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Multiple slip effects on MHD unsteady viscoelastic nano-fluid flow over a permeable stretching sheet with radiation using the finite element method

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Abstract

The current study investigates the impact of multiple slips on Jeffrey fluid model for unsteady magnetohydrodynamic viscoelastic buoyant nanofluid in the presence of Soret and radiation over a permeable stretching sheet. Appropriate transformations are utilized to obtain the relevant nonlinear differential system. The obtained differential system is tackled numerically with the finite element method. Effect of the controlling parameters on dimensionless quantities such as velocity, temperature, concentration, and nano-fluid volume fraction profile, as well as on dimensionless numbers such as local Nusselt, Sherwood, nano-particle Sherwood, and the local friction coefficient is analyzed. The effect of multiple slips is examined and found that the boundary layer flow increases in the presence of multiple slips. Numerically obtained solutions are contrasted with the published literature and found to be in nice agreement. The present study has many applications in coating and suspensions, cooling of metallic plate, paper production, heat exchangers technology, and materials processing exploiting.

Introduction

In recent technological advances, the study of non-Newtonian materials has caught the attention of engineers and scientists. Broad stimulation of scientists is due to the wide use of liquids in technology and in industry. Examples of such applications are paints, colloids and suspensions, cosmetics, polymer solutions, foods, exotic lubricants, paper production, coal water, ketchup, glues, ink, blood, some oils, fiber technology and clay coating. Due to the diversity of non-Newtonian liquids, there is no law to explain the viscous and elastic properties of these liquids. Despite all these challenges, researchers have made valuable contributions to current literature on a variety of non-Newtonian fluids [1,2,3,4,5,6,7,8]. Among them, visco-elastic fluids are expected to be more important in the current research area due to their extensive applications in engineering and industrial production. Some of the most recent practices in this regard are in Kumar et al. The effect of non-linear thermal radiation on the flow of the mixed visco-elastic nano-liquid double diffusion convection boundary layer is discussed on a stretch film. Convective heat transfer and the MHD visco-elastic nanofluid flow induced by a stretch film are studied by Shit et al. [9]. Sheikholeslami et al. [10] studied the impact of the magnetic field and thermal radiation on the hydrothermal behavior of nanofluids of \(Fe_3O_4\)\(H_2O\). The impact of Lorentz forces on the flow of CuO–water nanofluids in a permeable housing is presented by CVFEM and Nanofluid fluxes and heat transfer in a cavity. magnetic field from Sheikholeslami et al. [11, 12]. The flow of the magnetohydrodynamic boundary layer (MHD) with a stagnation point on a visco-elastic liquid stretch film in the presence of thermal radiation was studied by Naryana et al. Application of the law of Darcy to the flow of nanofluids in a porous cavity below impact of Lorentz forces investigated by Sheikholeslami et al. Hussanan et al. [13] investigated the analytical solution for the suction and injection flow of a visco-elastic Cassco fluid past a stretch surface in the presence of viscous dissipation. Majeed et al. [14] investigated the analysis of heat transfer in a ferromagnetic visco-elastic fluid stream on a stretching sheet.

Non-Newtonian liquids have been discovered a lot of significant and helpful for innovative perspectives, for example, multi-grade oils, fluid cleansers, paints, polymer arrangements, and polymer softens are talked about in Elahi [15]. Moreover, ongoing advances in nanotechnology have prompted the improvement of another imaginative class of warmth move called nanofluids made by scattering nanoparticles. Non-Newtonian nanofluids are generally experienced in numerous modern and innovation applications, for instance melts of polymers, natural arrangements, paints, tars, black-tops, and pastes, and so on. The form of nanofluids is essentially the dispersion of solid nanoparticles in liquids such as water, oil or ethylene glycol. These nanoparticles are usually made from metals, oxides and carbon nanotubes. Nanoparticles can also be used in biomedical applications such as magnetic resonance imaging, photothermal therapy, drug delivery control, protein separation, biosensor, DNA detection and immune sensors. Choi [16] presented for the first time an innovative mixing technique using nanoparticles and basic liquids, with the aim of improving heat transfer by increasing the thermal conductivity of the liquid.

Boungiorno [17] then explained the reasons for improving the heat transfer of nanofluids and concluded that Brownian diffusion and heat development were the reasons for this improvement. Some recent articles about nanomaterials are presented in the Refs. [18,19,20,21,22,23]. Sulochana et al. [24] investigated the magnetohydrodynamic radiation fluid thin film flux of a kerosene nanofluid with the aligned magnetic field. Daniel et al. [25] investigated the effects of slip and convection conditions on the flow of MHD nanofluids on a porous non-linear stretch / shrink film. The nanofluid streams in a microchannel with oblique cross-flow injection are being studied by Shriniy et al. [26].

The effect of exponentially varying viscosity and permeability on the Blasius current of the nanotile fluid on an electromagnetic plate through a porous medium is presented by Hakkem et al. [27] Waqas et al. [28] discussed the interaction of thermal radiation in hydromagnetic visco-elastic nanomaterials subject to gyrotactic microorganisms. Maleki et al. [29] investigated heat transfer and nano-liquid flow on a porous plate with radiation and slip limit conditions. The effects of the second-order nano-fluid poiseuille call plan under the influence of Stefan blowing into a channel are discovered by Alamri et al. [30]. The effects of thermal radiation and slip on the flow of the MHD stagnation point of non-Newtonian nano fluid on a convection stretch surface are being studied by Besthapu et al. [31].

