ARTICLE

J. Cent. South Univ. (2019) 26: 1184-1204

DOI: https://doi.org/10.1007/s11771-019-4080-0

Radiative squeezing flow of unsteady magneto-hydrodynamic Casson fluid between two parallel plates

N. B. NADUVINAMANI¹, Usha SHANKAR^{1, 2}

1. Department of Mathematics, Gulbarga University, Kalaburagi 585106, Karnataka, India;

2. Department of Karnataka Power Corporation Limited, Raichur Thermal Power Station, Shaktinagar 584170,Karnataka, India

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: Present numerical study examines the heat and mass transfer characteristics of magneto-hydrodynamic Casson fluid flow between two parallel plates under the influence of thermal radiation, internal heat generation or absorption and Joule dissipation effects with homogeneous first order chemical reaction. The non-Newtonian behaviour of Casson fluid is distinguished from those of Newtonian fluids by considering the well-established rheological Casson fluid flow model. The governing partial differential equations for the unsteady two-dimensional squeezing flow with heat and mass transfer of a Casson fluid are highly nonlinear and coupled in nature. The nonlinear ordinary differential equations governing the squeezing flow are obtained by imposing the similarity transformations on the conservation laws. The resulting equations have been solved by using two numerical techniques, namely Runge-Kutta fourth order integration scheme with shooting technique and bvp4c Matlab solver. The comparison between both the techniques is provided. Further, for the different set physical parameters, the numerical results are obtained and presented in the form of graphs and tables. However, in view of industrial use, the power required to generate the movement of the parallel plates is considerably reduced for the negative values of squeezing number. From the present investigation it is noticed that, due to the presence of stronger Lorentz forces, the temperature and velocity fields eventually suppressed for the enhancing values of Hartmann number. Also, higher values of squeezing number diminish the squeezing force on the fluid flow which in turn reduces the thermal field. Further, the destructive nature of the chemical reaction magnifies the concentration field; whereas constructive chemical reaction decreases the concentration field. The present numerical solutions are compared with previously published results and show the good agreement.

Key words: squeezing flow; thermal radiation; heat generation or absorption; Casson fluid; Joule dissipation; magnetic field

Cite this article as: N. B. NADUVINAMANI, Usha SHANKAR. Radiative squeezing flow of unsteady magneto-hydrodynamic Casson fluid between two parallel plates [J]. Journal of Central South University, 2019, 26(5): 1184–1204. DOI: https://doi.org/10.1007/s11771-019-4080-0.

1 Introduction

In the field of theoretical or experimental heat and mass transfer studies, many of the researchers have obtained the semi-analytical and numerical solutions of the most realistic problems by considering the non-Newtonian behaviour of fluids under the influence of applied magnetic field. To find the suitable thermodynamic solution for the increasing heat and mass transfer phenomena in fluids, the selection of desired fluid like incompressible or compressible, polyphasic or monophasic, heterogeneous or homogeneous and Newtonian or non-Newtonian, is important. However, determination of the type of fluid like Newtonian or non-Newtonian plays a key role in finding the solution to the desired problem. From the available literature it is observed that, the non-Newtonian fluids have large number of applications in various branches of science and engineering, particularly in the field of polymer and rubber industries, bioengineering and biology, lubrication industries, petroleum and pharmaceutical industries etc.

Having the applications of non-Newtonian fluids in consideration, many researchers have shown their interest to investigate the heat and mass transfer characteristics of non-Newtonian fluids in different geometries with different effects. In this direction the few numerical approaches are made to analyze the thermal characteristics Newtonian/non- Newtonian fluids. SIAVASHI et al [1] numerically investigated the flow and heat transfer characteristics of nanofluid flow inside an annulus filled with porous medium under the influence of entropy generation effect using Darcy-Brinkman- Forchheimer using two-phase mixture model. From their investigation it is observed that, the addition of nanoparticles to base fluid medium enhances the performance number. Also, their study shows that, increasing Reynolds number and nanoparticle volume fraction increases the frictional entropy generation and decreases the thermal entropy generation process. SIAVASHI et al [2] discussed the free convection flow and heat transfer characteristics of double diffusive two-phase flow in a square enclosure with porous medium under the influence of thermal and solutal source. Their investigation reports that, the presence of two perpendicular rectangular sources inside the enclosure gives the suitable values of local heat and mass transfer rates and on the other hand, entropy generation effect is minimized. Further, SIAVASHI et al [3] extended their study to investigate the thermodynamic behaviour of two-phase flow of water with heavy oil in a three-dimensional reservoir using semi-analytical and the time- of-flight (TOF) methods. Their study shows that, the streamline approach is the best suitable method when compared to the classical models to solve the three-dimensional models with complex, fine-scale geological descriptions. GHASEMI et al [4] discussed the magneto-hydrodynamic free convection flow and heat transfer characteristics of Cu-water nanofluid inside a square enclosure with porous medium under the influence of viscous dissipation and temperature dependence of viscosity using lattice Boltzmann technique. From their analysis it is noted that, the behaviour of Nusselt number is observed to be wavy trend with respect to Hartmann number. Further, the influence of porous fins and Cu-water nanofluid on free convection flow and heat transfer behaviour with entropy generation in a cavity was studied by SIAVASHI et al [5]. It is noticed from their study that, the lower nanoparticle concentration intensifies the heat transfer rate. Further, the heat transfer behaviour of two-phase water flow in a fractured porous medium was studied by AHMADPOUR et al [6] using streamline method and IMPES techniques. Their study proved that, the streamline technique is more appropriate than the IMPES method. However, the modern thermodynamic industries effectively develop nanoparticles along with the lubricants to control the frictional heating in various moving parts of the machines. Since, the heat transfer characteristics of lubricants may drastically vary under the influence of different thermodynamic conditions. Therefore, the heat and mass transfer studies play an important role in assembling of highly efficient thermodynamic machines. It is noticed from the available literature that, the non-Newtonian fluids cannot be mathematically modeled by using the Newtonian fluid flow models. Thus, Casson fluid is one of the typical examples of non-Newtonian fluid. Human blood, honey, cream, palm oil, shampoo mayonnaise sauce and etc. are some of the suitable examples of Casson fluid. Thus, the following literature highlights the applications of non-Newtonian fluids in various branches of science engineering.

