Lie group analysis for MHD squeezing flow of viscous fluid saturated in porous media

Alexandria Engineering Journal (2019) 58, 1001–1010

H O S T E D BY

Alexandria University

Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com

ORIGINAL ARTICLE

Lie group analysis for MHD squeezing flow of viscous fluid saturated in porous media G. Magalakwe a, M.L. Lekoko a, K. Modise a, Chaudry Masood Khalique b,* a School of Mathematical and Statistical Sciences, North-West University, Potchefstroom Campus, Private Bag X6001, Potchefstroom 2520, South Africa b International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa

Received 2 March 2019; revised 23 August 2019; accepted 1 September 2019 Available online 31 October 2019

KEYWORDS Two-dimensional viscous laminar flow; Stable filtration process; Lie group analysis; Perturbation method

Abstract A two-dimensional laminar flow driven by fluid injection through porous surface which represent an incompressible fluid inside a filtration chamber during extraction of particles from the fluid is investigated. The study constructs a mathematical model that represents internal flow field during filtration process proficiently by using basic conservation laws of mass, momentum and energy. For better understanding of dynamics of the case study, solutions that lead to stable filtration process (balanced dynamical system) are obtained and consequently provide an insight of the important dynamics that lead to an optimal filtration process. The solution of partial differential equations representing the internal flow field similarity transformation based on Lie group method is utilized to reduce the system to ordinary differential equations. Thereafter, double perturbation method is employed to determine semi-analytical solutions of reduced system. Effects of various parameters that arise from the configuration (design) of the filter are presented graphically and analysed to show the connection between the case study and findings. Ó 2019 The Authors. Published by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).

1. Introduction It is vitally important to have mathematical representations (models) of various industrial phenomena that represent what take place in real life. Most industrial phenomena lead to three dimensional mathematical models, for a model to be realistic, * Corresponding author. E-mail addresses: [email protected] (G. Magalakwe), [email protected] (C.M. Khalique). Peer review under responsibility of Faculty of Engineering, Alexandria University.

hence yield complex models. On the other hand a simpler model can be obtained by careful consideration of the system configuration (design) that might lead to lower dimensions without loss of generality. The design of the system is one of the most important step during modelling process that prioritizes realistic models and appropriated boundary conditions. Finding solutions of such mathematical representations of industrial phenomena provide better understanding of the complicated processes and dynamics of such phenomena. In the pioneering study by Berman [1], Navier-Stokes equations were investigated to study the effect of wall porosity on a steady-state laminar flow inside a non-deformable channel.

https://doi.org/10.1016/j.aej.2019.09.002 1110-0168 Ó 2019 The Authors. Published by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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Nomenclature 2h Cp Cfx g Gr h_ H0 J k N Nux P P Pr Re R s T Tw Th t t Uw u u Vw

distance between walls, m specific heat, J/kg K skin friction coefficient gravitational acceleration, m/s2 Grashoff number wall dilation rate, m/s magnetic field strength N/m A dimensionless Joule heating parameter permeability of porous medium m2 Stuart number local Nusselt number dimensionless pressure pressure, Nm/2 Prandtl number permeation Reynolds number dimensionless porosity variable kinematic viscosity, m2/s temperature of the fluid, K temperature at wall, K temperature of the fluid at a certain height from the wall, K time, s dimensionless time axial velocity at the wall, m/s axial velocity, m/s dimensionless axial velocity normal velocity at the wall, m/s

The author used perturbation method to obtained closed form solutions of the axial velocity and pressure drop which indicate that both velocity and pressure depend on fluid injection and normal space variable. Dauenhauer et al. [2] investigated an unsteady state regime of the flow [1] in a deformable channel with wall suction or injection. Authors found numerical solutions by using Runge-Kutta along with Jacobian method to find unknown initial guesses for the numerical scheme. Boutros et al. [3] found analytical solution of flow field [2] using Lie group analysis. Matebese et al. [4] investigated the effect of variable magnetic field on the flow field studied by [3]. The mathematical models studied in [2–4] have direct application to filtration process since particles can be restricted from entering the fluid chamber through the surface of the chamber while the deformable channel is squeezed to increase outflow. This article seeks to investigate filtration process further by looking at various forces affecting the flow inside the fluid chamber. Due to this application many researchers have studied various models to understand the flow inside a deformable channel in the past and recent, to gain better insight of the dynamics of such flows. Finding solutions of such models became prominent part of research over the years. These two arms of research provide engineers, physicists, etc with theoretical insight information of what to expect from the final product. Most recently, researchers use different methods to investigate squeezing flow due to contraction of walls and heat transfer [2–9] to understand the flow dynamics and temperature distribution inside a deformable channel. Domairry and

v v x x y y

normal velocity, m/s dimensionless normal velocity dimensionless axial coordinate axial coordinate, m normal coordinate, m dimensionless normal coordinate

Greek symbols q density, kg=m3 W dimensionless stream function a dimensionless wall dilation rate / porosity  W stream function, m2 =s r electrical conductivity of fluid, S/m l magnetic permeability, H/m h dimensionless temperature function k thermal diffusion b thermal expansion Superscript 0 differentiation with respect to y . differentiation with respect to t Subscript w wall condition h fluid height

