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Journal of Thermal Analysis and Calorimetry

, Volume 135, Issue 2, pp 1021–1030 | Cite as

Thermally radiated squeezed flow of magneto-nanofluid between two parallel disks with chemical reaction

  • Ikram Ullah
  • Muhammad WaqasEmail author
  • Tasawar Hayat
  • Ahmed Alsaedi
  • M. Ijaz Khan
Article

Abstract

This research is exhibited to visualize the squeezed flow of magneto-nanoliquid between two parallel disks. Transportations of heat and mass are characterized through thermal radiation and chemical reaction. Suitable similarity variables lead to dimensionless problem. The homotopic technique is adopted to find out the solutions. Behavior of sundry variables is declared graphically. Physical quantities of curiosity such as skin friction and Nusselt number at both disks are estimated and elaborated. Our investigation depicts that thermal field is augmented via radiation and Brownian diffusion variables. Besides, comparative table is also designed to validate our present outcomes with previous limiting study.

Keywords

Thermal radiation Squeezing flow Nanoparticles Parallel disks Chemical reaction 

List of symbols

u, v, w

Velocity components

x, r, z

Space coordinates

T

Temperature

Th

Upper-disk temperature

K

Second-grade parameter

B0

Uniform magnetic field strength

K1

Reaction rate

DB

Brownian diffusion coefficient

DT

Thermophoretic diffusion coefficient

kf

Thermal conductivity

α*

Thermal diffusivity

α1

Normal stress moduli

w0

Suction/blowing velocity

S

Suction/blowing parameter

Nb

Brownian motion parameter

Nt

Thermophoresis parameter

Derivative via η

f

Dimensionless velocity

θ

Dimensionless temperature

ϕ

Dimensionless concentration

Sq

Squeezing parameter

γ

Chemical reaction parameter

\({\mathcal{L}}_{\text{f}} ,{\mathcal{L}}_{\uptheta} ,{\mathcal{L}}_{\upphi}\)

Linear operator

M

Magnetic parameter

qr

Radiative heat flux

Le

Lewis number

Pr

Prandtl number

Cf0, Cf1

Skin friction at lower and upper disks

Nu0, Nu1

Nusselt number at lower and upper disks

Sh0, Sh1

Sherwood number at lower and upper disks

Rd

Radiation parameter

σ

Electrical conductivity

υ

Kinematic viscosity

ρf

Nanofluid density

η

Transformed coordinate

cp

Specific heat at constant pressure

Rer

Local Reynolds number

μ

Dynamic viscosity

(ρc)f

Fluid heat capacity

p

Pressure

(ρc)p

Nanoparticles effective heat capacity

Nf, Nθ, Nϕ

Nonlinear operators

\(B_{\text{i}}^{ * }\) = (i = 1–8)

Constants

\(\hbar_{\text{f}} ,\hbar_{\uptheta} ,\hbar_{\upphi}\)

Nonzero auxiliary parameters

σ1

Stefan–Boltzmann constant

m1

Mean absorption coefficient

Introduction

Recently considerable interest has been devoted regarding analysis of squeezed flow owing to its prompt improvement in fluid dynamics and its implication in numerous practical and industrial utilizations like polymer processing, food/chemical engineering, injection and comparison casting, biophysical and distinct others. Flow inside nasogastric and syringes tubes is also a kind of squeezing flows. Initial analysis about squeezed flows is elaborated by Stefan [1] who established basic modeling of such flows considering lubrication assumption. Fluid inertia characteristics in squeezed films are elaborated by Kuzma [2]. Siddiqui et al. [3] modeled transient MHD squeezed flow via channel. Analysis of magneto-squeezed flow by infinite channel is described by Sweet et al. [4]. Porous medium impact in squeezed unsteady Casson material flow with magneto-hydrodynamics is examined by Khan et al. [5]. Kumar et al. [6] evaluated variable conductivity characteristics in nonlinear magneto-squeezed material flow. Analysis of radiated squeezed non-Newtonian (Eyring–Powell) material chemically reacted flow is scrutinized by Adesanya et al. [7] and Balazadeh et al. [8].

Heat transportation has major contribution in numerous areas of industry. Besides, high-performance chilling is needed widely in industrial systems. Nanomaterials are sprouting materials that demonstrate thermal features outstanding in comparison with conventional material. Rheological characteristics of nanomaterials together with thermophysical features are a significant matter. There exist distinct commonly utilized base materials like lubricants, bio-liquids, refrigerants, polymeric solutions, ethylene and triethylene glycols having aptitude to rise the thermal competence of standard heat transporting materials. The nanomaterials have ample utilizations for illustration of power generator, melt-spinning, airplanes, micro-machines, micro-reactors, micro-machines chilling and ventilation of papers, glass-fiber production and numerous others. A fascinating characteristic of nanomaterial mechanism is the reality that thermal and Brownian agitation surpasses any settling movement subject to gravity. This characteristic illustrates that theoretical aspect about nanomaterial occurs only when the nanoparticles are smaller enough. Buongiorno [9] recommended that heat augmentation of nanomaterial is predominantly affected by thermal or Brownian agitation. Numerous researches have been elaborated through nanomaterial consideration via distinct physical characteristics (see Refs. [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]).

Visualizing non-Newtonian material flows is turn out to be growingly significant in the material processing. In food process industries for illustration, food materials’ internal flows through devices delicately modify ultimate properties like texture and taste. Viscoelastic materials like gels, bio-fluids, polymer solutions are a class of non-Newtonian materials that own both elastic and viscous properties. The scrolling and stretching of molecular chains yield elastic character of such materials [21]. To elaborate the flow characteristics of viscoelastic materials, several models [22, 23, 24, 25, 26, 27] have been introduced. Here, we intend to utilize viscoelastic (second grade) material for formulation which elaborates normal stress characteristics. Few researches related to such material flows are stated through Refs. [28, 29, 30, 31, 32, 33].

