Steady and unsteady flow of a second grade MHD fluid in a porous medium with Hall current effects is studied. Assuming an à priori known vorticity proportional to the stream function up to an additive uniform stream, exact solutions for velocity field are obtained corresponding to different choices of pertinent flow parameters. Graphical results are presented to depict the influence of pertinent flow parameters on the considered MHD flow.

MHD flow analysis of differential type fluids have been the subject of numerous experimental, mathematical and numerical studies over the past few decades due to their diverse applications in for example exploration geophysics, hydrology, cooling system designs and MHD generators, see Refs. 1–11 and references therein.

The constitutive equations of the differential type fluids, that are non-Newtonian (and thus strongly non-linear) in nature, possess non-linearity both in inertial and viscous parts. It is an elusive task to obtain closed form analytic solutions to these flow problems, in fact, they can rarely be solved exactly. Consequently, numerical and asymptotic solutions are sought that are bound to approximation errors and simplifying assumptions. The exact solutions, if available, are important since they describe complete flow field profiles and rheological effects throughout the natural domain and can be used to synthesize stability, consistency and convergence of numerical and asymptotic prototypes.

In this paper, a specific differential type flow problem is dealt with. We consider two dimensional incompressible MHD flow of a second grade fluid in a porous medium subject to Hall current effects. The aim of this investigation is to invoke an inverse method using an à priori known vorticity function, thereby extrapolating the results in Ref. 12 to the case of MHD porous media with Hall effect. Vorticity is assumed to be proportional to the stream function up to an additive uniform stream. Steady as well as unsteady exact solutions to stream functions and velocity field with various choices of emerging flow parameters are derived.

The inverse method described by Neményi13 provides a valuable tool to find exact solutions. In fact, additionally known physical and geometric characteristics of the underlying non-Newtonian flow field are exploited therein to greatly simplify the governing equations. Subsequently, the problem of finding an exact solution becomes tractable. This technique has already been used to find exact solutions to flow problems, for instance, by Benharbit and Siddiqui,14 Chandna and Oku-Ukpong,15 Hui,16 Kaloni and Huschilt,17 Labropulu,12 Fetecau et al.,9 and Aziz et al.10 Taylor18 was the first to cater to flow problems, for finding out exact solutions, by assuming a vorticity proportional to the stream function. The work was subsequently extended by Kovasznay19 and Lin and Tobak20 by slightly modifying the local vorticity assumed by Taylor up to a uniform stream perturbation.

To put the present work in proper context, we emphasize some significant differences with prior work. For second grade fluids, several attempts have been made to find analytic and closed form solutions to velocity field and stream functions for various physical and geometrical configurations. Coscia and Galdi1 discussed classical issues such as existence, uniqueness and stability of solutions to steady second grade flows in regular domains. Dunn & Fosdick26 also discussed stability of flow problems. Ting21 obtained solutions for unsteady second grade fluids in bounded regions under the condition that the highest order derivative term is positive. In Ref. 22, Hayat et al. provided some unsteady solutions considering particular forms of the stream functions. Similar results were obtained by Siddiqui and Kaloni9 and Siddiqui et al.10 by varing the form of stream functions. In Ref. 23 series solutions to the second grade flow problem with convective boundary conditions were proposed. Steady state solutions for second grade fluids were obtained by Zhang et al.24 using a specific vorticity. Labropulu12 also considered a specific vorticity, different from that discussed in Ref. 24 and presented both steady as well as unsteady solutions. Khan et al.4 studied MHD fluid flow problems in porous media with specific geometric configuration using Fourier transform methods. Mohyuddin and Ellahi, studied similar flow problems in free media in Ref. 5, subsequently corrected in Ref. 6, and presented the solutions using special stream functions. However, in prior work, no attempt has been made, at least not to the knowledge of the authors, to resolve both steady and unsteady second grade MHD fluids in porous media using a vorticity proportional to the stream function up to an additive uniform stream depending upon both spatial variables; see (17) below. The results presented in this paper, actually extrapolate those provided by Labropulu,12 which can be extracted by approaching the Hall parameter and the porosity constant to zero. Vafai and Tien32 analyzed the effects of a solid boundary and the inertial forces on flow and heat transfer in porous media. The boundary and inertial effects are characterized in terms of three dimensionless groups, and these effects are shown to be more pronounced in highly permeable media, high Prandtl-number fluids, large pressure gradients, and in the region close to the leading edge of the flow boundary layer. Moreover, setting second grade flow parameters equal to zero, we recover results for Newtonian fluids.