Hatami [5] used differential transformation method (DTM) to investigate squeezing Cu-water nanofluid between parallel plates. The study found that when squeezing the internal flow, the Nusselt number increases with increase in nanoparticle volume fraction and Eckert number. Authors also found that an increase in squeeze number and the Eukert number leads to the decrease in thermal boundary layer. Ghadikolaei et al. [6] studied the effects of thermal radiation and Joule heating on an unsteady MHD Eyring-Powell squeezing flow in a stretching channel. Balazadeh et al. [7] further studied transient EyringPowell squeezing flow in a stretching channel due to magnetic field using DTM. The studies in [6,7], reported that the velocity decrease due to Lorentz force generation and temperature decrease with the increase in thermal radiation. Ghadikolaei et al. [8] studied three dimensional squeezing flow of mixture base fluid suspended by hybrid nanoparticle dependent on the shape factor. The influence of different parameters variations on the velocity and temperature were analysed graphically. Also the skin friction and Nusselt number were investigated when the channel height increase and decrease respectively. Authors [9], studied flow [8] by replacing ethylene glycol-water by ethylene glycol and considered rotating channel with nonlinear thermal radiation. The study found that oscillating behaviour of horizontal linear velocity was due to the change in rotation parameter. Another issue which is key to the current study (filtration) is magnetic field integrated on the system to collect particles such as steel, iron, etc. and parameters that arise from magnetic field such as Lorentz force and Joule heating. Here are some of the

Lie group analysis for MHD squeezing flow of viscous fluid research work done to understand the effects of magnetic field on the system. The study [10] investigated laminar flows in a semi-porous channel in the presence of a uniform magnetic field using homotopy analysis. Ziabakhsh and Domairry [11] studied the effects of magnetic field on a mixed convection flow in a lid-driven cavity with linearly heated wall using lattice Boltzmann method. Their results indicate that heat transfer decreases with an increase in Hartmannn number for various Richardson numbers and direction of the magnetic field. The joule heating effects on the natural convection flow along a vertical wavy surface was investigated [12]. From their study, authors concluded that an increase in Joule heating parameter leads to a decrease in the local rate heat transfer and the increase in local skin friction coefficient. Also the flow rate decreases and the boundary layer grows thick when the effects of the magnetic field is increased. Mohyud-Din et al. [13] studied the three dimensional heat and mass transfer with magnetic effects for a flow of nanofluid between two parallel plates in a rotating system. Convective laminar wall jet flow of viscous nanofluid with passive control model was studied by Zaidi et al. [14], inspired by the increase in demand for of an efficient heating/cooling system for automotive and process industries. They concluded that Lorents force plays a critical role in normalizing flow and increasing temperature. Authors in [15] investigated the effects of variable Lorentz force on nanofluid in a movable parallel plate. They observed that an increase in the brownian motion parameter leads to an increase in temperature while the effects of increasing Brownian motion leads to a decrease on concentration, moreover an increase in thermophoresis leads to a decrease in temperature and increase in concentration. The study by Chadikolaei et al. [16], investigated the magnetohydrodynamic boundary layer flow and heat transfer of an incompressible Iron (II,III) oxide-ethylene glycol nanoparticle on micropolar fluid with suspended dust particles. They studied the effects of various parameter on velocity and temperature in two phases of the fluid. The study of Erying-Powell nanofluid fluid flow over a rotating disk was performed by [17]. Authors in [17], found that temperature shows dual behaviour when thermophoresis parameter and Prandtl number changes. Their study also found that the temperature increases with the increase in thermophoresis parameter and decrease with increase in Prandtl number. In [18], authors investigated diffusion-thermo and thermaldiffusion consequences in a viscous flow between deformable permeable walls. This study found analytical solutions for velocity field, temperature and concentration. The studies of heat and mass transfer studied in a convergent/divergent channels were studied by [19,20]. Other studies related to MHD phenomena has been discussed by various researchers [21–25]. Many researchers studied squeezing flow due to deformation of the channel. Also the effects of surface and body forces were investigated. A literature review shows that no attempt has been made to study the use of permeable surface, porous medium and wall dilation to investigate the dynamics of filtration process. Motivated by the above mentioned works, the current case study extends the flow studied by [1–4] in three directions by taking into consideration various forces affecting flow field during filtration process and their importance in finding optimal outflow, thus, more filtrates. The first direction is related to the influence of a constant magnetic field on fluid

1003 whereas the second investigates effect of porous medium on axial velocity for optimal outflow (filtrates). Lastly, effects of heat transfer are taken into consideration. The main aim of this work is to explore a practical problem by studying flow dynamics during filtration process. 2. Mathematical representation of problem Consider an unsteady 2 D flow of viscous, incompressible and magneto-hydrodynamic (MHD) fluid in porous medium bounded by two deformable walls, which has same permeability parallel to axial direction. The length of the walls are assumed to be infinitely greater than the distance between them. The parallel plates are located at a distance y ¼ hðtÞ and y ¼ hðtÞ from centre of the chamber. The chamber is closed on left and open on right side to allow outflow. Due to the nature of the walls being permeable, fluid mass is able to flow in at the surface and flow out of the open face on the right while filtering particles. The inflow of fluid through these walls is minimal since walls are weakly permeable. Furthermore, the permeable nature of the walls and the porous nature of the medium allow the fluid into the system through the walls and pores of medium, this is called injection. The schematic representation of the above specified flow is represented in the Fig. 1 below. The two-dimensional flow of the viscous, incompressible and magneto-hydrodynamic fluid inside a filtration chamber is governed by the following equations [1–3,26] and boundary conditions: @ u @ v þ ¼ 0; @ x @ y

