Unsteady nanofluid flow over an inclined stretching surface with convective boundary condition and anisotropic slip impact

Consider the unsteady, laminar, three-dimensional flow of nanofluid over a semi-infinite inclined surface embedded in a porous medium and subject to anisotropic slip and convective boundary condition impacts. The surface is considered to be linearly stretched in the x-orientation with a velocity bx and the y-orientation is inclined at an angle Ω to the horizontal line, whilst the z-orientation is perpendicular to the plate surface. In addition, it is assumed that the plate surface is maintained by convective heat transfer at a certain value T_f, while the temperature of the ambient nanofluid is T_∞ such that T_f> T_∞. The flow model and physical coordinate system are displayed in Fig. 1. The thermophysical properties of a nanofluid are presented in Table 1.

Table 1. Thermophysical properties of water and Cu nanoparticles at 25^o [11].

The nanofluid properties is constant except the density in the buoyancy terms of the balance of momentum equations in x- and y-orientations. Under the above assumptions, the boundary layer equations governing the convective flow and heat transfer of the present investigation are [31]

$\frac{\partial u}{\partial x}+\frac{\partial w}{\partial z}=0$,              (1)

$\begin{align}  & \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+w\frac{\partial u}{\partial z}= \\ & \frac{1}{{{\rho }_{nf}}}\left[ {{\mu }_{nf}}\frac{{{\partial }^{2}}u}{\partial {{z}^{2}}}+{{g}^{*}}{{(\rho \beta )}_{nf}}\cos \Omega (T-{{T}_{\infty }})-\frac{{{\mu }_{nf}}}{K}u \right] \\\end{align}$          (2)

$\begin{align}  & \frac{\partial v}{\partial t}+w\frac{\partial v}{\partial z}= \\ & \frac{1}{{{\rho }_{nf}}}\left[ {{\mu }_{nf}}\frac{{{\partial }^{2}}v}{\partial {{z}^{2}}}-\frac{{{\mu }_{nf}}}{K}v+{{g}^{*}}{{(\rho \beta )}_{nf}}\sin \Omega (T-{{T}_{\infty }}) \right] \\\end{align}$           (3)

$\frac{\partial w}{\partial t}+w\frac{\partial w}{\partial z}=\frac{1}{{{\rho }_{nf}}}\left[ -\frac{\partial p}{\partial z}+{{\mu }_{nf}}\frac{{{\partial }^{2}}w}{\partial {{z}^{2}}}-\frac{{{\mu }_{nf}}}{K}w \right]$,           (4)

$\frac{\partial T}{\partial t}+w\frac{\partial T}{\partial z}={{\alpha }_{nf}}\frac{{{\partial }^{2}}T}{\partial {{z}^{2}}}$,           (5)

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Figure 1. Flow model and physical co-ordinate system

The appropriate boundary conditions for this problem are defined as follows,

$u(t,x,0)=bx+{{N}_{1}}{{\mu }_{nf}}\frac{\partial u}{\partial z}$,$v(t,x,0)={{N}_{2}}{{\mu }_{nf}}\frac{\partial v}{\partial z}$,$w(t,x,0)=0$,$-{{k}_{nf}}\frac{\partial T}{\partial z}={{h}_{f}}({{T}_{f}}-T)$,$\begin{align}  & u(t,x,\infty )=0,v(t,x,\infty )=0, \\ & T(t,x,\infty )={{T}_{\infty }},\frac{\partial w}{\partial z}(t,x,\infty )=0 \\\end{align}$,            (6)

where x, y, and z are the Cartesian coordinates. u, v, w, P and T are the fluid velocity components in the x-, y-, and z- orientations, pressure and nanofluid temperature, respectively. g^*, b, t, and W are the gravitational acceleration, constant having units of inverse time, dimensional time, ambient concentration, and the inclination angle, respectively. N₁ and N₂ are the velocity slip coefficients in the x and y orientations, respectively. h_f is the heat transfer coefficient. b_nf is the thermal expansion coefficient of the nanofluid; ρ_nf is the is the effective density of the nanofluid, μ_nf is the effective dynamic viscosity of the nanofluid and α_nf is the thermal diffusivity of the nanofluid, (ρC_p)_nf is the heat capacitance of nanofluid, (ρβ)_nf is the thermal expansion coefficient of the nanofluid which are given by [11];

