Magnetohydrodynamic flow of continuous dusty particles and non-Newtonian Darcy fluids between parallel plates

Abstract

The coupled unsteady power-law conducting fluid flow and continuous dusty viscous fluid flow under the influence of magnetic field are solved using the finite difference method. The resistive forces of Darcy porous medium and the external uniform magnetic field are applied on the flow. The second-order accurate finite difference schemes are applied on the coupled governing equations to transform the non-linear partial differential equations to linearized system of algebraic equations. This system is solved iteratively using the generalized Thomas algorithm. Some results are introduced to study the convergence and stability of the present works. The effects of non-Newtonian fluid, continuous dusty particles, Darcy model on the velocity field, and friction factor of both the fluid and dust particles phases are demonstrated.

Keywords

Continuous dusty fluid unsteady magnetohydrodynamic flow non-Newtonian Darcy fluid parallel plates friction factor finite difference method generalized Thomas algorithm

Introduction

The two-phase magnetohydrodynamic (MHD) fluid flow is very important in engineering applications such as aerodynamics equipment and MHD generators. The efficiency of these devices is affected by magnetic, Darcy resistance forces and dust particles. The high particle concentration leads to higher particle-phase viscous stresses and can be accounted for endowing the particle phase by the so-called particle-phaseviscosity.^1–5 The dusty viscous fluid flow between parallel plates has been studied using perturbation method under the influence of volume fraction of the particles.^1,3 Unsteady flow of dusty conducting fluid through pipes has been studied analytically and numerically under the influence of magnetic field.^2,4 Velocities, skin friction factors, and volumetric flow rates are computed and discussed. Madhura and Swetha⁵ studied the influence of volume fraction of dust particles on flow through porous rectangular channel. They used Laplace transform and finite difference method (FDM) to obtain velocities and skin friction factors for fluid as well as dust particles.

The non-Newtonian fluids in porous medium have many engineering applications such as nuclear reactors, solar devices, polymer processing industries, food industries, and petroleum’s reservoirs. The dusty flow of non-Newtonian fluids is studied by many authors.^6–11 Non-Newtonian viscoelastic and temperature-dependent viscosity effects on hydromagnetic dusty fluid flow between parallel plates are studied.^6–8 These problems have been solved analytically and numerically to compute velocities and friction factors under influences of magnetic and porous medium resistances. Steady two-dimensional dusty fluid flow over surfaces has been studied.^9–11 FDM has been applied on the governing equations to obtain shear stresses.

The aim of present work is to study and compute the effects of continuous dusty viscous particles and conducting power-law fluid and Darcy resistance force on the velocities and skin friction factors of both fluid phase and particle phase. Second-order accurate finite difference schemes are applied to solve the coupled non-linear differential equations of fluid and dust particles. A linearization technique is applied on these equations to transform them to linearized ones. Iterations are used to achieve convergence and stability of linearized governing equations with boundary conditions.

Formulation of the problem

The dust particles are assumed to be electrically non-conducting, spherical in shape, and uniformly distributed throughout the fluid. The two plates are assumed to be electrically non-conducting. A constant pressure gradient is applied in the x-direction and a uniform magnetic field B_o is applied in the positive y-direction. Due to the infinite dimensions in the x and z directions, the physical quantities do not change in these directions and therefore we have ∂/∂x = ∂/∂z = 0 and the problem becomes unsteady one-dimensional (Figure 1).

Figure 1.

Problem geometry and boundary conditions.

The momentum equation for fluid phase is given by^6,12

ρ \frac{D v}{Dt} = - \nabla P + \nabla \cdot \underline{\underline{τ}} - KN (v - v_{p}) + f_{b}

(1a)

where $K = 6 π μ_{0}$ is the Stokes constant and $f_{b}$ is the body force containing the Lorentz and Darcy forces. In other words

f_{b} = Jx B_{o} - \frac{μ_{0}}{k_{p}} v

(1b)

The Hall current and ion slip are neglected, since the magnetic Reynolds number is very large. Thus, the current density J is given by¹²

J = σ (E + Jx B_{o})

(2)

Solving equation (2) for J yields

Jx B_{o} = - σ B_{o}^{2} u i

(3)

In terms of equation (3), equation (1a) may be rewritten as follows^6,7

\begin{matrix} ρ \frac{\partial u}{\partial t} & = - \frac{dP}{dx} + \frac{\partial}{\partial y} [μ \frac{\partial u}{\partial y}] \\ - (σ B_{o}^{2} + \frac{μ_{0}}{k_{p}}) u - KN (u - u_{p}) \end{matrix}

(4)

where $u = u (y, t)$ is the velocity component in x-direction.