ð2:1Þ
2 2  2 2  @ u @ u @ u 1 @ P @ u @ u s/ rl H0 þ u þ v þ s  u ¼ 0; ð2:2Þ þ u þ @t @ x @ y q @ x @ x2 @ y2 k q
 2 2 @ v @ v @ v 1 @ P @ v @ v s/ þ u þ v þ s þ þ v  gbðT  Tw Þ ¼ 0; ð2:3Þ @t @ x @ y q @ y k @ x2 @ y2
2  @T @T @T @ T @2 T rl2 H20 2 þ u þ v k þ u ¼ 0: ð2:4Þ  @t @ x @ y @ x2 @ y2 qcp

Here u is axial velocity, v is normal velocity, T is temperature, q is density, P is pressure, s is kinematic viscosity, g is gravitational acceleration, b is thermal expansion, k is thermal diffusivity, t is time, / is porosity parameter, k is permeability of porous medium, r is electrical conductivity, l is magnetic permeability, H0 is magnetic field strength and cp is specific heat. The boundary conditions are ðiÞ

uð x; hðtÞ; tÞ ¼ 0;

ðiiÞ

@ uð x;0;tÞ @ y

ðiiiÞ

u ¼ 0

¼ 0;

vð x; hðtÞ; tÞ ¼ Vw ;

vð x; 0; tÞ ¼ 0;

@Tð x;0;tÞ @ y

Tð x; hðtÞ; tÞ ¼ Tw ;

¼ 0;

ð2:5Þ

at x ¼ 0:

The velocity components in terms of stream function  x; y; tÞ are Wð u ¼

 x; y; tÞ @ Wð ; @ y

v ¼ 

 x; y; tÞ @ Wð : @ x

ð2:6Þ

Invoking y ¼ y=hðtÞ into (2.6) we obtain u ¼

 x; y; tÞ 1 @ Wð ; h @y

v ¼ 

 x; y; tÞ @ Wð : @ x

ð2:7Þ

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Schematic representation of the system and velocity profile.

Fig. 1

Substituting (2.7) into momentum Eqs. (2.2), (2.3)energy Eq. (2.4) together with the non-dimensional quantities given below u ¼ Vuw ;

v ¼ Vvw ;

t ¼ tVhw ;

a ¼ hsh ;

_

y y ¼ hðtÞ ;

x x ¼ hðtÞ ;

H N ¼ rhl ; qVw 2

2

1 R



W ¼ hVWw ;

s/h ¼ kV ; w



P ¼ qVP2 ;

ð2:8Þ

w

h h ¼ TTT ; w Th

3.1. Lie group analysis To obtain a balanced filtration system (stable), the system must be in such a way that fluid temperature is constant throughout the chamber, i.e., T  Tw . This equilibrium effect of temperature results in a small Gr . The dimensionless system (2.9)–(2.11) admits the following six symmetries:

yield the following dimensionless system: X1 ¼

Wyt þ Wy Wxy  Wx Wyy þ Px  R1e ½aWy þ ayWyy þ Wxxy þ Wyyy  þ R1 Wy þ NWy ¼ 0; Wxt þ Wy Wxx  Wx Wxy  Py  R1e ½ayWxy þ Wxyy þ Wxxx 

ð2:9Þ

þ R1 Wx þ Gr h ¼ 0; ð2:10Þ
 @h @h @h 1 @2h @2h Re 2 þ Wy  Wx   J Wy ¼ 0: þ @ t @x @y Pr Re @x2 @y2 Pr ð2:11Þ In Pr ¼

addition lCp k

to

non-dimensional

h ÞL ; Re ¼ hVm w ; Gr ¼ gbðTwmT 2

3

and

quantities rhV l2 H2

J ¼ qCp ðTww T0h Þ

(2.8), are

Prandtl number, Reynolds number, Grashof number and Joule heating parameter respectively. Similarly boundary conditions (2.5) becomes ðiÞ Wy ¼ 0; Wx ¼ 1; h ¼ 1; at y ¼ 1; ðiiÞ Wyy ¼ 0; Wx ¼ 0; hy ¼ 0; at y ¼ 0;

ð2:12Þ

ðiiiÞ Wy ¼ 0 at x ¼ 0:

@W ; @y

v¼

@W : @x

X5 ¼ F2 ðtÞ

X2 ¼ h @ ; @y

@ ; @ t

ð2:13Þ

3. Solution methodology We use Lie group analysis [27–32] to transform (2.9)–(2.12) to system of ordinary differential equations. Thereafter double perturbation is employed to find semi-analytical solutions for the problem under investigation.