This paper is organized as follows. In Section II, mathematical modeling of the flow problem is presented. The underlying assumptions are detailed and the equations governing the flow problem are derived. The differential equation satisfied by the stream function for the given vorticity function is also obtained (Sections II B and II C). In Section III the steady state problem is resolved for stream function and expressions for velocity components are derived. Section IV deals with the unsteady problem for stream function and subsequently for velocity components. Pertinent cases are discussed and the solutions are derived. Section V delineates numerical results for some of streamline flow patterns by varying flow parameters. The paper is concluded in Section VI.

In this section, we state basic equations and assumptions governing the flow. We also introduce some notations and the flow problem for stream function in order to obtain the velocity components.

Consider two dimensional incompressible flow of a second grade fluid in Cartesian xy–plane. The fluid is conducting under the application of an applied magnetic field and it flows through a porous medium together with Hall current effects. The medium has constant permeability μm > 0 and porosity ϕ where 0 < ϕ < 1. The motion of the fluid is governed by the following constitutive equations:

div V = 0 ,
(1)
ρ V t + V V = div T + J × B + R d ,
(2)
div B = 0 , curl B = μ e J , curl E = B t ,

where ρ is the (constant) density of fluid, V is the velocity vector, 𝕋 is the stress tensor, E is the total electric field, J is the current density, B is the total magnetic field, Rd is the modified Darcy’s resistance and μe is the magnetic permeability of electrons. The total electric and induced magnetic fields are assumed to be negligible and will be omitted. Further, the applied magnetic field is assumed to be of the form

B = B induced + B applied = B applied = ( 0 , 0 , B 0 ) .

We recall that for a second grade fluid, the Cauchy stress tensor 𝕋 is given by

T = p I + ν A 1 + α 1 A 2 + α 2 A 1 2 ,
(3)

where p represents the hydrostatic pressure, 𝕀 is the identity tensor, ν is the dynamic viscosity, and parameters α1 and α2 are the viscoelasticity and cross-viscosity of the fluid, respectively. The strain and acceleration tensors 𝔸1 and 𝔸2 are defined by the expressions25 

A 1 = V + V t , and A 2 = A 1 t + ( V ) A 1 + A 1 V + V t A 1 ,
(4)

respectively, where the superscript t indicates the transpose of the second order tensor ∇V. The Clausius-Duhem inequality and the supposition of minimum Helmholtz free energy in equilibrium impose the following thermodynamic stability restrictions on normal stress moduli26–28 

α 1 0 , α 1 + α 2 = 0 .
(5)

As pointed out in Ref. 1, the mathematical problem is well posed even without constraint (5). However, we will keep these thermodynamic compatibility conditions for simplicity.

The generalized Ohm’s law,29 suggests that for a strong magnetic field

J + ω e τ e B 0 J × B = σ [ E + V × B + 1 e n e p e ] ,
(6)

where ωe is the cyclotron frequency of electrons, τe is the electron collision time, ne is the number density of electrons, σ is the electric conductivity, e is the electron charge and pe is the electron pressure. The term involving electron pressure gradient, ∇pe, in (6) is negligible and will be omitted. Moreover, physical constraints dictate that the product of electron collision time with cyclotron frequency of electron ωe is of order unity whereas the product of ionic collision time τi and ionic cyclotron frequency ωi is very small, that is,

ω e τ e o 1 and ω i τ i 1 .

See Ref. 30 for a brief and comprehensive note on relative orders of magnitude of τe, τi, ωe and ωi. Consequently, we find that

J × B = σ B 0 2 ( 1 + i h ) 1 + h 2 V ,
(7)

where h = ωeτe is the Hall parameter.31 

It is known that the pressure gradient is a resistance measure to the flow in the bulk. The modified Darcy’s law, using the relationship between pressure gradient and Darcian velocity Vd (given by Vd = ϕV) suggests that Rd (being the Darcian resistance to the flow in solid matrix) for second grade fluids has the form

R d = ν μ m 1 + α 1 ν t ϕ V .
(8)

We refer to Refs. 2, 11, and 32 for detailed discussions on modified Darcy’s law.