The apparent viscosity $μ$ of power-law fluid is given by⁹

μ = μ_{0} {| \frac{\partial u}{\partial y} |}^{n - 1}

(5)

Newton’s second law including stresses gives the motion of the dust particles in the following form^1,2

ρ_{p} \frac{\partial u_{p}}{\partial t} = μ_{p} \frac{\partial^{2} u_{p}}{\partial y^{2}} + KN (u - u_{p})

(6)

It is assumed that the pressure gradient is impulsively applied at $t$ = 0 and the fluid starts its motion from rest. Thus

t \leq 0 : u = u_{p} = 0

(7a)

For $t > 0$ , the no-slip condition at the plates results in

t > 0, y = - h : u = u_{p} = 0

(7b)

t > 0, y = h : u = u_{p} = 0

(7c)

Introducing the following dimensionless variables and parameters

\begin{matrix} (\hat{x}, \hat{y}) = \frac{(x, y)}{h}, \hat{t} = \frac{t μ_{0}}{ρ h^{2}}, \hat{u} = \frac{u ρ h}{μ_{0}}, {\hat{u}}_{p} = \frac{u_{p} ρ h}{μ_{0}}, \\ Re = {(\frac{ρ h^{2}}{μ_{0}})}^{n - 1}, Ha = B_{o} h \sqrt{\frac{σ}{μ_{0}}}, s = \frac{ρ h^{2}}{k_{p}}, \\ Q = \frac{ρ_{p} N}{ρ}, \hat{P} = \frac{P ρ h^{2}}{μ_{o}^{2}}, α = - \frac{d \hat{P}}{d \overset{⌢}{x}}, \\ β = \frac{μ_{p}}{μ_{0}} \frac{ρ}{ρ_{p}}, L_{1} = \frac{ρ K h^{2}}{μ_{0} ρ_{p}} \end{matrix}

Equations (4)–(7) take the following form, and hats are dropped for simplicity

\frac{\partial u}{\partial t} = α + \frac{1}{Re} \frac{\partial}{\partial y} [μ \frac{\partial u}{\partial y}] - (H_{a}^{2} + s^{2}) u - L_{1} Q (u - u_{p})

(8)

μ = {| \frac{\partial u}{\partial y} |}^{n - 1}

(9)

\frac{\partial u_{p}}{\partial t} = β \frac{\partial^{2} u_{p}}{\partial y^{2}} + L_{1} (u - u_{p})

(10)

t \leq 0 : u = u_{p} = 0

(11a)

t > 0, y = - 1 : u = u_{p} = 0

(11b)

t > 0, y = 1 : u = u_{p} = 0

(11c)

The method of solution (FDM)

The governing equations (8)–(11) with condition (11) represent an initial-boundary value problem of non-linear partial differential equations. Consider the following substitutions

v = \frac{\partial u}{\partial y}, v_{p} = \frac{\partial u_{p}}{\partial y}, and λ = \frac{\partial μ}{\partial y}

(12)

Governing equations (8)–(10) are rewritten as follows

\begin{matrix} \frac{\partial u}{\partial t} & = α + \frac{1}{Re} [\bar{μ} \frac{\partial v}{\partial y} + \bar{λ} v] \\ - (H_{a}^{2} + s^{2}) u - L_{1} Q (u - u_{p}) \end{matrix}

(13)

μ = {| v |}^{n - 1}, \bar{λ} = \frac{\partial \bar{μ}}{\partial y}

(14)

\frac{\partial u_{p}}{\partial t} = β \frac{\partial v_{p}}{\partial y} + L_{1} (u - u_{p})

(15)

where bar notation refers to the iterated terms which transform the system equations (12)–(15) to a linearized one. These equations with condition (11) represent a system of linearized partial differential equations which is solved numerically using FDM. The Crank–Nicolson implicit method¹² is used at two successive time levels where the finite difference equations relating the variables are obtained by writing the equations at the mid-point of the computational cell as shown in Figure 2.