Motivated by the aforementioned studies, the present paper aims to investigate investigates the effect of multiple slips on unsteady two-dimensional magnetohydrodynamic boundary layer flow of viscoelastic buoyant nanofluid with thermal radiation and Soret effect over a stretching sheet. Governed by an appropriate similarity transformation procedure partial differential equations are transformed into ordinary differential equations. The resulting ODE’s is numerically solved by hybrid approach consisting of finite element method [32,33,34,35,36,37,38,39,40]. The consequences obtained were comprehensively discussed in tabulation and graphical representation.

Mathematical formulation

Consider the unsteady two-dimensional MHD boundary layer of an electrically conductive liquid immersed in visco-elastic floating nanofluid with multiple slips and thermal radiation on a linear stretching sheet. The flow of conductive fluid is caused by stretching the sheet in the direction \(U (x, t) = ax / (1- \lambda t)\), where a is the stretch ratio and \(\lambda\) is the positive constant. Consider that there is no flux of nanoparticles on the wall and that the surface extends in the direction of y. Suppose \(T_w\), \(\psi _w\) and \(C_w\) define the temperature, dissolved concentration and friction of the nanoparticles on the stretch sheet as follows:

$$\begin{aligned} T_w &= T_\infty + T_0 \left( \frac{ax}{2\nu (1-\lambda t)^2}\right) \\ C_w &= C_\infty + C_0\left( \frac{ax}{2\nu (1-\lambda t)^2}\right) \\ \phi _w &= \phi _\infty + \phi _0\left( \frac{ax}{2\nu (1-\lambda t)^2}\right) \end{aligned}$$

where \(T_0\), \(C_0\) and \(\phi _0\) are the reference temperature, the reference solute concentration and the reference concentration of the nanoparticles, and \(T_\infty\) is the ambient temperature, \(C_\infty\) the concentration solutal and \(\phi _\infty\) is supposed to be the concentration of nanoparticles. In the current study, \(B(x) = B_0 x^{- 1/2}\) summarizes the magnetic field input, where \(B_0\) is a uniform magnetic field strength. According to the above hypothesis, the conservation of mass, linear momentum conservation, energy conservation, salt concentration and volume fraction of nanoparticles can be obtained as follows:

$$\begin{aligned}&\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=0, \end{aligned}$$
(1)
$$\begin{aligned}&\frac{\partial u}{\partial t}+ u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\nonumber \\&\quad = \frac{\nu }{1+\lambda ^*} \left[ \frac{\partial ^2 u}{\partial y^2} + \lambda _t\left( u\frac{\partial ^3 u}{\partial x \partial y^2}+ \frac{\partial u}{\partial x}\frac{\partial ^2 u}{\partial y^2} \right. \right. \nonumber \\&\qquad \left. \left. - \frac{\partial u}{\partial y} \frac{\partial ^2 u}{\partial x \partial y} + v \frac{\partial ^3 u}{\partial y^3} \right) \right] \nonumber \\&\quad -\frac{\sigma B^2(x)u}{\rho }+ g\beta _T (T-T_\infty ) + g\beta _C (C-C_\infty )\nonumber \\&\qquad + g\beta _\phi (\phi -\phi _\infty ) \end{aligned}$$
(2)
$$\begin{aligned}&\frac{\partial T}{\partial t}+ u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \left( 1+ \frac{16 T^3_\infty \sigma ^*}{3k^* \kappa } \right) \frac{\partial ^2 T}{\partial y^2} \nonumber \\&\qquad +\tau \left[ D_B\frac{\partial \psi }{\partial y}\frac{\partial T}{\partial y}+\frac{D_T}{T_\infty }\left( \frac{\partial T}{\partial y}\right) ^2\right] -\frac{\sigma B^2(x)u}{\rho } \end{aligned}$$
(3)
$$\begin{aligned}&\frac{\partial C}{\partial t}+ u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=D_s\frac{\partial ^2 C}{\partial y^2} + D_{CT} \frac{\partial ^2 T}{\partial y^2} \end{aligned}$$
(4)
$$\begin{aligned}&\frac{\partial \phi }{\partial t}+ u\frac{\partial \phi }{\partial x}+v\frac{\partial \phi }{\partial y}=D_B\frac{\partial ^2 \phi }{\partial y^2} + \frac{D_T}{T_\infty } \frac{\partial ^2 T}{\partial y^2} \end{aligned}$$
(5)

with respect to the boundary conditions

$$\begin{aligned}&u =U(x,t)+U_{slip}, v=v_w, T=T_w(x,t)+T_{slip},\nonumber \\&C=C_w(x,t)+C_{slip}, \nonumber \\&\phi =\phi _w(x,t)+\phi _{slip} \quad \ at\ y \rightarrow 0 \end{aligned}$$
(6)
$$\begin{aligned}&u\rightarrow 0, T\rightarrow T_\infty , C \rightarrow C_\infty , \phi \rightarrow \phi _ \infty , \quad as \quad y \rightarrow \infty , \end{aligned}$$
(7)