The aforestated researches witnessed that squeezed flows are formulated only for viscous materials. Therefore, we intend to formulate the non-Newtonian (viscoelastic) material squeezed flow with magneto-hydrodynamics. The additional novel features considered for present analysis include thermal radiation, chemical reaction and nanomaterial Brownian and thermophoretic diffusions. Homotopic scheme [34, 35, 36, 37, 38, 39, 40] is utilized for computations of resulting nonlinear systems. Besides, the thermal and solutal distributions corresponding to influential factors are interpreted graphically and numerically.

Description of the problem

Two-dimensional unsteady squeezing flow of second-grade nanoliquid between two parallel disks in the presence of magnetic field is addressed. An electrically conducted liquid is considered. It is also assumed that induced magnetic field is neglected due to low magnetic Reynolds number. Energy and concentration expressions accounted the terms radiation and first-order chemical reaction. The parallel disks are distanced at \(h(t) = H(1 - \alpha t)^{1/2}\). The flow configuration of present system is interpreted in Fig. 1. The governing basic expressions for flow can be written as [36, 37, 38, 39, 40, 41]:
$$\frac{\partial u}{\partial r} + \frac{u}{r} + \frac{\partial w}{\partial r} = 0,$$
(1)
$$\begin{aligned} & \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial r} + w\frac{\partial u}{\partial z} \\ & \quad = - \frac{1}{{\rho_{\text{f}} }}\frac{\partial p}{\partial r} + \frac{\mu }{{\rho_{\text{f}} }}\left( {\frac{{\partial^{2} u}}{{\partial r^{2} }} + \frac{{\partial^{2} u}}{{\partial z^{2} }} + \frac{1}{r}\frac{\partial u}{\partial r} - \frac{u}{{r^{2} }}} \right) \\ & \quad \quad + \frac{{\alpha_{1} }}{{\rho_{\text{f}} }}\left( {\begin{array}{*{20}l} {\frac{2}{r}\frac{{\partial^{2} u}}{\partial t\partial r} - \frac{2}{{r^{2} }}\frac{\partial u}{\partial t} + 2\frac{{\partial^{3} u}}{{\partial t\partial r^{2} }} + \frac{{\partial^{3} u}}{{\partial t\partial z^{2} }}} \hfill \\ {\quad +\, \frac{{\partial^{3} w}}{\partial t\partial r\partial z} + 2\frac{{u^{2} }}{{r^{3} }} - 2\frac{w}{{r^{2} }}\frac{\partial u}{\partial z} - \frac{1}{r}\left( {\frac{\partial u}{\partial z}} \right)^{2} } \hfill \\ {\quad -\, \frac{\partial u}{\partial z}\frac{{\partial^{2} w}}{{\partial z^{2} }} + w\frac{{\partial^{3} u}}{{\partial z^{3} }} - 2\frac{u}{{r^{2} }}\frac{\partial u}{\partial t} + \frac{\partial u}{\partial r}\frac{{\partial^{2} u}}{{\partial z^{2} }}} \hfill \\ {\quad +\, \frac{\partial w}{\partial r}\frac{{\partial^{2} w}}{{\partial z^{2} }} + \frac{1}{r}\left( {\frac{\partial w}{\partial r}} \right)^{2} + 2\frac{w}{r}\frac{{\partial^{2} u}}{\partial z\partial r}} \hfill \\ {\quad +\, \frac{\partial w}{\partial r}\frac{{\partial^{2} u}}{\partial z\partial r} + \frac{\partial w}{\partial z}\frac{{\partial^{2} w}}{\partial z\partial r} - \frac{\partial u}{\partial r}\frac{{\partial^{2} w}}{\partial z\partial r} + u\frac{{\partial^{3} u}}{{\partial r\partial z^{2} }}} \hfill \\ {\quad +\, w\frac{{\partial^{3} u}}{{\partial r\partial z^{2} }} + 2\frac{u}{r}\frac{{\partial^{2} u}}{{\partial r^{2} }} + \frac{\partial u}{\partial r}\frac{{\partial^{2} u}}{{\partial r^{2} }} + \frac{\partial u}{\partial z}\frac{{\partial^{2} w}}{{\partial r^{2} }}} \hfill \\ {\quad +\, 2\frac{\partial w}{\partial r}\frac{{\partial^{2} w}}{{\partial r^{2} }} + 2w\frac{{\partial^{3} u}}{{\partial z\partial r^{2} }} + u\frac{{\partial^{3} w}}{{\partial z\partial r^{2} }} + 2u\frac{{\partial^{3} u}}{{\partial r^{3} }}} \hfill \\ \end{array} } \right) - \frac{{\sigma B^{2} u}}{{\rho_{\text{f}} }}, \\ \end{aligned}$$
(2)
$$\begin{aligned} & \frac{\partial w}{\partial t} + u\frac{\partial w}{\partial r} + w\frac{\partial w}{\partial z} \\ & \quad = - \frac{1}{{\rho_{\text{f}} }}\frac{\partial p}{\partial z} + \frac{\mu }{{\rho_{\text{f}} }}\left( {\frac{{\partial^{2} w}}{{\partial r^{2} }} + \frac{{\partial^{2} w}}{{\partial z^{2} }} + \frac{1}{r}\frac{\partial w}{\partial r}} \right) \\ & \quad \quad + \frac{{\alpha_{1} }}{{\rho_{\text{f}} }}\left( {\begin{array}{*{20}l} {\frac{1}{r}\frac{{\partial^{2} u}}{\partial t\partial z} + \frac{1}{r}\frac{{\partial^{2} w}}{\partial t\partial r} + \frac{{\partial^{3} w}}{{\partial t\partial r^{2} }} + \frac{{\partial^{3} u}}{\partial t\partial r\partial z} + 2\frac{{\partial^{3} w}}{{\partial t\partial z^{2} }}} \hfill \\ {\quad -\, \frac{1}{r}\frac{\partial