Transient MHD Darcy-Forchheimer of Williamson-Casson flow with CCS: Application of wastewater treatment

Abstract

With regards to the Cattaneo-Christoph (CCS) speculation, the thermal and mass transfer of a MHD Williamson-Casson ferrofluid flow through a permeable medium obeying the Darcy-Forchmeier law through an extended chamber or slab is carefully considered. The chemical reaction and Buongiorno nanofluid model were modified in the model to describe the nanoscale properties of liquid particles. Heat sources can be divided into two categories: linear and exponential space-dependent convection heat sources. The entire governing equations are changed to common differential terms, using a reasonable similarity change. These equations and their associated boundary conditions are calculated numerically via the fourth-order Runge-Kutta method. Comparison was made with the analytical solution in a special case, and very good agreement was reached. Economic success with deeper water purification can be attained by developing models for wastewater treatment facilities, conducting experiments and calculations on them, and making predictions for their nature in order to improve their cleaning efficiency, Prevent issues from arising during construction, operation, and reconstruction. The structural parameters of the quencher were physically modeled using mathematical modeling; It is possible to establish a cost-effective and effective filter model for industry and small settlements using the calculations and vertical filter model created in the article. Furthermore, it will be feasible to produce. Our mathematical procedure assure that: the deposit’s mass will be diminished through treated water under attractive field impacts; The magnetic field openness with nanometer-sized particles modifies the physical and synthetic properties of water particles bringing about exceptional characteristics; the water concentration will be decreased with the increase in the bending modulus; This means that in wastewater treatment the sediment mass will decrease.

Keywords

Casson and Williamson nanofluids MHD Cattaneo–Christov speculation exponential heat source Darcy-Forchheimer

Introduction

The science of fluid dynamics has been and continues to be a pioneer in understanding the rheology of fluids, which is critical for improving industrial, engineering, biomedical and nanotechnology processes and products,^1–4 especially non-Newtonian fluids. Non-Newtonian fluids have emerged in many diverse industries such as the petroleum and chemical industries. It has a wide range of applications, from understanding blood flow in the human body to improving the flow of oil in pipelines. Among the problems that researchers and engineers face is wastewater treatment. Modern wastewater may contain unsafe mixtures such as heavy metals, microorganisms, natural organisms, microplastics, oil, and infections.⁵ The rheological properties of non-Newtonian fluids differ from those of Newtonian fluids. This difference led to the diversity of these fluids, including Casson fluid. Casson fluid⁶ is mentioned as a non-Newtonian viscous substance that has multiple uses. It is characterized by yield stress, and its viscosity approaches zero at very high shear rates. Due to the many benefits of this liquid, a large number of studies have appeared that dealt with the study of this liquid, such as the study of Rasool et al.,⁷ Jamshid et al.,⁸ Dawlat et al.^9,10 Among these non-Newtonian fluids is Williamson fluid,¹¹ which has elastic and viscous properties. In this research, researchers believe that combining Casson fluid with another non-Newtonian fluid such as Williamson fluid has the ability to effectively characterize dusty water, especially in wastewater treatment processes. In addition, studying the flow of Williamson-Casson fluids containing solid particles is crucial to understanding fluid behavior in wastewater treatment, especially when applying MHD magneto-hydrodynamics.^12–14 MHD is the most capable strategy for recovering spent water and producing energy by using wastewater, sewage and slurry as a compressible conductive fluid. Many specialists and researchers have focused on the effect of MHD on non-Newtonian fluids. Ullah Awan et al.¹⁵ proposed the use of Williamson nanofluids on rubber sheets to study radiation with sink and thermal effects. Patil et al.¹⁶ have presented a study on the heat and mass exchange of magnetic-Williamson nanofluid flow across a stretched permeable surface. The study includes the effects of thermal radiation and chemical reactions. The effects of thermal radiation, electromagnetic force, and chemical interactions on heat and mass transfer of a Williamson nanofluid on a cylinder moving at a specified speed through a permeable medium were studied by Rao and Deka.¹⁷

An electrical MHD flow of a Williamson Nano Casson fluid toward the expansion plate implemented with mass flow is performed using Jawad et al.¹⁸ Affected by sliding speed, Patil et al.¹⁹ demonstrated MHD radiative flux with microorganisms for a non-Newtonian Casson-Williamson mixture nanofluid on a greatly extended permeability surface. Eswara Rao et al.²⁰ explored the effects of thermal and mass diffusion and magnetic field under Brownian effects and thermophoresis effects on a Casson–Williamson mixture on a stretched plate. The modern value of Casson’s fluid can be expanded when combined with Williamson’s fluid, so a few scientists have led these tests, and research papers can be traced in the references.^21–27

Abbas et al.²⁸ highlighted the Williamson nanofluid flow over a non-linear extending sheet cultivated in a permeable medium. Previous studies have focused on examining the flow of Williamson nanofluid over a non-linear extending sheet embedded in a permeable medium. One commonly used modification to account for inertia effects in Darcian flow is the Darcy-Forchheimer model, which includes a term proportional to the square of velocity in the energy equation to consider the impact of inertia.²⁹ Several researchers have^30–32 investigated the Darcy-Forchheimer model for non-Newtonian fluid flow, including Casson-Williamson nanofluid flow with nonlinear heat radiation, suction, and heat usage.³³ Other studies have explored the Darcy-Forchheimer flow of Williamson liquid over a Riga plate.³⁴ Ur Rehman et al. gathered the energy conditions with unsteady non-Newtonian Williamson constitutive equations for nan- particles and tackle them mathematically.³⁵ Additionally, the Darcy-Forchheimer Williamson nanofluid model over an extended surface under convective conditions has been investigated using intelligent backpropagated brain networks and the Levenberg-Marquardt approach.³⁶ Influences of MHD on mass-heat transfer in the fluid of kind a third-grade over an exponentially slanted extending sheet fixed in a permeable medium with Darcy-Forchheimer regulation impact have also been proposed by Abbas et al.³⁷