X3 ¼

X6 ¼ F1 ðtÞ

@ ; @h

X4 ¼

@ : @x

@ ; @W

Lie symmetries X1 and X2 are the only generators that leaves the system and the boundary conditions invariant. It follows from these two symmetries that W ¼ Wðx; y; tÞ; h ¼ hðx; y; tÞ; P ¼ Pðx; y; tÞ are invariant under X1 and X2 , if UW ¼ XðW  Wðx; y; tÞÞ ¼ 0; Uh ¼ Xðh  hðx; y; tÞÞ ¼ 0; UP ¼ XðP  Pðx; y; tÞÞ ¼ 0; where X is Lie symmetry operator. Thus the general solutions are given by W ¼ hðyÞHðx; yÞ; P ¼ Cðx; yÞ; h ¼ sðx; yÞ:

ð3:14Þ

Substituting (3.14) into (2.9) and multiplying by K ¼ R1e , we have 3

Also, the relation between the stream function and the velocity (2.7), yields the following non-dimensional relation u¼

@ ; @ t

1 , H

letting

2

d h d h K dy 3 þ ½aKy  hK1  3KK2  dy2   þ aK  2aKyK2  hK3 þ hK4  KK5  3KK6 þ R1 þ N dh dy  2   dh 1 K1 dy þ aKK2 þ R K2 þ NK2  aKK6 y  KK9  KK10 h

þ½K7  K8 h2 þ H1

dC ; dx

ð3:15Þ

where K1 ¼ Hx ; K5 ¼ HHxx ; K9 ¼

Hxxy H

;

Hy H H ; K3 ¼ xH y H H H H K6 ¼ Hyy ; K7 ¼ yH xy ; H K10 ¼ Hyyy :

K2 ¼

;

K4 ¼ Hxy ; K8 ¼

Hx Hyy H

;

ð3:16Þ

Lie group analysis for MHD squeezing flow of viscous fluid Solving K1 ¼ Hx for Hðx; yÞ from (3.16) gives Hðx; yÞ ¼ K1 x þ K11 ðyÞ:

ð3:17Þ

Using the above Eq. (3.17) into W ¼ hðyÞHðx; yÞ from (3.14) we get W ¼ ðK1 x þ K11 ðyÞÞhðyÞ:

ð3:18Þ

The first derivative of (3.18) with respect to y together with (2.12) (iii) gives K11 ðyÞhðyÞ ¼ K12 ; where K12 is an arbitrary constant. Thus Eq. (3.18) becomes W ¼ xGðyÞ þ K12

ð3:20Þ

ð3:21Þ

which implies that K12 ¼ 0 from (3.19). Thus W ¼ xGðyÞ:

ð3:22Þ

Substituting the above stream function (3.22) into (2.13) yields the velocity components as u dG ¼ ; v ¼ GðyÞ: ð3:23Þ x dy The above velocity components confirm that normal flow is injected into the filter chamber and axial velocity changes along the normal direction per length of the chamber. This information confirms the parabolic flow field inside the chamber. Differentiating equation obtained from substituting (3.22) in (2.10) with respect to x, yields Pxy ¼

1 Gr hx : h2

ð3:24Þ

Similarly by substituting (3.22) into (2.9) and invoking (3.24) yields h4  3  2 3 2 2 d G þ a y ddyG3 þ 2 ddyG2 þ Re G ddyG3  RRe ddyG2  Re ddyG2 N dy4 i ð3:25Þ d2 G x þ h12 Gr hx ¼ 0: Re dG dy dy2

Using (3.22) and h ¼ sðx; yÞ from (3.14) in (2.11), we obtain
 @s dG @s 1 @2s @2s Re þ ð3:26Þ  J x2 G0 ðyÞ2 ¼ 0;  G x Pr @x dy @y Pr Re @x2 @y2 and the boundary conditions (2.12) become Gð0Þ ¼

dGð1Þ d2 Gð0Þ ds ¼ ðx; 0Þ ¼ 0; Gð1Þ ¼ sðx; 1Þ ¼ 1: ¼ dy dy dy2 ð3:27Þ

Equating powers of h from (3.25) and thereafter using the value of h from Eq. (3.14) yields Gr sx ¼ 0. This result implies that s ¼ EðyÞ and Gr  0 which satisfy dynamics of the stable filtration system. Since Gr  0, Eq. (3.24) yields Pxy ¼ 0 which confirms that the flow is laminar. Since the driving force is only in the axial direction at a constant rate, using the fact that Gr  0, for a stable system and temperature s ¼ EðyÞ, (3.25)– (3.27) yields

d2 G dy2

2

 Re ddyG2 N

d2 G dy2

¼ 0;
 @E 1 @2E Re GðyÞ þ J x2 G0 ðyÞ2 ¼ 0; þ Pr @y Pr Re @y2 Re dG dy

ð3:28Þ

ð3:29Þ

and the boundary conditions Gð0Þ ¼

ð3:19Þ

by letting GðyÞ ¼ K1 hðyÞ. Using P ¼ Cðx; yÞ from (3.14), and (3.17) into (3.15) yields K11 ¼ 0;

1005 h 3 i 2 3 d4 G þ a y ddyG3 þ 2 ddyG2 þ Re G ddyG3  RRe dy4

dGð1Þ d2 Gð0Þ @Eð0Þ ¼ ¼ ¼ 0; Gð1Þ ¼ Eð1Þ ¼ 1: dy @y dy2 ð3:30Þ

From the above system, velocity GðyÞ and temperature EðyÞ are both functions of y only, thus, by comparing powers of x from Eq. (3.29) yields the following equations:
 @E 1 @2E þ ¼ 0; ð3:31Þ x0 :GðyÞ @y Pr Re @y2 Re x2 :J G0 ðyÞ2 ¼ 0: ð3:32Þ Pr Based on the design of the system under investigation, Joule heating parameter J becomes zero, since the system injects fluid and the velocity of the system (filtrates) is non constant, thus Re and G0 ðyÞ cannot be zero. By using the assumption Gr  0, the same results can be obtained from the Joule heat 2