u}{\partial z}\frac{\partial w}{\partial z} + \frac{w}{r}\frac{{\partial^{2} u}}{{\partial z^{2} }} + \frac{\partial u}{\partial z}\frac{{\partial^{2} u}}{{\partial z^{2} }} + 2\frac{\partial w}{\partial z}\frac{{\partial^{2} w}}{{\partial z^{2} }}} \hfill \\ {\quad +\, 2w\frac{{\partial w^{3} }}{{\partial z^{3} }} + \frac{1}{r}\frac{\partial u}{\partial z}\frac{\partial u}{\partial r} + \frac{1}{r}\frac{\partial w}{\partial z}\frac{\partial w}{\partial r} + \frac{\partial w}{\partial r}\frac{{\partial^{2} u}}{{\partial z^{2} }}} \hfill \\ {\quad -\, \frac{1}{r}\frac{\partial u}{\partial r}\frac{\partial w}{\partial r} + \frac{u}{r}\frac{{\partial^{2} u}}{\partial z\partial r} - \frac{\partial w}{\partial z}\frac{{\partial^{2} u}}{\partial z\partial r} + 2\frac{\partial u}{\partial r}\frac{{\partial^{2} u}}{\partial z\partial r}} \hfill \\ {\quad +\, \frac{w}{r}\frac{{\partial^{2} w}}{\partial z\partial r} + \frac{\partial u}{\partial z}\frac{{\partial^{2} w}}{\partial z\partial r} + w\frac{{\partial^{3} u}}{{\partial r\partial z^{2} }} + 2u\frac{{\partial^{3} w}}{{\partial r\partial z^{2} }}} \hfill \\ {\quad +\, \frac{\partial u}{\partial z}\frac{{\partial^{2} u}}{{\partial r^{2} }} - \frac{\partial w}{\partial r}\frac{{\partial^{2} u}}{{\partial r^{2} }} + \frac{u}{r}\frac{{\partial^{2} w}}{{\partial r^{2} }} + \frac{\partial w}{\partial z}\frac{{\partial^{2} w}}{{\partial r^{2} }}} \hfill \\ {\quad +\, u\frac{{\partial^{3} u}}{{\partial z\partial r^{2} }} + w\frac{{\partial^{3} w}}{{\partial z\partial r^{2} }} + u\frac{{\partial^{3} w}}{{\partial r^{3} }}} \hfill \\ \end{array} } \right), \\ \end{aligned}$$
(3)
$$\begin{aligned} \frac{\partial T}{\partial t} + u\frac{\partial T}{\partial r} + w\frac{\partial T}{\partial z} = & \alpha^{ * } \left( {\frac{{\partial^{2} T}}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial T}{\partial r} + \frac{{\partial^{2} T}}{{\partial z^{2} }}} \right) \\ & + \left( {\rho c} \right)_{\text{p}} /\left( {\rho c} \right)_{\text{f}} \left( {\begin{array}{*{20}l} {D_{\text{B}} \left( {\frac{\partial C}{\partial r}\frac{\partial T}{\partial r} + \frac{\partial C}{\partial z}\frac{\partial T}{\partial z}} \right)} \hfill \\ {\quad +\, \frac{{D_{\text{T}} }}{{T_{\text{m}} }}\left( {\left( {\frac{\partial T}{\partial r}} \right)^{2} + \left( {\frac{\partial T}{\partial z}} \right)^{2} } \right)} \hfill \\ \end{array} } \right) \\ & - \frac{1}{{(\rho c)_{\text{f}} }}\frac{{\partial q_{\text{r}} }}{\partial z}, \\ \end{aligned}$$
(4)
$$\begin{aligned} \frac{\partial C}{\partial t} + u\frac{\partial C}{\partial r} + w\frac{\partial C}{\partial z} = & D_{\text{B}} \left( {\frac{{\partial^{2} C}}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial C}{\partial r} + \frac{{\partial^{2} C}}{{\partial z^{2} }}} \right) \\ & + \frac{{D_{\text{T}} }}{{T_{\text{m}} }}\left( {\frac{{\partial^{2} T}}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial T}{\partial r} + \frac{{\partial^{2} T}}{{\partial z^{2} }}} \right) \\ & - K_{1} (C - C_{\text{h}} ), \\ \end{aligned}$$
(5)
$$\begin{aligned} u & = 0,\quad w = - w_{0} ,\quad T = T_{\text{w}} ,\quad C = C_{\text{w}} \;{\text{at}}\;z = 0, \\ u & = 0,\quad w = \frac{\partial h}{\partial t},\quad T = T_{\text{h}} ,\quad C = C_{\text{h}} \;{\text{at}}\;z = h\left( t \right). \\ \end{aligned}$$
(6)
Here (u, w) are the fluid velocities in (r, z)-directions, respectively, p the pressure, σ the electrical conductivity, ν the kinematic viscosity, ρf the density of base liquid, α1 the normal stress moduli, μ the dynamic viscosity, T the temperature, (ρc)f the heat capacity of liquid, (ρc)p the nanoparticles effective heat capacity, Th the upper-disk temperature, C the concentration, \(\alpha^{ * } = k_{\text{f}} /\left( {\rho c} \right)_{\text{f}}\) the thermal diffusivity, DB the coefficient of Brownian diffusion, k the thermal conductivity, K1 the reaction rate and DT the coefficient of thermophoresis diffusion. Through Rosseland’s approximation, the radiative heat flux qr is
$$q_{\text{r}} = - \frac{{4\sigma_{1} }}{{3m_{1} }}\frac{{\partial (T^{4} )}}{\partial z},$$
(7)
where m1 designates the coefficient of mean absorption and σ1 presents the Stefan–Boltzmann. Considering
$$T^{4} \cong - 3T_{\infty }^{4} + 4T_{\infty }^{3} T,$$
(8)
we have from Eq. (7)
$$\frac{{\partial q_{\text{r}} }}{\partial z} = - \frac{{16\sigma_{1} T_{\infty }^{3} }}{{3m_{1} }}\frac{{\partial^{2} T}}{{\partial z^{2} }}.$$
(9)
Fig. 1