Building upon previous research, the goal of the current investigation is to better understand the unsteady two-dimensional magnetohydrodynamic Darcy-Forchheimer flow of Williamson and Casson nanofluids. The effects of thermophoresis/Brownian motion are considered, and the Cattaneo-Christov heat-mass flux speculation (CCS) is employed to model the energy and mass equations. Based on heat absorption, an exponential heat source, and heat-generating properties, the thermal transport of the nanofluid is examined. Utilizing activation energy relations, the concentration equation is adjusted. The study examines the influence of various governing model parameters, such as velocity $V (u, 0, w)$ , temperature $\tilde{T}$ , nanofluid concentration ${\tilde{C}}_{n}$ , local skin friction ${\tilde{S}}_{f}$ , local Nusselt number ${\tilde{N}}_{u}$ , and local Sherwood number ${\tilde{S}}_{h}$ , using tables and charts.

Problem formulation

Formulation of mathematical model for cooling system which can be considered as a circular cylinder electromagnetic device installed to recycle and treat hard water from wastewater. This gadget can be displayed as an unsteady Darcy-Forchheimer two-layered MHD flow of Williamson-Casson nanofluids under the CCS of intensity and mass transition over an extending cylinder, as displayed in Figure 1.

Figure 1.

Graphic summary of the flow of the problem.

The assumption is made that the cylinder, having a specific radius, undergoes movement with a constant linear velocity $ς_{0} = \frac{a z}{(1 - ϑ t)},$ where $a, ϑ >> 0$ .The fluid velocities are denoted by $(u, w)$ which are measured in the direction $(z, r)$ axes, respectively. The current model of the Williamson nanofluid is based on the effects of thermophoresis as well as Brownian motion. We express the temperature and the concentration of the surface and the surrounding liquid the symbols $({\tilde{T}}_{w}, {\tilde{T}}_{\infty})$ and $({\tilde{C}}_{w}, {\tilde{C}}_{\infty})$ , respectively. Here, ${\tilde{T}}_{w} >> {\tilde{T}}_{\infty}$ and ${\tilde{C}}_{w} >> {\tilde{C}}_{\infty}$ . A magnetic field was applied in the direction of radiation with a strength of $B (t) = \frac{B_{0}^{*}}{\sqrt{1 - ϑ t}}$ , while neglecting the induced magnetic field due to the small Reynolds number, where the magnetic field strength is denoted by symbol $B_{0}^{*}$ . $T^{*}$ is Cauchy stress tensor for the Williamson viscosity type, as described in references,^11,12 can be expressed as:

T^{*} = - PI + ϕ,

(1)

ϕ = (μ_{\infty} + \frac{μ_{0} - μ_{\infty}}{1 - Γ \overset{\cdot}{γ}}) A_{1},

(2)

where pressure is denoted $P,$ while the identity vector is denoted $I$ and $ϕ$ is extra stress tensor, $A_{1}$ is the first Rivlin-Erickson tensor, and $Γ$ is constant ( $Γ >> 0$ ). $μ_{0}$ and $μ_{\infty}$ are the viscosity at zero and infinity shear rates, respectively. Further, $\overset{\cdot}{γ}$ is defined as

\overset{\cdot}{γ} = \sqrt{\frac{π}{2}},

(3)

Where

π = \frac{1}{2} trace (A_{1}^{2}),

(4)

If we consider that the value of $μ_{\infty} = 0$ , $Γ \overset{\cdot}{γ} << 1 .$ and apply binomial expansion to equation (2) then the equation becomes (2) reduced to:

ϕ = μ_{0} (1 + Γ \overset{\cdot}{γ}) A_{1} .

(5)

The type of Casson fluid is listed as^6–8

τ_{ij} = {\begin{matrix} (μ_{B} + \frac{P_{y}}{\sqrt{2 π_{c}}}) 2 {\overset{\cdot}{γ}}_{ij}, π ≺ π_{c}, \\ (μ_{B} + \frac{P_{y}}{\sqrt{2 π}}) 2 {\overset{\cdot}{γ}}_{ij}, π ≻ π_{c} . \end{matrix}

(6)

When $P_{y}$ , the Casson fluid yield stress is shown by $P_{y} = \frac{μ_{B} \sqrt{2 π}}{β}$ . If $π ≻ π_{c}$ in a Casson fluid flow, then $μ_{0} = μ_{B} + \frac{P_{y}}{\sqrt{2 π}} .$ As a result,

μ_{0} = \frac{μ_{B}}{ρ} (1 + \frac{1}{β})

(7)

since the kinematic viscosity is dependent on the plastic dynamic viscosity, Casson number, and fluid density. Where, respectively, $τ_{ij}, {\overset{\cdot}{γ}}_{ij}, π = {\overset{\cdot}{γ}}_{ij} {\overset{\cdot}{γ}}_{ij}$ and $π_{c}$ are components of the stress tensor, the strain tensor’s rate, the product of the strain tensors’ rates, and the model’s critical value.

The Cattaneo-Christov stove speculation is used to explain the use of a modified version of Fourier’s law, where $\tilde{q}$ and $\tilde{j}$ represent heat and mass flows, respectively.^19,25

\tilde{q} + Γ_{t} (\partial_{t} \tilde{q} + V . \nabla \tilde{q} + (\nabla . V) \tilde{q} - \tilde{q} . \nabla V) = - k \nabla \tilde{T},

(8)

\tilde{j} + Γ_{c} (\partial_{t} \tilde{j} + V . \nabla \tilde{j} + (\nabla . V) \tilde{j} - \tilde{j} . \nabla V) = - D_{B} \nabla \tilde{C} .

(9)

Equations (8) and (9) represent the expressions for heat and mass flux, where the relaxation time is associated with thermal and solute levels. By setting ( $Γ_{t}$ , $Γ_{c}$ ) to zero in the mentioned equations, we can obtain the classical Fourier’s law. In the case of an incompressible fluid, equations (8) and (9) take a simplified form.