${{\rho }_{nf}}=(1-\phi ){{\rho }_{f}}+\phi {{\rho }_{s}}$, ${{\mu }_{nf}}=\frac{{{\mu }_{f}}}{{{(1-\phi )}^{2.5}}}$, ${{\alpha }_{nf}}=\frac{{{k}_{nf}}}{{{\left( \rho {{C}_{p}} \right)}_{nf}}}$,

${{\left( \rho {{C}_{p}} \right)}_{nf}}=(1-\phi ){{\left( \rho {{C}_{p}} \right)}_{f}}+\phi {{\left( \rho {{C}_{p}} \right)}_{s}}$${{\left( \rho \beta  \right)}_{nf}}=(1-\phi ){{\left( \rho \beta  \right)}_{f}}+\phi {{\left( \rho \beta  \right)}_{s}}$.            (7)

Here, f is the solid volume fraction parameter, m_f is the dynamic viscosity of the basic fluid, β_f  and β_s are the thermal expansion coefficients of the base fluid and nanoparticle, respectively, ρ_f  and ρ_s are the densities of the basic fluid and nanoparticle, respectively, k_nf is the effective thermal conductivity of nanofluid which is given as;

$\frac{{{k}_{nf}}}{{{k}_{f}}}=\frac{\left( {{k}_{s}}+2{{k}_{f}} \right)-2\phi \left( {{k}_{f}}-{{k}_{s}} \right)}{\left( {{k}_{s}}+2{{k}_{f}} \right)+\phi \left( {{k}_{f}}-{{k}_{s}} \right)}$,            (8)

where, k_s is the thermal conductivity of the nanoparticles and k_f  is the thermal conductivity of base fluid.

Introducing the following non-dimensional quantities;

$\tau =bt$,$\eta =z/\sqrt{{{\nu }_{f}}t}$,$u=bx{f}'(\tau ,\eta )+\Gamma \cos \Omega g(\tau ,\eta )$,$v=\Gamma \sin \Omega h(\tau ,\eta )$,$w=-\sqrt{{{b}^{2}}{{\nu }_{f}}t}f$, $\Gamma =\frac{{{g}^{*}}{{\beta }_{f}}({{T}_{f}}-{{T}_{\infty }})}{b}$,$\theta (\tau ,\eta )=\left( T-{{T}_{\infty }} \right)/\left( {{T}_{f}}-{{T}_{\infty }} \right)$,$P={{\rho }_{f}}b{{\nu }_{f}}G(\tau ,\eta )$,$Da=Kb/{{\upsilon }_{f}}$, ${{\delta }_{1}}={{N}_{1}}{{\mu }_{f}}\sqrt{b/{{\nu }_{f}}}$, ${{\delta }_{2}}={{N}_{2}}{{\mu }_{f}}\sqrt{b/{{\nu }_{f}}}$, $\delta ={{\delta }_{2}}/{{\delta }_{1}}$, $Bi={{h}_{f}}a/{{\operatorname{Re}}^{1/2}}{{k}_{f}}$, $\Pr ={{\upsilon }_{f}}/{{\alpha }_{f}}$.            (9)

In view of the Eqn. (9), the basic field of Eqns. (1)-(6) with Eqns (7)-(8) can be expressed in dimensionless form as;

${{\Gamma }_{1}}{f}'''+\frac{\eta }{2}{f}''+\tau \left( f{f}''-{{{{f}'}}^{2}}-\frac{{{\Gamma }_{1}}}{Da}{f}' \right)-\tau \frac{\partial {f}'}{\partial \tau }=0$,         (10)

${{\Gamma }_{1}}{g}''+\frac{\eta }{2}{g}'+\tau \left( f{g}'-{f}'g-\frac{{{\Gamma }_{1}}}{Da}g+{{\Gamma }_{2}}\theta  \right)-\tau \frac{\partial g}{\partial \tau }=0$,        (11)

${{\Gamma }_{1}}{h}''+\frac{\eta }{2}{h}'+\tau \left( f{h}'-\frac{{{\Gamma }_{1}}}{Da}h+{{\Gamma }_{2}}\theta  \right)-\tau \frac{\partial h}{\partial \tau }=0$,           (12)

${{\Gamma }_{3}}(\phi ){G}'+{{\Gamma }_{1}}{f}''+\frac{\eta }{2}{f}'-\frac{1}{2}f+\tau \left( f{f}'-\frac{{{\Gamma }_{1}}}{Da}f \right)-\tau \frac{\partial f}{\partial \tau }=0$            (13)