Flow configuration

Upon using Eq. (9), the energy equation can be converted to the following form:
$$\begin{aligned} \frac{\partial T}{\partial t} + u\frac{\partial T}{\partial r} + w\frac{\partial T}{\partial z} & = \alpha^{ * } \left( {\frac{{\partial^{2} T}}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial T}{\partial r} + \frac{{\partial^{2} T}}{{\partial z^{2} }}} \right) \\ & \quad + \frac{{\left( {\rho c} \right)_{\text{p}} }}{{\left( {\rho c} \right)_{\text{f}} }}\left( {\begin{array}{*{20}l} {D_{B} \left( {\frac{\partial C}{\partial r}\frac{\partial T}{\partial r} + \frac{\partial C}{\partial z}\frac{\partial T}{\partial z}} \right)} \hfill \\ {\quad +\, \frac{{D_{\text{T}} }}{{T_{\text{m}} }}\left( {\left( {\frac{\partial T}{\partial r}} \right)^{2} + \left( {\frac{\partial T}{\partial z}} \right)^{2} } \right)} \hfill \\ \end{array} } \right) \\ & \quad + \frac{1}{{(\rho c)_{\text{f}} }}\frac{{16\sigma_{1} T_{\infty }^{3} }}{{3m_{1} }}\frac{{\partial^{2} T}}{{\partial z^{2} }}. \\ \end{aligned}$$
(10)
The non-dimensional variables can be defined as:
$$\begin{aligned} u & = \frac{\alpha r}{{2\left( {1 - \alpha t} \right)}}f^{\prime } (\eta ),\quad w = - \frac{\alpha H}{{\sqrt {1 - \alpha t} }}f(\eta ),\quad \eta = \frac{z}{{H\sqrt {1 - \alpha t} }}, \\ \theta \left( \eta \right) & = \frac{{T - T_{\text{h}} }}{{T_{\text{w}} - T_{\text{h}} }},\quad \phi (\eta ) = \frac{{C - C_{\text{h}} }}{{C_{\text{w}} - C_{\text{h}} }}. \\ \end{aligned}$$
(11)
After elimination of pressure gradient expressions (2)–(6) and (10) lead to
$$f^{iv} - \left( {S_{\text{q}} \left( {\eta f^{\prime \prime \prime } + 3f^{\prime \prime } - 2ff^{\prime \prime \prime } } \right)} \right) + \left( {\frac{K}{2}\left( {\eta f^{v} + 5f^{iv} - 2ff^{{^{v} }} } \right) - M^{2} f^{\prime \prime } = 0} \right),$$
(12)
$$\left. {\left( {1 + \frac{4}{3}{\text{Rd}}} \right)\theta^{\prime \prime } + \mathop {Pr}\limits S_{\text{q}} \left( {f\theta^{\prime } - \eta \theta^{\prime } } \right) + \mathop {Pr}\limits N_{\text{b}} \theta^{\prime } \phi^{\prime } + \mathop {Pr}\limits N_{\text{t}} \theta^{{{\prime }2}} = 0,} \right\}$$
(13)
$$\phi^{\prime \prime } + \mathop {Pr}\limits Le\,S_{\text{q}} \left( {f\phi^{\prime } - \eta \phi^{\prime } } \right) + \frac{{N_{\text{t}} }}{{N_{\text{b}} }}\theta^{\prime \prime } - \mathop {Pr}\limits Le\,\gamma \phi = 0,$$
(14)
$$f(0) = S,\quad f^{\prime } (0) = 0,\quad \theta (0) = 1,\quad \phi (0) = 1,$$
(15)
$$f(1) = \frac{1}{2},\quad f^{\prime } (1) = 0,\quad \theta (1) = 0,\quad \phi (1) = 0.$$
(16)
Here \(Pr\left( { = \frac{\nu }{{\alpha^{ * } }}} \right)\) signifies the Prandtl number, \(N_{\text{b}} \left( { = \frac{{\tau D_{\text{B}} }}{\nu }\left( {C_{\text{w}} - C_{\text{h}} } \right)} \right)\) the Brownian motion parameter, \(Le\left( { = \frac{{\alpha^{ * } }}{{D_{\text{B}} }}} \right)\) the Lewis number, \(S\left( { = \frac{{w_{0} }}{\alpha H}} \right)\) the blowing/suction parameter, \(N_{\text{t}} \left( { = \frac{{\tau D_{\text{T}} }}{{\nu T_{\text{m}} }}\left( {T_{\text{w}} - T_{\text{h}} } \right)} \right)\) the thermophoresis parameter, \(K\left( { = \frac{{\alpha \alpha_{1} }}{\mu (1 - \alpha t)}} \right)\) the second-grade parameter, \(\gamma \left( { = \frac{{(1 - \alpha t)K_{1} }}{\alpha }} \right)\) the chemical reaction parameter, \({\text{Rd}}\left( { = \frac{{4\sigma_{1} T_{\infty }^{3} }}{{km_{1} }}} \right)\) the radiation parameter, \(M\left( { = HB_{0} \sqrt {\frac{\sigma }{\mu }} } \right)\) the magnetic parameter and \(S_{\text{q}} \left( { = \frac{{\alpha H^{2} }}{2\nu }} \right)\) the squeezing parameter. Skin frictions corresponding to lower and upper disks are:
$$C_{{{\text{f}}0}} = \frac{{\left. {\tau_{\text{rz}} } \right|_{{{\text{z}} = 0}} }}{{\rho \left( {\frac{\alpha H}{{2\left( {1 - \alpha t} \right)^{1/2} }}} \right)^{2} }},$$
(17)
$$C_{{{\text{f}}0}} = \frac{{\left. {\tau_{\text{rz}} } \right|_{{{\text{z}} = 0}} }}{{\rho \left( {\frac{\alpha H}{{2\left( {1 - \alpha t} \right)^{1/2} }}} \right)^{2} }},$$
(18)
and
$$C_{{{\text{f}}1}} = \frac{{\left. {\tau_{\text{rz}} } \right|_{{{\text{z}} = {\text{h}}\left( {\text{t}} \right)}} }}{{\rho \left( {\frac{\alpha H}{{2\left( {1 - \alpha t} \right)^{1/2} }}} \right)^{2} }},$$
(19)
$$\begin{aligned} \tau_{\text{rz}} & = \mu \left( {\frac{\partial u}{\partial z} + \frac{\partial w}{\partial r}} \right) \\ & \quad + \alpha_{1} \left( {\begin{array}{*{20}l} {\frac{{\partial^{2} u}}{\partial t\partial z} + \frac{{\partial^{2} w}}{\partial t\partial r} + u\left( {\frac{{\partial^{2} u}}{\partial r\partial z} + \frac{{\partial^{2} w}}{{\partial r^{2} }}} \right) + w\left( {\frac{{\partial^{2} u}}{{\partial z^{2} }} + \frac{{\partial^{2} w}}{\partial z\partial r}} \right)} \hfill \\ {\quad -\, \frac{\partial w}{\partial z}\frac{\partial u}{\partial r} + \frac{\partial w}{\partial z}\frac{\partial w}{\partial r} + \frac{\partial u}{\partial z}\frac{\partial u}{\partial r} - \frac{\partial w}{\partial r}\frac{\partial u}{\partial r}} \hfill \\ \end{array} } \right). \\ \end{aligned}$$
(20)
The dimensionless forms of skin frictions are:
$$\begin{aligned} \frac{{H^{2} }}{{r^{2} }}Re_{\text{r}} C_{{{\text{f}}0}} = & \left( {1 + \frac{3}{2}K} \right)\;f^{\prime \prime } \left( 0 \right), \\ \frac{{H^{2} }}{{r^{2} }}Re_{\text{r}} C_{{{\text{f}}1}} = & \left( {1 + \frac{3}{2}K} \right)\;f^{\prime \prime } \left( 1 \right), \\ \end{aligned}$$
(21)
with
$$Re_{\text{r}}^{ - 1} = \frac{2\nu }{{r\alpha H\left( {1 - \alpha t} \right)^{1/2} }}.$$
(22)
Local Nusselt numbers at lower and upper disks are:
$$\begin{aligned} Nu_{{{\text{r}}0}} & = - \frac{H}{{\left( {T_{\text{w}} - T_{\text{h}} } \right)}}\left. {\frac{\partial T}{\partial z}} \right|_{{{\text{z}} = 0}} + (q_{\text{r}} )_{{{\text{z}} = 0}} \\ & = - \frac{1}{{\sqrt {1 - \alpha t} }}\left( {1 + \frac{4}{3}{\text{Rd}}} \right)\theta^{\prime } \left( 0 \right), \\ \end{aligned}$$
(23)
$$\begin{aligned} Nu_{{{\text{r}}1}} & = - \frac{H}{{\left( {T_{\text{w}} - T_{\text{h}} } \right)}}\left. {\frac{\partial T}{\partial z}} \right|_{{{\text{z}} = {\text{h}}\left( {\text{t}} \right)}} + (q_{\text{r}} )_{{{\text{z}} = {\text{h}}\left( {\text{t}} \right)}} \\ & = - \frac{1}{{\sqrt {1 - \alpha t} }}\left( {1 + \frac{4}{3}{\text{Rd}}} \right)\theta^{\prime } \left( 1 \right). \\ \end{aligned}$$
(24)
Local Sherwood numbers at lower and upper disks are:
$$Sh_{{{\text{r}}_{0} }} = - \frac{H}{{\left( {C_{\text{w}} - C_{\text{h}} } \right)}}\left. {\frac{\partial C}{\partial z}} \right|_{{{\text{z}} = 0}} = - \frac{1}{{\sqrt {1 - \alpha t} }}\phi^{\prime } (0),$$
(25)
$$Sh_{{{\text{r}}_{1} }} = - \frac{H}{{\left( {C_{\text{w}} - C_{\text{h}} } \right)}}\left. {\frac{\partial C}{\partial z}} \right|_{{{\text{z}} = {\text{h}}\left( {\text{t}} \right)}} = - \frac{1}{{\sqrt {1 - \alpha t} }}\phi^{\prime } (1).$$
(26)