The configuration of the present flow problem in xy–plane suggests that the velocity field has the form

V = u x , y , t , v x , y , t , 0 ,

where u and v are, respectively, the velocity components along x − and y–directions. Note that Equation (2) together with (3), (4), (7) and (8) becomes

ρ V t + V V = div p I + ν A 1 + α 1 A 2 + α 2 A 1 2 σ B 0 2 ( 1 + i h ) 1 + h 2 V ν ϕ μ m 1 + α 1 ν t V .
(9)

The aim of this paper is to find expressions for the velocity components u and v using a specific vorticity function.

Let us define the modified pressure p ̂ and viscosity ν ̂ by

p ̂ ρ 2 ( u 2 + v 2 ) + p α 1 u 2 u + v 2 v 1 4 ( 3 α 1 + 2 α 2 ) tr A 1 2 ,
(10)
ν ̂ μ m σ B 0 2 ( 1 + i h ) 1 + h 2 ν ϕ .
(11)

Consequently, after simple manipulation, Equations (9) and (1), together with (10) and (11), yield

u x + v y = 0 ,
(12)
p ̂ x + ρ u t v κ = ν + α 1 t 2 u α 1 v 2 κ ν ̂ μ m ϕ α 1 μ m t u ,
(13)
p ̂ y + ρ v t + u κ = ν + α 1 t 2 v + α 1 u 2 κ ν ̂ μ m ϕ α 1 μ m t v ,
(14)

where κ = vxuy = − ∇2ψ, with ψ being the stream function. Here we have used the fact that the velocity field is divergence-free which leads to the existence of a scalar field ψ, that relates to the velocity components through the following relations:

u ( x , y , t ) = ψ y and v ( x , y , t ) = ψ x .
(15)

Therefore, Equations (12), (13) and (14), by virtue of (15) and after eliminating the modified pressure p ̂ , yield

ρ t 2 ψ ( ψ , 2 ψ ) ( x , y ) + α 1 ( ψ , 4 ψ ) ( x , y ) + ν ̂ μ m + ϕ α 1 μ m t ( 2 ψ ) = ν + α 1 t 4 ψ ,
(16)

where we have used the compatibility relation 2 p ̂ x y = 2 p ̂ y x and the notation

( f , g ) ( x , y ) = f x g y f y g x .

In the sequel, we take the density of fluid to be unity without loss of generality so that the stress moduli, α1 and α2, the dynamic viscosity, ν, and the conductivity σ are density normalized and we continue using ν, σ and α1 by abuse of notations.

In subsequent sections we shall find the stream functions satisfying (16), which together with (15) will lead us to the velocity components.

In the rest of this investigation, vorticity is assumed to be proportional to stream function up an additive uniform stream, that is, let the vorticity be given by

2 ψ a ψ c x b c y 2 ,
(17)

where a ≠ 0, b and c are real numbers.

By virtue of (17), Equation (16) for the steam function becomes

β ψ t + 2 b c γ y ψ x c γ ψ y ν a ψ + ν ̂ μ m ψ + ν a c x + b y 2 ν ̂ c μ m x + b y 2 + 2 ν b c = 0 ,
(18)

where

β γ + ϕ α 1 μ m and γ 1 a α 1 .
(19)

In this section, we consider a time-independent stream function in (18), that is, ψ t = 0 , in which case, ψ obeys the equation

+ 2 b c γ y ψ x c γ ψ y ν a ψ + ν ̂ μ m ψ + ν a c x + b y 2 ν ̂ c μ m x + b y 2 + 2 ν b c = 0 .
(20)

Remark 3.1.

It can readily be seen thatψvanishes forc = 0. Therefore, for non-trivial solutions,cmust not be zero.