$\frac{{{\Gamma }_{4}}}{\Pr }\left( \frac{{{k}_{nf}}}{{{k}_{f}}} \right){\theta }''+\frac{\eta }{2}{\theta }'+\tau f{\theta }'-\tau \frac{\partial \theta }{\partial \tau }=0$,           (14)

where Eq. (1) is identically satisfied. In Eqs. (10)-(14), a prime denotes partial differentiation with respect to h, and the parameters G₁, G₂, G₃ and G₄ are given by;

${{\Gamma }_{1}}(\phi )=\frac{1}{{{(1-\phi )}^{2.5}}[1-\phi +\phi ({{\rho }_{s}}/{{\rho }_{f}})]}$,

${{\Gamma }_{2}}(\phi )=\frac{\left[ 1-\phi +\phi \left( {{(\rho \beta )}_{s}}/{{(\rho \beta )}_{f}} \right) \right]}{1-\phi +\phi ({{\rho }_{s}}/{{\rho }_{f}})}$,

${{\Gamma }_{3}}(\phi )=\frac{1}{\left[ 1-\phi +\phi ({{\rho }_{s}}/{{\rho }_{f}}) \right]}$,

${{\Gamma }_{4}}(\phi )=\frac{1}{\left[ 1-\phi +\phi \left( {{(\rho {{C}_{p}})}_{s}}/{{(\rho {{C}_{p}})}_{f}} \right) \right]}$.         (15)

The transformed boundary conditions become

$f(\tau ,0)=0$,${f}'(\tau ,0)=1+\frac{{{\delta }_{1}}}{{{(1-\phi )}^{2.5}}}{{\tau }^{-1/2}}{f}''(\tau ,0)$$g(\tau ,0)=\frac{{{\delta }_{1}}}{{{(1-\phi )}^{2.5}}}{{\tau }^{-1/2}}{g}'(\tau ,0)$, $G(\tau ,0)=0$

$h(\tau ,0)=\frac{{{\delta }_{2}}}{{{(1-\phi )}^{2.5}}}{{\tau }^{-1/2}}{h}'(\tau ,0)$,

$\frac{{{k}_{nf}}}{{{k}_{f}}}{\theta }'(\tau ,0)=-Bi{{\tau }^{1/2}}\left[ 1-\theta (\tau ,0) \right]$, ${f}'(\tau ,\infty )=g(\tau ,\infty )=h(\tau ,\infty )=\theta (\tau ,\infty )=0$.         (16)

In the above equation parameters; Da, d₁, d₂, d, Bi, Pr are respectively the Darcy number, slip factors, slip ratio , Biot number, Prandtl number and dimensionless time.

The quantities of the physical interest are the local skin friction coefficients in the x- and y-orientations and local Nusselt number which are an important parameters commonly used in fluid mechanics. The non-dimensional forms of these quantities are defined as [31];

${{C}_{fx}}=\frac{2}{{{(1-\phi )}^{2.5}}\operatorname{Re}_{x}^{1/2}\sqrt{\tau }}\left( {f}''\left( \tau ,0 \right)+\frac{G{{r}_{x}}}{\operatorname{Re}_{x}^{2}}\cos \Omega {g}'\left( \tau ,0 \right) \right)$            (17)

${{C}_{fy}}=\frac{2G{{r}_{x}}}{{{(1-\phi )}^{2.5}}\operatorname{Re}_{x}^{5/2}\sqrt{\tau }}\sin \Omega {h}'\left( \tau ,0 \right)$,        (18)

$N{{u}_{x}}=\frac{{{q}_{w}}x}{{{k}_{f}}({{T}_{f}}-{{T}_{\infty }})}=-\operatorname{Re}_{x}^{1/2}{{\tau }^{-1/2}}\left( \frac{{{k}_{nf}}}{{{k}_{f}}} \right){\theta }'\left( \tau ,0 \right)$,         (19)

where $G{{r}_{x}}={{g}^{*}}{{\beta }_{f}}({{T}_{f}}-{{T}_{\infty }}){{x}^{3}}/{{\upsilon }_{f}}^{2}$ and ${{\operatorname{Re}}_{x}}=b{{x}^{2}}/{{\upsilon }_{f}}$ are the local Grashof and Reynolds number, respectively.

It is noteworthy to mention that by substituting $Da,Bi\to \infty $, f=0 and d₁=d₂=0 in Eqs. (10)-(15), the problem is reduced to the transient flow of regular fluid and heat transfer over an inclined stretching sheet which is discussed previously in [31].