where u and v are the speed components, x and y, \(\nu\), \(\sigma\), \(\rho\), respectively, the kinetic viscosity, electrical conductivity, and fluid viscosity. \(D_B\), \(D_T\), \(D_s\) are respectively Brownian diffusion, thermophoretic diffusion and sol solution. To solve the Eqs. (1)–(7) we have introduced the following transformations for agreements:

$$\begin{aligned} \eta &= \sqrt{\frac{a}{\nu (1-\lambda t)}}y, \quad \psi = \sqrt{\frac{a\nu }{(1-\lambda t)}}xf(\eta ),\nonumber \\ \theta (\eta )&= \frac{T-T_\infty }{T_w - T_\infty }, S(\eta ) &= \frac{C-C_\infty }{C_w - C_\infty },\quad \gamma (\eta ) = \frac{\phi -\phi _\infty }{\phi _w - \phi _\infty } \end{aligned}$$
(8)

Given the transformation equation, the partial non-linear differential equations (1)–(6) transform into the following non-linear ODE’s system:

$$\begin{aligned}&f''' + \beta (2f'f''' -f''^2- f f^{iv})\nonumber \\&\quad +(1+\lambda ) ( ff'' -f'^2 - A\left[ \frac{\eta }{2}f'' + f' \right] \nonumber \\&\quad -Mf' + \lambda _1 \theta +\lambda _2 S +\lambda _3 \gamma )= 0, \end{aligned}$$
(9)
$$\begin{aligned}&(1+R)\frac{1}{Pr}\theta '' - f'\theta + f\theta ' - A\left( \frac{\eta }{2}\theta ' + 2\theta \right) \nonumber \\&\quad + Nb\gamma '\theta ' + Nt\theta '^2 -Mf' = 0 \end{aligned}$$
(10)
$$\begin{aligned}&\frac{1}{Sc}S'' - f'S + fS' -A\left( \frac{\eta }{2}S' + 2S\right) + S_r \theta '' = 0 \end{aligned}$$
(11)
$$\begin{aligned}&\gamma '' - Le \left[ f'\gamma -f\gamma ' + A\left( \frac{\eta }{2}\gamma ' + 2\gamma \right) \right] + \frac{Nt}{Nb}\theta '' = 0 \end{aligned}$$
(12)

and the transformed boundary conditions Eqs. (6) and (7) are:

$$\begin{aligned}&f(0)=f_w, f'(0)=1+S_f f''(0), \theta (0)=1+ S_t\theta '(0),\nonumber \\&S(0)=1+S_pS'(0),\nonumber \\&\gamma (0)=1+S_g\gamma '(0), \end{aligned}$$
(13)
$$\begin{aligned}&f'(\infty )\rightarrow 0, \theta (\infty )\rightarrow 0, S(\infty )\rightarrow 0, \gamma (\infty ) \rightarrow 0, \end{aligned}$$
(14)

The primes show the differentiation with respect to \(\eta\). The parameters in Eqs. (9)–(12) \(M=\frac{\sigma B^2_0}{\rho c_o}\), \(Pr=\frac{\nu }{\alpha }\), \(Nb=\frac{\tau D_B (\psi _w - \psi _\infty )}{\nu }\), \(Nt=\frac{\tau D_T (T_w-T_\infty )}{\nu T_\infty }\), \(Sc=\frac{\nu }{D_B}\), \(Le= \frac{\sigma }{D_B}\), \(R = \frac{16\sigma ^* T_\infty ^3}{3k_fK^*}\), \(Sr=D_T T_0/\nu C_0\) are magnetic, Prandtl, Brownian, thermophoresis, Schmidt, Lewis number, thermal radiation and Soret number, respectively. \(A= \lambda _t/a\) is the unsteady parameter and \(\lambda _t\) is the retardation time, \(\lambda _1= g\beta _TT_0/{a\nu }\), \(\lambda _2= g\beta _C C_0/{a\nu }\), \(\lambda _3= g\beta _\phi \phi _0/{a\nu }\) are the buoyancy parameters and \(\beta = \lambda _t a/(1-\lambda t)\) is the Deborah number, \(S_f\) is the hydrodynamic slip, \(S_t\) the thermal slip, \(S_p\) the solutal slip and \(S_g\) is considered to be the nano-particle slip condition. Expression for physical quantities of interest are local skin friction coefficient \(C_f\), Nusselt number Nu, the Sherwood number \(Sh_x\) and the nano-particle Sherwood number \(Sh_{x,n}\) are,

$$\begin{aligned}&C_f = \frac{\tau _w}{\frac{1}{2}\rho U_w^2}, Nu= \frac{xq_w}{\kappa (T_w-T_\infty )},\nonumber \\&Sh_x=\frac{xq_m}{D_s (C_w-C_\infty )}, Sh_{x,n}=\frac{xq_{np}}{D_B(\phi _w-\phi _\infty )}, \end{aligned}$$
(15)
$$\begin{aligned}&\tau _w = \mu \left( \frac{\partial u}{\partial y}\right) _{y=0} + \lambda _t\left( u\frac{\partial ^2 u}{\partial x \partial y}+ 2\frac{\partial u}{\partial x}\frac{\partial u}{\partial y} + v \frac{\partial ^2 u}{\partial y^2}\right) ,\nonumber \\&q_w=-\alpha \left( \frac{\partial T}{\partial y}\right) _{y=0},\nonumber \\&q_m=-D_s\left( \frac{\partial C}{\partial y}\right) _{y=0}, q_{np}= -D_B\left( \frac{\partial \phi }{\partial y}\right) _{y=0}. \end{aligned}$$
(16)