SITHOLE et al [7] investigated the heat mass transfer characteristics of second-grade nanofluid about a stretching sheet with MHD effect under the influence of viscous dissipation and nonlinear thermal radiation impacts, numerically using spectral local linearization technique. Further, their study accounts the heterogeneous and homogenous chemical reactions along with convective boundary condition. From their discussion it is noticed that, the intensified entropy generation is observed for larger values of Reynolds and magnetic numbers. Further, an increment in second grade nanofluid parameter increases the thermal and velocity fields in the flow region. DHLAMINI et al [8] discussed the influence of binary chemical reaction and activation energy on a time-dependent mixed convection flow of nanofluid past an infinitely long flat plate with Brownian motion and thermophoresis effects under the presence of viscous dissipation impacts numerically via spectral quasilinearization method. From their investigation it is observed that, the unsteady parameter decreases the velocity and temperature fields; whereas the concentration field increases. Further, it is remarked that, the concentration field magnified for the increasing values of thermophoresis and activation energy parameters. The flow, heat and mass transfer characteristics of couple stress nanofluid past a stretching sheet with Brownian motion and thermophoresis effects under the influence of heat generation, thermal radiation and magneto-porous medium was investigated by SITHOLE et al [9] via spectral quasi-linearization technique. From their study it is evident that, the increasing couple stress fluid parameter upsurges the local skin-friction coefficient. Further, the magnifying Brownian motion and thermophoresis parameters increase the temperature of the fluid considerably. DAS et al [10] numerically investigated heat and mass transfer characteristics of Casson fluid with entropy generation analysis of homogenous and heterogeneous chemical reactions in the presence of magnetic field using spectral quasi linearization technique. From their analysis it is revealed that, an increment in Brinkman number decreases the Bejan number. Since, an upsurge in Brinkman number diminishes the entropy generation. Further, GOQO et al [11] analyzed the influence of thermal radiation and entropy generation on viscous nanofluid flow past a wedge with non-isothermal characteristics via spectral quasi linearization method. Their study demonstrates that, entropy generation is considerably intensified by increasing the viscous dissipation and thermal radiation parameters. Further, it is stated that, irreversibility behaviour is suppressed for the decreasing values of viscous dissipation and thermal radiation parameters.

The thermodynamic study of squeezing flow in different geometries has attracted the attention of several researchers due to its numerous practical applications in various industries. In particular, the squeezing flow between parallel and orthogonally moving circular plates occurs very frequently in many practical situations. However, most of the mechanical apparatus in machines works under the influence of moving pistons. In such cases two plates show the squeezing movement normal to each other. Also, many of the hydraulic lifters, engines and electric motors use the concept of squeezing flow in their operations. Hence, the study of squeezing flow between parallel plates has become one of the most active research areas in the field of computational fluid dynamics. Further, the phenomena of squeezing flow have large number of applications in field of medicine. For an instance, the fluid flow inside the syringes and narrow gastric tubes is considered as squeezing flow example. Especially these types of flows have advantages in modeling of a synthetic materials or chemicals passage inside living bodies. More details about applications and advantages can be found in the available literature [12–15].

OYELAKIN et al [16] studied the impacts of thermal radiation and heat source on time- dependent heat and mass transfer characteristics of Casson nanofluid flow past a stretching sheet with Brownian motion and thermophoresis under the presence of Dufour and Soret effects using spectral relaxation technique. Their investigation shows that, an increment in Dufour number decays the temperature field, whereas, increment in Soret number upsurges the concentration distribution in the flow region. HAROUN et al [17] numerically investigated the unsteady mixed convection flow and hear transfer characteristics of MHD nanofluid past a stretching/shrinking sheet under the influence of viscous dissipation, heat generation and Soret and Dufour effects using spectral relaxation scheme. Further, the influence of viscous dissipation and chemical reaction on magneto-hydrodynamic flow of nanofluid about a stretching or shrinking sheet with Soret effect was analyzed by KAMESWARAN et al [18] via bvp4c Matlab solver. It is noticed from their study that, the local wall shear stress decreases in the case of Ag-water nanofluid when compared to Cu-water nanofluid. Further, the effect of thermal radiation and ramped temperature on free convection flow dusty fluid over an impulsively moving plate under the influence of magnetic field was studied by NANDKEOLYAR et al [19] using Laplace transform method. It is evident from their investigation that, the particle and fluid velocity decayed for the increasing values of magnetic field. MAKANDA et al [20] discussed the effects of MHD and thermal radiation on free convection flow of Casson fluid over a cylinder with partial slip under the influence of non-Darcy porous medium using bi-variate quasilinearization technique. From their investigation it is observed that, the velocity field decreases for the increasing values of magnetic number.

However, following are the some suitable examples of research done in the area of squeezing flows. The fundamental research work on squeezing flows was made by STEFAN [21]. Based on the assumptions of lubrication theory, STEFAN formulated the basic mathematical model for squeezing flows in detail under the appropriate thermodynamic conditions. Later a number of researchers have given their attention to studying the characteristics of heat and mass transfer in squeezing flow with different geometries. REYNOLDS [22] extended the STEFAN’S work to elliptic plates. Also, ARCHIBALD [23] studied the similar problem for rectangular plates. Since, then a number of scientists and engineers studied and analyzed the STEFAN’s work in various flow configurations in a better way [24–28]. MUSTAFA et al [29] investigated the transient squeezing flow of non-Newtonian viscous incompressible fluid between parallel plates by considering homotopy analysis method. Also, they reported that, for the higher values of Prandtl and Eckert numbers, the temperature field enhances rapidly in the flow region.

