Analysis of MHD Fluid Flow and Heat Transfer Inside an Inclined Deformable Filter Chamber: Lie Group Method

This paper investigates MHD fluid flow and distribution of heat inside a filter chamber during a process of filtering particles from the fluid. A flow model of MHD viscous incompressible fluid inside a filter is studied to seek semi-analytical solutions which are analysed to find flow and heat dynamics that lead to optimal outflow (maximum filtrates) during filtration. Lie group method is used to reduce a system of four partial differential equations describing fluid flow and temperature distribution inside the filter chamber to a system of two ordinary differential equations. The reduced system is then solved by perturbation process to obtain semi-analytical solutions for flow velocity and temperature variation inside the chamber. To understand the flow dynamics and heat distribution of the underlying case study better, the influence of different parameters during filtration resulting from the filter design, flow dynamics and heat effects are graphically presented and analysed in order to identify a combination of flow parameters that yields the best filtration process. The findings show that to maximise filtrates production, more fluid injection is required during filtration. Moreover, from the findings, it is evident that the temperature increase inside the chamber arising from the Joule heating effect is ideal since it increases internal work done and hence increases filtrates production.


Introduction
Studies investigating the extraction of particles pertaining to flow injection through a permeable surface and porous medium have gained attention due to their importance in various manufacturing industries, such as water purification and petroleum processes, to name a few.
For manufacturing industries to have a better understanding of such complicated manufacturing processes, it is vital to have mathematical models representing the dynamics of such processes, which can be solved analytically or numerically.The analytical or numerical solutions of such models help scientists with a theoretical understanding of such processes and what to anticipate as an end product.Various research works have been conducted to investigate different parameters that are relevant to the underlying research work of laminar fluid flow due to the injection of fluid through permeable surfaces.In many experimental, theoretical and computational studies, such flows have been the subject of interest.Berman [1] investigated an incompressible viscous fluid flow inside a permeable channel in two dimensions to introduce the channel flow study.The perturbation method was used to obtain an analytical solution of the twodimensional flow.Further research on this subject was subsequently carried out to understand more about the flow dynamics inside permeable channels for small and large injection or suction; see references [2][3][4].
The above-mentioned studies did not address wall effects caused by the deformation of a channel.To advance these studies, Uchida and Aoki [5] studied the effects of wall contraction or expansion caused by deformation on an unsteady fluid flow inside a deformable pipe.The focus of their study was to understand the unsteady blood flow due to forced valve contraction or expansion.Dauenhauer and Majdalani [6] also looked at unsteady flows in deformable rectangular permeable channels which expand uniformly.Dauenhauer and Majdalani in [7] further established the analytical solution of the Navier-Stokes equations for an unsteady incompressible flow inside a deformable permeable channel.To advance the solution of the flow in [7] using different approach, Majdalani et al. [8] used the perturbation method to find the flow analytically.Similarity transformation based on Lie symmetry approach was used [9] to further study the flow in [7,8] for small fluid permeation when walls contract or expand slowly.
In the works mentioned above, the study of flow inside permeable deformable channel was done without taking into consideration various forces which play an influential role on the flow dynamics.To account for the change in fluid flow momentum due to work done by surface area, effects of porous media and permeability on the flow, which affect the internal pressure, the current paper seeks to study these forces since they play an important role in many permeable surface and porous medium applications.To further advance the flow model studied by [8,9], Mahmood et al. [10] studied various forces affecting the flow studied by [8,9].The authors used homotopy perturbation and shooting methods to analyse the effects of surface forces on the flow field.The findings in [10] show that a larger porosity parameter decreases surface force effects, thus decreasing the fluid flow.When porous effects become minimal, the results correlate with existing non-porous results.