Figure 2.

The computational cell.

The infinite domain of solution ( $- 1 < y < 1$ and $t > 0$ ) is divided into m-subintervals in y-direction such that the mesh size is $Δ y = 2 / m$ with counter $i$ = 1, 2, 3, …, $m$ + 1 and mesh of size $Δ t$ in time variable with counter $j$ = 1, 2, 3, …, ∞. The linearized system coupled with linearized partial differential equations (12)–(15) is transformed to a system algebraic equations using the following central difference and average schemes at mid-point of the cell ( $i$ + 1/2, $j$ + 1/2) shown in Figure 2¹²

{\frac{\partial u}{\partial t} |}_{i + 1 / 2}^{j + 1 / 2} = \frac{u_{i + 1}^{j + 1} - u_{i + 1}^{j} + u_{i}^{j + 1} - u_{i}^{j}}{Δ t} + o [{(Δ t)}^{2} + {(Δ y)}^{2}]

(16)

{\frac{\partial v}{\partial y} |}_{i + 1 / 2}^{j + 1 / 2} = \frac{v_{i + 1}^{j + 1} + v_{i + 1}^{j} - v_{i}^{j + 1} - v_{i}^{j}}{Δ y} + o [{(Δ t)}^{2} + {(Δ y)}^{2}]

(17)

{u |}_{i + 1 / 2}^{j + 1 / 2} = \frac{u_{i + 1}^{j + 1} + u_{i + 1}^{j} + u_{i}^{j + 1} + u_{i}^{j}}{4} + o [{(Δ t)}^{2} + {(Δ y)}^{2}]

(18)

Similar difference schemes are used for other dependent variables. Finally, the resulting block tri-diagonal system is solved using generalized Thomas algorithm.¹²

The skin friction factor is an important fluid flow parameter because it is very important in the engineering applications as they are important factors that can be used to improve the shape and efficiency of equipment in aerodynamics. This parameter is computed after solving the governing equations.

The skin friction factor is defined as follows^13,14

{\overset{⌢}{C}}_{f} = \frac{2 ρ τ_{w} h^{2}}{μ_{0}^{2}}

(19)

The dimensionless form of skin friction factors for fluid and dust particles is written, respectively, as follows

C_{f} (t) = \frac{R_{e} {\overset{⌢}{C}}_{f}}{2} = {(\frac{\partial u}{\partial y})}_{y = - 1} = v | \begin{matrix} j \\ i = 1 \end{matrix}

(20)

C_{fp} (t) = \frac{R_{e} {\overset{⌢}{C}}_{f p}}{2} = {(\frac{\partial u_{p}}{\partial y})}_{y = - 1} = v_{p} | \begin{matrix} j \\ i = 1 \end{matrix}

(21)

Error analysis

The central differences and averages at mid-point of cell in Figure 2 have a truncation error (TE) of second-order accuracy, that is, $o ((Δ t)^{2} + (Δ y)^{2})$ . This error tends to zero as $Δ t \to 0$ and $Δ y \to 0$ shown in Tables 1 –4.

Table 1.

Sample of convergence of present results (pseudo-fluid).

	$u (0, 1)$			$u_{p} (0, 1)$
$(Δ t)^{2}$	$(Δ y)^{2} =$ 4 × 10⁻² (m = 10)	$(Δ y)^{2} =$ 4 × 10⁻⁴ (m = 100)	$(Δ y)^{2} =$ 1.6 × 10⁻⁵ (m = 500)	$(Δ y)^{2} =$ 4 × 10⁻² (m = 10)	$(Δ y)^{2} =$ 4 × 10⁻⁴ (m = 100)	$(Δ y)^{2} =$ 1.6 × 10⁻⁵(m = 500)
10⁻²	1.371690	1.342755	1.342306	0.463135	0.456136	0.456067
10⁻⁴	1.369958	1.341008	1.340728	0.461952	0.454671	0.454601
10⁻⁶	1.363140	1.340955	1.340675	0.461216	0.454624	0.454554
10⁻⁸	1.369936	1.340937	1.340709	0.461958	0.454628	0.454559

$(n = R = 0.5, β = H_{a} = s = R_{e} = 1, α = 5, L_{1} = 1.25, Q = 0.4)$ .

