**1. Introduction**

The mechanism of drag and heat loss reduction [1] has been the focus of intensive analysis due to its application in the prevention of loss of mechanical energy. Drag and heat loss reduction may create energy savings, processing time reduction, enhancement in thermal rating, and make equipment more durable. Several well-known methods have been proposed by researchers to reduce the drag and heat loss in physical systems out of them utilization of stretching/shrinking surfaces [2] and enhancing the thermal conductivity of the involved fluid are famous [3].

Nanofluids, an achievement of researchers and scientists of the developing world of nanotechnology, exploit the thermal conductivity of solids to enhance the thermal conductivity of a fluid by adding nano-sized solid particles. Materials commonly used for nanoparticles include oxides such as alumina, silica, titania and copper oxide, and metals such as copper and gold. Carbon nanotubes and diamond nanoparticles have also been used to realize nanofluids. Nanoparticles vary from 1 to 100 nm in diameter. Thermal conductivity can be increased up to two times by adding small amount of nanoparticles. Popular base fluids include water and organic fluids such as ethanol and ethylene glycol. The volumetric fraction of the nanoparticles is usually below 5%.

A wide range of nanofluids exist in nature, like blood, which is a complex biological compound, made up of different nanoparticles that perform various functions at molecular level. A number of natural processes occurring in atmosphere and biosphere have wide variety of composition of different fluids and nanoparticles. Manufacturing and industrial waste materials are also composed of nanoscale particles and fluids. Various self-assembly processes for nanostructures generate from the addition of nanoparticles in base fluid. Considering the wide-ranging uses of nanofluid in industry and science, and the model of nanofluid presented by Buongiorno [4], many experimentalists and researchers have showed great interest in the study of nanofluids in the last few years [5–12].

Keeping the fact in view that the unsteady flows are more generalized, and the applications of nanofluids and stretching surfaces in drag and heat loss reduction, this article analyzes the unsteady flow of nanofluid over a moving surface. The study of flow over a linearly stretching sheet was initiated by Crane [13]. He derived the analytical solution of two-dimensional momentum equations. This notable work of Crane [13] has been studied by many researchers in many directions. Some recent works on the topic of stretching/shrinking surfaces are References [14–18] and the references given therein.

In 1997, Todd [19] introduced a new family of unsteady boundary layer flow over a moving surface emerging from a moving slot. He proposed a new set of transformations containing the Blasius–Rayleigh–Stoke variable to write the governing unsteady partial differential equations in similar form. Fang et al. [20] conducted the heat-transfer analysis for this boundary layer flow. In this article, we carry out the numerical analysis of unsteady flow of nanofluid past a movable surface emerging from a moving slot by converting the governing coupled unsteady partial differential equations into similar form using the transformation involving the Blasius–Rayleigh–Stoke variable. The results are presented graphically and the effects of nanoparticles on skin friction, Nusselt number and Sherwood number are discussed in detail. Dual solutions are observed for a specific range of moving slot parameter and are found to be altered due to the presence of nanoparticles. Furthermore, the numerical data is used to write the correlation expressions for certain important flow quantities by performing linear regression. Correlation expressions enable the readers to obtain the values of numerical results for different values of involved parameters from analytical expressions.

#### **2. Mathematical Formulation**

Consider the unsteady two-dimensional flow and heat transfer of an incompressible viscous-based nanofluid over a heated moving semi-infinite plate. The surface is emerging out along the *x*-axis from a moving slot (see Figure 1 for geometry of the problem). At time *t* = 0, the fluid is at rest. The governing boundary layer [21] equations are given as:

$$\frac{\partial \mathcal{U}}{\partial X} + \frac{\partial V}{\partial Y} = 0 \tag{1}$$

$$\frac{\partial \mathcal{U}}{\partial t} + \mathcal{U}\frac{\partial \mathcal{U}}{\partial X} + V\frac{\partial \mathcal{U}}{\partial Y} = \nu \frac{\partial^2 \mathcal{U}}{\partial Y^2},\tag{2}$$

$$\frac{\partial T}{\partial t} + \mathcal{U} \frac{\partial T}{\partial X} + V \frac{\partial T}{\partial Y} = \sigma \frac{\partial^2 T}{\partial Y^2} + \varepsilon \left( D\_B \frac{\partial T}{\partial Y} \frac{\partial C}{\partial Y} + \frac{D\_T}{T\_\infty} \left( \frac{\partial T}{\partial Y} \right)^2 \right), \tag{3}$$

$$\frac{\partial \mathcal{C}}{\partial t} + \mathcal{U} \frac{\partial \mathcal{C}}{\partial X} + V \frac{\partial \mathcal{C}}{\partial Y} = D\_B \frac{\partial^2 \mathcal{C}}{\partial Y^2} + \frac{D\_T}{T\_{\infty}} \frac{\partial^2 T}{\partial Y^2} \tag{4}$$