KHAN et al [30] thermodynamically analyzed the squeezing flow characteristics of Casson fluid between parallel plates. From their study it is noticed that, the axial velocity field is a decreasing function of squeezing number. Also, a similar problem was extended by KHAN et al [31] for the circular parallel plates by considering the flow and heat transfer characteristics of Casson fluid. The numerical results presented in Ref. [31] were based on homotopy perturbation method and it is noticed that, the temperature field is an increasing function of Prandtl number and Eckert number. Further, MOHYUD-DIN et al [32] studied squeezing flow between circular parallel plates by considering the differential transform and Runge-Kutta methods. However, the magneto-hydrodynamic viscous incompressible squeezing flow between two long infinite parallel plates was studied by KHAN et al [33]. KHAN et al [34] discussed the influence of permeability and magnetic field on squeezing flow of Casson fluid between two parallel plates by using the homotopy perturbation method. Their study reports that, the magnetic number and porous parameter have the same effect on velocity profile irrespective of their sign in the flow region. The influence of magnetic field on viscous incompressible electrically conducting Casson fluid squeezing flow between two parallel plates was investigated by AHMED et al [35] by using variation of parameters method. From the literature [35] it is noticed that, the magnetic field normalizes the velocity in the flow region.

SIAVASHI et al [36] numerically investigated the free convective flow and heat transfer characteristics of Al₂O₃ nanoparticles and entropy generation effect in porous medium in circular annulus with conductive porous layer via two-phase mixture model. From their investigation it is observed that, for the fully developed porous cavity, the ratio k_eff/k_f=16 is recommended. Further, their study reports that, for the higher values of performance number (PE), the lower entropy generation is noticed in fully developed porous cavities. The flow and heat transfer characteristics of kerosene-alumina nanofluid between two parallel plates under the influence of magnetic field was investigated by MAHMOODI et al [37] via differential transform method. From their semi-analytical study it is remarked that, the increasing values of nanofluid volume fraction, magnetic number and viscosity parameter enhances the local momentum transfer coefficient. Also, their discussion reports that, the local heat transfer rate magnifies with increasing values of nanofluid volume fraction and magnetic number. SARI et al [38] numerically analyzed the flow and heat transfer of copper/water nanofluid between converging-diverging plane walls via Adomian decomposition and Runge-Kutta methods. Their study accounts the Jeffery-Hamel flow of nanofluids and thermodynamic properties. From their discussion it is noticed that, local heat transfer rate decreases for the increasing values of channel half-angle and Eckert number, whereas, Nusselt number decreases for the magnifying values of Reynolds number. Further, the attendance of Cu nanoparticles in base fluid medium intensifies the heat transfer rate between non-parallel walls and this situation leads to the increment in the local Nusselt number for the enhancing values of nanoparticles volume fraction. MAJID et al [39] investigated the laminar forced convection flow and heat transfer characteristics of water-Al₂O₃ nanofluid and entropy generation impact at the hydrodynamic and thermal entry regions in concentric annulus with constant heat flux on the walls using CFD technique along with Patankar algorithm. Their study states that, for the larger values of Reynolds number, the nanoparticles concentration strongly influences the heat transfer and entropy generation minimization. KU-ER- BAN-JIANG et al [40] experimentally investigated the heat transfer characteristics of Cu nano particles at different concentrations in various flow regimes with 300

However, the influence of variable magnetic field on electrically conducting viscous incompressible fluid flow between two parallel plates under the presence of porous medium was investigated by SHAH et al [42]. Further, KUMAR et al [43] discussed the effect of aligned magnetic field on Casson fluid squeezing flow between parallel plates. Recently, OJJELA et al [44] investigated the effects of magnetic field on entropy generation and heat and mass transfer characteristics of Casson fluid flow between parallel disks. From their investigation it is observed that, Hartmann number increases the entropy generation whereas Bejan number decreases.

Further, the influence of nonlinear radiation parameter on Casson fluid squeezing flow between two parallel disks with impermeable upper wall and permeable lower wall was studied by MOHYUD- DIN et al [45]. From their discussion it is noticed that, the temperature field is a decreasing function of nonlinear radiation parameter. Also, a similar problem was extended by KHAN et al [46] by considering the thermal radiation effect on Casson fluid squeezing flow between two parallel disks. The unsteady magneto-hydrodynamic viscous incompressible squeezing flow between two parallel plates was discussed by KUMAR et al [47]. It is observed from their study that, the increasing Hartmann number decreases the temperature and velocity fields in the flow region. Further, KUMAR et al [48] studied the Casson fluid squeezing flow with nonlinear thermal radiation effect under the influence of thermophoresis and Brownian motion. From their study it is clearly observed that, the physical parameter governing the Brownian motion decreases the concentration field whereas thermophoresis parameter enhances the concentration field in the flow region. Recently, SRINIVAS et al [49] studied the heat and mass transfer characteristics of pulsatile magneto- hydrodynamic Casson fluid flow in porous channel under the influence of thermal radiation effect.

Thus, from the above mentioned literature, it is worthy to note that, the present problem has a large number of engineering and biomedical and applications in various industries. Also, it is noticed that, many of the researchers have considered the magnetic field and thermal radiation effects in squeezing flow of Casson fluid with heat transfer phenomena only. However, the heat and mass transfer characteristics of Casson fluid squeezing flow between two parallel plates with Joule dissipation, internal heat generation or absorption under the influence of homogenous first order chemical reaction has not been reported in the literature. Hence, the authors have motivated with this literature gap and made an effort to analyze this particular heat and mass transfer problem. However, the considered physical problem is of great interest in the field of engineering and biomedical sciences. Thus, in this paper, the radiative squeezing flow of unsteady magneto-hydrodynamic Casson fluid between two parallel plates is investigated by imposing the suitable similarity transformations under the laws of fluid motion. The resulting coupled highly nonlinear partial differential equations are solved by using Runge-Kutta fourth order integration scheme with shooting technique.