1 2

0 b mGr which confirms minimal effect parameter [12], J ¼ qcprðT w T1 Þ

of the Joule heat during stable operation. From the above results system (3.28)–(3.30) becomes h 3 i 2 3 2 2 d4 G þ a y ddyG3 þ 2 ddyG2 þ Re G ddyG3  RRe ddyG2  Re ddyG2 N dy4 ð3:33Þ d2 G ¼ 0; Re dG 2 dy dy
 @E 1 @2E þ GðyÞ ¼ 0; ð3:34Þ @y Pr Re @y2 and the boundary conditions Gð0Þ ¼

dGð1Þ d2 Gð0Þ @Eð0Þ ¼ ¼ ¼ 0; Gð1Þ ¼ Eð1Þ ¼ 1: dy @y dy2

ð3:35Þ

3.2. Semi-analytical solution In this subsection we find solutions of Eqs. (3.33), (3.34) with the corresponding boundary conditions (3.35) using double perturbation method. For more information the reader is referred to the studies [3]. The solutions of axial velocity and temperature can be represented by decreasing perturbation series as G ¼ G1 þ Re G2 þ R2e G3 þ OðR3e Þ;

ð3:36Þ

where G1 ¼ G10 þ aG11 þ a2 G12 þ Oða3 Þ; G2 ¼ G20 þ aG21 þ a2 G22 þ Oða3 Þ and

ð3:37Þ

G3 ¼ G30 þ aG31 þ a2 G32 þ Oða3 Þ:

Similarly E ¼ E1 þ Re E2 þ R2e E3 þ OðR3e Þ; where

ð3:38Þ

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G. Magalakwe et al.

E1 ¼ E10 þ aE11 þ a2 E12 þ Oða3 Þ E2 ¼ E20 þ aE21 þ a2 E22 þ Oða3 Þ; and E3 ¼ E30 þ aE31 þ a2 E32 þ Oða3 Þ:

and ð3:39Þ

Substituting (3.36) and (3.38) into (3.33) and (3.34) gives momentum and energy equations in terms of Re which yields six equations from equating like powers of Re . Thereafter substituting (3.37) and (3.39) into those resulting equations yields equations in terms of the second perturbation parameter a which yields system ordinary differential equations by equating powers of a. Solving the resulting system yields

EðyÞ ¼ 1:

ð3:47Þ

We note that the solution GðyÞ of [3] is retrieved when the pore size of the medium becomes extremely large and magnetic field becomes extremely small. This verifies the correctness of the present case study, since in the absence of a magnetic field and porous medium this work reduces to the special case presented in [3]. The steady state regime during filtration, leads to the following interesting physical quantity:

i 1 h 2 2 2 yfa ð13  25y2 Þðy2  1Þ þ 210aðy2  1Þ  1400ðy2  3Þg ; 2800 h 2 1 G2 ðyÞ ¼ 232848000R yðy2  1Þ ð831600fRðy2  7N þ 2Þ  7g   2310a 2y2 fð240N  227ÞR þ 240g þ ð552N þ 681ÞR þ 65Ry4 þ 552 þa2 ½35y4 fð3905N  6561ÞR þ 3905g þ 2y2 ½f133595N þ 50481gR þ 133595  3½f29953N þ 114111gR þ 29953 þ 12600Ry6 ; G1 ¼

ð3:40Þ

ð3:41Þ

2  2 yðy2 1Þ 2 2 2 6 4 G3 ¼ 1271350080000R 2 1260a R f1001N ð5y  9Þð25y  37Þ  26Nð875y þ 18305y

þ293y2  51137Þ  4060y8 þ 63133y6 þ 357696y4 þ 427177y2 þ 394166g þ 26Rf77Nð5y2  9Þ ð25y2  37Þ  875y6  18305y4  293y2 þ 51137 þ 1001ð5y2  9Þð25y2  37ÞgÞa2 105Ry8 fð6510N  46873ÞR þ 6510g  42y6 fR½350Nfð1339N  7698ÞR þ 2678g þ 3099111R 2694300 þ 468650g þ 14y4 fR½900Nfð6552N  10585ÞR þ 13104g  2957491R  9526500

ð3:42Þ

þ5896800g  y fRð84N½ð12621055N þ 3260532ÞR þ 2524210  95806709R þ 273884688Þ þ106016820gR þ 3Rf42N½ð245908N þ 2413431Þ þ 491816 þ 100425529R þ 101364102g  þ783825R2 y10 þ 30984408 þ 491400ðRf7y4 ½ð55N  102ÞR þ 55  2y2 f77N½ð10N  23ÞR þ 20   þ530Rg þ 77Nfð44N þ 69ÞR þ 88g þ 28Ry6  1406R þ 1771ð2y2 þ 3Þ þ 308ð11  5y2 Þ : 2

(i) Skin friction

and E1 ¼ 1;

E2 ¼ 0;