\tilde{q} + Γ_{t} (\partial_{t} \tilde{q} + V . \nabla \tilde{q} - \tilde{q} . \nabla V) = - k \nabla \tilde{T},

(10)

\tilde{j} + Γ_{c} (\partial_{t} \tilde{j} + V . \nabla \tilde{j} - \tilde{j} . \nabla V) = - D_{B} \nabla \tilde{C} .

(11)

With the application of the assumptions of the boundary layer orders $O (z) = O (1) = O (u)$ and $O (r) = O (1) = O (w)$ . Terms with the order $O (δ)$ are neglected and $O (1)$ are retained. Where $δ$ is the thickness of the boundary layer.

Hence, the equations that govern the problem are given below³⁸:

\partial_{z} (ru) + \partial_{r} (rw) = 0,

(12)

\begin{matrix} \partial_{t} u + u \partial_{z} u + w \partial_{r} u = ν (1 + \frac{1}{β}) \frac{1}{r} \partial_{r} (r \partial_{r} u) \\ + \frac{ν Γ}{\sqrt{2}} (\frac{1}{r} \partial_{r} (r (\partial_{r} u)^{2}) - \frac{σ B^{2} (t)}{ρ} u - (\frac{ν}{K_{p}} + \frac{c_{p}}{\sqrt{K_{p}}} u) u, \end{matrix}

(13)

\begin{array}{l} \partial_{t} \tilde{T} + u \partial_{z} \tilde{T} + w \partial_{r} \tilde{T} + Γ_{t} (\partial_{t t}^{2} \tilde{T} + \partial_{t} u \partial_{z} \tilde{T} + 2 u \partial_{t z}^{2} \tilde{T} + \partial_{t} w \partial_{r} \tilde{T} + 2 w \partial_{t r}^{2} \tilde{T} \\ + 2 u w \partial_{t z}^{2} \tilde{T} + w^{2} \partial_{r r}^{2} \tilde{T} + u^{2} \partial_{z z}^{2} \tilde{T} + u \partial_{z} u \partial_{z} \tilde{T} + w \partial_{r} u \partial_{z} \tilde{T} + u \partial_{z} w \partial_{r} \tilde{T} + w \partial_{r} w \partial_{r} \tilde{T}) \\ = \frac{k}{ρ c_{p}} r^{- 1} \partial_{r} (r \partial_{r} \tilde{T}) + τ (D_{B} \partial_{r} \tilde{T} \partial_{r} \tilde{C} + \frac{D_{T}}{T_{\infty}} {(\partial_{r} \tilde{T})}^{2}) + \frac{Q_{0}}{ρ c_{p}} (\tilde{T} - {\tilde{T}}_{\infty}) + \frac{Q_{0}^{*}}{ρ c_{p}} ({\tilde{T}}_{w} - {\tilde{T}}_{\infty}) \\ \exp (- n ({(\frac{a}{ν (1 - ϑ t)})}^{\frac{1}{2}} \frac{r^{2} - R^{2}}{2 R})), \end{array}

(14)

\begin{matrix} \partial_{t} \tilde{C} + u \partial_{z} \tilde{C} + w \partial_{r} \tilde{C} + Γ_{c} (\partial_{tt}^{2} \tilde{C} + \partial_{t} u \partial_{z} \tilde{C} + 2 u \partial_{tz}^{2} \tilde{C} \\ + \partial_{t} w \partial_{r} \tilde{C} + 2 w \partial_{tr}^{2} \tilde{C} + 2 uw \partial_{tz}^{2} \tilde{C} + w^{2} \partial_{rr}^{2} \tilde{C} + u^{2} \partial_{zz}^{2} \tilde{C} \\ + u \partial_{z} u \partial_{z} \tilde{C} + w \partial_{r} u \partial_{z} \tilde{C} + u \partial_{z} w \partial_{r} \tilde{C} + w \partial_{r} w \partial_{r} \tilde{C}) \\ = D_{B} r^{- 1} \partial_{r} (r \partial_{r} \tilde{C}) + \frac{D_{T}}{T_{\infty}} r^{- 1} \partial_{r} (r \partial_{r} \tilde{T}) \\ - K_{r}^{2} (\tilde{C} - {\tilde{C}}_{\infty}) {(\frac{\tilde{T}}{{\tilde{T}}_{\infty}})}^{n} \exp (\frac{- E_{a}}{k \tilde{T}}) . \end{matrix}

(15)

The corresponding boundary conditions for the given problem are³⁹

\begin{matrix} u = ς_{0} = \frac{a z}{1 - ϑ t}, w = ε_{0} = - \frac{ε_{0}^{*}}{\sqrt{1 - ϑ t}}, \tilde{T} = {\tilde{T}}_{w}, \tilde{C} = {\tilde{C}}_{w} at r = R, \\ u \to 0, \tilde{T} \to {\tilde{T}}_{\infty}, \tilde{C} \to {\tilde{C}}_{\infty} at r \to \infty . \end{matrix}}

(16)