2 Non-Newtonian Casson fluid model

The rheological equation governing the non-Newtonian viscous incompressible isotropic Casson fluid [35, 45] is considered in terms of standard stress and strain relationship of the following form.

            (1)

where τ_ij is the (i, j)th component of stress tensor; π=e_ij·e_ij describes the product of deformation components; e_ij indicates the deformation rate on the (i, j)th component; π_c shows the critical value. The dynamic plastic viscosity of Casson fluid is denoted by μ_B and p_y represents the yield stress.

In the present paper, viscous incompressible, two-dimensional, radiative squeezing flow of unsteady, magneto-hydrodynamic Casson fluid between two parallel plates is considered. The influences of thermal radiation, heat generation or absorption, Joule heating and chemical reaction effects on flow are accounted in the governing equations. The flow configuration is modeled in such way that, the plates are separated by a distance where l is the initial location (when time t=0). However, the parallel plates are squeezed when α>0 till they reach t=1/α and plates are separated when α<0. Cartesian coordinate system is considered to describe this problem, in which, x-axis is considered along the axial flow direction and y-axis is taken normal to the axial direction. Figure 1 clearly describes the geometry and coordinate system of the considered problem. Thermal radiation and heat generation or absorption effects are considered in the energy equation. Additionally, the viscous and Joule dissipation effects, heat generation due to the frictional forces produced by shear stress in the flow region, are also accounted. These effects are quite natural and are significant in the case of fluids with high viscosity or high speed fluid flows. For the higher values of Eckert number (>>1) usually this type of behaviour is noticed [29]. In addition to this, the time- dependent first order homogeneous chemical reaction effect is also accounted in mass diffusion equation. With these assumptions, the governing equations for mass, momentum, energy and mass transfer in unsteady two dimensional flow of a viscous incompressible Casson fluid are given by the following equations [29–33, 42, 46].

                              (2)

          (3)

            (4)

  (5)

                        (6)

In the above Eq. (3), the last term on the right hand side signifies the magnetic field. It is observed that, the magnetic field can be used as a control phenomenon in many flows as it normalizes the flow behaviour. Further, the second term on the right hand side of Eq. (5) (energy equation) is the viscous dissipation term which is always positive and represents a source of heat due to friction between the fluid particles. Further, the irreversible process by means of which the work done by a fluid on adjacent layers due to the action of shear forces is transformed into heat is defined as viscous dissipation. Viscous dissipation is of interest for many applications: significant temperature rises are observed in polymer processing flows such as injection molding or extrusion at high rates. Aerodynamic heating in the thin boundary layer around high speed aircraft raises the temperature of the skin. Boundary layer flows with internal heat generation over a stretching sheet continue to receive attention because of its many practical applications in a broad spectrum of engineering systems.

Figure 1 Flow configuration and coordinate system of problem

u and v are the velocity components along x and y directions; ρ is the density; ν is the kinematic viscosity; β indicates the Casson fluid parameter; σ is the electrical conductivity; B₀ is the magnetic field; k is the thermal conductivity; α is the squeezing rate; c_p is the specific heat capacity; σ^* is the Stefan-Boltzmann constant; k^* is the absorption coefficient; Q^* is the coefficient of heat generation or absorption; T, C are the temperature and concentration of the fluid; T₀, C_o are the temperature and concentration of the fluid in the beginning time; k₁ is the coefficient of chemical reaction and D_m is the coefficient of mass diffusion.

The relevant boundary conditions governing the flow Eqs. (2)–(6) are as follows:

at y=h(t)   (7a)

at y=0         (7b)

However, Eqs. (2)–(6) along with conditions (7a) and (7b) governing the Casson fluid squeezing flow are coupled highly nonlinear in nature and are not amenable to any analytical methods. Hence, in the system of partial differential equations,Eqs. (2)–(6) are reduced to ordinary differential equations by using the following similarity transformations [29–33, 50].

       (8)

where

By introducing Eq. (8), into Eqs. (2)– (7), the resultant equations are as follows:

               (9)

                      (10)

    (11)

Similarly, the boundary conditions in view of Eq. (8) are given as follows:

at η=0 (12a)

at η=1    (12b)

The superscript symbol prime in the above equations denotes the differentiation with respect to the similarity variable η. The non-dimensional control parameters occurring in Eqs. (9)–(11) are defined as follows:

(squeezing number);

(Hartmann number);

(Hartmann number);

(Prandtl number);

(radiation parameter);

(heat source/sink parameter);

(Schmidt number);

and (chemical reaction parameter).

It is important to note that, the squeezing number describes the motion of parallel plates. In the present case S>0 corresponds to the movement of plates away from one another; on the other hand S<0 corresponds to the movement of plates moving close together (it is usually known as squeezing flow). Prandtl and Eckert numbers are used to regulate the temperature field. Also, the absence of viscous dissipation effects is governed by Ec=0. Further, the time-dependent chemical reaction parameter decides the type of the chemical reaction. In view of this reason, the destructive, constructive and no chemical reactions are governed by Kr<0, Kr>0 and Kr=0 respectively. Additionally, it is observed that, the present unsteady squeezing flow problem has no direct solution. Thus, in the present investigation, Runge-Kutta integration scheme with shooting technique is used to produce the approximate similarity solution for the present problem. For this purpose, the following numerical solution procedure is described.