Homotopic solutions

The initial estimations and operators are expressed as follows:
$$\begin{aligned} f_{0} \left( \xi \right) & = \left( { - 1 + 2S} \right)\xi^{3} - \frac{1}{2}\left( { - 3 + 3S} \right)\xi^{2} + S, \\ \theta_{0} \left( \xi \right) & = 1 - \xi ,\quad \phi_{0} (\xi ) = 1 - \xi , \\ \end{aligned}$$
(27)
$${\bar{\mathbf{L}}}_{\text{f}} \left( f \right) = \frac{{{\text{d}}^{4} f}}{{{\text{d}}\xi^{4} }},\quad {\bar{\mathbf{L}}}_{\uptheta} \left( \theta \right) = \frac{{{\text{d}}^{2} \theta }}{{{\text{d}}\xi^{2} }},\quad {\bar{\mathbf{L}}}_{{\varphi }} \left( \phi \right) = \frac{{{\text{d}}^{2} \phi }}{{{\text{d}}\xi^{2} }},$$
(28)
The above operators obey
$$\begin{aligned} & {\bar{\mathbf{L}}}_{\text{f}} \left[ {B_{1}^{ * } + B_{2}^{ * } \xi + B_{3}^{ * } \xi^{2} + B_{4}^{ * } \xi^{3} } \right] = 0, \\ & {\bar{\mathbf{L}}}_{\uptheta} \left[ {B_{5}^{ * } + B_{6}^{ * } \xi } \right] = 0, \\ & {\bar{\mathbf{L}}}_{{\varphi }} \left[ {B_{7}^{ * } + B_{8}^{ * } \xi } \right] = 0. \\ \end{aligned}$$
(29)
Here \(B_{i}^{ * }\) (i = 1–8) are the arbitrary constants. The zeroth-order deformation problems can be put into the forms:Here Þ ∈ [0, 1] shows the embedding parameter, \(\hbar_{\text{f}}\), \(\hbar_{\uptheta}\) and \(\hbar_{\upphi}\) the auxiliary variables, and Nf, \({\mathbf{N}}_{\uptheta}\) and \({\mathbf{N}}_{\upphi}\) the nonlinear operators.
The mth-order deformation problems are:
$${\bar{\mathbf{L}}}_{\text{f}} \left[ {f_{\text{m}} \left( \xi \right) - \chi_{\text{m}} f_{{{\text{m}} - 1}} \left( \xi \right)} \right] = \hbar_{\text{f}} {\mathbf{R}}_{\text{m}}^{\text{f}} \left( \xi \right),$$
(38)
$${\bar{\mathbf{L}}}_{\uptheta} \left[ {\theta_{\text{m}} \left( \xi \right) - \chi_{\text{m}} \theta_{{{\text{m}} - 1}} \left( \xi \right)} \right] = \hbar_{\uptheta} {\mathbf{R}}_{\text{m}}^{\theta } \left( \xi \right),$$
(39)
$${\bar{\mathbf{L}}}_{\upphi} \left[ {\phi_{\text{m}} \left( \xi \right) - \chi_{\text{m}} \phi_{{{\text{m}} - 1}} \left( \xi \right)} \right] = \hbar_{\upphi} {\mathbf{R}}_{\text{m}}^{\upphi} \left( \xi \right),$$
(40)
$$f_{\text{m}} \left( 0 \right) = f_{\text{m}}^{\prime } \left( 0 \right) = 0,\quad \theta_{\text{m}} \left( 0 \right) = \phi_{\text{m}} (0) = 0,$$
(41)
$$f_{\text{m}} \left( 1 \right) = f_{\text{m}}^{\prime } \left( 1 \right) = 0,\quad \phi_{\text{m}} \left( 1 \right) = \theta_{\text{m}} \left( 1 \right) = 0,$$
(42)
$$\begin{aligned} {\mathbf{R}}_{\text{m}}^{\text{f}} \left( \xi \right) = & f_{{{\text{m}} - 1}}^{{\prime {\text{v}}}} - S_{\text{q}} \left( {\xi f_{{{\text{m}} - 1}}^{\prime \prime \prime } + 3f_{{{\text{m}} - 1}}^{\prime \prime } - 2\mathop \sum \limits_{k = 0}^{m - 1} f_{{{\text{m}} - 1 - {\text{k}}}} f_{\text{k}}^{\prime \prime \prime } } \right) \\ & + \frac{K}{2}\left( {\xi f_{{{\text{m}} - 1}}^{\text{v}} + 5f_{{{\text{m}} - 1}}^{\text{iv}} - 2\mathop \sum \limits_{k = 0}^{m - 1} f_{{{\text{m}} - 1 - {\text{k}}}} f_{\text{k}}^{\text{iv}} } \right) - M^{2} f_{{{\text{m}} - 1}}^{\prime \prime } , \\ \end{aligned}$$
(43)
$${\mathbf{R}}_{\text{m}}^{\uptheta} \left( \xi \right) = \frac{1}{Pr}\left( {1 + \frac{4}{3}Rd} \right)\theta_{{{\text{m}} - 1}}^{\prime \prime } + S_{\text{q}} \left( {\mathop \sum \limits_{k = 0}^{m - 1} f_{{{\text{m}} - 1 - {\text{k}}}} \theta_{\text{k}}^{\prime } - \eta \theta_{{{\text{m}} - 1}}^{\prime } } \right) + N_{\text{b}} \mathop \sum \limits_{k = 0}^{m - 1} \theta_{{{\text{m}} - 1 - {\text{k}}}}^{\prime } \phi_{\text{k}}^{\prime } + N_{\text{t}} \mathop \sum \limits_{k = 0}^{m - 1} \theta_{{{\text{m}} - 1 - {\text{k}}}}^{\prime } \theta_{\text{k}}^{\prime } ,$$
(44)
$${\mathbf{R}}_{\text{m}}^{\upphi} \left( \xi \right) = \phi_{{{\text{m}} - 1}}^{\prime \prime } + \frac{{N_{\text{t}} }}{{N_{\text{b}} }}\theta_{{{\text{m}} - 1}}^{\prime \prime } + \mathop {Pr}\limits \;Le\;S_{\text{q}} \left( {\mathop \sum \limits_{k = 0}^{m - 1} f_{{{\text{m}} - 1 - {\text{k}}}} \phi_{\text{k}}^{\prime } - \eta \phi_{{{\text{m}} - 1}}^{\prime } } \right) - \mathop {Pr}\limits \;Le\;\gamma \phi ,$$
(45)
$$\chi_{\text{m}} = \left\{ {\begin{array}{*{20}l} {0,} \hfill & {m \le 1,} \hfill \\ {1,} \hfill & {m > 1.} \hfill \\ \end{array} } \right.$$
(46)
The general solutions \((f_{\text{m}} ,\theta_{\text{m}} ,\phi_{\text{m}} )\) through special solutions \((f_{\text{m}}^{ * } ,\theta_{\text{m}}^{ * } ,\phi_{\text{m}}^{ * } )\) are:
$$f_{\text{m}} \left( \xi \right) = f_{\text{m}}^{ * } \left( \xi \right) + B_{1}^{ * } + B_{2}^{ * } \xi + B_{3}^{ * } \xi^{2} + B_{4} \xi^{3} ,$$
(47)
$$\theta_{\text{m}} \left( \xi \right) = \theta_{\text{m}}^{ * } \left( \xi \right) + B_{5}^{ * } + B_{6}^{ * } \xi ,$$
(48)
$$\phi_{\text{m}} \left( \xi \right) = \phi_{\text{m}}^{ * } \left( \xi \right) + B_{7}^{ * } + B_{8}^{ * } \xi ,$$
(49)
where the constants \(B_{\text{i}}^{ * }\) (i = 1–8) after using Eqs. (40) and (41) are:
$$\begin{aligned} B_{1}^{ * } & = \left. {f_{\text{m}}^{ * } \left( \xi \right)} \right|_{{\upxi = 0}} ,\quad B_{2}^{ * } = \left. {\frac{{\partial f_{\text{m}}^{ * } \left( \xi \right)}}{\partial \xi }} \right|_{{\upxi = 0}} ,\\ B_{3}^{ * } & = - 3\left. {f_{\text{m}}^{ * } \left( \xi \right)} \right|_{\xi = 1} + \left. {\frac{{\partial f_{\text{m}}^{ * } \left( \xi \right)}}{\partial \xi }} \right|_{{\upxi = 1}} - 3B_{1}^{ * } - 2B_{2}^{ * } , \\ B_{4}^{ * } & = 2\left. {f_{\text{m}}^{ * } \left( \xi \right)} \right|_{{\upxi = 1}} - \left. {\frac{{\partial f_{\text{m}}^{ * } \left( \xi \right)}}{\partial \xi }} \right|_{{\upxi = 1}} + 2B_{1}^{ * } + B_{2}^{ * } , \\ B_{5}^{ * } & = - \left. {\theta_{\text{m}}^{ * } \left( \xi \right)} \right|_{{\upxi = 0}} ,\quad B_{6}^{ * } = \left. {\theta_{m}^{ * } \left( \xi \right)} \right|_{{\upxi = 0}} - \left. {\theta_{\text{m}}^{ * } \left( \xi \right)} \right|_{{\upxi = 1}} ,\\ B_{7}^{ * } & = - \left. {\phi_{\text{m}}^{ * } \left( \xi \right)} \right|_{{\upxi = 0}} ,\quad B_{8}^{ * } = \left. {\phi_{\text{m}}^{ * } \left( \xi \right)} \right|_{{\upxi = 0}} - \left. {\phi_{\text{m}}^{ * } \left( \xi \right)} \right|_{\xi = 1} . \\ \end{aligned}$$
(50)