Plugging the values from (8) into (15)-(16), we can get

$$\begin{aligned}&C_f (Re_x)^{\frac{1}{2}}= 2(1+3\beta f''(0)), \quad Re_x^{-1/2}Nu=-(1+R)\theta '(0),\nonumber \\&Re_x^{-1/2}Sh_x=-S'(0),\quad Re_x^{-1/2}Sh_{x,n}=-\gamma '(0) \end{aligned}$$
(17)

Where \(Re_x=U_wx/\nu\) is the Reynolds number. The ordinary differential equations(ODE) are highly nonlinear, which are solved numerically by Hybrid finite element technique.

Finite element method solutions

The current problem is solved by using the finite element method(FEM). Due to the numerical integration, the error is minimized by using the hybrid technique inherent in FEM. As a result, this approach is expected to yield better and more effective results. The steps used in FEM are:

  • Discretization of domain into small elements.

  • Selection of appropriate shape function.

  • Development of finite element equations.

  • Assemble the element equations to obtain global equations.

  • Incorporation of the boundary conditions.

  • Solve the simultaneous equations for the unknowns .

  • Interpolation of results (Fig. 1).

Fig. 1
figure1

Flow chart of the finite element method

we assume

$$\begin{aligned} f'=h. \end{aligned}$$
(18)

plugging Eq. (18) into Eqs. (9)–(14), we get

$$\begin{aligned}&h'' + \beta (2hh'' -h'^2- f h''') +(1+\lambda ) ( fh' -h^2 \nonumber \\&\quad - A\left[ \frac{\eta }{2}h' + h \right] \nonumber \\&\quad -Mh + \lambda _1 \theta +\lambda _2 S +\lambda _3 \gamma )= 0, \end{aligned}$$
(19)
$$\begin{aligned}&(1+R)\frac{1}{Pr}\theta '' - h\theta + f\theta ' - A\left( \frac{\eta }{2}\theta ' + 2\theta \right) \nonumber \\&\quad \quad + Nb\gamma '\theta ' + Nt\theta '^2 -Mh = 0, \end{aligned}$$
(20)
$$\begin{aligned}&\frac{1}{Sc}S'' - f'S + fS' -A\left( \frac{\eta }{2}S' + 2S\right) + S_r \theta '' = 0, \end{aligned}$$
(21)
$$\begin{aligned}&\gamma '' - Le \left[ h\gamma -f\gamma ' + A\left( \frac{\eta }{2}\gamma ' + 2\gamma \right) \right] + \frac{Nt}{Nb}\theta '' = 0, \end{aligned}$$
(22)

with following boundary condition

$$\begin{aligned}&f(0)=f_w, h(0)=1+S_f h'(0), \theta (0)=1+ S_t\theta '(0), \quad S(0)=1+S_pS'(0),\nonumber \\&\gamma (0)=1+S_g \gamma '(0), \end{aligned}$$
(23)
$$\begin{aligned}&h(\infty )\rightarrow 0, \theta (\infty )\rightarrow 0, S(\infty )\rightarrow 0, \gamma (\infty ) \rightarrow 0, \end{aligned}$$
(24)

For calculation purposes, the parameter \(\eta\) at \(\infty\) is chosen large enough. The numerical solution therefore has no discernible variation for \(\ eta\) greater than \(\eta _ {max}\). Depending on the limit condition, \(\eta _{max}\) is set to \(\eta _{max}=\) 10.

Variational formulations

The variational form associated with Eqs. (13)–(18) over a quadratic element \(\Omega _e = (\eta _e, \eta _{e+1})\) is given by

$$\begin{aligned}&\int _{\eta _e}^{\eta _{e+1}} s_1 \{ f'-h\}d\eta = 0, \end{aligned}$$
(25)
$$\begin{aligned}&\int _{\eta _e}^{\eta _{e+1}} s_2 \{h'' + \beta (2hh'' -h'^2- f h''')\nonumber \\&\quad +(1+\lambda ) ( fh' -h^2 - A\left[ \frac{\eta }{2}h' + h \right] \end{aligned}$$
(26)
$$\begin{aligned}&-Mh + \lambda _1 \theta +\lambda _2 S +\lambda _3 \gamma )\}d\eta = 0, \end{aligned}$$
(27)
$$\begin{aligned}&\int _{\eta _e}^{\eta _{e+1}} s_3 \{(1+R)\frac{1}{Pr}\theta '' - h\theta + f\theta ' - A\left( \frac{\eta }{2}\theta ' + 2\theta \right) \nonumber \\&\quad + Nb\gamma '\theta ' + Nt\theta '^2 -Mh\}d\eta = 0, \end{aligned}$$
(28)
$$\begin{aligned}&\int _{\eta _e}^{\eta _{e+1}} s_4 \{ \frac{1}{Sc}S'' - hS + fS' -A\left( \frac{\eta }{2}S + 2S\right) + S_r \theta ''\}d\eta = 0, \end{aligned}$$
(29)
$$\begin{aligned}&\int _{\eta _e}^{\eta _{e+1}} s_5 \{\gamma '' - Le \left[ h\gamma -f\gamma ' + A\left( \frac{\eta }{2}\gamma ' + 2\gamma \right) \right] + \frac{Nt}{Nb}\theta ''\}d\eta = 0, \end{aligned}$$
(30)

where \(s_1, s_2, s_3, s_4\) and \(s_5\) functions are of arbitrary form or test functions.