3 Numerical solution procedure

The ordinary differential equations Eqs. (9)– (11) with conditions (12), governing the radiative squeezing flow of unsteady magneto-hydrodynamic Casson fluid between two parallel plates are coupled and nonlinear in nature. Hence, to solve these complex flow equations, numerically stable Runge-Kutta fourth order integration algorithm with standard shooting technique [51] is used. In this method, first, we decompose the coupled higher order ordinary differential equations into a set of first order ordinary differential equations in the following form:

       (13)

(14)

   (15)

Further, we have considered that, F(η)=F₀(η), θ(η)=θ₀(η) and f(η)=f₀(η). However, to solve flow Eqs. (13)–(15), the necessary boundary conditions on velocity (F, F′), temperature (θ) and concentration (f) fields are respectively given by the following equations.

               (16)

                       (17)

                        (18)

The considered numerical procedure in this paper begins by reducing the boundary value problem (BVP) into an initial value problem (IVP) by assuming the suitable values of unknowns quantities F₁(0), F₃(0), θ₀(0), f₀(0). The reduced IVP is then solved by utilizing the standard Runge-Kutta fourth order integration scheme. For the better numerical accuracy, the step size is chosen as h=0.01. Generally, in the shooting technique the convergence criterion mainly depends on appropriate guesses of the initial conditions. Further, the iteration process terminates when the relative difference between the present and the previous iterative values of velocity, temperature and concentration fields matches. In order to get the accurate numerical results the convergence criterion is taken as 10^–5. However, once the convergence criterion is approached, the system of ordinary differential equations is integrated using Runge- Kutta fourth order integration scheme.

4 Results and discussion

4.1 Verification of numerical results

To confirm the accuracy of the present numerical scheme and correctness of the obtained numerical solutions, the results are compared with those of MUSTAFA et al [29]. This comparison is shown in Table 1. From Table 1 it is observed that, the present numerical results are exactly matching with the semi-analytical results of Ref. [29]. Also, it is noticed from Table 1 that, the absolute values of wall shear stress are enhanced for the increasing values of squeezing number, whereas, the Nusselt and Sherwood numbers decrease. Also, it is obvious that, the negative values of the Nusselt number indicate the flow of heat from the surface of parallel plates to the fluid between the plates. Further,Table 2 illustrates the accuracy of the Runge-Kutta Method which is used in comparison with the bvp4c Matlab solver and Homotopy analysis method (HAM). The numerical results show the excellent agreement between the Runge-Kutta shooting method (RK-SM), HAM and bvp4c methods and further show the consistency of the RK-SM in obtaining the solution of nonlinear problems.

Table 1 Comparison of momentum, heat and mass transport coefficients between present numerical result (RK-SM) and analytical (HAM) results obtained by MUSTAFA et al [29] for different values of S when Ha→0, R→0, Q→0 and β→∞

Table 2 Convergence test results for fixed values of Ha=0.5, Pr=Sc=0.7 and β=R=Q=Ec=0.1

4.2 Influence of control parameters on flow behaviour

To describe the physical insight of the present problem in depth, the thermodynamic flow behaviour of skin-friction coefficient, Nusselt and Sherwood numbers along with velocity, temperature and concentration profiles in the flow region for different set physical parameters namely, squeezing parameter (S), Casson fluid parameter (β), Hartmann number (Ha), radiation parameter (R), heat generation or absorption parameter (Q), Eckert number (Ec), Prandtl number (Pr), chemical reaction parameter (Kr) and Schmidt number (Sc) are investigated. For better understanding of the numerical results, the computer generated numerical data are presented in the form of graphs and tables.

1) Effect of squeezing number

The thermodynamic behaviour of velocity, temperature and concentration profiles for different values of squeezing number (S) is depicted in Figures 2–9 with fixed values of β=0.8, Ha=R=Q=Ec=Kr=0.1, Pr=0.1 and Sc=0.7. The movement of the parallel plates moving away from one another (S>0) is illustrated by Figures 2, 4, 6 and 8. Similarly, the movement of the parallel plates coming close to one another (S<0) is described by Figures 3, 5, 7 and 9 in the flow region. For positive values of squeezing number, radial velocity field (F) decreases from η=0 to η=1 and radial velocity profile increases from η=0 to η=1 for negative values of S. These changes in radial velocity fields are clearly shown in Figures 2 and 3, respectively. This increment in velocity field is due to the reason that, when parallel plates move apart from one another, fluid is sucked into the channel consequently which increases the velocity field. In other case, when plates move close to one another, liquid inside the channel is emitted out which gives the liquid dropping inside the channel and hence velocity of the fluid decreases. However, squeezing number is a function of velocity field in the flow region.

Figure 2 Effect of S>0 on F(η)

Figure 3 Impact of S<0 on F(η)

Figure 4 Effect of S>0 on F′(η)

Figure 5 Influence of S<0 on F′(η)

Figure 6 Effect of S>0 on θ(η)

Figure 7 Influence of S<0 on θ(η)

Figure 8 Impact of S>0 on f(η)

Further, Figures 4 and 5 describe the influence of squeezing number on axial velocity field (F′) for both positive and negative values of S. It is interesting to note from Figure 4 that, there exists a critical value (η_C) of η, such that F′ decreases for η≤η_C and increases for η≥η_C for the positive values of S. From Figure 4 it is clear that η_C=0.45. Further, from Figure 5 it is noticed that, the velocity field increases in the region 0≤η≤0.42 and opposite trend is observed in the other portion 0.42≤η≤1.0 for S<0. Also, it is noticed that, the axial velocity field is magnified in the neighbourhood of the lower plate and ends in the vicinity of the upper plate. From Figure 5 it is noticed that, at η=η_C=0.42 all the velocity curves are coincided. This means that, at η_C=0.42, the squeezing number S has the same effect on velocity field in the flow region. More clearly, axial velocity field is increased when S>0 and decreased when S<0. Due to these changes in axial velocity near the boundaries, cross flow behaviour occurs at the central portion of channel which is clearly shown in Figures 4 and 5. Also, as per the industrial use, the power required to generate the movement of the parallel plates is considerably reduced for the negative values of squeezing number.