E3 ¼ 0:

ð3:43Þ

Finally, we obtain the solution of equation axial velocity and temperature distribution as

Cfx ¼ 

2lð@u=@yÞy¼1 ¼ 2R1=2 G00 ð0Þ; ex qUw

i 1 h 2 2 2 yfa ð13  25y2 Þðy2  1Þ þ 210aðy2  1Þ  1400ðy2  3Þg þ G¼  2800 h 2 1 yðy2  1Þ ð831600fRðy2  7N þ 2Þ  7gþ Re 232848000R   2310 2y2 fð240N  227ÞR þ 240g þ ð552N þ 681ÞR þ 65Ry4 þ 552 þa2 ½35y4 fð3905N  6561ÞR þ 3905g þ 2y2 ½f133595N þ 50481gR þ 133595  3½f29953N þ 114111gR þ 29953 þ 12600Ry6 þ  2  yðy2 1Þ 2 2 2 2 6 4 R2e 1271350080000R 2 1260a R f1001N ð5y  9Þð25y  37Þ  26Nð875y þ 18305y

ð3:48Þ

ð3:44Þ

ð3:45Þ

þ293y2  51137Þ  4060y8 þ 63133y6 þ 357696y4 þ 427177y2 þ 394166g þ 26Rf77Nð5y2  9Þ ð25y2  37Þ  875y6  18305y4  293y2 þ 51137 þ 1001ð5y2  9Þð25y2  37ÞgÞa2 105Ry8 fð6510N  46873ÞR þ 6510g  42y6 fR½350Nfð1339N  7698ÞR þ 2678g þ 3099111R 2694300 þ 468650g þ 14y4 fR½900Nfð6552N  10585ÞR þ 13104g  2957491R  9526500 þ5896800g  y2 fRð84N½ð12621055N þ 3260532ÞR þ 2524210  95806709R þ 273884688Þ þ106016820gR þ 3Rf42N½ð245908N þ 2413431Þ þ 491816 þ 100425529R þ 101364102g  þ783825R2 y10 þ 30984408 þ 491400ðRf7y4 ½ð55N  102ÞR þ 55  2y2 f77N½ð10N  23ÞR þ 20   þ530Rg þ 77Nfð44N þ 69ÞR þ 88g þ 28Ry6  1406R þ 1771ð2y2 þ 3Þ þ 308ð11  5y2 Þ

ð3:46Þ

Lie group analysis for MHD squeezing flow of viscous fluid   where sw ¼ l @u @y

1007

is the wall shear stress. y¼1

(ii) Nusselt number xqw xE0 ð0Þ ¼ ; kðTw  Th Þ kðTw  Th Þ   where qw ¼ k @T is the surface heat flux. @y

Nux ¼

ð3:49Þ

y¼1

Using the velocity GðyÞ and temperature EðyÞ from (3.44) and (3.47), the above quantities become Cfx  0; Nux ¼ 0:

ð3:50Þ ð3:51Þ

The above results show that at the walls, the skin friction is extremely small, thus, it does not have any effect on the system during stable operation. Also the results reveal that during stable operation, the Nusselt number does not have any effects when temperature is constant within the system. Thus, confirm laminar nature of the flow. These findings are ideal during filtration process. 4. Results and discussion In this section the graphical representations of the self-axial velocity u=x is shown to investigate optimal filtrates. An analysis is carried out to interpret the effect of self-axial velocity subject to some physical parameters such as the wall dilation a, permeation Reynolds number Re , the porosity variable R, the Stuart number N. Temperature distribution is also presented. The analysis gives more insight information about the dynamics of the filtration process that yields optimal outflow (area under the graph) of filtrates. The area as an average flow applies to all figures.

Fig. 3 Impact of a on velocity profile, where blue = 1, pink = 0:5, black = 0, green = 0:5 and red = 1 at R ¼ 0:25; Re ¼ 1 and N ¼ 2:5.

The squeezing (wall dilation) effect of the above velocity profiles reflects the same parabolic velocity behaviour as in [5,7] between the walls. 4.2. Effects of Reynolds number inside the filtration chamber Figs. 4 and 5 show that velocity increases when more fluid is injected inside the chamber. Also, the increase in Stuart number leads to a reverse flow close to the walls which is not ideal during filtration process (to obtain optimal filtrates). Fig. 5,

4.1. Effects of wall dilation Results from Fig. 2 and 3 illustrate that when the chamber (wall dilation) increases the axial velocity for both small and average pore size increases. Thus, more fluid outflow. Area between the graphs u=x for different values of a and axis u=x ¼ 0 depicts that average outflow for both flow fields (filtrates) are relatively the same for small and average pore size.

Fig. 2 Impact of a on velocity profile, where blue = 1, pink = 0:5, black = 0, green = 0:5 and red = 1 at R ¼ 0:5; Re ¼ 1 and N ¼ 2:5.

Fig. 4 Impact of Re on velocity profile, where blue = 5, pink = 4, black = 3, green = 2 and red = 1 at R ¼ 0:5; a ¼ 1 and N ¼ 2:5.

Fig. 5 Impact of Re on velocity profile, where blue = 5, pink = 4, black = 3, green = 2 and red = 1 at R ¼ 0:5; a ¼ 1 and N ¼ 5.