Finite element formulations

The equations of the finite element model is obtained by replacing the finite element approach of the following form in Eqs. (19)–(23).

$$\begin{aligned} f &= \sum _{j=1}^3 f_j \psi _j,\ h= \sum _{j=1}^3 h_j \psi _j,\ \theta = \sum _{j=1}^3 \theta _j \ \psi _j, \nonumber \\ S &= \sum _{j=1}^3 S_j \psi _j,\ \gamma =\sum _{j=1}^3 \gamma _j \psi _j, \end{aligned}$$
(31)

with \(s_1=s_2=s_3=s_4=s_5=\psi _i (i=1,2)\), where the shape function \(\psi _i\) for a line element \(\Omega _e = (\eta _e,\eta _e+1)\) are given by

$$\begin{aligned} \psi _1 &= \frac{(\eta _{e+1}-\eta )(\eta _{e+1}+\eta _e-2\eta )}{(\eta _{e+1}-\eta _e)^2},\ \psi _2= \frac{4(\eta -\eta _e)(\eta _{e+1}-\eta )}{(\eta _{e+1}-\eta _e)^2},\nonumber \\ \psi _3 &= -\frac{(\eta -\eta _e)(\eta _{e+1}+\eta _e-2\eta )}{(\eta _{e+1}-\eta _e)^2}, \end{aligned}$$
(32)

The model equations of the finite element method are therefore given by

$$\begin{aligned} \begin{bmatrix} F^{11}&F^{12}&F^{13}&F^{14}&F^{15}\\ F^{21}&F^{22}&F^{23}&F^{24}&F^{25}\\ F^{31}&F^{32}&F^{33}&F^{34}&F^{35}\\ F^{41}&F^{42}&F^{43}&F^{44}&F^{45}\\ F^{51}&F^{52}&F^{53}&F^{54}&F^{55} \end{bmatrix} \begin{bmatrix} f \\ h \\ \theta \\ S\\ \gamma \end{bmatrix} = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \\ b^4 \\ b^5 \end{bmatrix} \end{aligned}$$
(33)