Figure 9 Effect of S<0 on f(η)

Also, the thermodynamic variations noticed in the temperature profile (θ) for both S>0 and S<0 in flow region are illustrated in Figures 6 and 7, respectively. For the positively increasing values of squeezing number, the temperature field is eventually suppressed, which is shown in Figure 6. This decay in thermal field is owing to the reason that, the larger value of squeezing number decreases the squeezing force on the flow and consequently temperature field decreases. On the other hand, from Figure 7 it is noticed that, the temperature field is magnified for S<0. It is obvious that, the temperature field is relatively high when plates move close to one another, since the magnifying value of squeezing number is closely related to the decaying of the kinematic viscosity, the distance between the parallel plates and the speed at which plates move. However, it is clear that, temperature profile behaves like a monotonically increasing function of S<0. However, Figures 8 and 9 describe the influence of S on concentration profile (f) in the flow region. Enhanced concentration field is noticed for S>0 as shown in Figure 8. Also, from Figure 9 it is observed that, the concentration field decreases for S<0. Further, from Figure 9 it is noticed that, the squeezing number has a diminishing effect for increasing values of η. Thus, the concentration profile is a function of squeezing number.

2) Influence of Casson fluid parameter

The impact of Casson fluid parameter on velocity, temperature and concentration profiles is described in Figures 10–13 with fixed S=Ha=0.5, R=Q=Kr=0.1, Sc=0.7, Pr=0.7 and Ec=0.8. From Figure 10 it is noticed that, the radial velocity profile increases for increasing values of β. This variation in normal component of velocity is mainly due to the fact that, under the impact of applied stresses, a small upsurge in Casson fluid parameter causes the decaying of fluid viscosity and produces the relatively less opposition to the fluid flow between the plates. This situation is responsible for the enhancement of velocity field in the flow region. Also, it is noticed that, β has no significant effect on radial velocity profile in the regions 0≤η≤0.1 and 0.9≤η<1.0 as compared to the other portion of the channel.

Further, the influence of β on axial velocity profile is illustrated in Figure 11. It is observed from Figure 11 that, the increasing values of β increases the axial velocity field in the region η<0.5 and reverse trend is noticed in the other portion of the channel η>0.5. Further, owing to these variations in axial velocity field at the boundaries, a cross flow trend appears at the central region of the channel.

Figure 10 Impact of β on F(η)

Figure 11 Effect of β on F′(η)

Figure 12 Influence of β on θ(η)

Figure 13 Effect of β on f(η)

Further, the influence of β on temperature profile (θ) is described in Figurer 12. Figure 12 clearly demonstrates that, an increment in Casson fluid parameter eventually decreases the temperature field in the flow region. Further, the thermal boundary layer thickness suppresses for increasing values of Casson fluid parameter. However, an increment in elasticity stress variable causes the thinning of thermal boundary layer. It is clear that, the thinner thermal boundary layer is equivalent to the smaller thermal diffusivity values, which shows the larger temperature gradient in the neighbourhood of parallel plates. Moreover, temperature field is a decreasing function of β and it decreases from η=0 to η=1.0. Since, β is explicitly occurring in the thermal equation (refer Eq. (10)), temperature field can be easily normalized by increasing the β values. Additionally, it is noticed from Figure 13 that, all the curves in concentration profile are coincided with one another, which indicates that, β has no considerable effect on concentration profile, because β has no explicit occurrence in the mass diffusion equation (refer Eq. (11)).

3) Effect of Hartmann number

The influence of Hartmann number on flow behaviour is described in Figures 14–17 with fixed values of Pr=0.7, Sc=0.7 and β=S=R=Q= Kr=Ec=0.1. Figure 14 clearly illustrates that, an increment in Hartmann number decreases the normal component of velocity profile in the flow region. It is quite obvious to expect that, a small upsurge in Hartmann number is responsible for the enhancement of the Lorentz forces associated with magnetic field. These magnified Lorentz forces offer more and more resistance to fluid flow inside the channel. Therefore, it is expected that, the velocity field will decrease for increasing values of Hartmann number. Further, from Figure 15 it is noticed that, the axial velocity profile decreases in the region η<0.5 for increasing value of Ha and opposite trend is noticed in the region η>0.5. At the point near to η≈0.5 the influence of Ha on velocity profile is almost the same (refer Figure 15). Thus, cross flow trend is noticed at the central portion, the channel due to these variations in the axial velocity at the boundaries.

Figure 14 Impact of Ha on F(η)

Figure 15 Influence of Ha on F′(η)

Figure 16 Effect of Ha on θ(η)

Figure 17 Impact of Ha on f(η)

However, the variations noticed in the temperature profile (θ) for different values of Ha are shown in Figure 16. From this figure it is observed that, temperature field is suppressed for the increasing values of Ha. In this particular case thinner temperature boundary layer is observed because an upsurge in Hartmann number gives the decrease in elasticity stress variable, which in turn is responsible for the decaying of the thermal field in the flow region. Moreover, the temperature field is an increasing function of Ha. Further, from Figure 17 it is very important to note that, all the concentration curves governing the concentration field are concurred because Ha has no considerable effect on concentration profile. Ha has no explicit occurrence in concentration equation (refer Eq. (11)).

4) Influence of radiation parameter

The effect of radiation parameter on temperature profile is described in Figure 18 with fixed values of S=Ha=0.5, β=Q=Kr=0.1, Sc=0.7 and Ec=1.0, Pr=0.7. Figure 18 clearly portrays that, an upsurge in thermal radiation parameter eventually diminishes on the temperature profile in the flow region. This fact can be depicted through the relation Thus, in view of the relationa small upsurge in R causes the decay of absorption coefficient k^*, hence temperature profile decreases. Also, the slope of temperature curves close to the wall indicates that the heat flows from the surface plates to the fluid. Thus, from Figure 18 it is conclude that, the numerical results obtained in this figure are reasonable and are acceptable. Also, physically, an increment in thermal radiation parameter gives the greater temperature value which may be useful in many of the thermodynamic industries.