In order to gain a clear physical insight of this investigation, the impact of the Darcy number Da, solid volume fraction parameter f, slip factors δ₁ and δ₂, and Biot number Bi on the profiles of nanofluid velocity components, pressure and temperature as well as the local skin-friction coefficients in the x and y orientations C_fx and C_fy, and the local Nusselt number Nu_x are presented in Figs. 3-8. The current numerical investigation is carried out for copper-water nanofluid as working fluid and the value of Prandtl number Pr of  base fluid (water) is kept constant at 6.2.

Figs. 3(a)-3(e) visualize the impact of the Darcy number on the profiles of nanofluid velocity components in x-orientation ${f}'(\tau ,\eta )$ and $g(\tau ,\eta )$, velocity component in y-orientation $h(\tau ,\eta )$, pressure $G(\tau ,\eta )$, and temperature $\theta (\tau ,\eta )$, respectively. It is noteworthy to mention from the definition of Da, that the value of Darcy number measures the extent of the permeability of porous medium, that is the existence of a porous medium in the flow provides resistance to flow, thus, this resistive force leads to a considerable decelerate the motion of the nanofluid along the stretched surface. This is appeared in Figs. 3(a)-3(e) by increasing the Darcy number Da from 0.001, 0.01,1 (very high permeability) to 10 (weak permeability) clearly induces a pronounced enhancement in the x-velocity components (${f}'$and g) and y-velocity component h, i.e., accelerates the flow. In addition, the variations in velocities are maximized some distance from the wall, towards the free stream. As such the nanofluid acceleration towards the edge of the boundary layer is less impeded by wall effects here. Also, evolution in Da implies to damp the Darcian drag force, due to the inverse relationship in Eqs. (10)-(13); porous drag force is therefore progressively lowered with an evolution in permeability, i.e., Darcy number, which serves to promote the flow velocities in the regime. Subsequently, these promotes in the flow have a tendency to a robust dwindling in both of temperature and pressure profiles. In addition, it is interesting to note that an increment in the values of Da are followed by corresponding rise in all hydrodynamics boundary layers and slight diminution in thermal boundary layers thickness.

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Figure 3. Impact of Darcy Number Da on (a) fluid velocity of x-orientation (${f}'$), (b) fluid velocity of x-orientation (g), (c) fluid velocity of y-orientation (h), (d) pressure profiles (G), (e) fluid temperature ($\theta$)

Figs 4(a)-4(d) demonstrate the impact of Darcy number Da upon variation in skin-friction coefficients in the x- and y- orientations C_fx and C_fy and local Nusselt number Nu_x, respectively, against the dimensionless time t. As seen from the definitions of C_fx, C_fy and Nu_x, they are directly proportional to ${f}''(\tau ,0)$,${g}'(\tau ,0)$,${h}'(\tau ,0)$and$-{\theta }'(\tau ,0)$, respectively. it is manifested from these Figs. that there are two opposite behaviors for the local skin-friction coefficients and local Nusselt number. These behaviors are clarified by the reduction in the skin-friction coefficient in the x-orientation ${f}''(\tau ,0)$ and a huge enhancement in either the skin-friction coefficients in the x- and y-orientations ${g}'(\tau ,0)$,${h}'(\tau ,0)$ and the Nusselt number $-{\theta }'(\tau ,0)$ as a result of rising the Darcy number Da.

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Figure 4. Impact of Darcy Number Da on (a) the skin-friction coefficient in the x-orientation f''(τ,0), (b) skin-friction coefficient in the x-orientation g'(τ,0), (c) skin-friction coefficient in the y-orientation h'(τ,0), (d) wall heat transfer -θ'(τ,0)

This is due to the fact that the increment in the flow velocity components close to the wall with decreasing the nanofluid temperature as Da enlarges, causing the wall slope of the fluid linear velocity and the negative wall slope of the temperature profiles to grow. On other hand, both the local skin-friction coefficients and Nusselt number become lower at the dimensionless time t=0 and they are larger with high values of t.

Figs. 5(a)-5(e) depict the influences of slip factors δ₁ and δ₂ on dimensionless nanofluid velocity components, pressure and temperature profiles, respectively. It is apparent from these Figs. and based on the definition of slip ratio that an elevation in the slip factors δ₁ and δ₂ results in a considerable reduction in both the x-velocity component ${f}'$ and pressure profiles G, while a reverse trend occurred with the profiles of velocity of x-orientation (g), fluid velocity of y-orientation (h), and nanofluid temperature q. This is due to the fact that a rising the values of the velocity slip parameters have a tendency to accelerate the flow along the y-orientation which is inclined to the horizontal line, whereas a reverse trend occurred along the stretched surface in the x-orientation. This, in turn, produces a sufficient depress in x-velocity component and a strong promotes occurs in the fluid velocity profiles in y-orientation and temperature. There will be a corresponding increase in the hydrodynamics and thermal boundary layer thickness. These behaviors are clearly shown in Figures 5(a)-5(e).