where *U* and *V* are the velocity components in *X* and *Y* directions. *T* is the fluid temperature, *C* is the nanoparticles volume fraction, *ν* is the kinematic viscosity, *σ* is the thermal diffusivity of the fluid, *ε* is the ratio of heat capacities of the nanoparticles (*ρc*)*<sup>p</sup>* and base fluid (*ρc*)*<sup>f</sup>* , *DB* and *DT* are the Brownian and thermophoretic diffusion coefficients respectively. For water nanofluids at room temperature with nanoparticles of 1–100 nm diameters, the Brownian diffusion coefficient ranges from 4 × <sup>10</sup>−<sup>10</sup> to 4 × <sup>10</sup>−<sup>12</sup> m2/s. For alumina/water and copper/water (*ρc*)*<sup>p</sup>* is 3.1 and 3.4 MJ/m3 respectively. The thermophoretic diffusion is equal to 6 × <sup>10</sup>−<sup>5</sup> for aluminum/water nanofluid and 6 × <sup>10</sup>−<sup>6</sup> for copper/water nanofluid.

**Figure 1.** Systematic diagram of the problem. δ, δc, δ<sup>T</sup> represent the thicknesses of momentum, thermal and nanoparticles concentration boundary layers respectively.

The corresponding boundary conditions are:

$$\begin{aligned} \mathcal{U}(X,Y,t) &= 0, \; V(X,Y,t) = 0, \; T(X,Y,t) = 0, \; \mathcal{C}(X,Y,t) = 0 \; at \; t = 0, \\ \mathcal{U}(X,Y,t) &= \mathcal{U}\_W, \; V(X,Y,t) = 0, \; T(X,Y,t) = T\_W, \; \mathcal{C}(X,Y,t) = \mathcal{C}\_W \; at \; Y = 0, \\ \mathcal{U}(X,Y,t) &\to 0, \; T(X,Y,t) \to T\_{\infty}, \; \mathcal{C}(X,Y,t) \to \mathcal{C}\_{\infty} \; as \; Y \to \infty. \end{aligned} \tag{5}$$

Since the unsteady flow is a generalized case of steady flow, Todd [19] generalized the Blasius and Rayleigh–Stokes variables to get similar equations for the boundary layer flow of viscous fluid over a moving surface, termed as the Blasius–Rayleigh–Stokes variable:

$$
\eta = \Upsilon / \sqrt{\cos(\alpha)\nu t + \sin(\alpha)(\nu X / \ell L\_W)}.\tag{6}
$$

This variable depicts that the slot at *Y* = 0 is moving with a constant speed −*Uw* cot *α*. To obtain similarity solutions for the system of Equations (1)–(5), we introduce the following similarity variables

$$\begin{array}{ll} \psi(\mathbf{x}, \mathbf{y}, t) = lI\_W \sqrt{\cos(\alpha) \nu t + \sin(\alpha)(\nu \mathbf{x} / lL\_W)} f(\eta), \\ \theta(\eta) = \frac{T - T\_{\infty}}{T\_W - T\_{\infty}}, \quad \phi(\eta) = \frac{C - C\_{\infty}}{C\_W - C\_{\infty}}. \end{array} \tag{7}$$

in the governing equations to get the following ordinary differential equations:

$$f'''' + \frac{1}{2}(\cos a)\eta f'' + \frac{1}{2}(\sin a)f f'' = 0,\tag{8}$$

$$
\theta'' + \frac{\text{Pr}}{2}((\cos a)\eta + (\sin a)f)\theta' + \text{N}\_b\theta'\phi' + \text{N}\_l\theta'^2 = 0,\tag{9}
$$

$$
\phi'' + \frac{\mathcal{L}\varepsilon}{2}((\cos a)\eta + (\sin a)f)\phi' + (\frac{N\_t}{N\_b})\theta'' = 0,\tag{10}
$$

subject to boundary conditions:

$$\begin{array}{c} f(\eta) = 0, \ f'(\eta) = 1, \ \theta(\eta) = 1, \ \phi(\eta) = 1 \text{ at } \eta = 0, \\\ f'(\eta) \to 0, \ \theta(\eta) \to 0, \ \phi(\eta) \to 0 \text{ as } \eta \to \infty, \end{array} \tag{11}$$

where prime represents the differentiation with respect to variable *η*. Pr is Prandtl number, *Nt* is thermophoresis diffusion parameter, *Nb* is Brownian diffusion parameter and *Le* is Lewis number given by the following expressions:

$$\Pr = \frac{\nu}{\sigma'} \text{ N}\_b = \frac{\varepsilon D\_B (\mathbb{C}\_W - \mathbb{C}\_\infty)}{\sigma}, \text{ N}\_l = \frac{\varepsilon D\_T (T\_W - T\_\infty)}{T\_W \sigma}, \text{ L} \\ \varepsilon = \frac{\nu}{D\_B}. \tag{12}$$

The range of the parameters of interest, namely thermophoresis diffusion parameter and Brownian diffusion parameter is given as: *Nb*∈ (0.0, 0.5) and *Nt*∈ [0.0, 0.5).