5) Impact of heat source/sink parameter

The influence of heat generation or absorption parameter (Q) on temperature profile is illustrated in Figures 19 and 20 with fixed values of S=Ha=0.5, Pr=0.7, Sc=0.7 and β=R=kr=0.1, Ec=1.0. It is observed from Figure 19 that as Q increases, the temperature field increases. Also, the thickness of thermal boundary layer increases for increasing values of Q. It is expected that, during the heat generation process more temperature is usually released into the working fluid. Owing to this reason, the temperature profile upsurges as heat generation parameter increases. Also, due to the exothermic chemical reactions, the temperature field increases. However, from Figure 20 it is observed that, the temperature profile decreases for the negatively increasing values of Q.

Figure 18 Influence of R on θ(η)

Figure 19 Effect of Q>0 on θ(η)

Figure 20 Impact of Q<0 on θ(η)

6) Influence of Eckert number (Ec) and δ

Figures 21 and 22 are presented to describe the effects of Ec and δ on temperature profile (θ) for fixed values of S=Ha=0.5, β=R=Q=0.1, and Pr=Sc=0.7, Kr=0.1. From Figure 21 it is noticed that, temperature field is enhanced for increasing values of Ec. The physical justification behind this fact is that the presence of frictional forces in the fluid causes the release of heat energy into the fluid which gives the intensified temperature field in the flow region. This situation is frequently explained in terms of viscous dissipation term, usually an Eckert number [29]. However, Ec has direct impact on temperature field since it has explicit occurrence in energy equation (refer Eq. (10)). The thickness of the thermal boundary layer decreases for increasing values of Ec. From the above observation, it is noticed that, temperature field is an increasing function of Ec.

Figure 21 Effect of Ec on θ(η)

Figure 22 Influence of δ on θ(η)

Furthermore, the effect of δ on temperature profile is described in Figure 22. From this figure it is noticed that, temperature field increases for increasing values of δ.

7) Effect of Prandtl number

The influence of Pr on temperature profile is described in Figure 23 with fixed values of S=Ha=0.5, β=R=Q=Kr=0.1, Sc=0.7. It is observed from Figure 23 that, temperature field increases for increasing values of Pr. This increment in temperature profile is mainly due to decrease in the thermal conductivity values. However, dissipation effects are also responsible for the increase of temperature field in the flow. Further, the thickness of temperature boundary layer decreases [29] for increasing values of Pr. This decrement is mainly due to the fact that, the magnified Pr values greatly decrease the thermal diffusivity which in turn causes the decreasing of the temperature boundary layer thickness. Generally, it is known that, Pr<1 associated with the liquid materials with low viscosity and high thermal conductivity whereas Pr>1 corresponds to the high-viscosity materials like oils etc. From this observation it is concluded that, temperature field is an increasing function of Pr.

Figure 23 Impact of Pr on θ(η)

8) Influence of chemical reaction parameter

The effect of chemical reaction parameter on concentration profile is illustrated in Figures 24 and 25 with fixed values of S=Ha=0.5, β=R=Q=Ec=0.1 and Pr=0.7, Sc=0.7. Generally, it is observed that,in many of the cases the decreased concentration field is noticed for destructive chemical reaction. In view of this, Figure 24 confirms the results obtained in Ref. [29]. Thus, Figure 24 clearly illustrates that, for the destructive chemical reaction concentration distribution decreases. Further, the influence of generative homogeneous first order chemical reaction (Kr<0) on concentration field is shown in Figure 25. It is observed that, the concentration field increases for constructive chemical reaction. Thus, the concentration field is a function of chemical reaction parameter Kr.

Figure 24 Effect of Kr>0 on f(η)

Figure 25 Influence of Kr<0 on f(η)

9) Effect of Schmidt number

The influence of Schmidt number (Sc) on concentration profile with fixed values of S=Ha=0.5, β=R=Q=Ec=Kr=0.1 and Pr=0.7 is described in Figure 26. Figure 26 clearly illustrates that, the increasing Schmidt number decreases the concentration field in the flow region. This fact can be justified as follows: a small upsurge in Schmidt number diminishes the coefficient of mass diffusion, which in turn causes the decreases of concentration field in the flow region. Further, it is observed that, the concentration field is a decreasing function of Sc. Also, the concentration boundary layer thickness decreases for increasing values of Sc.

Figure 26 Impact of Sc on f(η)

5 Momentum, heat and mass transport coefficients

Due to the increased industrial applications of heat and mass transfer phenomena, it is necessary to study the behaviour of wall shear stress, heat and mass transfer rates. In view of this reason, the numerical values of skin-friction coefficient F″(1), Nusselt number θ′(1) and Sherwood number f′(1) are tabulated in Tables 3–5. Further, the following dimensional equations of skin-friction, Nusselt and Sherwood numbers are used to derive the non- dimensional equations.

                (19)

           (20)

                   (21)

Using Eq. (8), the dimensional Eqs. (19)–(21) are reduced to the following non-dimensional form:

              (22)

                (23)

                       (24)

To describe the behaviour of momentum, heat and mass transport rates for different values of flow parameters, Tables 3–5 are presented. Table 3 demonstrates that, momentum transport coefficient (F″(1)) decreases for S>0 (plate moving apart) and it increases for S<0 (plate coming together). Also, it is noticed that, when the plates go apart, the increasing value of β decreases the skin-friction coefficient at the wall. Further, the increasing value of Ha decreases the momentum transport rate.