Table 2.

Sample of convergence of present results (Newtonian fluid).

	$u (0, 1)$			$u_{p} (0, 1)$
$(Δ t)^{2}$	$(Δ y)^{2} =$ 4 × 10⁻² (m = 10)	$(Δ y)^{2} =$ 4 × 10⁻⁴ (m = 100)	$(Δ y)^{2} =$ 1.6 × 10⁻⁵ (m = 500)	$(Δ y)^{2} =$ 4 × 10⁻² (m = 10)	$(Δ y)^{2} =$ 4 × 10⁻⁴ (m = 100)	$(Δ y)^{2} =$ 1.6 ×10⁻⁵ (m = 500)
10⁻²	1.248934	1.238871	1.238777	0.404365	0.403985	0.403981
10⁻⁴	1.247549	1.237878	1.237785	0.403489	0.403127	0.403123
10⁻⁶	1.247538	1.237868	1.237776	0.403480	0.403120	0.403116
10⁻⁸	1.247445	1.237863	1.237771	0.403428	0.403129	0.403126

$(n = R = 0.5, β = H_{a} = s = R_{e} = 1, α = 5, L_{1} = 1.25, Q = 0.4)$ .

Table 3.

Sample of convergence of present results (dilatant fluid).

	$u (0, 1)$			$u_{p} (0, 1)$
$(Δ t)^{2}$	$(Δ y)^{2} =$ 4 × 10⁻² (m = 10)	$(Δ y)^{2} =$ 4 × 10⁻⁴ (m = 100)	$(Δ y)^{2} =$ 1.6 × 10⁻⁵ (m = 500)	$(Δ y)^{2} =$ 4 × 10⁻² (m = 10)	$(Δ y)^{2} =$ 4 × 10⁻⁴ (m = 100)	$(Δ y)^{2} =$ 1.6 × 10⁻⁵ (m = 500)
10⁻²	1.163586	1.169189	1.169448	0.368170	0.370231	0.370250
10⁻⁴	1.161883	1.168393	1.168648	0.367208	0.369594	0.369616
10⁻⁶	1.161924	1.168412	1.168644	0.367201	0.369594	0.369610
10⁻⁸	1.162178	1.168389	1.168698	0.367219	0.369600	0.369623

$(n = 1.5 R = 0.5, β = H_{a} = s = R_{e} = 1, α = 5, L_{1} = 1.25, Q = 0.4)$ .

Table 4.

Sample of convergence of skin friction factor $C_{f} (t)$ $(Δ t)^{2}$ = 10⁻⁸.

t	$n$ = 0.5			$n$ = 1
	m = 20(Δy)² = 4 × 10⁻²	m = 100(Δy)² = 4 × 10⁻⁴	m = 800(Δy)² = 1.6 × 10⁻⁶	m = 10(Δy)² = 4 × 10⁻²	m = 100(Δy)² = 4 × 10⁻⁴	m = 800(Δy)² = 1.6 × 10⁻⁶
0.2	3.349308	3.461747	3.463402	2.163507	2.160250	2.162499
0.4	4.475874	4.607999	4.609512	2.654840	2.647173	2.648285
0.6	5.024476	5.169221	5.170519	2.845719	2.837475	2.838665
0.8	5.310278	5.463451	5.464571	2.923008	2.915216	2.915860
1	5.464259	5.623108	5.624089	2.955638	2.948306	2.949151
1.2	5.549590	5.711578	5.712466	2.969855	2.962969	2.963418
1.4	5.597368	5.761288	5.762108	2.976393	2.969716	2.970391
1.6	5.624577	5.789483	5.790259	2.979434	2.972918	2.973260
1.8	5.639818	5.805582	5.806324	2.980889	2.974484	2.975048
2	5.648477	5.814812	5.815537	2.981505	2.975278	2.975536
2.2	5.653568	5.820122	5.820838	2.981783	2.975684	2.976164
2.4	5.656411	5.823190	5.823897	2.981950	2.975899	2.976100
2.6	5.658119	5.824959	5.825660	2.981999	2.976020	2.976430
2.8	5.659010	5.825980	5.826685	2.981951	2.976091	2.976430
3	5.659541	5.826570	5.827271	2.981984	2.976121	2.976430
3.2	5.659946	5.826912	5.827598	2.981972	2.976139	2.976430
3.4	5.660074	5.827107	5.827780	2.981975	2.976139	2.976430
3.6	5.660284	5.827221	5.827881	2.981975	2.976139	2.976430
3.8	5.660194	5.827273	5.827941	2.981975	2.976146	2.976430
4	5.660278	5.827293	5.827973	2.981975	2.976146	2.976430
Steady state	5.760396	5.827397	5.828109	2.982602	2.976486	2.976431