Remark 3.2.
Forγ = 0, the differential equation (20) for the stream function becomes the algebraic equation
ν a ψ + ν ̂ μ m ψ + ν a c x + b y 2 ν ̂ c μ m x + b y 2 + 2 ν b c = 0 ,
that can be solved readily forψgiving
ψ = ( c x + b c y 2 ) 2 b c ν μ m ν ̂ a ν μ m .
Therefore, using (15), we arrive at
u = 2 bcy and v = c .

In rest of this section, we let γ ≠ 0 and derive the velocity components in the stationary case. Consider the transformations ξ = x + by2 and η = y and note that

ξ , η x , y = 1 0 .

Therefore (20) takes the form

ψ η + a μ m ν ν ̂ c γ μ m ψ = a μ m ν ν ̂ μ m γ ξ + 2 ν b μ m a μ m ν ν ̂ ,

which is a linear differential equation for ψ with respect to independent variable η. Therefore, its solution can be given by

ψ ( ξ , η ) = c ξ + 2 ν b c μ m μ m ν a ν ̂ + f ξ exp a μ m ν ν ̂ c μ m γ η ,
(21)

where f ξ is an arbitrary function. In order to identify function f(ξ), we evaluate the vorticity from (21) and equate the expression with (17). Substituting the value of ψ from (21) into (17), we obtain

2 ψ = a c ξ + 2 ν b c μ m a μ m ν ν ̂ + f ξ exp a μ m ν ν ̂ c μ m γ η a c ξ , = 2 a b c μ m ν a μ m ν ν ̂ + a f ξ exp a μ m ν ν ̂ c μ m γ η .
(22)

On evaluating vorticity ∇2ψ using (21) and then substituting the expression into (22) we find

1 + 4 b 2 η 2 f ξ + 2 f ξ b 1 2 η a μ m ν ν ̂ c μ m γ = f ( ξ ) a μ m ν ν ̂ 2 c 2 μ m 2 γ 2 a 2 b c ν ̂ a μ m ν ν ̂ exp a μ m ν ν ̂ c μ m γ η .
(23)

In view of the independence of ξ and η, we note that either f ξ = 0 or b = 0, so that the following cases may arise.

When f′(ξ) = 0 in (23), the stream function and the velocity components are respectively given by

ψ ( x , y ) = c x + b y 2 + 2 b c ν μ m a μ m ν ν ̂ + A 0 exp a μ m ν ν ̂ y c μ m γ ,
(24)
u ( x , y ) = 2 b c y A 0 a μ m ν ν ̂ c μ m γ exp a μ m ν ν ̂ y c μ m γ ,
(25)
v ( x , y ) = c ,
(26)

where A0 is a non-zero arbitrary constant.

If b = 0, then f(ξ) = f(x) and satisfies the second order differential equation

d 2 f d x 2 ( x ) + f ( x ) a μ m ν ν ̂ 2 c 2 μ m 2 γ 2 a = 0 .
(27)

Note that the form of the solution to Equation (27) depends on the sign of the discriminant

a c 2 μ m 2 γ 2 a μ m ν ν ̂ 2 .

Therefore, three situations arise. For each case, we first determine the stream functions and then the corresponding velocity components.

  1. Positive discriminant: a c 2 μ m 2 γ 2 a μ m ν ν ̂ 2 > 0 .

    As the discriminant is positive, the characteristic equation for (27) possesses two distinct real roots
    r 1 = a c 2 μ m 2 γ 2 a μ m ν ν ̂ 2 c μ m γ ,
    (28)
    r 2 = a c 2 μ m 2 γ 2 a μ m ν ν ̂ 2 c μ m γ .
    (29)
    Therefore, f(x) can be given by
    f ( x ) = A 1 e r 1 x + A 2 e r 2 x ,
    where A1 and A2 are arbitrary constants. This allows us to write the stream function by virtue of (21) as
    ψ ( x , y ) = c x + A 1 e r 1 x + A 2 e r 2 x exp a μ m ν ν ̂ c μ m γ y ,
    (30)
    and subsequently the velocity components by (15) as
    u ( x , y ) = a μ ν ν ̂ y c a μ m γ A 1 e r 1 x + A 2 e r 2 x exp a μ m ν ν ̂ a c μ m γ y ,
    (31)
    v ( x , y ) = c a c 2 μ m 2 γ 2 a μ m ν ν ̂ 2 c μ m γ A 1 e r 1 x + A 2 e r 2 x exp a μ m ν ν ̂ a c μ m γ y , = c r 1 A 1 e r 1 x + A 2 e r 2 x exp a μ m ν ν ̂ a c μ m γ y ,
    (32)
    where r1 and r2 are given by (28) and (29).
  2. Null discriminant: a c 2 μ m 2 γ 2 a μ m ν ν ̂ 2 = 0 .