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shows two regions above the line u=x ¼ 0 and below which indicates the effect of propulsion force and drag force respectively. Hence less permeates propel fast. 4.3. Effects of porosity variable inside the filtration chamber Fig. 6, illustrates that when the chamber volume increases while fluid permits at an extremely lower speed with an average magnetic strength, variation of porosity has no effects during filtration process. Thus, the small pore size should be taken under this situation. Fig. 7 on the other hand indicates that with more injection, the effect of magnetic field yields reverse flow. The fluid flow is pressure driven, thus, the increase in chamber size decreases internal pressure, as a result decrease outflow.

Fig. 8 Impact of N on velocity profile, where blue = 5, pink = 3:5, black = 2:5, green = 1:5 and red = 0:5 at R ¼ 0:5; a ¼ 1 and Re ¼ 5.

4.4. Effects of Stuart number inside the filtration chamber Figs. 8 and 9 illustrate that expansion and contraction of the chamber yield the increase and the decrease of drag force respectively. Expansion leads to more effects of magnetic field on the fluid bulk compared to contraction since the surface walls move closer to the magnets during expansion. 4.5. Temperature distribution inside the chamber Fig. 10 indicates constant temperature for filtration process (when the system is stable).

Fig. 9 Impact of N on velocity profile, where blue = 5, pink = 3:5, black = 2:5, green = 1:5 and red = 0:5 at R ¼ 0:5; a ¼ 1 and Re ¼ 5.

Fig. 6 Impact of R on velocity profile, where blue = 1, pink = 0:75 and black = 0:5 at Re ¼ 1; a ¼ 1 and N ¼ 5.

Fig. 10

Temperature distribution profile.

5. Concluding remarks

Fig. 7 Impact of R on velocity profile, where blue = 1, pink = 0:75 and black = 0:5 at Re ¼ 5; a ¼ 1 and N ¼ 5.

In this work, Lie group analysis along with double perturbation method is used to study the internal flow. The configuration and design of the filter revealed the important parameters which affect the internal flow during filtration process. Also proficient mathematical formation of the case study is obtained from basic conservation laws of mass, momentum and energy. The findings from the study indicate that to have an optimal outflow the following dynamics are critical:

Lie group analysis for MHD squeezing flow of viscous fluid (1) Wall dilation rate: The increase in wall dilation rate creates more space inside the chamber. Thus the pressure difference between the internal and external pressure increases in such a way that the system allows more injection of fluid inside chamber, which results in more flow out of the chamber. (2) Reynolds: The flow injection into the filter chamber has a positive effect during filtration process. The more fluid mass into the chamber leads to more permeates out of the system. (3) Porosity: The variation of pore size of the medium does not affects fluid flow for weak injection along with average magnetic field and expansion. The average outflow is more when the fluid injection is minimal. (4) Magnetic field: Increasing magnetic load zone leads to a reverse flow away from the centre of the chamber and hence decreases the average outflow. To counter this negative effect, more injection is needed to results in a net force towards the chamber since both magnetic field and the injection act normal to the surface. (5) Temperature: Temperature is constant throughout the chamber, hence its effect during filtration plays an important role based on the following reasons. Firstly, it increases the internal pressure, thus, increases the driving force which is needed to have more filtrates. Secondly, it decreases the drag force by decreasing the fluid density, thus, yields more flow output. Lastly effects of buoyancy force which leads to turbulent flow becomes minimal due to the temperature effects. Hence, variation of temperature is not ideal for a stable filtration process. (6) Skin friction: During stable operation the effects of skin friction is minimal. At this stage the fluid is less dense due to the effect of temperature from the walls which results in minimal drag force within the system. (7) Nusselt: Also stable operation leads to no influence of Nusselt number. The system at this point has no heat flux, thus fluid layers have same density. (8) Joule heat parameter: When the temperature of the fluid approximates the temperature of the walls this leads to extremely small Grashoff number Gr . As a result, there is no effect of joule heat due to magnetic field.

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[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

Acknowledgements Authors would like to thank CSIR and North-West University South Africa for their financial support. References [1] A.S. Berman, Laminar flow in channels with porous walls, J. Appl. Phys. 24 (1953) 1232–1235. [2] E.C. Dauenhauer, J. Majdalani, Exact self similarity solution of the Navier-Stokes equations for a deformable channel with wall suction or injection, AIAA 3588 (2001) 1–11. [3] Z. Boutros, B. Minas, A. Badran, S. Hassan, Lie group method solution for two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability, Appl. Math. Model. 31 (2007) 1092–1108. [4] B. Matebese, A. Adem, C.M. Khalique, T. Hayat, Twodimensional flow in a deformable channel with porous

[17]

[18]

[19]

