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In this paper, an investigation of the effects of some physical parameters and Hall current on magneto hydrodynamics (MHD) fluid flow with heat flux over a porous medium was carefully examined, taking into consideration Hall effects where the temperature and concentration are assumed to be oscillating with time. Furthermore, perturbation method is used in solving the governing equations. The profiles of velocity, temperature and concentration are presented graphically, going into the problem the primary and secondary velocity are presented and compute for some physical parameters such as mass Grashof number (
*Gc*), Schmidt number
*Sc*, Prandtl number (
*Pr*) viscoelastic parameter (
*K*
_{1}) and hall current parameter (
*m*). Results indicated that primary velocity increases with increase in values of
*Gc* on one hand and on the other hand it decreases with increase in the values of
*Pr*,
*K*
_{1} and
*m*. Secondary velocity demonstrated opposite trend.

The problems of fluid flow continue to attract the attention of so many researchers in engineering science, applied mathematics and applied geophysics. Heat and mass transfer is used in aerodynamic, extrusion of plastic sheets and other engineering processes which include chemical engineering and petroleum engineering [

Consider the flow of incompressible memory fluid in an infinite plane with heat and mass transfer, under the influence of an induced magnetic field and constant suction. The x-axis is taken along the plane in the upward direction and a straight line perpendicular to that of the y-axis. All fluid properties are assumed constant. Since the fluid is conducting, the magnetic Reynolds number is much less than unity and hence the induced magnetic field is not neglected.

The equations governing the flow under Boussineqs approximation are:

Continuity equation:

∂ v ∂ y = 0

Momentum equation

∂ u ′ ∂ t + υ 0 ∂ u ′ ∂ y ′ = υ ∂ 2 u ′ ∂ y 2 − υ u ′ K * + g β * ( C ′ 0 − C ′ d ) + g β * ( T ′ 0 − T ′ d ) − σ β 0 2 ρ ( 1 + m 2 ) ( u + m w ) − K 1 { ∂ 3 u ′ ∂ t ∂ y 2 } (1)

∂ w ′ ∂ t ′ + υ 0 ∂ w ′ ∂ y ′ = υ ∂ 2 w ′ ∂ y ′ 2 − υ w ′ K * − σ β 0 2 ρ ( 1 + m 2 ) ( w − m u ) − K 1 { ∂ 3 w ′ ∂ t ∂ y ′ 2 } (2)

Energy equation:

∂ T ′ ∂ t ′ + υ 0 ∂ T ′ ∂ y ′ = k ρ c p ∂ 2 T ′ ∂ y ′ 2 − 1 ρ c p ∂ q r ∂ y + 1 ρ c p ( ∂ u ′ ∂ y ′ ) 2 (3)

Concentration equation:

∂ C ′ ∂ t ′ + υ 0 ∂ C ′ ∂ y ′ = D ∂ 2 C ′ ∂ y ′ 2 − K C ′ (4)

The initial boundary conditions are:

u = 0 , w = 0 , T = T d + ( T 0 − T d ) ε e i ω t , C = C d + ( C 0 − C d ) ε e ι ω t at y = 0 u = 0 , w = 0 , T = 0 , C = 0 at y = d } (5)

where u is the velocity of the fluid in the x-direction and v in the y-direction, T is the temperature of the fluid, C is the concentration of the fluid, g is the acceleration due to gravity, β * are the kinematic viscosity, K is the thermal conductivity and Cp is the specific heat capacity of the fluid at constant pressure. t is the time, σ is the electrical conductivity of the fluid and μ e is the magnetic permeability. T_{0} is the temperature of the plate and T_{d} is the temperature of the fluid far away from plate. C_{0} is the concentration of the plate and C_{d} is the concentration of the fluid far away from the plate. M is magnetic number, K_{1} is viscoelastic parameter, ῴ is the frequency of oscillation, m is hall current parameter.

Introducing the following non-dimensional quantities,

w ′ 1 = U e w 1 , ∂ w ′ 1 = U e d w 1 , ∂ 2 w ′ 1 = U e d 2 w 1 , y = y ′ d , t = t υ 0 υ ⇒ t ′ = υ t υ 0 , ∂ t ′ = υ υ 0 d t , θ = T ′ − T ′ d T ′ 0 − T ′ d , M = σ β 0 2 υ ρ υ 0 2 , K * = K 0 υ 0 2 υ 2 C = C ′ − C ′ d C ′ 0 − C ′ d , G c = υ β * υ ( C ′ ω − C ′ ∞ ) U υ 0 2 } (6)

The term ∂ q r ∂ y represents the radiative heat flux. By using Rosseland approximation, the radiation heat flux q r = − 4 σ ∗ ∂ T 4 3 a R ∂ y , where and Stephen and

Boltzmann constants and mean absorption coefficient respectively. We assume that the temperature difference within the flow is such that may be expanded in a Taylor’s series.