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Figure 5. Impact of slip factor δ₁ on (a) fluid velocity of x-orientation (${f}'$), (b) fluid velocity of x-orientation (g), (c) fluid velocity of y-orientation (h), (d) pressure profiles (G), (e) fluid temperature ($\theta$)

The variations of the skin-friction coefficients and local Nusselt number with different values of slip factors δ₁ and δ₂ are respectively, shown in Figs 6(a)-6(d). It is noteworthy to mention that an augmentation in the slip factors δ₁ and δ₂ implies a noticeable reduction in all the local skin-friction coefficient and Nusselt number. This result was predicted because the increment in the velocity slips elucidate a considerable acceleration in the flow along the y-orientation with maximization in the hydrodynamics and thermal boundary layers thickness, as depicted in Figures 5(a)-5(e).

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Figure 6. Impact of slip factor δ₁ on (a) the skin-friction coefficient in the x-orientation ${f}''(\tau ,0)$, (b) skin-friction coefficient in the x-orientation ${g}'(\tau ,0)$, (c) skin-friction coefficient in the y-orientation ${h}'(\tau ,0)$, (d) wall heat transfer $-{\theta }'(\tau ,0)$

Figs. 7(a)-7(e) elucidate the influences of solid volume fraction parameter f and Biot number Bi on the nanofluid velocity components, pressure and temperature profiles, respectively. it is noteworthy to mention that a raise in the solid volume fraction parameter f results in a considerable decrease in all the velocity components and pressure profiles. Moreover, the nanofluid temperature reduces near the stretched surface with the enhancement in the solid volume fraction parameter f, whereas the opposite behavior occurs with temperature after finite distance in the free stream. This consistency with the physical behavior that increasing the volume of nanoparticles implies a high thermal conductivity of nanofluid (see Table 1), consequently the thermal boundary layer thickness increases. The cause for this outcome is that an increment in the volume fraction produces a high-energy transport through the flow associated with the irregular motion of the ultrafine particles. On other side, the enhancement in the Biot number Bi causes a huge increase in either the fluid velocity of x-orientation (g), the fluid velocity of y-orientation (h) or temperature profiles. The cause for this behavior is that, a large value in Biot number yields a great surface convective which in turn provides more heat to the stretching surface and as a consequence of the temperature difference between the surface and the nanofluid will increase. As an outcome, the velocity as well as the wall temperature and thermal boundary layer thickness magnifies due to the enlargement in the values of Bi. Moreover, it is noteworthy to mention that the changes in Bi cause no effects on the behaviors in both the x-velocity components ${f}'$ and pressure profiles G due to the governing equations (11), (14) and (15) which are uncoupled from the other equations.

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Figure 7. Impact of solid volume fraction parameter f on (a) the fluid velocity of x-orientation (${f}'$), (b) fluid velocity of x-orientation (g), (c) fluid velocity of y-orientation (h), (d) pressure profiles (G), (e) fluid temperature ($\theta$)

Finally, Figs. 8(a)-8(d) exhibit the impact of the solid volume fraction f and Biot number Bi on the local skin-friction coefficients and Nusselt number. It is manifested that an increment in the solid volume fraction tends to damp the skin-friction coefficients and the Nusselt number, whilst the reverse behavior happens with the increase of Biot number Bi. This phenomenon is true even in the increase of solid volume fraction f, which, results in a increase in the thermal boundary layer thickness, associated with considerable decrease in the wall shear stress adjacent to the stretching surface with a decrease in the wall temperature gradient, and hence produces a sufficient reduction in the local Nusselt number. Furthermore, It is manifested that an increasing in the value of Biot number Bi has a tendency to increase both the local skin friction coefficients and local Nusselt number. The reason for this behavior is that as Bi increases, the cold nanofluid on stretched surface is convectively heated and thus, the flow velocity rises, which in turn enhances in the gradients of velocity components and temperature.

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Figure 8. Impacts of solid volume fraction parameter f and Biot number Bi on (a) the skin-friction coefficient in the x-orientation ${f}''(\tau ,0)$, (b) skin-friction coefficient in the x-orientation ${g}'(\tau ,0)$, (c) skin-friction coefficient in the y-orientation ${h}'(\tau ,0)$, (d) wall heat transfer $-{\theta }'(\tau ,0)$