$(R = 0.5, β = H_{a} = s = R_{e} = 1, α = 5, L_{1} = 1.25, Q = 0.4)$ .

Tables 1 –3 illustrate a sample of convergence of present solution depending on the influences of $u (0, 1)$ and $u_{p} (0, 1)$ by mesh sizes ( $Δ t$ and $Δ y$ ). Three flow indices: $n$ = 0.5 (pseudo-fluid), 1 (Newtonian fluid), and 1.5 (dilatant fluid) are listed in these tables to show their effects on convergence of present results. It is observed that $u (0, 1)$ and $u_{p} (0, 1)$ converge to four decimal places when $(Δ t)^{2} = 10^{- 8}$ and $(Δ y)^{2} = 1.6 \times 10^{- 5}$ .

Table 4 shows a sample of convergence of skin friction factor $C_{f} (t)$ at two flow indices ( $n$ = 0.5 and 1) when $(Δ t)^{2} = 10^{- 8}$ and $(Δ y)^{2} = 4 \times 10^{- 2}, 4 \times 10^{- 4}$ , and $1.6 \times 10^{- 6}$ . It is observed that $C_{f} (t)$ is convergent and stable since it converges to the steady-state solution near $t$ = 4. It is also observed that 99% of steady-state friction factor $C_{f}$ occurs at $t$ = 1.455 and 1.185 for $n$ = 0.5 and 1, respectively. Thus, the friction factor for Newtonian fluid is faster than pseudofluid as it reaches the steady-state values.

Figure 3 shows the comparison between the present results with Attia¹⁵ for $H_{a} = 0$ , $n = 1$ , and $β$ → 0. It is shown that the present results are of good agreement with Attia.¹⁵

Figure 3.

Comparison of time development of central plane velocities with Attia:¹⁵ (a) fluid velocity and (b) particle velocity.

Results and discussions

Computations are made for $α = 5$ , $R_{e} = 1$ , $s = 1$ , $R = 0.5$ , $L_{1} = 1.25$ , and $Q = 0.4$ . Samples of the present solution are shown in Figures 4 –12 to illustrate the behaviors of velocities and friction factors. The results are also listed in Tables 5 –11 when accurate velocities and friction factors are needed in engineering design and comparisons with experimental and analytical works.

Figure 4.

Effect of power law index n on time development of (a) fluid velocity: u(0, t), (b) particle velocity: u_p(0, t), (c) fluid friction factor: C_f(t), and (d) particle friction factor: C_fp(t) with H_a = β = 1.

Figure 5.

Effect of viscosity ratio β on time development of (a) fluid velocity: u(0, t), (b) particle velocity: u_p(0, t), (c) fluid friction factor: C_f(t), and (d) particle friction factor: C_fp(t) with H_a = n = 1.

Figure 6.

Effect of Hartmann number H_a on time development of (a) fluid velocity: u(0, t), (b) particle velocity: u_p(0, t), (c) fluid friction factor: C_f(t), and (d) particle friction factor: C_fp(t) with β = n = 1.

Figure 7.