where \(F_{mn}\) and \(b_m\) (m, n = 1, 2, 3, 4, 5) are defined as

$$\begin{aligned} F^{11}_{ij} &= \int _{\eta _e}^{\eta _{e+1}} \psi _i\frac{d\psi _j}{d\eta }d\eta ,\nonumber \\ F^{12}_{ij} &= -\int _{\eta _e}^{\eta _{e+1}} \psi _i \psi _j d\eta , \ F^{13}_{ij}=F^{14}_{ij}=0,F^{15}_{ij}= F^{21}_{ij}=0,\nonumber \\ F^{22}_{ij} &= -\int _{\eta _e}^{\eta _{e+1}} \frac{d\psi _i}{d\eta } \frac{d\psi _j}{d\eta }d\eta -\beta \int _{\eta _e}^{\eta _{e+1}}\bar{h} \frac{d\psi _i}{d\eta } \frac{d\psi _j}{d\eta }d\eta \nonumber \\&-2 \beta \int _{\eta _e}^{\eta _{e+1}} \bar{h}' \psi _i \frac{d\psi _j}{d\eta }d\eta +\beta \int _{\eta _e}^{\eta _{e+1}} \frac{d\psi _i}{d\eta }\bar{f} \frac{d^2\psi _j}{d\eta ^2}d\eta \nonumber \\&+ \beta \int _{\eta _e}^{\eta _{e+1}} \psi _i\bar{f}' \frac{d^2\psi _j}{d\eta ^2}d\eta + (1+\lambda )\int _{\eta _e}^{\eta _{e+1}} \psi _i\bar{f} \frac{d^2\psi _j}{d\eta ^2}d\eta \nonumber \\&- (1+\lambda ) \int _{\eta _e}^{\eta _{e+1}}\bar{h}\psi _i \psi _jd\eta \nonumber \\&- A \frac{\eta }{2}(1+\lambda )\int _{\eta _e}^{\eta _{e+1}} \psi _i \frac{d\psi _j}{d\eta }d\eta \nonumber \\&\quad - 2A(1+\lambda ) \int _{\eta _e}^{\eta _{e+1}}\psi _i \psi _jd\eta -M(1+\lambda ) \int _{\eta _e}^{\eta _{e+1}}\psi _i \psi _jd\eta \nonumber \\ F^{23}_{ij} &= (1+\lambda ) \lambda _1\int _{\eta _e}^{\eta _{e+1}}\psi _i \psi _jd\eta , F^{24}_{ij} =(1+\lambda ) \lambda _2\int _{\eta _e}^{\eta _{e+1}}\psi _i \psi _jd\eta , \nonumber \\ F^{25}_{ij} &=(1+\lambda ) \lambda _3\int _{\eta _e}^{\eta _{e+1}}\psi _i \psi _jd\eta ,\nonumber \\ F^{31}_{ij} &= 0,\quad F^{32}_{ij} =-M \int _{\eta _e}^{\eta _{e+1}}\psi _i \psi _jd\eta ,\nonumber \\ F^{33}_{ij} &= -(1+R)\frac{1}{Pr}\int _{\eta _e}^{\eta _{e+1}} \frac{d\psi _i}{d\eta } \frac{d\psi _j}{d\eta }d\eta \nonumber \\&\quad - \int _{\eta _e}^{\eta _{e+1}} \bar{h}\psi _i \psi _j d\eta + \int _{\eta _e}^{\eta _{e+1}}\bar{f} \psi _i \frac{d\psi _j}{d\eta }d\eta \nonumber \\&\quad -A \frac{\eta }{2}\int _{\eta _e}^{\eta _{e+1}} \psi _i \frac{d\psi _j}{d\eta }d\eta - 2A\int _{\eta _e}^{\eta _{e+1}}\psi _i \psi _jd\eta \nonumber \\&\quad + Nb\int _{\eta _e}^{\eta _{e+1}}\bar{\gamma }' \psi _i \frac{d\psi _j}{d\eta }d\eta \nonumber \\&+ Nt\int _{\eta _e}^{\eta _{e+1}}\bar{\theta }' \psi _i \frac{d\psi _j}{d\eta }d\eta ,\quad F^{34}_{ij}=0, F^{35}_{ij} = 0, \nonumber \\ F^{41}_{ij} &= F^{42}_{ij}=0,\nonumber \\ F^{43}_{ij} &= -Sr\int _{\eta _e}^{\eta _{e+1}} \frac{d\psi _i}{d\eta } \frac{d\psi _j}{d\eta }d\eta ,\nonumber \\ F^{44}_{ij} &= -\frac{1}{Sc}\int _{\eta _e}^{\eta _{e+1}} \frac{d\psi _i}{d\eta } \frac{d\psi _j}{d\eta }d\eta + \int _{\eta _e}^{\eta _{e+1}}\bar{f} \psi _i \frac{d\psi _j}{d\eta }d\eta \nonumber \\&- \int _{\eta _e}^{\eta _{e+1}}\bar{h} \psi _i \psi _j{d\eta }d\eta \nonumber \\&-A \frac{\eta }{2}\int _{\eta _e}^{\eta _{e+1}} \psi _i \frac{d\psi _j}{d\eta }d\eta - 2A\int _{\eta _e}^{\eta _{e+1}}\psi _i \psi _jd\eta ,\nonumber \\ F^{45}_{ij} &= F^{51}_{ij} = F^{52}_{ij}=F^{54}_{ij}=0, F^{53}_{ij}= -\frac{Nt}{Nb}\int _{\eta _e}^{\eta _{e+1}} \frac{d\psi _i}{d\eta } \frac{d\psi _j}{d\eta }d\eta \quad \nonumber \\ F^{55}_{ij} &= -\int _{\eta _e}^{\eta _{e+1}} \frac{d\psi _i}{d\eta } \frac{d\psi _j}{d\eta }d\eta + Le \int _{\eta _e}^{\eta _{e+1}}\bar{f} \psi _i \frac{d\psi _j}{d\eta }d\eta \nonumber \\&\quad -Le \int _{\eta _e}^{\eta _{e+1}}\bar{h} \psi _i \psi _j{d\eta }d\eta \nonumber \\&-Le A \frac{\eta }{2}\int _{\eta _e}^{\eta _{e+1}} \psi _i \frac{d\psi _j}{d\eta }d\eta - 2LeA\int _{\eta _e}^{\eta _{e+1}}\psi _i \psi _jd\eta , \end{aligned}$$
(34)

and

$$\begin{aligned} b^1_i &= 0, \ b^2_i=-\left( \psi \frac{dh}{d\eta }\right) ^{\eta _e+1}_{\eta _e},\nonumber \\ b^3_i &= -(1+R)\frac{1}{Pr}\left( \psi \frac{d\theta }{d\eta }\right) ^{\eta _e+1}_{\eta _e},\nonumber \\ b^4_i &= -\frac{1}{Sc}\left( \psi \frac{d S}{d\eta }\right) ^{\eta _e+1}_{\eta _e}- Sr\left( \psi \frac{d S}{d\eta }\right) ^{\eta _e+1}_{\eta _e},\nonumber \\ b^5_i &= -\left( \psi \frac{d\phi }{d\eta }\right) ^{\eta _e+1}_{\eta _e}-\frac{Nt}{Nb}\left( \psi \frac{d\theta }{d\eta }\right) ^{\eta _e+1}_{\eta _e}, \end{aligned}$$
(35)

where \(\bar{f}=\sum _{j=1}^3 \bar{f}_j \psi _j,\ \bar{h}= \sum _{j=1}^3 \bar{h}_j \psi _j,\ \bar{\theta }'= \sum _{j=1}^3 \bar{\theta }'_j \psi _j\) and \(\bar{\phi }' = \sum _{j=1}^3 \bar{\phi }'_j \ \psi _j\) are supposed to be known. Hence after assembling all the element equations, we get the order of \(722\times 722\) matrix. The resulting system is nonlinear, therefore an iterative scheme is utilized in the solution. After the boundary conditions are applied, the remaining system equations are solved by gaussian elimination method and will repeat this process until the desired accuracy of 0.00005 obtained.