In current heat transfer research, the topic of flows influenced by magnetic field strength through porous media is rapidly rising due to their importance in engineering applications.Due to this fact, the study of heat transfer through porous media has drawn the attention of many researchers.To understand the effects of heat transfer, more research work was done to study the influence of parameters such as magnetic field, Joule heating, radiation, and buoyancy on the flow through deformable channels.
In order to have an insight into the heat effect on various flows inside a channel, Sheikholeslami et al. [11] studied the effects of variable magnetic effects on a two-phase flow and heat transfer inside parallel walls numerically.The study discovered that the effects of magnetic force increase Lorentz force, thus decreasing axial fluid flow.The study also indicated that the increase in magnetic force and squeezing parameter leads to an increase in the absolute value of skin friction.A nano-fluid flow inside two parallel horizontal circulating plates was studied by Sheikholeslami et al. [12].In their study, they found that the rise in magnetic parameters increases thermal boundary layer thickness, which correlates with the findings in [11].
Sharma and Sinha [13] analysed the impacts of viscous dissipation and Joule effects on an unsteady MHD fluid flow and heat distribution through a porous stretching medium.Their research found that the temperature of fluid increases when radiation effects increase.Also, they found that the increased heat flux within the fluid bulk is caused by the heat source and dissipation parameter; thus, these two parameters increase the temperature of the fluid while the Nusselt effects are decreased in the system.The investigation of chemically radiative MHD natural convective flow and heat transfer on an inclined moving porous plate was done by Balamurugan et al. [14].It was found that the radiation parameter increase leads to an increase in fluid velocity and a decrease in temperature.Vijaykumar and Keshava Reddy [18] studied how radiation absorption and Joule heating parameter affect a chemically reactive MHD fluid flow influenced by heat source and diffusion.The authors used similarity transformation and thereafter solved the transformed governing equations numerically using the fourth-order Runge-Kutta method together with the shooting method.
Lot of research work has been done to advance the effectiveness of Lie group analysis to transform a complicated model into simpler models without changing the physics represented by those models.The use of symmetry analysis to study (1+1)-dimensional Hénon-Lane-Emden system was done by Muatjetjeja [15] to investigate the physical importance of the conservation of the Hénon-Lane-Emden system.Lie group method was used to obtain exact solutions of the Kadomtsev-Petviashvili equation by Adem [16].The author obtained soliton solutions which were shown to have dissimilar physical structures.Freire and Muatjetjeja [17] used Lie symmetry analysis to study Lane-Emden-Klein-Gordon-Fock system.The authors used Noether symmetry classification to classify parameters affecting the Lane-Emden-Klein-Gordon-Fock model and also derive the conservation laws of this model.
Magalakwe et al. [19] studied filter chamber internal flow and heat distribution with the aim of finding solutions of momentum and heat effects which lead to a stable filtration process.Their research found that buoyancy force and Joule heating become minimal when the internal fluid temperature approximates the filter chamber temperature.Focusing on the extraction of particles using a permeable surface and porous medium during filtration, the current study further investigates the work of [19] in four folds by considering different forces affecting the internal flow behaviour and distribution of heat during a process of filtering particles from the fluid.The first fold explores the impact on the axial velocity of an inclined filter chamber.The study further examines the effects of the Joule heating and buoyancy force.Lastly, the current study considers the effects of temperature variation during the filtration process on the flow field.