Here, $ρ$ , $ρ c_{p}$ $ν, k$ and $σ$ are the density, the volumetric heat capacity, the kinematic viscosity, the thermal conductivity, and the electrical conductivity of the nanofluids, respectively. They are defined according to permeability of the porous medium, $\frac{c_{p}}{\sqrt{K_{p}}}$ the non-uniform inertia coefficient of the porous medium, $τ$ the ratio of the effective heat capacity of the nanofluid, $D_{B}$ is the Brownian diffusion coefficient, $D_{T}$ is the thermophoretic diffusion coefficient, $K_{r}^{2}$ is the chemical reaction rate, $ε_{0}$ is expressed by $ε_{0} = - \frac{ε_{0}^{*}}{\sqrt{1 - ϑ t}}$ . As now known, $ε_{0}$ express the mass transfer at the surface with and $ε_{0} << 0$ for injection and $ε_{0} >> 0$ being for suction. To make the equations more manageable, they are transformed into a dimensionless form by introducing certain dimensionless quantities as defined below

\begin{matrix} η = {(\frac{a}{ν (1 - ϑ t)})}^{1 / 2} \frac{r^{2} - R^{2}}{2 R}, w = - {(\frac{a ν}{(1 - ϑ t)})}^{1 / 2} \frac{R}{r} F (η), \\ u = \frac{a z}{1 - ϑ t} F' (η), θ (η) = \frac{\tilde{T} - {\tilde{T}}_{\infty}}{{\tilde{T}}_{w} - {\tilde{T}}_{\infty}}, φ (η) = \frac{\tilde{C} - {\tilde{C}}_{\infty}}{{\tilde{C}}_{w} - {\tilde{C}}_{\infty}} . \end{matrix}}

(17)

Equation (12) is satisfied automatically and equations (13)–(16) yield

\begin{matrix} (1 + β^{- 1}) ((1 + 2 \tilde{α} η) F ″' + 2 \tilde{α} F ″) + 2 Ω {(1 + 2 \tilde{α} η)}^{3 / 2} F ″ F ″' \\ + 3 α Ω {(1 + 2 \tilde{α} η)}^{1 / 2} {F ″}^{2} \\ - (\frac{s}{2} η F ″ + s F' + {F'}^{2} - F F ″ + (\tilde{M} + \tilde{λ} + f^{*} F') F') = 0, \end{matrix}

(18)

\begin{matrix} (1 + 2 \tilde{α} η) θ ″ + 2 \tilde{α} θ' + Q θ + Q_{E}^{*} \exp (- n η) + \\ pr (\begin{matrix} (F - \frac{s}{2} η) θ' - β_{T} ((\begin{matrix} 2^{- 2} (s η)^{2} - s η F + \\ {F ″'}^{2} \end{matrix}) θ ″ + (\begin{matrix} \frac{3}{4} s η (s - \frac{2}{3} F') \\ + (F' - \frac{3}{2} s) F \end{matrix}) θ') \\ + N_{b} (1 + 2 \tilde{α} η) θ' φ' + N_{t} (1 + 2 \tilde{α} η) {θ'}^{2} \end{matrix}) \\ = 0, \end{matrix}

(19)

\begin{matrix} (1 + 2 \tilde{α} η) φ ″ + 2 \tilde{α} φ' - L_{e} K_{c} (1 + δ θ)^{n} \exp (\frac{- E}{1 + δ θ}) φ \\ - L_{e} ((F - \frac{s}{2} η) φ' - β_{c} ((\begin{matrix} \frac{1}{4} (s η)^{2} - s η F + \\ F^{2} \end{matrix}) φ ″ + (\begin{matrix} \frac{3}{4} s^{2} η - \frac{1}{2} s η F' \\ - (\frac{3}{2} s - F') F \end{matrix}) φ')) \\ + \frac{N_{t}}{N_{b}} ((1 + 2 \tilde{α} η) θ ″ + 2 \tilde{α} θ') = 0, \end{matrix}

(20)

with

\begin{matrix} F (η)_{η = 0} = F_{w} = \frac{ε_{0}^{*}}{\sqrt{a ν}}, F' (η)_{η = 0} = 1, θ (η)_{η = 0} = 1, φ (η)_{η = 0} = 1, \\ F' (η)_{η \to \infty} \to 0, θ (η)_{η \to \infty} \to 0, φ (η)_{η \to \infty} \to 0 . \end{matrix}} .

(21)

Where the curvature parameter is denoted by $\tilde{α} = {(\frac{ν (1 - ϑ t)}{a R^{2}})}^{1 / 2}$ ,the unsteadiness parameter is indicated by $s = \frac{γ}{a}$ , the Williamson parameter is implied by $Ω = {(\frac{Γ^{2} a^{3}}{2 ν {(1 - ϑ t)}^{3}})}^{1 / 2} z$ , the activation energy parameter is signified by $E = \frac{E_{a}}{k {\tilde{T}}_{\infty}}$ , the magnetic parameter is demarcated by $\tilde{M} = \frac{σ B_{0}^{2}}{ρ a}$ , the porosity parameter is denoted by $\tilde{λ} = \frac{ν (1 - ϑ t)}{K_{p} a}$ ,the local inertial coefficient is signified by $f^{*} = \frac{c_{p} z}{\sqrt{K_{p}}},$ Chemical reaction parameter attended by $K_{c} = \frac{K_{r}^{2} (1 - ϑ t)}{a},$ Prandtl number is indicated by $pr = \frac{μ c_{p}}{k},$ thermal and mass relaxation times parameters are represented by $β_{t} = \frac{a Γ_{t}}{(1 - ϑ t)}$ and $β_{c} = \frac{a Γ_{c}}{(1 - ϑ t)},$ respectively, Lewis number is indicated by $L_{e} = \frac{ν}{D_{B}},$ the temperature difference parameter is indicated by $δ = \frac{{\tilde{T}}_{w} - {\tilde{T}}_{\infty}}{{\tilde{T}}_{\infty}},$ thermophoresis parameter is signified by $N_{t} = \frac{τ D_{T} ({\tilde{T}}_{w} - {\tilde{T}}_{\infty})}{ν {\tilde{T}}_{\infty}},$ Brownian motion parameter is denoted $N_{b} = \frac{τ D_{B} ({\tilde{C}}_{w} - {\tilde{C}}_{\infty})}{ν},$ sink and thermal source parameters are represented by $Q = \frac{Q_{0} (1 - ϑ t)}{a ρ c_{p}}$ and $Q_{E}^{*} = \frac{Q_{0}^{*} (1 - ϑ t)}{a ρ c_{p}},$ respectively.