Convergence analysis

It is quite clear that derived series solutions comprise the auxiliary variables \(\hbar_{\text{f}} ,\hbar_{\uptheta}\) and \(\hbar_{\upphi}\) which are very important in controlling the convergence. Convergence region is the zone parallel to \(\hbar\)-axis. In order to acquire the sustainable values of \(\hbar_{\text{f}} ,\hbar_{\uptheta}\) and \(\hbar_{\upphi}\), the \(\hbar\)-curves are depicted at 20th order of estimations. Figures 2 and 3 evidently portray that the convergence region lies inside the ranges \(- \,0.89 \le \hbar_{\text{f}} \le - \,0.1\), \(- \,1.2 \le \hbar_{\uptheta} \le - \,0.11\) and \(- \,1.2 \le \hbar_{\upphi} \le - \,0.2\) for lower disk (η = 0) and \(- \,0.7 \le \hbar_{\text{f}} \le - \,0.0.18\), \(- \,0.85 \le \hbar_{\uptheta} \le - \,0.2\) and \(- \,0.9 \le \hbar_{\upphi} \le - \,0.19\) for upper disk (η = 1).
Fig. 2

ℏ-curves for f, θ and ϕ at lower disk

Fig. 3

ℏ-curves for f, θ and ϕ at upper disk

Analysis of the outcomes

Here we manifest in this section that how influential variables influence the non-dimensional temperature and concentration. Figure 4 signifies that how magnetic variable affects the nanoliquid temperature θ(η). One can observe that more resistance is offered to the fluid particles due to the strength in Lorentz force. Thus, increasing trend in temperature is seen for larger M. Curves of θ(η) against Sq are plotted in Fig. 5. Less temperature distribution is perceived for larger Sq. In fact, stronger squeezing variable Sq decays the kinematics viscosity, which thins the thermal field. Variation in temperature θ(η) for varying Pr is depicted in Fig. 6. An augmentation in Pr shows decay in temperature. Physically, an increment in Pr diminishes the thermal diffusivity, which substantially decays the temperature distribution. Features of thermophoresis parameter Nt on θ(η) are outlined in Fig. 7. It is pointed out that an increase in Nt improves the liquid thermal state. Physically, higher Nt estimations lead to stronger thermophoretic force, which pushes the nanoparticles from hot toward the cold zone. In Fig. 8, it is narrated that higher Nb augments the nanoparticles temperature θ(η). One noted that stronger Nb pronounces the nanoparticles collision which generates more heat and eventually rises θ(η). Figure 9 is sketched to see the characteristics of Rd on θ(η). It is pointed out that higher Rd enhances the thermal boundary layer. Physically in radiation process, more heating to the active liquid results in an increment of θ(η). Behavior of Nt on solutal field ϕ(η) is disclosed in Fig. 10. It is examined that an enhancement in ϕ(η) is noticed for larger Nt. In fact, an increment in thermophoresis affect generated by thermal gradient causes sizeable mass flux on disk which rises the concentration. Variation in concentration field against Nb is portrayed in Fig. 11. Here, larger Nb decays the concentration field. Higher Nb causes to augment the motion of nanoparticles and ultimately the viscosity of nanoliquid decays. That is why ϕ(η) decays through Nb. Figure 12 exhibits the feature of Le on concentration distribution. Concentration field is enhanced for larger Le. In fact, Le and Nb have inverse link with each other. Higher estimations of Le decay the Brownian diffusion, which consequently declines the concentration. Figure 13 presents that larger Sq leads to higher ϕ(η). The variation of concentration field with chemical reaction parameter is disclosed in Fig. 14. It is observed that an increase in the value of chemical reaction parameter decreases the concentration of species in the boundary layer. This is due to the fact that chemical reaction in this system results in consumption of the chemical and hence results in decrease in concentration profile. Tables 1 and 2 show that 15th order of estimations is adequate regarding the convergence for both lower and upper disks. Surface drag forces at lower and upper disks are computed in Table 3. It is observed that surface drag forces at both disks are enhanced via second-grade parameter K, while reverse feature is found for suction parameter S. Table 4 demonstrates the numerical data of local Nusselt numbers at the lower and upper disks via S, M, Sq, Nt, Nb, Le and Pr. It is analyzed that local Nusselt numbers at both disks are reduced via Sq and Le, while reverse behavior is observed for higher M and Pr. Table 5 depicts the comparison of presented investigation with Hayat et al. [41] in a limiting sense for M and Sq. Here good agreement is noted.
Fig. 4