Table 1 Comparison of \(-f''(0)\) for various values of M when \(f_w=A=S_f=0\) and Pr when \(M=f_w=S_f=S_t=A=\lambda _1=\lambda _2=R=0\)
Table 2 Comparison of \(-f''(0)\) for various values of A when \(M=f_w=S_f=\lambda _1=\lambda _2=0\)
Table 3 Comparison of \(-\theta '(0)\) for various values of Pr (\(M=f_w=S_\theta =\sigma =\lambda _1=\lambda _2=\lambda _3=R=S_f=0;\))

Results

The velocity distribution decreases with the increment of magnetic parameter M with suction, injunction and no suction in the presence of hydrodynamic slip \(S_f\) and absence of hydrodynamic slip \(S_f\) are depicted in Fig. 2. With the enhancing of viscoelastic parameter \(\beta\) and presence/absence of unsteadiness parameter A effects on increasing the velocity profile with both cases of hydrodynamic and no hydrodynamic slip are discussed in Fig. 3.

Fig. 2
figure2

Effect of M and \(f_w\) on velocity distribution with slip and no slip condition \(S_f\)

Fig. 3
figure3

Effect of \(\beta\) and A on velocity distribution with slip and no slip condition \(S_f\)

Fig. 4
figure4

Effect of \(\lambda _1\) and A on velocity distribution with slip and no slip condition \(S_f\)

Fig. 5
figure5

Effect of \(\lambda _2\) and R on velocity distribution with slip and no slip condition \(S_f\)

Fig. 6
figure6

Effect of \(\lambda _3\) and A on velocity distribution with slip and no slip condition \(S_f\)

Fig. 7
figure7

Effect of R and M on temperature profile with slip and no slip condition \(S_t\)

Fig. 8
figure8

Effect of A and R on temperature profile with slip and no slip condition \(S_t\)

Fig. 9
figure9

Effect of \(\lambda _1\) and \(f_w\) on temperature profile with slip and no slip condition \(S_t\)

Fig. 10
figure10

Effect of Nt and \(f_w\) on temperature profile with slip and no slip condition \(S_t\)

Fig. 11
figure11

Effect of Sc and M on concentration profile with slip and no slip condition \(S_p\)

Fig. 12
figure12

Effect of Sr and A on concentration profile with slip and no slip condition \(S_p\)

Fig. 13
figure13

Effect of Le and A on nanofluid volume fraction profile with slip and no slip condition \(S_g\)

In the absence and presence of the unsteady parameter A, the radiation R and the hydrodynamic slip condition \(S_f\), the velocity profile is enhanced as the buoyancy parameters \(\lambda _1\), \(\lambda _2\) and \(\lambda _3\) increases, the results are described graphically in Figs. 4, 5 and 6. The influence of thermal radiation R and magnetic parameter M with thermal slip and no thermal slip condition are presented in the Fig. 7, applying the magnetic field heats up the fluid and thus reduces the heat and mass transfer rates from the wall causing increases in fluid temperature. We noticed from Fig. 7 that the temperature profile increases as the radiation parameter R value increases. Figures 8 and 9 depict that the temperature decreases as the effect of unsteadiness A increases with the presence and absence of the radiation parameter R and thermal slip condition \(S_t\). The temperature profile increases as the thermophoresis parameter increase with presence and absence of suction \(f_w\) and thermal slip parameter \(S_t\), as described in Fig. 10. The effect of the Schmidt number Sc and the magnetic parameter M on the concentration distribution is shown in Fig. 11. We observed that as the Schmidt number increases, the concentration distribution retard with the presence and absence of solute slip \(S_p\). Figure 12 presents the characteristics of the Sr Soret number and the A instability with the presence and absence of an absolute \(S_p\) slip on the concentration profile. It is clear from the figure that the intensification of the Soret number has reinforced the concentration profile and the associated concentration boundary layer.