Physical quantities ${\tilde{C}}_{n}$ , ${\tilde{N}}_{u}$ and ${\tilde{S}}_{h}$ are defined as follows

{\tilde{C}}_{n} = \frac{τ_{w}}{ρ u_{0}^{2}}, {\tilde{N}}_{u} = \frac{z q_{w}}{k ({\tilde{T}}_{w} - {\tilde{T}}_{\infty})} and {\tilde{S}}_{h} = \frac{z j_{w}}{D_{B} ({\tilde{C}}_{w} - {\tilde{C}}_{\infty})},

(22)

where

τ_{w} = μ ((1 + β^{- 1}) + \frac{Γ}{\sqrt{2}} \partial_{r} u) (\partial_{r} u)_{r = R},

q_{w} = \partial k (\partial_{r} \tilde{T})_{r = R}, j_{w} = - D_{B} (\partial_{r} \tilde{C})_{r = R} .

Applying equation (17) to the physical quantities, we get

\begin{matrix} R_{e}^{1 / 2} {\tilde{C}}_{n} = (1 + β^{- 1} + Ω F ″ {(η)}_{η = 0}) F ″ (η)_{η = 0}, \\ R_{e}^{- 1 / 2} {\tilde{N}}_{u} = - θ' (η)_{η = 0}, R_{e}^{- 1 / 2} {\tilde{S}}_{h} = - φ' (η)_{η = 0} . \end{matrix}} .

(23)

Numerical approach

The dimensionless problems (18–20) and boundary conditions (21) are solved for their numerical solutions using the shooting technique with the fourth-order Runge-Kutta method. This numerical technique can easily handle many kinds of problems. Also, its convergence is easy to achieve. For the computer program, the Mathematica software is running on a PC. The numerical procedure consists of these steps as follow^40–44:

1- The boundary value problem is transformed into an initial value problem because the system consisting of equations (18)–(20) contains one-third-order equation and two second-order equations. Therefore, to obtain the solution, seven initial conditions are required. However, in equation (21), only three initial conditions are provided. As a result, three suitable initial conditions are incorporated.

2- The numerical solution is obtained using the Runge-Kutta method.

3- The calculated solution is iteratively refined until it satisfies the convergence criterion with a computational tolerance of $10^{- 6}$ .

4- If the computed solution fails to meet the convergence criterion, new initial guesses are introduced

Equation (18) in a special case $β \to \infty, \tilde{α} = Ω = s = \tilde{λ} = f^{*} = 0$ with boundary conditions

$F (η)_{η = 0} = F_{w}, F' (η)_{η = 0} = 1$ and $F' (η)_{η \to \infty} \to 0$ has an analytic solution⁴⁵ as:

\begin{matrix} F (η) = \frac{1}{Λ} (2 [1 - 2 e^{- \frac{1}{2} η Λ}] + Λ F_{w}), Λ = F_{w} \\ + \sqrt{4 [1 + \tilde{M}] + {F_{w}}^{2}}, \end{matrix}

(24)

in this special case, a very good agreement was obtained when comparing the numerical results with the analytical solution as shown in Table 1.

Table 1.

The amount of agreement between numerical and analytical values for $R_{e}^{1 / 2} {\tilde{C}}_{n}$ .

$F_{w}$	$\tilde{M}$	Numerical result	Analytical result⁴⁵	Error (%)
−6.0	0.1	−0.1780497164141408	−0.17804971641414058	$1.24709 \times 10^{- 15}$
−5.5		−0.19321253055232165	−0.19321253055228738	$1.77412 \times 10^{- 13}$
−5.0		−0.21108834234631632	−0.21108834234519191	$5.32671 \times 10^{- 12}$
−4.5		−0.23243831748424737	−0.23243831746128188	$9.88025 \times 10^{- 11}$
2.0		−2.4491376746393083	−2.449137674618944	$8.31483 \times 10^{- 12}$
2.5		−2.881716887188228	−2.8817168872080727	$6.88637 \times 10^{- 12}$
3.0		−3.3303005217700075	−3.3303005217723127	$6.9221 \times 10^{- 13}$
4.0	0.0	−4.236067977499337	−4.23606797749979	$1.06932 \times 10^{- 13}$
	0.5	−4.3452078799116025	−4.345207879911716	$2.61637 \times 10^{- 14}$
	1	−4.449489742783356	−4.449489742783178	$4.01223 \times 10^{- 14}$
	2	−4.6457513110645845	−4.645751311064591	$1.33827 \times 10^{- 15}$
	5	−5.162277660168387	−5.16227766016838	$1.37641 \times 10^{- 15}$
	10	−5.872983346207417	−5.872983346207417	0

Where, $E r r o r % = | \frac{A n a l y t i c a l r e s u l t - N u m e r i c a l r e s u l t}{A n a l y t i c a l r e s u l t} |$ .

Results and discussion

In ongoing years, mathematical modeling of wastewater treatment processes has turned into an acknowledged apparatus in designing practice and is broadly utilized by counseling organizations and managing organizations. Utilization of numerical models goes from examination to treatment plant plan, activity, control, and investigating. Mathematical displaying of the wastewater treatment process assumes a significant part in the administration of a compelling treatment strategy for existing plants. Mathematical reproduction can be utilized to upgrade comprehension of the perplexing connections between physical, chemical in environmental engineering. According to mathematical demonstrating, a mockup can be made through actual displaying of the primary boundaries of the quencher, and the results of tests did in lab depend on the mathematical results.⁴⁶ Magnetic water treatment (MWT) is a generally new method in environmental administration. Magnetic field openness with nanometer-sized particles modifies the physical and synthetic properties of water particles bringing about exceptional characteristics. Magnetized water has shown different properties with potential applications in various fields of ecological administration. Scale avoidance/end, soil upgrade, plant development, crop yield, water saving, and wastewater treatment is a portion of these applications. Attractive treatment of water rebuilds the water atoms into little, uniform, and hexagonally organized bunch facilitating their movement through the ways in plant and creature cell films. What’s more, harmful specialists can’t enter the MW structure. The viability of magnetic fields for water treatment applications is as yet a dubious inquiry, and the important peculiarities can’t be made sense of, so mathematical description will help in more explanation of nanofluid behavior. The Circular cylinder electromagnetic device can be introduced in a framework to reuse and treat hard water from water and wastewater which can be demonstrated as unsteady Darcy-Forchheimer two-dimensional MHD flow of Williamson-Casson nanofluids under CCS over an extending cylinder.