Hence, expanding T 4 about T d and neglecting higher order terms, we get

T ′ 4 = T ′ d 4 + 4 T d 3 T ∗ = 4 T d 3 T ′ − 3 T ′ d 4

We assume the following solutions:

u ′ = u ′ 0 + ε u ′ 1 e i ω t w ′ = w ′ 0 + ε w ′ 1 e i ω t θ ′ = θ ′ 0 + ε θ ′ 1 e i ω t C ′ = C ′ 0 + ε C ′ 1 e i ω t } (7)

∂ 2 u 0 ∂ y 2 + ∂ u 0 ∂ y − L u 0 − J w 0 + G c C 0 = 0 (8)

where L = ( 1 K s + M 1 + m 2 ) , J = ( M m 1 + m 2 )

d 2 w 0 d y 2 − d w 0 d y − L w 0 + J u 0 = 0 (9)

Combining (8) and (9) using complex variable method, we have,

d 2 F d y 2 − d F d y − F L − F J = − G c C 0 (10)

where F = ( u 0 + i w 0 ) and i = − 1

d 2 F d y 2 − d F d y − P 1 F = − G c C 0 (11)

where P 1 = ( L + J )

⇒ ( 1 − i ω K 1 ) d 2 u 1 d y 2 − d u 1 d y − L n u 1 + J w 1 = − G c C ′ 1 (12)

where L n = ( L + i ω )

⇒ ( 1 − i ω K 1 ) d 2 w 1 d y 2 − d w 1 d y − ( L + i ω ) w 1 + J u 1 = 0

⇒ ( 1 − i ω K 1 ) d 2 w 1 d y 2 − d w 1 d y − L 0 w 1 + J u 1 = 0 (13)

Using the method of complex variable, combining (12) and (13) to have,

P 2 d 2 H d y 2 − d H d y − P 3 H = − G c C 1 (14)

where H = ( u 1 + i w 1 ) , P 2 = ( 1 − i ω K 1 ) , L = ( L 0 + i ω ) .

The new boundary conditions are,

u ′ 0 = u ′ 1 = 0 , w ′ 0 = w ′ 1 = 0 , θ ′ 0 = θ ′ 1 = 1 , C ′ 0 = C ′ 1 = 1 , at y = 0 u ′ 0 = u ′ 1 = 0 , w ′ 0 = w ′ 1 = 0 , θ ′ 0 = θ ′ 1 = 0 , C ′ 0 = C ′ 1 = 0 , at y = 1 } (15)

To solve for mass diffusion, therefore we assume:

Concentration to be

C ( y , t ) = C 0 + C 1 ε exp ( i ω t ) C ( y , t ) = A 1 exp ( m 1 y ) + A 2 exp ( m 2 y ) + ( A 3 exp ( m 3 y ) + A 4 exp ( m 4 y ) ) ε exp ( i ω t ) (16)

To solve for the momentum equation then

F ( h ) = A 5 exp ( m 5 y ) + A 6 exp ( m 6 y ) (17)

d 2 H d y 2 − P 4 d H d y − P 5 H = 0 (18)

H ( y ) = A 7 exp ( m 7 y ) + A 8 exp ( m 8 y ) + D 3 exp ( m 3 y ) + D 4 exp ( m 4 y ) (19)

Therefore, the solution of primary velocity is assumed to be

F ( u 0 + u 1 ) = u 0 + u 1 ε e i ω t

F ( u 0 + u 1 ) = A 5 exp ( m 5 y ) + A 6 exp ( m 6 y ) + D 1 exp ( m 1 y ) + D 2 exp ( m 2 y ) + A 7 exp ( m 7 y ) + A 8 exp ( m 8 y ) + D 3 exp ( m 3 y ) + D 4 exp ( m 4 y ) ε e i ω t (20)

Then, the secondary velocity also is as follows:

H ( w 0 + w 1 ) = w 0 + w 1 ε e i ω t

H ( w 0 + w 1 ) = i { A 5 exp ( m 5 y ) + A 6 exp ( m 6 y ) + D 1 exp ( m 1 y ) + D 2 exp ( m 2 y ) + A 7 exp ( m 7 y ) + A 8 exp ( m 8 y ) + D 3 exp ( m 3 y ) + D 4 exp ( m 4 y ) ε e i ω t } (21)

From the energy equation we have,

∂ T ′ ∂ t ′ + υ 0 ∂ T ′ ∂ y ′ = k ρ c p ∂ 2 T ′ ∂ y ′ 2 − 1 ρ c p ∂ q r ∂ y + 1 ρ c p ( ∂ u ′ ∂ y ′ ) 2

Substituting the value of ∂ T ′ yields

∂ θ ∂ t + ∂ θ ∂ y = 1 P r ∂ 2 θ ∂ y 2 + 4 R ∂ 2 θ 3 P r ∂ y 2 + E c ( ∂ u ′ ∂ y ′ ) 2

where E c = υ ρ C p ( T ′ 0 − T ′ d )

∂ θ ∂ t + ∂ θ ∂ y = E P r ∂ 2 θ ∂ y 2 + E c ( ∂ u ′ ∂ y ′ ) 2

where E = ( 1 + 4 R 3 )

∂ θ ∂ t + ∂ θ ∂ y = E P r ∂ 2 θ ∂ y 2 + E c ( ∂ u ′ ∂ y ′ ) 2 (22)

We also assumed solution to be

θ = θ 0 + θ 1 ε e i ω t u = u 0 + u 1 ε e i ω t }

Therefore, the solution of the energy equation is as follows:

θ = { A + B exp ( P r E 2 y ) + D 11 exp ( 2 m 5 y ) + D 12 exp ( 2 m 6 y ) + D 13 exp ( 2 m 1 y ) + D 14 exp ( 2 m 2 y ) + D 15 exp ( m 5 + m 6 ) y + D 16 exp ( m 5 + m 1 ) y + D 17 exp ( m 5 + m 2 ) y + D 18 exp ( m 6 + m 1 ) y + D 19 exp ( m 6 + m 2 ) y + D 20 exp ( m 1 + m 2 ) y }

+ { A 9 exp ( m 9 y ) + A 10 exp ( m 10 y ) + D 21 exp ( m 5 + m 7 ) y + D 22 exp ( m 5 + m 8 ) y + D 23 exp ( m 5 + m 3 ) y + D 24 exp ( m 5 + m 4 ) y + D 25 exp ( m 6 + m 7 ) y + D 26 exp ( m 6 + m 8 ) y + D 27 exp ( m 6 + m 3 ) y + D 28 exp ( m 6 + m 4 ) y + D 29 exp ( m 1 + m 7 ) y + D 30 exp ( m 1 + m 8 ) y

+ D 31 exp ( m 1 + m 3 ) y + D 32 exp ( m 1 + m 4 ) y + D 33 exp ( m 2 + m 7 ) y + D 34 exp ( m 2 + m 8 ) y + D 35 exp ( m 2 + m 3 ) y + D 36 exp ( m 2 + m 4 ) y } ε exp ( i ω t ) (23)

Figures 1-10 represent velocity profile for the flow.

Investigation of effects of some physical parameters and hall current on MHD fluid flow with heat flux over a porous medium is studied by transforming the governing partial differential equations into ordinary differential equations which are then solved using perturbation techniques. The result of the flow variables indicates that the fluid temperature is reduced by increasing Prandtl number (Pr) and radiation parameter (R). Concentration is reduced with increase in Schmidt number (Sc) and chemical reaction parameter (K). The primary velocity decrease with increasing prandtl (Pr), radiation parameter and hall-current while the opposite trend is observed in secondary velocity. The primary velocity increases with increase in mass Grashoof number (Gr) and thermal Grasshoof number (Gc) also the reverse is the case in secondary velocity. The primary velocity decreases with increase in M, s and Sc.

The authors declare no conflicts of interest regarding the publication of this paper.

Sarki, M.N., Ahmed, A. and Uwanta, I.J. (2021) Investigation of the Effects of Some Physical Parameters and Hall Current on MHD Fluid Flow with Heat Flux over a Porous Medium. Advances in Pure Mathematics, 11, 652-664. https://doi.org/10.4236/apm.2021.117043