Mathematical Formulation of the Case Study
As shown in Fig. 1, a two-dimensional electrically conducting and radiative MHD laminar flow of a viscous incompressible fluid inside an inclined permeable filter chamber influenced by a uniform magnetic field of strength B 0 acting perpendicular to the surface walls is considered.It is assumed that fluid concentration is homogeneous/uniform.The left end of the filter chamber is closed with an insulated wall, while the other end on the right is open to allow outflow.The filter chamber is also bounded by two permeable walls above and below the internal flow, which are maintained at a constant wall temperature T w to minimise viscous effects.The filter is such that fluid can permit through the surface out or into the chamber through the permeable walls.The filter chamber deforms uniformly at a time-dependent rate ḣ(t), when it contracts or expands.The fluid emits radiation immediately as it permits through the chamber walls with uniform velocity V w .The chamber is such that the upper surface wall and the lower surface wall are a vertical chamber space y = h(t) and y = −h(t) away from the centre, respectively.
Filtration process flow dynamics and internal heat transfer representation is given by the following mathematical model: where ū( x, ȳ, t) is axial-velocity, v( x, ȳ, t) is normal velocity, T ( x, ȳ, t) is temperature, ρ is density, P( x, ȳ, t) is pressure, q r ( x, ȳ, t) is radiative flux, ν is kinematic viscosity, g is gravitational acceleration, β is thermal expansion, B 0 is induced magnetic field, φ is porosity parameter, k is permeability of porous medium, k is thermal conductivity, c p is specific heat capacity, γ is the angle of inclination and σ denotes the electrical conductivity.
The boundary conditions of flow velocity and temperature are The ū( x, ȳ, t) and v( x, ȳ, t) in terms of stream function ¯ are which satisfies the equation of continuity (1).
Scaling the height of a chamber by introducing the normal non-dimensional chamber vertical space y = ȳ/h(t) into the Eq. ( 6) gives Substituting the velocities ū( x, ȳ, t) and v( x, ȳ, t) from Eq. ( 7) into Eqs.( 2)-( 4) together with the following non-dimensional quantities gives the following non-dimensional equations of momentum and energy: where α = h ḣ ν is the wall deformation rate, R e = hV w ν is the permeation Reynolds number, ρV w is the Stuart number.The dimensionless form of boundary conditions are The dimensionless velocity components from (7) become

Method of Solution
In this section, we use similarity transformation based on Lie group analysis [20][21][22][23][24][25][26], which transforms the system of partial differential Eqs. ( 9)-( 11) together with conditions (12) to two ordinary differential equations and their boundary conditions.Thereafter semi-analytical solutions for the case study are obtained using perturbation.

Lie Symmetry Analysis
The following five Lie point symmetries are admitted by the dimensionless momentum and energy Eqs. ( 9)-( 11): The dynamics of the velocity, pressure and temperature of the case study are given by = (x, y, t), θ = θ(x, y, t), P = P(x, y, t) remain unchanged under X 1 and X 2 , since the two are the only symmetries satisfying the conditions where X is a Lie group generator.The Lie point symmetry X 2 gives unsteady solutions, which lead to an unsteady filtration process, whereas the Lie point symmetry X 1 gives solutions that lead to a steady filtration process while maintaining the dynamics of the case study.The characteristic Eq. ( 14) yields The above invariant conditions (15) yield the following steady-state velocity, pressure and temperature: Substitution of Eq. ( 16) into Eq.( 9) and multiplying by 1 H and thereafter letting gives Integrating H x = B 1 yields and substituting this result into the stream function of ( 16) yields Differentiating (20) with respect to y, using the condition (12) (iii), and thereafter solving the resultant equation, yields where B 12 is a constant of integration.Using Eq. ( 21) into ( 20), yields where G(y) = B 1 (y)h(y).Substituting the pressure P = (x, y) and velocity (19) into momentum equation along x (18), yields Equation (19), when B 11 = 0 gives and from Eq. ( 21), we get B 12 = 0. Since B 12 = 0, the non-dimensional stream velocity (22) takes the form Substituting (25) into Eq.( 13) gives the axial flow and normal flow velocities as The above equation confirms that the flow inside the chamber is laminar.It should be noted that due to the symmetric nature of the system, the axial-velocity profile is parabolic.Differentiating Eq. ( 10) with respect to x when the stream function is given by ( 25), yields Similarly, invoking (25) into ( 9) and differentiating the resulting equation with respect to y, gives Using Eq. ( 27) in (28), yields which, after differentiating twice with regard to x, becomes The filter design and the dynamics allow fluid injection into the filter chamber, density variation due to temperature effects and inclination of the chamber, hence causing the fluid inside the chamber to flow.Thus, R e , G r and cos(γ ) from Eq. ( 30) are non-zero parameter and therefor Equation (31) yields heat flux as where both A and B depends on y only.Equating powers of x obtained from substituting (32) into (29), yields the following equations: x 0 : R e G r B(y) cos(γ ) = 0, (33) Since R e , G r and cos(γ ) are non-zero parameters, Eq. (33), yields Thus, the system of Eqs. ( 33) and (34) yield the following ordinary differential equation representing momentum Using steady state temperature θ = τ (x, y) from ( 16) and stream function ( 25) into (11) gives the internal temperature variation as Substituting B(y) = 0 into Eq.(32) gives Equation (37) and condition (iii) at x = 0, yields internal temperature as which confirms that internal temperature distribution is parabolic.
According to (37), temperature distribution per length is given by which shows that heat effects from the surface wall diffuse along the y-direction only.
Substituting θ = τ (x, y) from ( 16) and ( 39) into (36), yields Thus, ordinary differential equations representing internal flow and temperature are given by with the following boundary conditions:

Semi-analytical Solution
This subsection uses the perturbation method by representing the velocity and temperature as perturbation series using injection along with wall deformation and Grashof, respectively, as perturbation quantities to obtain semi-analytical solutions of Eqs.(41)-(42) together with boundary conditions (43).For more information, the reader is referred to the references [8,9].The perturbation series of the internal velocity and temperature are as follows: where Substituting ( 44) and ( 45) into (41) and (42) yields sixteen equations and the appropriate boundary conditions by equating like powers of perturbation quantities R e , α and G r .Solving the resulting equations subject to conditions at the boundaries yield the solutions of the flow velocity and temperature distribution, respectively as

Physical Quantities
During filtration, the steady state regime yields the following two important physical parameters: (i) Skin friction where τ w = μ ∂u ∂ y y=1 is shear stress at the wall.(ii) Local Nusselt number where is the heat transfer rate at the wall.46) and (47) respectively, the above quantities become The Eq. ( 50) indicates that the skin friction on the walls is extremely low, so it does not impact the system during stable operation.

Results and Discussion
In this section, graphical representations of velocity variation, temperature distributions, and local Nusselt number are analysed to find specific dynamics that lead to optimal production of filtrates during the filtration process.The analytical procedure and the obtained results are validated with the results from the studies [8,9,19].The comparison of the present case study with Magalakwe et al. [19] is done when γ = 0; in other words, when the filter chamber is horizontal and also when there is no effect of radiation (R = 0).The results obtained show a decent agreement with those reported by [19].These analytical procedures can be successfully used to calculate as well as discriminate between the influences of numerous parameters [27][28][29], which could be utilized to represent a number of novel, complex characteristics that appear across various scientific disciplines.

Effects of Various Parameters on Flow Velocity Variation
This subsection analyses the graphical representations of the internal flow to study parameters which affect flow during filtration.

Effects of Deformation Rate Figures
2 and 3 illustrate both fluid injection and suction when the chamber expands (α > 0), expanding the walls increases the flow velocity at centre and decreases the flow towards the walls of the chamber.When the walls contract (α < 0), the volume of the chamber decreases, thus flow velocity decreases at centre and increases at the chamber walls.This increase in fluid flow when the system allows fluid injection and expansion is caused by the ability of fluid particles to move freely due to the additional space inside the chamber.This analysis correlates with the results found in [6].According to Figs. 2 and 3 the fluid flow is faster when fluid is injected into chamber compared to when fluid is sucked.From the figures it can be noted that it is not ideal to suck fluid out during filtration, since it reduces the outflow.