    When the discriminant is zero, the solution f(x) to (27) takes the form
    f ( x ) = B 1 + B 2 x ,
    where B1 and B2 are arbitrary constants. Thus, we have the stream function from (21) as
    ψ ( x , y ) = c x + B 1 + B 2 x exp a μ m ν ν ̂ c μ m γ y ,
    (33)
    and the velocity components from (15) are therefore,
    u ( x , y ) = a μ m ν ν ̂ c μ m γ B 1 + B 2 x exp a μ m ν ν ̂ c μ m γ y ,
    (34)
    v ( x , y ) = c B 2 exp a μ m ν ν ̂ c μ m γ y .
    (35)
  3. Negative discriminant: ( a c 2 μ m 2 γ 2 a μ m ν ν ̂ 2 < 0 . )

    Finally, when the discriminant is negative, the characteristic equation has two distinct and conjugate complex roots ±ir, where
    r = a c 2 μ m 2 γ 2 + a μ m ν ν ̂ 2 c μ m γ .
    (36)
    Then it follows that
    f ( x ) = C 1 cos ( r x ) + C 2 sin ( r x ) ,
    so that the stream function can be found as
    ψ ( x , y ) = c x + ( C 1 cos ( r x ) + C 2 sin ( r x ) ) exp a μ m ν ν ̂ c μ m γ y ,
    (37)
    whereas the velocity components are given by
    u ( x , y ) = a μ m ν ν ̂ c μ m γ ( C 1 cos ( r x ) + C 2 sin ( r x ) ) exp a μ m ν ν ̂ c μ m γ y ,
    (38)
    v ( x , y ) = c a c 2 μ m 2 γ 2 + a μ m ν ν ̂ 2 c μ m γ ( C 1 cos ( r x ) + C 2 sin ( r x ) ) × exp a μ m ν ν ̂ c μ m γ y , = c r ( C 1 cos ( r x ) + C 2 sin ( r x ) ) exp a μ m ν ν ̂ c μ m γ y .
    (39)
    Here C1 and C2 are arbitrary constants and r is as in (36).

In this section, we derive expressions for the unsteady velocity components, by considering a time dependent stream function in the case when b = 0, that is, we now deal with the following equation,

β ψ t c γ ψ y a μ m ν ν ̂ μ m ψ + ( a μ m ν ν ̂ ) c μ m x = 0 .
(40)

Remark 4.1.

Notice that several situations arise according to different choices for the parametersβ,γandc. Whenβ ≠ 0, we have following cases.

  1. (c ≠ 0, γ ≠ 0): We can transform (40) using a change of variables ξ = y + c γ β t , to
    ψ y + a μ m ν ν ̂ c μ m γ ψ = a μ m ν ν ̂ μ m γ x .
    (41)
  2. (c ≠ 0, γ = 0): For this choice of the parameters, (40) reduces to
    ψ t a μ m ν ν ̂ μ m β ψ = ( a μ m ν ν ̂ ) c μ m β x ,
    (42)
    which is somewhat similar to (41), with proper attention to the constant coefficients and a change of roles for independent variablesyandt.
  3. (c = 0): In this case, independently from the choice ofγ, Equation (40) reduces to
    ψ t a μ m ν ν ̂ μ m β ψ = 0 .
    (43)
    The Equation (43) also corresponds to (42) and thus to (41) with a specific choice ofc = 0.

Therefore, whenβ ≠ 0, it is sufficient to discuss the solution for (41) and the other cases follow immediately.