There is a difference between the flow and heat transfer processes of pure water and wastewater treated by electromagnetic device cooling system. Examination results show that the temperature of pure water and saline wastewater arrived at 373.15 K at 226.3 mm. Compared with pure water; saline wastewater has somewhat lower speed, and higher fluid film temperature under similar circumstances.⁴⁷ Figure 2 depicts the relationship between velocity and the modulus of curvature $\tilde{α}$ , as increasing the modulus of curvature increases the velocity of nanofluids in both (a) Casson fluid and (b) Williamson fluid, and this increase is more noticeable as we move away from the wall in Casson fluid than in Williamson fluid. This means that in wastewater treatments by cooling system, the increasing modulus of curvature $\tilde{α}$ will improve the efficiency of wastewater, and transformed into pure water suitable for various purposes. The effect of the bending modulus on the temperature of the nanofluids in the Casson and Williamson fluids is shown in Figure 3, where the temperature of the Casson case (a) increases with the increase in the bending modulus, whereas the temperature of the Williamson case (b) decreases on the wall and gradually disappears. Hence the treatment will improved in Williamson case better than Casson case. Furthermore, as shown in Figure 4, a rise in the modulus of curvature leads to a decrease in concentration in both fluids, implying that the sediment mass in magnetically treated water will decrease. This is evidence that the water concentration will be decreased with the increase in the bending modulus. This means that in wastewater treatment the sediment mass will decrease. It should be noted that the solid line graphics in Figures 2 to 4 represent the Casson fluid $(Ω = 0, β = 0.2)$ , whereas the dashed line images represent the Williamson fluid $(Ω = 0.2, β \to \infty)$ . It is noted that the graphic analysis from Figures 5 to 10 is performed for the flow confined by an extended cylinder $(\tilde{α} = 0.5; dashed line)$ and the flow due to the flat plate $(\tilde{α} = 0; solid line)$ . Figures 5 and 6 aim to highlight the influence of the $(Ω)$ Williamson coefficient and Casson coefficient $(β)$ on the velocity, respectively. The velocity in Casson fluid at $(Ω = 0)$ is higher than that of Williamson and begins to decrease as the Williamson coefficient increases. Thus, the width of the momentum boundary layer thickness can be reduced by increasing the dependence of viscosity on the shear rate. Also, the viscosity of the Williamson-Casson nanofluid decreases as a function of the shear rate faster than the viscosity of the Williamson-Casson nanofluid Forchheimer resistance behavior on flow for both the cylinder and the sheet. The viscosity of the wastewater is higher than the pure water, so in this treatment it’s much better to consider Williamson-Casson nanofluid flow model.⁴⁸ Figures 7 and 8 show the velocity $F' (η)$ and temperature $θ (η)$ variation basins of the Williamson-Casson nanofluid as functions of the unsteadiness parameter $s$ ; as the parameter rises $a$ , the velocity declines in contrast to the temperature. As the unsteadiness parameter rises $s$ , the rate of sheet and cylinder $a$ expansion also rises. As a result, the velocity decreases as the unsteadiness parameter rises, in order to enhance the treatment of wastewater, it’s much better to depend on small unsteadiness parameter to enhance the flow velocity to get pure water. However, this effect is reflected in the thermal profile. The rate of expansion of the sheet and cylinder increases the thermal profile of the fluid flow. Additionally, Figure 8 shows how the Prandtl number affects the temperature change. The compositional difference in both situations $(\tilde{α} = 0 & \tilde{α} = 0.5)$ affects how hot the nanofluids are. The Prandtl number $(pr)$ is related to thermal diffusion in an inverse manner because of the low temperature of the nanofluid. When $pr >> 1$ , it has an impact on the flow behavior and cooling rate of an electrically conductive liquid and is crucial for momentum propagation. Through experimental of wastewater treatments it’s better to choose physical values which produce large values of Prandtl number in order to reduce nanofluid temperature distribution.

Figure 2.

Variation of $F' (η)$ with $\tilde{α}$ at $F_{w} = f^{*} = \tilde{λ} = 0.1, s = 0.01, \tilde{M} = 0.5$ : (a) Casson fluid $(Ω = 0, β = 0.2)$ and (b) Williamson fluid $(Ω = 0.2, β \to \infty)$ .

Figure 3.

Variation of $θ (η)$ with $\tilde{α}$ at $F_{w} = f^{*} = \tilde{λ} = 0.1, s = 0.01, \tilde{M} = 0.5$ , $\Pr = 1, Q = Q_{E}^{*} = β_{t} = N_{b} = N_{t} = E = 0.1, n = K_{c} = δ = 0.5, β_{c} = 0.4, L_{e} = 0.01$ : (a) Casson fluid $(Ω = 0, β = 0.2)$ and (b) Williamson fluid $(Ω = 0.2, β \to \infty)$ .

Figure 4.

Variation of $φ (η)$ with $\tilde{α}$ at $F_{w} = f^{*} = \tilde{λ} = 0.1, s = 0.01, \tilde{M} = 0.5$ $\Pr = 1, Q = Q_{E}^{*} = β_{t} = N_{b} = N_{t} = E = 0.1, n = K_{c} = δ = 0.5, β_{c} = 0.4, L_{e} = 0.01$ : (a) Casson fluid $(Ω = 0, β = 0.2)$ and (b) Williamson fluid $(Ω = 0.2, β \to \infty)$ .