Effects of Reynolds Number R e
Figures 4 and 5 indicate that towards the chamber walls, flow velocity increases during fluid injection (R e > 0) and decreases towards the centre.During fluid suction (R e < 0) towards the centre, the flow velocity increases and decreases towards the walls.The analysis and findings in [11] are in good agreement with the current analysis.When the chamber walls expand (see Fig. 4), the movement of fluid is faster compared to when the system contracts (see Fig. 5).The increase in chamber volume while the system allows fluid into the chamber leads to an increase in filtrates production.Both figures indicate that the system produces more filtrates at a lower speed for fluid injection on the other hand, the system produces less filtrates faster when there is fluid suction.and 8 indicate the flow moves faster at the centre and slower towards the walls when porosity increases for fluid injection.For suction (Fig. 7), it shows that the increase in porosity while sucking out fluid increases the velocity towards the walls while decreasing the flow towards the middle of the chamber.The increase in fluid flow when the system allows fluid injection while pore size increases is due to the ability of the fluid to freely pass through the pores when the pores increase in size.To increase filtrates outflow while filtering particles effectively, the system needs to allow fluid injection, expand to increase chamber volume and decrease pore size during the filtration process.

Effects of Inclination Angle
Figure 9 shows that inclination is needed to increase filtrates outflow since the fluid flow rate is slow when the chamber is not inclined.Figure 10 indicates that the system produces few filtrates faster without chamber inclination during suction.Both figures indicate that inclining the filter chamber allows the impact of gravity to act along the outflow direction, thus decreasing buoyancy effects and increasing filtrates production.To have an effective filtration process, fluid injection is needed while the chamber is inclined.Figure 10 also shows that when chamber inclination increases, the increase in inclination leads to reverse flow towards the walls, thus affecting flow velocity at the bottom of the filter chamber only.

Effects of Stuart Number N
Figure 11 shows that increasing magnetic strength while allowing fluid injection decreases flow at the centre and increases the flow towards the walls.Sucking fluid out of the chamber while increasing magnetic strength increases flow towards the centre while decreasing the flow towards the walls; see Figure 12.The integration of magnetic strength creates a force that opposes fluid flow called Lorentz force.This force becomes more effective when fluid is injected compared to when fluid is sucked.To deal with negative Lorentz force effects,  sufficient fluid injection is needed during the filtration process to maintain desirable outflow while preventing contaminated particles from entering the system.Figures 13 and 14 show the effects of buoyancy on fluid flow caused by variation of density due to temperature effects.Figure 14 indicates that flow increases at the centre while it decreases towards the walls when buoyancy increases for suction.Figure 13 depicts the decrease in flow at the centre while the flow increases towards the walls when buoyancy effects increase during fluid injection.Both figures show that the Grashof number affects the flow more at the bottom of the chamber.This is caused by dense particles moving towards the bottom while the less dense particles move upwards.This subsection analyses the graphical representations of the internal temperature to study parameters which affect heat distribution during filtration.Figures 15 and 16 indicate that when the system allows fluid injection, there is an increase in temperature due to the additional temperature that the inflow of particles possesses.During suction, the system temperature decreases since particles which possess temperature are sucked, thus decreasing internal temperature during operation.The additional increase and the decrease in temperature when particles are injected or sucked, respectively, are caused by the fact that the introduction of more particles during injection increases Joule heating effects, whereas the removal of particles during suction decreases Joule heating effects.Both graphs indicate that when the system operates with neither particle injection nor suction, the internal temperature remains constant since no fluid temperature is sucked or injected into the system.

Effects of Joule Heating Parameter J
Figure 17 shows that the internal temperature increases due to Joule heating effects when the system allows fluid injection.The increase in internal temperature is more for lower Joule heating effects since the energy due to work done by the movement of particles is more when Joule heating effects are low.Figure 18 shows that the internal temperature decreases due to Joule heating effects when fluid is sucked out of the filter chamber.

