\frac{\mathrm{\partial}u}{\mathrm{\partial}y}\right)}^{2}+\frac{\sigma}{{\rho}_{f}{c}_{p}}{\left(u{B}_{0}-{E}_{0}\right)}^{2}+\frac{{Q}_{0}}{{\rho}_{f}{c}_{p}}(T-{T}_{\mathrm{\infty}})}\\ {\displaystyle +\left[\tau {D}_{B}\frac{\mathrm{\partial}C}{\mathrm{\partial}y}\frac{\mathrm{\partial}T}{\mathrm{\partial}y}+\frac{\tau {D}_{T}}{{T}_{\mathrm{\infty}}}{\left(\frac{\mathrm{\partial}T}{\mathrm{\partial}y}\right)}^{2}\right]+\frac{{D}_{m}{k}_{0}}{{c}_{s}{c}_{p}}\frac{{\mathrm{\partial}}^{2}C}{\mathrm{\partial}{y}^{2}}}\\ {\displaystyle -\frac{1}{{\rho}_{f}{c}_{p}}\frac{\mathrm{\partial}{q}_{r}}{\mathrm{\partial}y},}\end{array}$$(3)
$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}C}{\mathrm{\partial}t}+v\frac{\mathrm{\partial}C}{\mathrm{\partial}y}+u\frac{\mathrm{\partial}C}{\mathrm{\partial}x}={D}_{B}\frac{{\mathrm{\partial}}^{2}C}{\mathrm{\partial}{y}^{2}}+\frac{{D}_{T}}{{T}_{\mathrm{\infty}}}\frac{{\mathrm{\partial}}^{2}T}{\mathrm{\partial}{y}^{2}}}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\displaystyle +\frac{{D}_{m}{k}_{0}}{{T}_{m}}\frac{{\mathrm{\partial}}^{2}T}{\mathrm{\partial}{y}^{2}}-{k}_{1}(C-{C}_{\mathrm{\infty}}),}\end{array}$$(4)

where *u* is the velocity component in the *x*- direction and *v* is the velocity component in the *y*-direction. The kinematic viscosity is *ν*, the relaxation time of the UCM fluid is *k*_{0}, the thermal diffusivity is *α*, the variable thermal conductivity is *κ*(*T*) and the chemical reaction parameter is *k*_{1}. The Brownian diffusion coefficient is *D*_{B}, while the thermophoresis diffusion coefficients is *D*_{T}. Here, *τ* is the effective heat capacity of the nanoparticle material divided by the heat capacity of the ordinary fluid, *T* is the fluid temperature and *C* is the nanoparticle volume fraction. The temperature and the nanoparticle concentration at the wall are *T*_{w} and *C*_{w}, respectively, and *T*_{∞} and *C*_{∞} denote the ambient temperature and concentration respectively. The velocity slip is proportional to the local shear stress. These equations are subject to the boundary conditions:
$$\begin{array}{}{\displaystyle u={u}_{w}(x,t)+{u}_{\text{slip}}(x,t),\phantom{\rule{thinmathspace}{0ex}}v={v}_{w}(x,t),}\\ {\displaystyle -{k}_{0}\sqrt{1-\lambda t}\frac{\mathrm{\partial}T}{\mathrm{\partial}y}={h}_{f}({T}_{w}-T),}\\ {\displaystyle {D}_{B}\frac{\mathrm{\partial}C}{\mathrm{\partial}y}+\frac{{D}_{B}}{{T}_{\mathrm{\infty}}}\frac{\mathrm{\partial}T}{\mathrm{\partial}y}=0\phantom{\rule{thinmathspace}{0ex}}\text{at}\phantom{\rule{thinmathspace}{0ex}}y=0,}\\ {\displaystyle u\to 0,T\to {T}_{\mathrm{\infty}},\frac{\mathrm{\partial}u}{\mathrm{\partial}y}\to 0,C\to {C}_{\mathrm{\infty}}\text{as}y\to \mathrm{\infty},}\end{array}$$(5)