Figure 5.

Variation of $F' (η)$ with $Ω$ at $F_{w} = f^{*} = \tilde{λ} = 0.1, s = 0.01, \tilde{M} = 0.5, β = 0.2$ : (a) sheet α = 0 and (b) cylinder α = 0.5.

Figure 6.

Variation of $F' (η)$ with $β$ at $F_{w} = f^{*} = \tilde{λ} = 0.1, s = 0.01, \tilde{M} = 0.5, Ω = 0.2$ : (a) sheet α = 0 and (b) cylinder α = 0.5.

Figure 7.

Variation of $F' (η)$ with $S$ at $F_{w} = f^{*} = \tilde{λ} = 0.1, \tilde{M} = 0.5, β = Ω = 0.2$ : (a) sheet α = 0 and (b) cylinder α = 0.5.

Figure 8.

Variation of $θ (η)$ with $\Pr$ and $s$ at $F_{w} = f^{*} = \tilde{λ} = 0.1, \tilde{M} = 0.5, β = Ω = 0.2$ , $Q = Q_{E}^{*} = β_{t} = N_{b} = N_{t} = E = 0.1, n = K_{c} = δ = 0.5, β_{c} = 0.4, L_{e} = 0.01$ : (a) sheet $α = 0$ and (b) cylinder $α = 0.5$ .

Figure 9.

Variation of $θ (η)$ with $Q$ and $Q_{E}^{*}$ at $F_{w} = f^{*} = \tilde{λ} = 0.1, s = 0.01, \tilde{M} = 0.5, β =$ $Ω = 0.2, β_{t} = N_{b} = N_{t} = E = 0.1, n = K_{c} = δ = 0.5, β_{c} = 0.4, L_{e} = 0.01, pr = 1$ : (a) sheet $α = 0$ and (b) cylinder $α = 0.5$ .

Figure 10.

Variation of $θ (η)$ with $N_{t}$ and $β_{t}$ at $F_{w} = f^{*} = \tilde{λ} = 0.1, s = 0.01, \tilde{M} = 0.5,$ $β = Ω = 0.2, Q = Q_{E}^{*} = N_{b} = E = 0.1, n = K_{c} = δ = 0.5, β_{c} = 0.4, L_{e} = 0.01, pr = 1$ : (a) sheet $α = 0$ and (b) cylinder $α = 0.5$ .

Figure 9 depicts the effect of the exponential heat sink coefficient $Q_{E}^{*}$ and the behavior of the heat source parameter $Q$ on the temperature distribution $θ (η)$ . This graph depicts how rising values of the exponential heat sink and heat source parameters affected the temperature distribution of the plate and cylinder. The Cattaneo-Christov speculation, incorporating relaxation time parameters, introduces a delay in the transmission of thermal waves, which in turn requires additional time for the transportation of mass and heat. Consequently, at higher values, the thermal reduction indicates changes in the rate of heat transfer as the thermal transfer coefficient $N_{t}$ increases as shown in Figure 10. The temperature field is enhanced through the expansion of the thermal dispersion coefficient. The phenomenon of thermal separation is influenced by the response of migrating nanofluid particles to the force generated by the thermal gradient. Increased thermophoresis leads to a more efficient transfer process. The significance of the Lewis number is depicted in Figure 11, where an improved evaluation of the Lewis number results in a larger focus field. This relationship is due to the connection between mass diffusivity and the Lewis number. The skin friction coefficient, as shown in Table 2, increases with higher magnetic parameter and inertia coefficient of the porous medium. This can be attributed to the generation of Lorentz force, which resists the motion of fluid particles caused by changes in the magnetic field. The efficiency of the wastewater treatments will be improved by applying small values of magnetic parameter⁴⁶ Tables 3 to 5 present the variations in the local Sherwood number and Nusselt number. Both of these physical quantities decrease for higher values of $N_{b}$ and $f^{*}$ . However, different trends in local mass transfer rate are observed with increasing values of $\tilde{M}$ and $β_{c}$ .

Figure 11.

Variation of $φ (η)$ with $L_{e}$ at $\tilde{α} = 0.8, F_{w} = f^{*} = \tilde{λ} = 0.1, s = 0.01, \tilde{M} = 0.5,$ $β = Ω = 0.2, Q = Q_{E}^{*} = N_{t} = N_{b} = β_{t} = E = 0.1, n = K_{c} = δ = 0.5, β_{c} = 0.4, pr = 1$ .

Table 2.

Skin friction coefficient values expressed in numbers (*refers to Newtonian case and **refers to the case of sheet).

$F_{w}$	$β$	$Ω$	$\tilde{α}$	$\tilde{λ}$	$s$	$\tilde{M}$	$f^{*}$	$R_{e}^{1 / 2} {\tilde{C}}_{n}$
0.1	0.2	0.2	0.2	0.1	0.01	0.5	0.1	2.26437
0.15								2.29609
0.2								2.32825
0.1	0.3							2.18526
	2							2.42556
	5							2.84893
	$\to \infty$ *	0*						2.0384*
	0.2	0.1						2.29545
		0.2						2.26437
		0.4						2.20388
		0.2	0**					1.65984**
			0.2					2.26437
			0.4					2.94045
			0.5					3.30501
			0.2	0.1				2.26437
				1				3.34943
				2				4.54691
				0.1	0.01			2.26437
					0.1			2.34435
					0.3			2.52325
					0.01	0.2		1.90288
						0.5		2.26437
						1		2.86778
						0.5	0.1	2.26437
							0.3	2.41471
							0.5	2.56523

Table 3.

Values of ${\tilde{N}}_{u}$ , and, ${\tilde{S}}_{h}$ that were determined numerically for various factors when, $pr = 7, Q_{E}^{*} = Q = β_{t} = N_{t} = N_{b} = E = 0.1, n = K_{c} = δ$ $= 0.5, β_{c} = 0.4, L_{e} = 0.01 .$ (*refers to Newtonian case and **refers to the case of sheet).