Variation of θ(η) through M

Fig. 5

Variation of θ(η) through Sq

Fig. 6

Variation of θ(η) through Pr

Fig. 7

Variation of θ(η) through Nt

Fig. 8

Variation of θ(η) through Nb

Fig. 9

Variation of θ(η) through Rd

Fig. 10

Variation of ϕ(η) through Nt

Fig. 11

Variation of ϕ(η) through Nb

Fig. 12

Variation of ϕ(η) through Le

Fig. 13

Variation of ϕ(η) through Sq

Fig. 14

Variation of ϕ(η) through γ

Table 1

Convergence at lower disk when K = 0.3 = K1, M = Nt = 0.2 = S, Nb = 0.5 = R, Sq = Le = Pr = 1.0 and γ = 0.3

Order of approximations

− f″(0)

− θ′(0)

− ϕ′(0)

1

1.8790

0.8358

1.0725

5

1.9130

0.7698

1.2264

10

1.9131

0.7696

1.2326

15

1.9131

0.7696

1.2330

20

1.9131

0.7696

1.2330

30

1.9131

0.7696

1.2330

40

1.9131

0.7696

1.2330

Table 2

Convergence at upper disk when K = 0.3 = K1, M = Nt = 0.2 = S, Nb = 0.5 = R, Sq = Le = Pr = 1.0 and γ = 0.3

Order of approximations

− f″(1)

− θ′(1)

− ϕ′(1)

1

1.81903

1.11083

1.03917

5

1.79625

1.17893

1.04539

10

1.79621

1.17907

1.04519

15

1.79621

1.17905

1.04523

20

1.79621

1.17905

1.04524

30

1.79621

1.17905

1.04524

50

1.79621

1.17905

1.04524

Table 3

Skin friction coefficients at the lower and upper disks via S, K, M and Sq

K

M

S

S q

C f0

C f1

0.1

0.2

0.2

1.0

2.192644

2.087126

0.2

   

2.426313

2.343562

0.3

   

2.659187

2.604499

0.3

0.0

0.2

1.0

2.657925

2.603901

 

0.5

  

2.665798

2.607639

 

0.9

  

2.683339

2.616023

0.3

0.2

0.0

1.0

4.498153

4.414397

  

0.3

 

1.759886

1.721182

  

0.5

 

0.8736101

0.8528635

0.3

0.2

0.2

0.6

2.596938

2.606282

   

1.0

2.659187

2.604499

   

1.5

2.736807

2.603390

Table 4

Local Nusselt numbers at the lower and upper disks via S, M, Sq, Nt, Nb, Le and γ

S

K

M

S q

N t

N b

Le

γ

θ′(0)

θ′(1)

0.0

0.3

0.2

1.0

0.2

0.5

1.0

1.0

0.77444

1.21516

0.6

       

0.61257

1.29577

1.1

       

0.50356

1.35632

0.2

0.1

0.2

1.0

0.2

0.5

1.0

1.0

0.63707

1.28291

 

0.5

      

0.63707

1.28291

 

1.2

      

0.63707

1.28291

0.8

0.3

0.0

1.0

0.2

0.5

1.0

1.0

0.56634

1.32075

  

0.6

     

0.56636

1.32075

  

1.2

     

0.56640

1.32074

0.2

0.3

0.2

0.0

0.2

0.5

1.0

1.0

0.69050

1.39050

   

1.0

    

0.63707

1.28291

   

2.0

    

0.58798

1.18403

0.2

0.3

0.2

1.0

0.2

0.5

1.0

1.0

0.63707

1.28291

    

0.6

   

0.50761

1.52494

    

1.0

   

0.35291

1.79063

0.2

0.3

0.2

1.0

0.2

0.4

1.0

1.0

0.67273

1.22579

     

0.9

  

0.50888

1.52875

     

1.4

  

0.37849

1.87469

0.2

0.3

0.2

1.0

0.2

0.5

0.6

1.0

0.63610

1.28133

      

0.9

 

0.63687

1.28250

      

1.4

 

0.63794

1.28465

0.2

0.3

0.2

1.0

0.2

0.5

1.0

0.0

0.76580

1.16552

       

0.5

0.63707

1.28291

       

1.0

0.48055

1.47280

0.0

0.3

0.2

1.0

0.2

0.5

1.0

1.0

0.77444

1.21516

Table 5

Comparative data of f″(1) for distinct values of M and Sq when Nt = Nb = γ=0

M

S q

f″(0)

Hayat et al. [41]

Present outcomes

0

0.1

2.97682

2.909145

1

 

3.02725

2.989285

2

 

3.17424

3.122340

3

 

3.40620

3.220214

1.0

0.1

3.02725

3.019870

 

0.2

3.00560

2.998422

 

0.3

2.98468

2.960234

Conclusions

Chemically reacted radiated squeezed flow of magneto-nanoliquid between two parallel disks is explored here. The main findings associated with this analysis are listed below:
  • Temperature has a direct link with Nt and Nb.

  • Temperature and concentration depict quite opposite trend against squeezing parameter Sq.

  • Higher M and Rd rise the nanoliquid temperature.

  • Concentration distribution diminishes for larger γ.

  • Skin frictions are enhanced via M and K, whereas reverse trend is seen for S.

  • Rate of mass transfer at lower plate augments via S, Le, Sq and Nt.

  • For K = 0 our results reduce to viscous fluid flow case.

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  • Ikram Ullah
    • 1
  • Muhammad Waqas
    • 1
    Email author
  • Tasawar Hayat
    • 1
    • 2
  • Ahmed Alsaedi
    • 2
  • M. Ijaz Khan
    • 1
  1. 1.Department of MathematicsQuaid-I-Azam UniversityIslamabadPakistan
  2. 2.Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia

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