Whenβ = 0, (40) becomes identical to the steady state equation (20) except that nowψdepends on time variabletalso and therefore, the solution can be expressed in terms of an arbitrary functionf(t, ξ) as in (21). However, the arbitrary functionfnow satisfies a second order partial differential equation instead of an ordinary equation.

Remark 4.2.

For brevity, in rest of this contribution, we letβ,γandcbe non-zero and only discuss the solution of (41).

With β, γ and c non-zero, Equation (41) can be viewed as first order linear differential equation in y. Therefore, we can write its solution in the form

ψ = c x + g ( ξ , x ) exp a μ m ν ν ̂ c μ m γ y ,
(44)

where g(ξ, x) is an arbitrary function.

In order to fix the arbitrary function g, we proceed in the similar fashion as in steady case, that is, we use (44) to obtain ∇2ψ and then equate the expression with (17). Therefore, we obtain the following equation for g

2 g ξ 2 + 2 g x 2 2 a μ m ν ν ̂ c μ m γ g ξ + a μ m ν ν ̂ 2 c 2 μ m 2 γ 2 a g = 0 .
(45)

We now set

η = ξ cos θ + x sin θ and g ( ξ , x ) = G η , ( π θ π ) .
(46)

Consequently, from (45) we derive

d 2 G d η 2 2 a μ m ν ν ̂ c μ m γ cos θ d G d η + a μ m ν ν ̂ 2 c μ m 2 γ 2 a G = 0 ,
(47)

for which we look for a solution of the form

G ( η ) = exp m a μ m γ η .
(48)

If such a solution does exist, m must satisfy the characteristic equation for (47) given by

m 2 2 m a μ m ν ν ̂ cos θ + a μ m ν ν ̂ 2 a c 2 μ m 2 γ 2 = 0 ,
(49)

whereas the roots of this equation are

m ± = a μ m ν ν ̂ cos θ ± a c 2 μ m 2 γ 2 a μ m K ν ν ̂ 2 sin 2 θ ,
(50)

whose nature depends upon the discriminant

a c 2 μ m 2 γ 2 a μ m ν ν ̂ 2 sin 2 θ .

Therefore, following three cases arise.

  1. Positive discriminant: ( a c 2 μ m 2 γ 2 a μ m ν ν ̂ 2 sin 2 θ > 0 )

    If the discriminant of (49) is positive then it possesses two real and distinct roots. Thus, (47) has a solution of the form
    G η = D 1 exp m + c μ m γ η + D 2 exp m c μ m γ η ,
    where the roots m± are given by (50) and D1 and D2 are arbitrary constants.
    The corresponding stream function and velocity components, respectively, have the following forms:
    ψ ( x , y , t ) = c x + exp a μ m ν ν ̂ c μ m γ y × D 1 exp m + c μ m γ η + D 2 exp m c μ m γ η
    (51)
    u ( x , y , t ) = exp a μ m ν ν ̂ c μ m γ y [ a μ m ν ν ̂ c μ m γ m + c μ m γ cos θ D 1 exp m + c μ m γ η + a μ m ν ν ̂ c μ m γ m c μ m γ cos θ D 2 exp m η c μ m γ ] ,
    (52)
    v ( x , y , t ) = c exp a μ m ν ν ̂ c μ m γ y × [ D 1 m + c μ m γ exp m + c μ m γ η + D 2 m c μ m γ exp m c μ m γ η ] sin θ .
    (53)
    where η = ξcosθ + xsinθ with ξ = y + c γ β t .
  2. Null discriminant: ( a c 2 μ m 2 γ 2 a μ m ν ν ̂ 2 sin 2 θ = 0 )

    When the discriminant is zero, the characteristic equation (49) has two real and repeated roots m = m+ = m. Therefore the solution takes the form
    G η = E 1 + E 2 η exp a μ m ν ν ̂ c μ m γ cos θ η ,
    (54)
    where E1 and E2 are arbitrary constants. Thus, the expressions for the stream function and velocity components are given by
    ψ ( x , y , t ) = c x + E 1 + E 2 η exp a μ m ν ν ̂ c μ m γ cos θ η y ,
    (55)
    and
    u ( x , y , t ) = E 2 cos θ a μ m ν ν ̂ c μ m γ sin 2 θ E 1 + E 2 η × e x p a μ m ν ν ̂ c μ m γ cos θ η y ,
    (56)
    v ( x , y , t ) = c E 2 + a μ m ν ν ̂ c μ m γ cos θ E 1 + E 2 η sin θ × exp a μ m ν ν ̂ c μ m γ cos θ η y ,
    (57)
    respectively, where η = ξcosθ + xsinθ with ξ = y + c γ β t .
  3. Negative discriminant: ( a c 2 μ m 2 γ 2 a μ m ν ν ̂ 2 sin 2 θ < 0 )