where *k*_{0} = *k*(1−*λ t*), *u*_{w}(*x*,*t*) = −*ax*(1−*λ t*)^{−1}, *u*_{slip}(*x*,*t*) = *ν* *N*_{1}*∂ u*/*∂ y*,
$\begin{array}{}{N}_{1}=N\sqrt{1-\lambda t}\end{array}$ is the slip velocity, *λ* is the unsteadiness parameter and *k*_{0}(> 0), *a*(> 0) are positive constants. This assumption is to allow for the possibility of a similarity solution. We introduce the similarity variables:
$$\begin{array}{}{\displaystyle \eta =y\sqrt{\frac{a}{\nu (1-\lambda t)}},\phantom{\rule{1em}{0ex}}\psi =\sqrt{\frac{a\nu}{1-\lambda t}}xf(\eta ),}\\ T(x,t)={T}_{\mathrm{\infty}}+\theta (\eta )({T}_{w}-{T}_{\mathrm{\infty}}),\\ C(x,t)={C}_{\mathrm{\infty}}+\varphi (\eta )({C}_{w}-{C}_{\mathrm{\infty}}),\end{array}$$(6)

where *ψ* is the stream function, defined by *v* = −*∂ψ*/*∂x* and *u* = *∂ψ*/*∂y*. Equations (1) - (4) are transformed to
$$\begin{array}{}{\displaystyle {f}^{\u2034}-{f}^{\prime 2}+f{f}^{\u2033}-A\left(\frac{\eta}{2}{f}^{\u2033}+{f}^{\prime}\right)}\\ -\beta \left({f}^{2}{f}^{\u2034}-2f{f}^{\prime}{f}^{\u2033}\right)-{M}^{2}{f}^{\prime}-{M}^{2}\beta f{f}^{\u2033}=0,\end{array}$$(7)
$$\begin{array}{}{\displaystyle \frac{1}{P{r}_{eff}}{\theta}^{\u2033}+\frac{1}{Pr}\left[{\theta}^{\u2033}\left(1+\epsilon \theta \right)+\epsilon {\theta}^{\prime 2}\right]+{E}_{c}{f}^{\u20332}}\\ {\displaystyle +{E}_{c}{M}^{2}{\left({f}^{\prime}-{E}_{1}\right)}^{2}+\left(f-A\frac{\eta}{2}\right){\theta}^{\prime}}\\ +{D}_{f}{\varphi}^{\u2033}+He\theta +Nt{{\theta}^{\prime}}^{2}+Nb{\theta}^{\prime}{\varphi}^{\prime}=0,\end{array}$$(8)
$$\begin{array}{}{\displaystyle {\varphi}^{\u2033}+{S}_{c}\left(f-A\frac{\eta}{2}\right){\varphi}^{\prime}+\frac{Nt}{Nb}\phantom{\rule{thinmathspace}{0ex}}{\theta}^{\u2033}-\gamma {S}_{c}\varphi +\frac{Nt}{Nb}\phantom{\rule{thinmathspace}{0ex}}{\theta}^{\u2033}}\\ +{S}_{c}{S}_{r}{\theta}^{\u2033}=0,\end{array}$$(9)

where the prime denotes derivatives with respect to *η*. The boundary conditions are
$$\begin{array}{}{f}^{\prime}(0)=-1+\delta {f}^{\u2033}(0),f(0)=s,{f}^{\u2033}(\mathrm{\infty})=0,{f}^{\prime}(\mathrm{\infty})=0,\\ {\theta}^{\prime}(0)=-{B}_{i}(-\theta (0)+1),\theta (\mathrm{\infty})=0,\\ Nb{\varphi}^{\prime}(0)+Nt{\theta}^{\prime}(0)=0,\varphi (\mathrm{\infty})=0,\end{array}$$(10)

where the parameter
$\begin{array}{}\delta =N\sqrt{a\nu}\end{array}$ is the non-dimensional velocity slip and *v*_{w}(0,*t*) is the wall mass transfer velocity given by
$$\begin{array}{}{\displaystyle {v}_{w}(0,y)=\frac{{v}_{0}}{\sqrt{1-\lambda t}},}\end{array}$$

where *v*_{0} is the constant mass flux velocity. Thus
$$\begin{array}{}{\displaystyle s=-\frac{{v}_{0}}{a\nu}=f(0),}\end{array}$$