Figure 13 describes the characteristics of the Lewis number and the instability of A on the nano-liquid slip profile without nano-liquid slip conditions. It is from the figure that the intensification the Lewis number reduces the volume fraction profile of nanofluid. Moreover, when we increase the value of the Lewis number, the concentration field is reduced because it is inversely proportional to the Brownie coefficient. The Brown diffusion coefficient is low for higher Lewis numbers and the Brown diffusion coefficient results in a decrease in the concentration field. Figure 14 is ready to take action, the Nb Brownian motion parameter and the \(f_w\) suction parameter with the presence and absence of nano-fluid slip on the friction profile of the nano-fluid volume. The figure clearly shows that the volume friction profile of the nano-fluid and the associated boundary layer thickness of the nanoparticle concentration accumulate for a higher parameter for Brownian motion. Figure 15 describe the influences of A, M, \(S_f\), R, Nt and \(S_t\), on velocity and temperature gradients. Effects of M, A and \(S_f\) on skin friction \((C_f)\) are portrayed in Fig. 15a. As previously seen, the fluid velocity decreases by increasing the magnetic parameter due to the Lorentz force caused by the magnetic field; as a result, the rubbing of the skin shows a behavior for the higher magnetic parameter, as shown in this figure. Skin friction diminish for M with increasing the unsteady parameter A. Figure 15b shows the impacts of Nt, \(S_t\) and R on temperature gradient (Nu). These plots show that the the heat transfer rate decreases with the growth of the thermophoresis parameter. This is due to the fact that a larger one the thermophoretic force drives nanoparticles with high thermal conductivity from the hottest region to the ambient temperature liquid. Nano-scale thermophoresis therefore has a considerable influence on the behavior of heat transfer at plate level. Further, increasing the values of R the temperature gradient increases.

Discussion

A theoretical study was conducted to investigate the nanofluidic flux induced by a stretching surface [46]. It was probably the first attempt to reflect the flow of nanofluids on a stretch sheet using the Buongiorno model. Later Makinde and Aziz [47] investigated the effects of convective heat transfer in the nanofluid flow on the boundary layer on a flat plate. Recently, Hashim and Khan [48] studied the heat and mass transfer characteristics for the flow of Carreau nanofluids past a stretching surface. Multiple slip effects on MHD unsteady flow heat and mass transfer impinging on permeable stretching sheet with radiation was discussed [42] and found that the existence of the hydrodynamic slip increases the velocity boundary layer. Here, we focused on describing the effects of different flow variables on velocity distribution \((f'(\eta ))\), temperature profile \((\theta (\eta ))\), concentration \((S(\eta ))\), nano-fluid volume fraction profile \((\gamma (\eta ))\), skin friction \((C_f)\), Nusselt number (Nu), Sherwood number \(Sh_x\) and nano-particle Sherwood number \((Sh_{x,n})\). The nonlinear systems (18)–(22) subject to conditions (23) and (24) are solved numerically by hybrid finite element method. Further the characteristics of magnetic M, Brownian Nb, thermophoresis Nt, Schmidt Sc, Lewis number Le, thermal radiation R,Soret number Sr, unsteady parameter A, Buoyancy parameters \(\lambda _1, \lambda _2, \lambda _3\), Deborah number or viscoelastic parameter \(\beta\), hydrodynamic slip \(S_f\), thermal slip \(S_t\), solutal slip \(S_p\) and the nano-particle slip condition \(S_g\) are presented in this section. The numerical method is validated with the result obtained by Mabood and Das [41], Fazle mabood and Stanford Shateyi [42], Mudassar et al.[49], Gireesha et al. [50], Ishak et al.[45], Ali [43] and Chamkha et al.[44] in terms of skin friction coefficient and an excellent agreement is obtained, the comparison is illustrated in Tables 1, 2 and 3.

Fig. 14
figure14

Effect of Nb and \(f_w\) on nano-fluid volume fraction profile with slip and no slip condition \(S_g\)

Fig. 15
figure15

Effect of M,\(S_f\) and A on skin friction coefficient \(C_f\), and effect of Nt, R and \(S_t\) on Nusselt number

Conclusions

Here we explored a mathematical model to simulate an unsteady two-dimensional magnetohydrodynamic viscoelastic nano-fluid flow of an incompressible electrically conducting fluid over a permeable stretching sheet in the presence of multiple slips, Soret, and thermal radiation effect. The major results are listed below:

  • The boundary layers increases in the presence of multiple slips.

  • The velocity distribution boosts via viscoelastic parameter \((\beta )\) and buoyancy parameters \((\lambda _1,\) \(\lambda _2\), \(\lambda _3)\), and decrease with increase of magnetic parameter (M).

  • Temperature decays through unsteady parameter (A) and buoyancy parameter (\(\lambda _1\)), while it enhanced with radiation (R) and thermophoresis parameter (Nt).

  • Concentration profile enhance via increasing the Soret number (Sr) and diminish with Schmidt number (Sc).

  • For larger estimation of Brownian motion parameter (Nb) the nano-fluid volume fraction profile enhance however it is reduced for Lewis number (Le).

  • Skin friction coefficient shows decreasing behaviour against unsteadiness (A)and magnetic parameter (M).

  • Nusselt number shows increasing behaviour against radiation parameter (R) and thermophoresis parameter (Nt).

  • Sherwood number (\(Sh_x\)) is increasing function of Schmidt number (Sc) and magnetic parameter (M).

  • Nano-particle Sherwood number (\(Sh_{x,n}\)) is enhanced as a function of thermophoresis parameter (Nt) with increasing value of unsteadiness parameter (A).

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Khan, S.A., Nie, Y. & Ali, B. Multiple slip effects on MHD unsteady viscoelastic nano-fluid flow over a permeable stretching sheet with radiation using the finite element method. SN Appl. Sci. 2, 66 (2020). https://doi.org/10.1007/s42452-019-1831-3

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Keywords

  • MHD
  • Multiple slip
  • Viscoelastic buoyant nanofluid
  • Radiation
  • Finite element method