    As with a negative discriminant the characteristic equation has two complex and conjugate roots, therefore the solution to (47) can be given by
    G η = exp a μ m ν ν ̂ cos θ c μ m γ η ( F 1 cos ( M η ) + F 2 sin ( M η ) ) ,
    (58)
    where F1 and F2 are arbitrary constants and
    M = a μ m ν ν ̂ 2 sin 2 θ a c 2 μ m 2 γ 2 c μ m γ .
    The stream function ψ x , y , t is
    ψ ( x , y , t ) = c x + exp a μ m ν ν ̂ c μ m γ cos θ η y ( F 1 cos ( M η ) + F 2 sin ( M η ) ) ,
    (59)
    and the velocity components are given by
    u ( x , y , t ) = { F 2 M cos θ F 1 a μ m ν ν ̂ c μ m γ sin 2 θ cos ( M η ) F 1 M cos θ + F 2 a μ m ν ν ̂ c μ m γ sin 2 θ sin ( M η ) } × exp a μ m ν ν ̂ c μ m γ cos θ η y ,
    (60)
    v ( x , y , t ) = sin θ { F 2 M + F 1 a μ m ν ν ̂ c μ m γ cos θ cos ( M η ) F 1 M F 2 a μ m ν ν ̂ c μ m γ cos θ sin ( M η ) } × exp a μ m ν ν ̂ c μ m γ cos θ η y c ,
    (61)
    where η = ξcosθ + xsinθ with ξ = y + c γ β t .

In this section, the influence of emerging flow parameters on the streamline flow pattern is delineated both in steady as well as unsteady case. Throughout this section, the electric conductivity σ and constant in the applied magnetic field B0 are initialized to unity.

In order to depict streamline flow patterns we fix integration constant A0 = 1, vorticity coefficients b = 10 and c = 5.

In Figure 1, variations in the profile of the stream function ψ, derived in Section III A are recorded with respect to dynamic viscosity ν, stress modulus α1, porosity ϕ and Hall parameter h. In Figures 1(a)-1(c) contour plots of ψ(x, y) versus y are provided keeping x fixed and varying ν, α1 and ϕ respectively. In Figure 1(d), surface plots of ψ(x, y) are provided by varying h.

FIG. 1.

ψ(x, y) versus y using stream function profile (24) when f′(ξ) = 0, and (a) with variations in dynamic viscosity ν, (b) with variations in second grade parameter α1, (c) with variations in the porosity ϕ, (d) Variations in the Hall parameter h.

FIG. 1.

ψ(x, y) versus y using stream function profile (24) when f′(ξ) = 0, and (a) with variations in dynamic viscosity ν, (b) with variations in second grade parameter α1, (c) with variations in the porosity ϕ, (d) Variations in the Hall parameter h.

Close modal

In Figure 1(a), we take Hall parameter h = 0.5, second grade modulus α1 = 0.1, magnetic permeability μm = 2, porosity ϕ = 0.9 and vorticity coefficient a = 5. With five different values of ν, streamline flow patterns are plotted. The variations in the flow patterns is evident with varying values of ν.

Figure 1(b) highlights the non-Newtonian behavior of the streamlines. Different values for the stress modulus α1 are assumed. We set a = 20 and ν = 0.01 and fix the other parameters as in Figure 1(b). The case of α1 = 0 relates to Newtonian fluids. It can be seen that the streamline patterns are greatly altered by varying α1 which indicates that the flow is not only dependent on the viscosity but also on the nature of elasticity of the fluid.