Table 3 Local skin-friction coefficient for various values of S, β, Ha with fixed Sc=0.7, Pr=0.7 and R=Q=Ec=Kr=0.1

The influence of physical parameters on heat transfer rate (θ′(1)) is illustrated with the help of Table 4. It is clear from Table 4 that, Nusselt number increases for the magnifying values of squeezing number. This is due to the decrease in the viscosity of the fluid in the channel. Further,the increasing value of Ha enhances the heat transport rate. This is due to the presence of Joule dissipation effects. However, due to the existence of viscous dissipation in the thermal equation, temperature field increases for the increasing values of Ec and δ.

Similarly, Table 5 describes the effect of various control parameters on mass transfer rate (f′(1)) in the flow region. Table 5 clearly illustrates that, for the increasing values of S the mass transport rate decreases at the wall. This is mainly due to the contraction of channel. Also, it is noticed that, mass transfer rate increases for increasing values of Sc. Further, f′(1) decreases for Kr>0 (due to the constructive nature) and it increases for Kr<0.

Table 4 Local Nusselt number for various values of S, R, Q, δ with fixed Sc=0.7, Pr=0.7, β=Ec=Kr=0.1 and Ha=0.5

Table 5 Local Sherwood number for various values of S, Sc, Kr with fixed Ha=0.5, Pr=0.7 and β=R=Q=Ec=0.1

This is due to the destructive nature of the first order homogenous chemical reaction.

6 Conclusions

Present paper reports the radiative squeezing flow of unsteady magneto-hydrodynamic non- Newtonian Casson fluid between two parallel plates with Joule heating and heat generation or absorption in the presence of homogeneous chemical reaction effects. In the present problem, flow occurs due to the movement of parallel plates. The non-Newtonian Casson fluid flow model results are obtained and the highly nonlinear coupled two-dimensional unsteady partial differential equations are solved by using standard Runge-Kutta fourth order integration algorithm with shooting technique. The numerical simulations are performed for the different set control parameters. From the above numerical simulations following important conclusions are drawn.

1) In view of industrial use, the power required to generate the movement of the parallel plates is considerably reduced for the negative values of squeezing number.

2) The influence of negative and positive squeezing parameter on velocity field is observed to be reversed.

3) The higher values of squeezing number diminish the squeezing force on the fluid flow, which in turn reduces the thermal field.

4) The thinner temperature boundary layer corresponds to the lower values of thermal diffusivity and it shows the higher values of temperature gradient for the increasing values of β.

5) Due to the presence of stronger Lorentz forces the temperature and velocity fields behave like decreasing functions of Hartmann number.

6) Temperature field is suppressed for the increasing values of thermal radiation parameter.

7) The destructive chemical reaction intensifies the concentration field and constructive chemical reaction decreases the concentration field.

The current numerical investigation has provided some remarkable insights of squeezing flow with non-Newtonian Casson fluid between two parallel plates. The numerical results presented in this paper may be useful in theoretical and experimental studies related to the squeezing flow phenomena.

Acknowledgements

The authors wish to express their gratitude to the reviewers who highlighted important areas for improvement in this earlier draft of the article. Their suggestions have served specifically to enhance the clarity and depth of the interpretation of results in the revised manuscript. One of the author Usha SHANKAR wishes to thank Karnataka Power Corporation limited, Raichur Thermal Power Station, Shaktinagar, for their encouragement.

Nomenclature

B₀

Uniform magnetic field

C

Dimensionless concentration

C_f

Skin-friction coefficient

c_p

Specific heat capacity at constant pressure

D_m

Diffusion coefficient

Ec

Eckert number

F′, F

Velocity components along x and y axes, respectively

Ha

Hartmann number

k

Thermal conductivity

k₁

Chemical reaction coefficient

k^*

Absorption coefficient

Kr

Chemical reaction parameter

Nu

Nusselt number

p

Pressure

Pr

Prandtl number

Q

Heat generation or absorption parameter

Q^*

Coefficient of heat source or sink

R

Radiation parameter

S

Squeezing number

Sc

Schmidt number

Sh

Sherwood number

T

Fluid temperature

u, v

Dimensional velocity components along

x and y directions

Greek letters

α

Squeezing rate

β

Casson fluid parameter

η

Similarity variable

θ

Dimensionless temperature

μ

Dynamic viscosity

ν

Kinematic viscosity

ρ

Fluid density

f

Dimensionless concentration

σ

Electrical conductivity

σ^*

Stefan-Boltzmann constant

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(Edited by YANG Hua)

中文导读

两平行板间非稳态Casson磁流体的辐射挤压流动

摘要：本文通过数值计算，研究了在均匀一阶化学反应条件下，受热辐射、内热产生或吸收以及焦耳耗散效应的影响，Casson流体在两平行板间的传热传质特性。建立流变Casson流体流动模型，将Casson流体的非牛顿行为与牛顿流体的非牛顿行为进行区分。Casson流体的非稳态二维挤压流动的控制偏微分方程具有高度非线性和耦合性质。对守恒定律进行相似变换，得到控制挤压流的非线性常微分方程。分别应用利用龙格-库塔四阶积分法和bvp4c Matlab求解法对所得方程进行求解，并将结果进行比较。此外，对于不同的物理参数集，得到其数值结果，并以图和表的形式给出计算结果。然而，考虑到工业应用，因压缩数为负值，平行板运动所需的功率大幅度减少。研究发现，由于存在较强的Lorentz力，Hartmann数的增强值最终会抑制温度场和速度场。此外，压缩数越大，流体的压缩力越小，热场越小。而且化学反应的破坏性放大了浓度场，而构造性化学反应降低了浓度场。本文给出的数值解与以往的计算结果进行了比较，显示出较好的一致性。

关键词：挤压流；热辐射；热产生或吸收；Casson流体；焦耳耗散；磁场

Received date: 2018-10-16; Accepted date: 2019-01-29

Corresponding author: N. B. NADUVINAMANI, PhD, Professor; Tel: +91-9448650611; E-mail: naduvinamaninb@yahoo.com.in; ORCID: 0000-0001-7235-1000