Influence of time on profiles of (a) fluid velocity profile: u and (b) particle velocity profile: u_p with n = 0.5.

Figure 8.

Influence of time on profiles of: (a) fluid velocity profile: u and (b) particle velocity profile: u_p with n = 1.

Figure 9.

Influence of time on profiles of (a) fluid velocity profile: u and (b) particle velocity profile: u_p with n = 1.5.

Figure 10.

Influence of time on profiles of (a) fluid velocity profile: u and (b) particle velocity profile: u_p with β = 0.05.

Figure 11.

Influence of time on profiles of (a) fluid velocity profile: u and (b) particle velocity profile: u_p with β = 0.5.

Figure 12.

Influence of time on profiles of (a) fluid velocity profile: u and (b) particle velocity profile: u_p with β = 5.

Table 5.

Variations of steady-state velocities $u (y)$ and $u_{p} (y)$ with $β$ at n = 0.5.

y	$u (y)$			$u_{p} (y)$
	β = 0.05	β = 0.5	β = 5	β = 0.05	β = 0.5	β = 5
−1	0	0	0	0	0	0
−0.8	0.8741	0.8213	0.7787	0.6499	0.2848	0.0472
−0.6	1.2900	1.2061	1.1360	1.0942	0.5189	0.0873
−0.4	1.4780	1.3790	1.2932	1.3449	0.6855	0.1172
−0.2	1.5437	1.4393	1.3473	1.4614	0.7831	0.1353
0	1.5529	1.4478	1.3549	1.4933	0.8150	0.1414
0.2	1.5437	1.4393	1.3473	1.4614	0.7831	0.1353
0.4	1.4780	1.3789	1.2932	1.3449	0.6855	0.1172
0.6	1.2900	1.2061	1.1359	1.0942	0.5189	0.0873
0.8	0.8741	0.8213	0.7787	0.6499	0.2848	0.0472
1	0	0	0	0	0	0

$(R = 0.5, H_{a} = s = R_{e} = 1, α = 5, L_{1} = 1.25, Q = 0.4)$ .

Table 6.

Variations of steady-state velocities $u (y)$ and $u_{p} (y)$ with $β$ at n = 1.

y	$u (y)$			$u_{p} (y)$
	β = 0.05	β = 0.5	β = 5	β = 0.05	β = 0.5	β = 5
−1	0	0	0	0	0	0
−0.8	0.5307	0.5113	0.4937	0.4407	0.2179	0.0376
−0.6	0.9046	0.8681	0.8348	0.7972	0.4075	0.0708
−0.4	1.1520	1.1026	1.0569	1.0479	0.5517	0.0965
−0.2	1.2930	1.2356	1.1820	1.1947	0.6411	0.1126
0	1.3388	1.2787	1.2224	1.2429	0.6714	0.1181
0.2	1.2930	1.2356	1.1820	1.1947	0.6411	0.1126
0.4	1.1520	1.1026	1.0569	1.0479	0.5516	0.0965
0.6	0.9046	0.8681	0.8348	0.7972	0.4075	0.0708
0.8	0.5307	0.5113	0.4937	0.4407	0.2179	0.0376
1	0	0	0	0	0	0

$(R = 0.5, H_{a} = s = R_{e} = 1, α = 5, L_{1} = 1.25, Q = 0.4)$ .

Table 7.

Variations of steady-state velocities $u (y)$ and $u_{p} (y)$ with $β$ at n = 1.5.

y	$u (y)$			$u_{p} (y)$
	β = 0.05	β = 0.5	β = 5	β = 0.05	β = 0.5	β = 5
−1	0	0	0	0	0	0
−0.8	0.4084	0.3984	0.3889	0.3565	0.1870	0.0330
−0.6	0.7346	0.7147	0.6956	0.6636	0.3533	0.0626
−0.4	0.9827	0.9539	0.9262	0.9004	0.4838	0.0858
−0.2	1.1520	1.1164	1.0821	1.0544	0.5675	0.1007
0	1.2279	1.1891	1.1516	1.1103	0.5967	0.1059
0.2	1.1520	1.1164	1.0821	1.0544	0.5675	0.1007
0.4	0.9827	0.9539	0.9262	0.9004	0.4838	0.0858
0.6	0.7346	0.7147	0.6956	0.6636	0.3533	0.0626
0.8	0.4084	0.3984	0.3889	0.3565	0.1870	0.0330
1	0	0	0	0	0	0

$(R = 0.5, H_{a} = s = R_{e} = 1, α = 5, L_{1} = 1.25, Q = 0.4)$ .