The effects of porosity and Hall parameters are depicted in Figures 1(c) and 1(d) respectively. We take ν = 0.05 for both graphs while μm = 2 in the first one whereas μm = 0.5 in the second one, keeping the other parameters fixed as before. The influence of Hall effect and the porosity are apparent in both results.

In this section, we present graphs for the streamlines in the case when b = 0 and the discriminant is positive, that is we consider the stream function given by (30) and contour plots are provided for simplicity. The arbitrary integration constants are fixed to A1 = 10 and A2 = 1 and the coefficients are taken as follows: a = 5, b = 0, c = 5 and μm = 2.

In Figure 2, variations in the profile of the steady state stream function ψ given by Equation (30) are recorded with respect to dynamic viscosity ν, stress modulus α1, porosity ϕ and Hall parameter h as in the previous section.

FIG. 2.

ψ(x, y) versus y using stream function profile (30) when b = 0, and (a) with variations in dynamic viscosity ν, (b) with variations in second grade parameter α1, (c) with variations in the porosity ϕ, (d) with variations in the Hall parameter h.

FIG. 2.

ψ(x, y) versus y using stream function profile (30) when b = 0, and (a) with variations in dynamic viscosity ν, (b) with variations in second grade parameter α1, (c) with variations in the porosity ϕ, (d) with variations in the Hall parameter h.

Close modal

In Figure 2(a), we take h = 0.5, α1 = 0.1 and ϕ = 0.6 whereas the values of ν are varied. In Figure 2(b)ν is taken to be 0.1 and the values of α1 are varied. In Figure 2(c) different values of ϕ are taken while α1 = 0.05. Finally, in Figure 2(d) plots are given for different values of h taking ϕ = 0.8 and α = 0.05. The rest of the parameters are fixed for all figures. The influence of the material parameters on the flow patterns is significant.

In this section, we draw graphs for the unsteady streamlines in the case when the discriminant is negative using stream function given by (59). The integration constants are fixed to F1 = F2 = 1 and the coefficients are taken as follows: a = 5, b = 10 and μm = 2. In Figures 3 and 4, θ = π/4, whereas in Figure 5, it is π/8. For brevity, we take h = 0 throughout this section.

FIG. 3.

ψ(x, y, t) versus y when discriminant is negative and (a) t = 12, (b) t = 10, (c) t = 8.

FIG. 3.

ψ(x, y, t) versus y when discriminant is negative and (a) t = 12, (b) t = 10, (c) t = 8.

Close modal
FIG. 4.

ψ(x, y, t) versus y with negative discriminant and (a) η = 0.001, (b) η = 0.003, (c) η = 0.005.

FIG. 4.

ψ(x, y, t) versus y with negative discriminant and (a) η = 0.001, (b) η = 0.003, (c) η = 0.005.

Close modal
FIG. 5.

ψ(x, y, t) versus y by varying second grade modulus α1 when the discriminant is negative.

FIG. 5.

ψ(x, y, t) versus y by varying second grade modulus α1 when the discriminant is negative.

Close modal

In Figure 3, temporal profile of the stream function ψ, given by Equation (59), is recorded at three different times: t = 8, t = 10 and t = 12. The other parameters are as follows: ν = 1, α1 = 0.05, ϕ = 0.8, c = 1 and μm = 2.

The influence of η on ψ(x, y, t) is depicted in Figure 4, where we take α = 0.05, c = 0.05, t = 8 and μm = 0.05. Other parameters are as for Figure 3.

Finally, in Figure 5, fixing t = 10, μm = 2 and ν = 1, graphs for the stream function ψ(x, y, t) for three different values of α1 are provided.

In this paper, rheological properties of a second grade fluid in a porous medium by solving a two dimensional nonlinear partial differential equation were presented. It is observed that expressions for the stream function and the velocity components, in both steady and unsteady flow problems, are significantly altered due to the presence of the material parameters of the second grade fluid, porosity and the Hall current effects. It is worthwhile mentioning that in the absence of Hall current effects and porosity in the medium, already known results can be recovered. The Newtonian flow field can be recovered by setting second grade parameters equal to zero. In order to observe the influence of certain material parameters on the flow, various streamline flow patterns were also delineated.

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