[20]

medium and variable magnetic field, Math. Comput. Appl. 31 (2010) 674–684. G. Domairry, M. Hatami, Squeezing Cu-water nanofluid flow analysis between parallel plates by DTM-Pade´ Method, J. Mol. Liq. 193 (2014) 37–44. S.S. Ghadikolaei, Kh. Hosseinzadeh, D.D. Ganji, Analysis of unsteady MHD Eyring-Powell squeezing flow in stretching channel with considering thermal radiation and Joule heating effect using AGM, Case Stud. Therm. Eng. 10 (2017) 579–594. N. Balazadeh, M. Sheikholeslami, D.D. Ganji, Z. Li, Semi analytical analysis for transient Eyring-Powell squeezing flow in a stretching channel due to magnetic field using DTM, J. Mol. Liq. 260 (2018) 30–36. S.S. Ghadikolaeia, Kh. Hosseinzadehb, D.D. Ganji, Investigation on three dimensional squeezing flow of mixture base fluid(ethylene glycol-water) suspended by hybrid nanoparticle (Fe3O4-Ag) dependent on shape factor, J. Mol. Liq. 262 (2018) 378–388. S.S. Ghadikolaei, Kh. Hosseinzadeh, M. Hatami, D.D. Ganji, M. Armin, Investigation for squeezing flow of ethylene glycol (C2H6O2) carbon nanotubes (CNTs) in rotating stretching channel with nonlinear thermal radiation, J. Mol. Liq. 263 (2018) 10–21. Z. Ziabakhsh, G. Domairry, Solution of the laminar viscous flow in a semi-porous channel in the presence of a uniform magnetic field by using the homotopy analysis method, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 1284–1294. G.H.R. Kefayati, M. Gorji-Bandpy, H. Sajjadi, D.D. Ganji, Lattice Boltzmann simulation of MHD mixed convection in a lid-driven square cavity with linearly heated wall, Scientia Iranica B 4 (2012) 1053–1065. N. Parveen, M.A. Alim, Joule heating effect on magnetohydrodynamic natural convection flow along a vertical wavy surface, J. Naval Archit. Mar. Eng. 9 (2012) 11–24. S.T. Mohyud-Din, Z.A. Zaidi, U. Khan, N. Ahmed, On heat and mass transfer analysis for the flow of a nanofluid between rotating parallel plates, Aerosp. Sci. Technol. 46 (2015) 514–522. S.Z. AliZaidi, S.T. Mohyud-Din, Convective heat transfer and MHD effects on two dimensional wall jetflow of a nanofluid with passive control model, Aerosp. Sci. Technol. 49 (2016) 225– 230. Kh. Hosseinzadeh, A.J. Amiri, S.S. Ardahaie, D.D. Ganji, Effect of variable lorentz forces on nanofluid flow in movable parallel plates utilizing analytical method, Case Stud. Therm. Eng. 10 (2017) 595–610. S.S. Ghadikolaei, Kh. Hosseinzadeh, D.D. Ganji, M. Hatami, Fe3O4-(CH2OH)2 nanofluid analysis in a porous medium under MHD radiative boundary layer and dusty fluid, J. Mol. Liq. 258 (2018) 172–185. M. Gholiniaa, Kh. Hosseinzadeha, H. Mehrzadib, D.D. Ganjia, A.A. Ranjbara, Investigation of MHD Eyring-Powell fluid flow over a rotating disk under effect of homogeneous-heterogeneous reactions, Case Stud. Therm. Eng. 13 (2019) 1–10. S. Srinivas, A. Subramanyam Reddy, T.R. Ramamohan, A study on thermal-diffusion and diffusion-thermo effects in a two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability, Int. J. Heat Mass Transf. 55 (2012) 3008–3020. U. Khan, N. Ahmed, S.T. Mohyud-Din, Thermo-diffusion, diffusion-thermo and chemical reaction effects on MHD flow of viscous fluid in divergent and convergent channels, Chem. Eng. Sci. 141 (2016) 17–27. S.T. Mohyud-Din, U. Khan, N. Ahmed, S.M. Hassan, Magnetohydrodynamic flow and heat transfer of nanofluids in stretchable convergent/divergent channels, Appl. Sci. 5 (2015) 1639–1664.

1010 [21] A. Shafiq, Z. Hammouch, A. Turab, Impact of radiation in a stagnation point flow of Walters’ B fluid towards a Riga plate, Therm. Sci. Eng. Prog. 6 (2018) 27–33. [22] A. Shafiq, T.N. Sindhu, Statistical study of hydromagnetic boundary layer flow of Williamson fluid regarding a radiative surface, Res. Phys. 7 (2017) 3059–3067. [23] F. Naseem, A. Shafiq, L. Zhao, A. Naseem, MHD Bioconvective flow of Powell Eyring nanofluid over stretched surface, AIP Adv. (USA) 7 (2017) 065013. [24] A. Naseem, A. Shafiq, L. Zhao, M.U. Farooq, Analytical investigation of third grade nanofluidic flow over a riga plate using Cattaneo-Christov model, Res. Phys. 9 (2018) 961–969. [25] M. Khan, R. Malik, A. Munir, A. Shahzad, MHD flow and heat transfer of Sisko fluid over a radially stretching sheet with convective boundary conditions, J. Brazilian Soc. Mech. Sci. Eng. 38 (4) (2016) 1–11.

G. Magalakwe et al. [26] I.G. Currie, Fundamental Mechanics of Fluids, fourth ed., CRC Press, New York, 2012. [27] L.V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982. [28] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1993. [29] G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer, New York, 1989. [30] H. Stephani, Differential Equations. Their Solutions Using Symmetries, Cambridge University Press, Cambridge, 1989. [31] N.H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, John Wiley and Sons, Chichester, 1999. [32] F.M. Mahomed, Recent trends in symmetry analysis of differential equations, Notices SAMS 33 (2002) 11–40.