Effects of Grashof Number G r
Figure 19 shows that the internal temperature increases due to Grashof number effects when the system allows fluid injection, while Fig. 20 shows that the internal temperature decreases due to Grashof number effects when fluid is sucked out of the filter chamber.The increase in internal temperature is more for lower Grashof number effects since a lower Grashof number

Effects of Radiation R
Figure 21 shows that the internal temperature increases due to radiation when the system allows fluid injection.Figure 22 shows that the internal temperature decreases due to radiation effects when fluid is sucked.The increase in internal temperature due to radiation is more for high radiation effects since fluid emits energy in the form of radiation, thus increasing internal temperature when the fluid emits energy.

Effects of Prandtl Number P r
Figure 23 shows the increase in internal temperature due to the increase in Prandtl number when the system allows fluid injection.This increase in temperature is caused by the thermal diffusivity dominance inside the filter chamber when the Prandtl number is high, which leads to the increase in heat diffusion inside the chamber, thus increasing the internal temperature.Figure 24 shows that the internal temperature decreases due to Prandtl number effects when fluid is sucked.This decrease in temperature is due to the removal of internal temperature in the form of particles sucked, which poses temperature.

Behaviour of Local Nusselt Number Under the Influence of Parameters Arising from Flow Dynamics and Heat Distribution
This subsection analyses the effect of different parameters affecting the local number of Nusselt (51) during the filtration process.
Figures 25 and 26 show that Nusselt number increases with the increase in fluid injection and Prandtl number along the chamber length, respectively.Similarly, varying R, J , and G r respectively indicate that the increase in radiation, Joule heating and Grashof while injecting leads to a decrease in Nusselt number along the chamber.

Concluding Remarks
In this paper, the analytical solutions of flow and temperature distribution inside an inclined filter chamber were obtained and analysed to understand the filtration process better.An in-depth graphical analysis of the effects of various parameters arising from the filter design, flow dynamics and temperature effects was conducted to optimise outflow during the filtration process.The findings of the current study show that to optimise filtrates (outflow) during the filtration process, the following conclusions are critical: (1) Wall deformation rate: Wall expansion is needed since it increases chamber volume, thus allowing more fluid inside the filter chamber which increases external pressure and, as a result, increases fluid injection.(2) Reynolds: Fluid injection is needed to allow more fluid inside the filter chamber, thus increasing filtrates during the filtration process.Also, the increase in fluid particles due to injection creates internal friction (Joule heating effects), which increases internal temperature and decreases viscous effects, thus leading to free movement of fluid out of the chamber and increased outflow.(3) Porosity: The filter chamber's medium pore size must be small to stop contamination from passing through the medium without affecting temperature distribution since the temperature is independent of the pore size.(4) Stuart: Magnetic load zone is needed to stop particles from entering the filter chamber.
(5) Prandtl: Prandtl number does not affect momentum during the filtration process; thus, Prandtl number can be such that it increases internal temperature while decreasing viscosity effect and allowing free internal flow.(6) Grashof: The buoyancy effects/Grashof number must be minimal to allow stable internal flow.(7) Inclination: The chamber needs to be inclined in such a way that it allows gravity to act in the direction of the outflow.(8) Radiation: Lower radiation is needed to increase temperature, thus increasing outflow.(9) Joule heating: Joule heating parameter needs to increase the internal temperature to allow free movement of fluid out of the filter chamber.(10) Axial pressure: The chamber walls need to expand in such a way that their movement increases internal axial pressure to push more fluid out of the chamber.(11) Skin friction: Skin friction effects need to be small to allow free movement at the walls during the filtration process.( 12) Local Nusselt number: The increase in local Nusselt number is needed to increase fluid injection.
To understand the filtration process better, one can study the transition from steady-state regime to an unsteady-state regime in the future.
Author Contributions All authors contributed equally to the manuscript.
Funding Open access funding provided by North-West University.The authors have no source of funding to declare.

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Fig. 1
Fig. 1 Schematic representation of the case study