Table 8.

Variations of steady-state central velocities $u (0)$ and $u_{p} (0)$ with $β$ and $n$ at $H_{a} = 0$

$n$	$u (0)$			$u_{p} (0)$
	β = 0.05	β = 0.5	β = 5	β = 0.05	β = 0.5	β = 5
0.25	3.0801	2.6412	2.3172	3.0102	1.5483	0.2535
0.5	2.4599	2.1924	1.9774	2.3598	1.2273	0.2052
0.75	2.0141	1.8575	1.7220	1.8949	0.9990	0.1709
1	1.7371	1.6386	1.5489	1.6066	0.8539	0.1484
1.25	1.5641	1.4964	1.4328	1.4260	0.7606	0.1336
1.5	1.4505	1.4005	1.3526	1.3067	0.6978	0.1234
1.75	1.3720	1.3330	1.2951	1.2235	0.6534	0.1161
2	1.3184	1.2858	1.2545	1.1638	0.6207	0.1106

$(R = 0.5, s = R_{e} = 1, α = 5, L_{1} = 1.25, Q = 0.4)$ .

Table 9.

Variations of steady-state central velocities $u (0)$ and $u_{p} (0)$ with $β$ and $n$ at $H_{a} = 1$ .

$n$	$u (0)$			$u_{p} (0)$
	β = 0.05	β = 0.5	β = 5	β = 0.05	β = 0.5	β = 5
0.25	1.7126	1.5739	1.4572	1.6754	0.9252	0.1597
0.5	1.5529	1.4478	1.3549	1.4933	0.8150	0.1414
0.75	1.4287	1.3501	1.2783	1.3487	0.7313	0.1278
1	1.3388	1.2787	1.2224	1.2429	0.6714	0.1181
1.25	1.2745	1.2270	1.1817	1.1665	0.6283	0.1111
1.5	1.2279	1.1891	1.1516	1.1103	0.5967	0.1059
1.75	1.1932	1.1606	1.1289	1.0679	0.5728	0.1020
2	1.1675	1.1395	1.1121	1.0352	0.5542	0.0989

$(R = 0.5, s = R_{e} = 1, α = 5, L_{1} = 1.25, Q = 0.4)$ .

Table 10.

Variations of steady-state skin friction factors $C_{f}$ and $C_{fp}$ with $β$ and $n$ with $H_{a} = 0$ .

$n$	$C_{f}$			$C_{fp}$
	β = 0.05	β = 0.5	β = 5	β = 0.05	β = 0.5	β = 5
0.25	22.6051	20.0449	18.4056	8.9306	3.0603	0.4602
0.5	8.9867	8.2090	7.6290	5.3982	2.2043	0.3507
0.75	5.2865	4.9696	4.7068	3.7353	1.7005	0.2823
1	3.7653	3.6052	3.4632	2.8889	1.4063	0.2402
1.25	2.9906	2.8957	2.8079	2.4106	1.2255	0.2133
1.5	2.5370	2.4739	2.4140	2.1133	1.1069	0.1951
1.75	2.2442	2.1989	2.1550	1.9140	1.0244	0.1822
2	1.9993	1.9684	1.9354	1.7627	0.9630	0.1726

$(R = 0.5, s = R_{e} = 1, α = 5, L_{1} = 1.25, Q = 0.4)$ .

Table 11.

Variations of steady-state skin friction factors $C_{f}$ and $C_{fp}$ with $β$ and $n$ .