$F_{w}$	$β$	$Ω$	$α$	$λ$	$s$	$M$	$f^{*}$	$R_{e}^{- 1 / 2} {\tilde{N}}_{u}$	$R_{e}^{- 1 / 2} {\tilde{S}}_{h}$
0.1	0.2	0.2	0.2	0.1	0.01	0.5	0.1	1.20734	0.248953
0.15								1.39135	0.249113
0.2								1.58737	0.249286
0.1	0.3							1.17197	0.248954
	2							0.954746	0.248872
	5							0.86317	0.248812
	$\to \infty$ *	0*						0.880968*	0.248828*
	0.2	0.1						1.2085	0.248953
		0.2						1.20734	0.248953
		0.4						1.20492	0.248953
		0.2	0**					0.604385**	0.0998652**
			0.2					1.20734	0.248953
			0.4					1.20061	0.364446
			0.5					1.19904	0.417347
			0.2	0.1				1.20734	0.248953
				1				1.15907	0.248955
				2				1.11156	0.248947
				0.1	0.01			1.20734	0.248953
					0.1			1.15026	0.248924
					0.5			0.984804	0.248838
					0.01	0.2		1.22511	0.248949
						0.5		1.20734	0.248953
						1		1.17972	0.248955
						0.5	0.1	1.20734	0.248953
							0.3	1.20171	0.248953
							0.5	1.19624	0.248952

Table 4.

Values of ${\tilde{N}}_{u}$ , and, ${\tilde{S}}_{h}$ that were determined numerically for various factors when, $E = 0.1, K_{c} = δ = 0.5, L_{e} = 0.001, F_{w} = f^{*} = \tilde{λ} = 0.1, s = 0.01, \tilde{M} = 0.5, β_{c} = 0.4,$ $, β = Ω = \tilde{α} = 0.2$ .

$pr$	$Q$	$Q_{E}^{*}$	$n$	$β_{t}$	$N_{b}$	$N_{t}$	$R_{e}^{- 1 / 2} {\tilde{N}}_{u}$	$R_{e}^{- 1 / 2} {\tilde{S}}_{h}$
1	0.1	0.1	0.5	0.1	0.1	0.1	0.55556	0.248301
3							0.92293	0.248669
7							1.20734	0.248953
10							1.28201	0.249028
7	0.1						1.20734	0.248953
	0.2						0.983048	0.248729
	0.3						0.714275	0.24846
	0.1	0.1					1.20734	0.248953
		0.2					0.755348	0.248501
		0.3					0.242881	0.247988
		0.1	0.1				0.61261	0.248358
			0.2				0.899564	0.248645
			0.4				1.14639	0.248892
			0.5				1.20734	0.248953
			0.2	0.1			1.20734	0.248953
				0.3			0.430836	0.248177
				0.5			−1.03345	0.246712
				0.1	0.1		1.20734	0.248953
					0.2		1.12108	0.24843
					0.3		1.03644	0.248257
					0.1	0.1	1.20734	0.248953
						0.2	0.850204	0.249198
						0.3	0.585375	0.249005

Table 5.

Values of ${\tilde{N}}_{u}$ , and ${\tilde{S}}_{h}$ that were determined numerically for various factors when, $pr = 7, Q = Q_{E}^{*} = β_{T} = N_{b} = N_{t} = 0.1, s = 0.01, n = 0.5$ $F_{w} = f^{*} = \tilde{λ} = 0.1$ $\tilde{M} = 0.5, β = Ω = \tilde{α} = 0.2, β_{c} = 0.4$ .

$L_{e}$	$K_{c}$	$δ$	$E$	$β_{c}$	$R_{e}^{- 1 / 2} {\tilde{N}}_{u}$	$R_{e}^{- 1 / 2} {\tilde{S}}_{h}$
0.001	0.5	0.5	0.1	0.4	1.20734	0.248953
0.01					1.20598	0.252656
0.1					1.19385	0.282960
0.001	0.1				1.20734	0.248954
	0.3				1.20734	0.248953
	0.5				1.20734	0.248953
	0.5	0.1			1.20734	0.248953
		0.3			1.20734	0.248953
		0.5			1.20734	0.248953
		0.5	0.1		1.20734	0.248953
			0.3		1.20734	0.248953
			0.5		1.20734	0.248953
			0.1	0.1	1.20734	0.248944
				0.2	1.20734	0.248947
				0.3	1.20734	0.248950
				0.4	1.20734	0.248953
				0.5	1.20734	0.248956

Conclusions

The influence of a magnetic field on Williamson-Casson nanofluids flow in a Darcy-Forchheimer porous medium has been presented. The imaging technique method using the fourth-order Runge-Kutta method was used to solve the nonlinear system of equations governing the problem. In a special case, a comparison with an analytic solution was obtained, and very good agreement was found. This research gives an important expansion of the mathematical analysis of wastewater treatment under magnetic effects. We concluded that:

1- The mass of the deposit will be decreased through treated water and magnetic field effects.

2- The magnetic field openness with nanometer-sized particles modifies the physical and synthetic properties of water particles bringing about exceptional characteristics.

3- The water concentration will be decreased with the increase in the bending modulus. This means that in wastewater treatment the sediment mass will decrease.

4- In order to enhance the treatment of wastewater, it’s much better to depend on small unsteadiness parameter to enhance the flow velocity to get pure water.

5- The efficiency of the wastewater treatments will be improved by applying small values of magnetic parameter

Future research

In present mathematical simulation emphasis the interactions between physical, chemical controlled parameters, we ignored the biological processes in environmental engineering such as variation of the bacterial density growth, hence in future research, we will focus on mathematical model of bacterial density and nutrient concentration during wastewater treatment as proposed by Rahimi et al.⁴⁹

Footnotes

Appendix

Handling Editor: Chenhui Liang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Taghreed H Al-Arabi

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