where wall mass suction occurs when *s* > 0 while wall mass injection occurs when *s* < 0. The non-dimensional parameters in the above equations are the Maxwell parameter *β*( = *k*_{0}*a*), the unsteadiness parameter *A* ( = *λ*/*a*), the magnetic field parameter
$\begin{array}{}{\displaystyle M\left(=\sqrt{\sigma {B}_{0}^{2}/{\rho}_{f}a}\right),}\end{array}$ *He*( = *Q*_{0}/*ρ*_{f} c_{p}) the heat generation parameter and *Q*( = *Q*_{0}(1−*λ t*)), *Nb* ( = *τ D*_{B}(*C*_{w}−*C*_{∞})/*ν*) is the Brownian motion parameter and *Nt*( = *τ D*_{T} (*T*_{w} − *T*_{∞})/*ν T*_{∞}) is the thermophoresis parameter. The Prandtl number is *Pr*( = *ν*/*α*), the Schmidt number is
$\begin{array}{}Sc\left(=\nu /{D}_{B}\right),P{r}_{eff}\left(=Pr/(1+\frac{4}{3}R)\right)\end{array}$ is the effective Prandtl number,
$\begin{array}{}{B}_{i}\left(={h}_{f}/{k}_{0}\sqrt{\frac{\nu}{a}}\right)\end{array}$ is the Biot number, *γ* = *k*_{1}(*C*_{w}−*C*_{∞}) is the reaction parameter where *γ* < 0 denotes a destructive reaction, *γ* = 0 indicates that there is no reaction and *γ* > 0 denotes a generative reaction.

Other important physical parameters are the variable thermal conductivity *κ*(*T*), the Eckert number *E*_{c}, the local electromagnetic parameter *E*_{1}, the Soret number *S*_{r}, the Dufour number *D*_{f} and the radiation parameter *R*. These are defined as
$$\begin{array}{}{\displaystyle \kappa (T)={K}_{\mathrm{\infty}}\left(1+\epsilon \frac{T-{T}_{\mathrm{\infty}}}{\mathrm{\Delta}T}\right),{E}_{c}=\left(\frac{{x}^{2}{a}^{2}}{{c}_{p}\mathrm{\Delta}T(1-\lambda t)}\right),}\end{array}$$(11)
$$\begin{array}{}{\displaystyle {E}_{1}=\left(\frac{{E}_{0}(1-\lambda t)}{a{B}_{o}x}\right),}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}{S}_{r}=\left(\frac{{D}_{m}{k}_{0}\mathrm{\Delta}T}{{T}_{m}\nu \mathrm{\Delta}C}\right),{D}_{f}=\left(\frac{{D}_{m}{k}_{0}\mathrm{\Delta}C}{{c}_{s}{c}_{p}\mathrm{\Delta}T}\right),}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}R=\frac{4{\sigma}_{1}{T}_{\mathrm{\infty}}^{3}}{{K}_{1}{\rho}_{f}{c}_{p}{\alpha}_{m}},}\end{array}$$(12)

where *ε* is a small parameter, *Δ T* = *T*_{w} − *T*_{∞} and the thermal conductivity parameter is *K*_{∞}. The important flow attributes, the Nusselt number *Nu*_{x} and skin friction coefficient *C*_{f} are described by
$$\begin{array}{}{\displaystyle {C}_{f}=\frac{{\tau}_{w}}{\rho {u}_{w}^{2}(x)},\phantom{\rule{1em}{0ex}}N{u}_{x}=\frac{x{q}_{w}}{k({T}_{f}-{T}_{\mathrm{\infty}})}.}\end{array}$$(13)

Here *q*_{w} and *τ*_{w} are the plate heat flux and the skin friction respectively, defined as
$$\begin{array}{}{\displaystyle {q}_{w}=-k{\left(\frac{\mathrm{\partial}T}{\mathrm{\partial}y}\right)}_{y=0},\phantom{\rule{thinmathspace}{0ex}}{\tau}_{w}=\mu {\left(\frac{\mathrm{\partial}u}{\mathrm{\partial}y}\right)}_{y=0},}\end{array}$$(14)

where *μ* is the coefficient of viscosity. Equations (13) may be written as
$$\begin{array}{}R{e}_{x}^{\frac{1}{2}}{C}_{f}={f}^{\u2033}(0),\end{array}$$(15)
$$\begin{array}{}R{e}_{x}^{-\frac{1}{2}}N{u}_{x}=-{\theta}^{\prime}(0),\end{array}$$(16)

where
$\begin{array}{}R{e}_{x}=\frac{{u}_{w}(x)x}{\nu}\end{array}$ is the local Reynolds number. Here, the Sherwood number is zero, due to the assumption of zero mass flux at the surface.

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