$n$	$C_{f}$			$C_{fp}$
	β = 0.05	β = 0.5	β = 5	β = 0.05	β = 0.5	β = 5
0.25	13.8981	13.020	12.4065	5.1242	1.8460	0.2912
0.5	6.2646	5.9381	5.6793	3.5442	1.4814	0.2432
0.75	4.0825	3.9167	3.7749	2.7536	1.2594	0.2126
1	3.1133	3.0138	2.9244	2.3045	1.1173	0.1924
1.25	2.5843	2.5176	2.4557	2.0246	1.0215	0.1784
1.5	2.2575	2.2092	2.1634	1.8372	0.9540	0.1683
1.75	2.0382	2.0012	1.9655	1.7045	0.9043	0.1608
2	1.8521	1.8237	1.7951	1.6023	0.8660	0.1549

$(R = 0.5, H_{a} = s = R_{e} = 1, α = 5, L_{1} = 1.25, Q = 0.4)$ .

Figure 4(a)–(d) illustrates the effect of power law index n on time development of central plane velocities $u (0, t)$ and $u_{p} (0, t)$ and skin friction factors $C_{f} (t)$ and $C_{fp} (t)$ at $H_{a} = β = 1$ . It is observed that $u (0, t)$ , $u_{p} (0, t)$ , $C_{f} (t)$ , and $C_{fp} (t)$ decrease with increasing n. It is also observed that $u$ and $C_{f}$ reach their steady states faster than $u_{p}$ , and $C_{fp}$ . In general, fluid phase reaches its steady state faster than the particle phase.

Figure 5(a)–(d) illustrates the $β$ on time development of central plane velocities $u (0, t)$ and $u_{p} (0, t)$ and skin friction factors $C_{f} (t)$ and $C_{fp} (t)$ at $H_{a} = n = 1$ . It is observed that $u$ , $u_{p}$ , $C_{f}$ , and $C_{fp}$ decrease with increasing $β$ , but the particle phase is influenced more than the fluid phase because $β$ is the main parameter that governs the dusty particle motion.

Figure 6(a)–(d) illustrates the effect of H_a on time development of central plane velocities $u$ and $u_{p}$ and skin friction factors $C_{f} (t)$ and $C_{fp} (t)$ at $β = n = 1$ . It is observed that u, u_p, $C_{f} (t)$ , and $C_{fp} (t)$ decrease with increasing H_a, because of its resistive effect, but the fluid phase reaches its steady state faster than the particle phase.

Figures 7(a) and (b), 8(a) and (b), and 9(a) and (b) show the velocity profiles $u (y, t)$ and $u_{p} (y, t)$ for different independent variables ( $- 1 < y < 1$ ) and ( $t$ = 0.2, 0.4, and 2) and power law indices ( $n$ = 0.5, 1, and 1.5). It is observed that velocities $u$ and $u_{p}$ increase with increasing time t, but they decrease with increasing index $n$ . It is also observed that a slug flow is formed for $u$ with more pronounced values at $n$ = 0.5.

Figures 10(a) and (b), 11(a) and (b), and 12(a) and (b) show the velocity profiles $u (y, t)$ and $u_{p} (y, t)$ for different independent variables ( $- 1 < y < 1$ ) and ( $t$ = 0.2, 0.4, and 2) and $β$ (=0.05, 0.51, and 5). It is observed that velocities u and u_p increase with increasing time t, but they decrease with increasing $β$ . It is also observed that the particle phase is influenced more than the fluid phase because $β$ is the main parameter that governs the dusty particle motion.

Conclusion

The continuous dusty viscous particle model and the non-Newtonian power-law fluid flow are considered between two parallel stationary plates. The conducting fluid was studied considering magnetic and Darcy resistance forces. The influences of power law index, viscosity ratio, and Hartmann number on the velocities and skin friction factors for both the fluid and particle phases are studied. The second-order accurate FDM has been used to compute the velocities and friction factors. The generalized Thomas algorithm is applied on linearized system of equations. It is observed that the particle phase is influenced by particle viscosity ratio more than the fluid phase because it is the main parameter that governs the dusty particle motion. It is also observed that the fluid phase reaches its steady state faster than the particle phase. An error analysis is studied and tabulated to achieve the convergence and stability of present results.

Footnotes

Appendix 1

Handling Editor: James Baldwin

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

Karem Mahmoud Ewis

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