Lie group analysis and similarity solutions for hydro-magnetic Maxwell fluid through a porous medium
© Mekheimer et al; licensee Springer. 2012
Received: 20 August 2011
Accepted: 13 February 2012
Published: 13 February 2012
The equations of two dimensional incompressible fluid flow for hydro-magnetic Maxwell fluid through a porous medium have been studied. Lie group analysis has been employed and the group invariant solutions are obtained. Solutions corresponding to translational and rotational symmetries are obtained. A boundary value problem for the translational symmetry is investigated and the results are also sketched graphically. The effects of physical parameters have been noticed.
MSC 2011: 53C11; 76S05.
Non-Newtonian fluid behavior, which is characterized by a nonlinear viscosity dependence on the strain, can be observed in many complex fluids, for example, polymers, dense colloidal dispersions, surfactant solutions, micellar solutions chemical, and petroleum industries . In addition to shear-thinning and shear-thickening behavior, a dynamic or even chaotic response can be found in some fluids subjected to a steady shear flow. Because of the difficulty to suggest a single model which exhibits all properties of non-Newtonian fluids, they cannot be described as simply as Newtonian fluids. Due to this fact many models of constitutive equations have been proposed and most of them are empirical or semi empirical . Amongst these the differential type fluid model gained considerable attention of many researchers. The flows of non-Newtonian fluids are not only important because of their technological significance but also in the interesting mathematical features presented by the equations governing the flow. However on the other hand there are much controversies on these models as well. Such fluids are also inadequate to describe the relaxation phenomena. For a complete and detailed discussion of the relevant issues for differential type fluids, we refer the readers to Dunn and Rajagopal  and Aksel .
The non-Newtonian fluids are mainly classified into three types namely differential, rate and integral. The simplest subclass of the rate type fluids is the Maxwell model . This fluid model can very well describe the relaxation time effects. Specifically the Maxwell fluid model has been used for the viscoelastic flows where the dimensionless relaxation time is small. However in some more concentrated polymeric fluids the Maxwell model is also useful for large dimensionless relaxation time. Some recent investigations dealing with the flows of Maxwell fluids are given in the references [6–9].
Modified Darcy's law for a Maxwell fluid including the Hall current has been used for the modeling. In fact, the Hall effect is important when the Hall parameter, which is the ratio between the electron-cyclotron frequency and the electronatom-collision frequency, is high. This happens, when the magnetic field is high or when the collision frequency is low. In most cases, the Hall term has been ignored in applying Ohm's law as it has no marked effects for small and moderate values of the magnetic field. However, the current trend in the application of magnetohydrodynamics is towards a strong magnetic field, so that the influence of electromagnetic force is noticeable. Under these conditions, the Hall current is important and it has marked effects on the magnitude and direction of the current density and consequently on the magnetic-force term. Therefore, it is of interest to study the influence of the Hall current on the flow.
In the Earth there are a large number of problems that can be described by the interaction of a low viscosity fluid (water, oil, gas, magma) in a permeable (possibly deformable) matrix. Darcy's Law is the classic, empirically derived equation for the flux of a low viscosity fluid in a permeable matrix. This equation assumes that flow in the pores or cracks of the medium is essentially laminar and provides the average flux through a representative area that is larger than the pore scale and smaller than the scale of significant permeability variation (if such a scale exists). Various approaches have been used to justify this rule from first principles (see e.g., Dagan ) but it generally seems to work.
In this article, we apply the so-called symmetry methods for a particular problem of fluid mechanics. The main advantage of such methods is that they can successfully be applied to nonlinear differential equations [11–13]. The similarity solutions are quite popular because they result in the reduction of the independent variables of the problem. The symmetry transformations method transform the given family of equations of n independent variables, say, to another family of equations of n - 1 independent variables, which can further be solved [14, 15]. The fundamental concepts of this approach can be found in [16–19]. In our case, the problem under investigation is (2 + 1)-nonlinear partial differential equations (PDEs). Hence, any similarity solution will transform the system of (n + 1)-nonlinear PDEs into a system of (n)-nonlinear PDEs and any similarity solution will transform the system of (2)-nonlinear PDEs into a system of ordinary differential equations (ODEs).
Many authors used Lie group analysis method to obtain the exact solutions for some problems in fluid mechanics. Yurusoy and Pakdemirli  investigated the boundary layer equations of a non-Newtonian fluid model in which the shear stress is an arbitrary function of the velocity gradient. Yurusoy et al.  have obtained the solution for the creeping flow of the second grade fluid. Also the two-dimensional equations of motions for the slowly flowing and heat transfer in second grade fluid in cartesian coordinates neglecting the inertial terms are considered by Yürüsoy . Shahzad et al.  found the analytical solution of a micropolar fluid by using the Lie group analysis. Recently, Mekheimer et al. studied the Lie group analysis and similarity solutions for a couple stress fluid with heat transfer , Lie point symmetries and similarity solutions for an electrically conducting Jeffrey fluid  and Lie point symmetries and similarity solution for a micro-polar fluid through a porous medium .
From discussion above, we attend to find the analytical (similarity) solutions for the flow problem of an incompressible hydro-magnetic Maxwell fluid through a porous medium using Lie group analysis. The problem is presented as follows, in Section 2, the equations governing two-dimensional motion of an incompressible, MHD Maxwell fluid are introduced. In Section 3, the basic idea of the Lie group analysis method are given and used to find the isovector field of our equations. The similarity solutions corresponding to translational and rotational symmetry are obtained in Sections 3.1 and 3.2. Also a boundary value problem for the similarity solutions corresponding to translational symmetry are obtained in Section 4. The graphs for a boundary value problem (magma flow) are plotted and discussed in Section 5. Finally a concluding remarks are pointed in Section 6.
2 Equations of motion
where are the fluid velocities in the directions, is the pressure, and is the time. Here , ρ, μ, φ, k, e, n e , σ, B0, and m are the relaxation time, density, coefficient of viscosity, porosity of the porous medium, permeability, electric charge, the number density of electrons, electrical conductivity of the fluid, magnetic field and Hall parameter respectively.
where is the Reynolds number, is the Hartmann number and L, U are the dimensionless length and velocity, respectively.
3 Lie group analysis and isovector fields
where D ij = D i (D j ) = D j (D i ) = D ji and .
where a i , i = 1,..., 5 are arbitrary constants, δ(t) is arbitrary function of the variable t only.
3.1 Translational symmetry
where c4 and c5 are arbitrary constants.
3.2 Rotational symmetry
where , and G3 are functions of ψ and t.
where ψ and ϕ are the same in (29).
4 Solutions for hydro-magnetic Maxwell fluid through a porous medium: (magmatic fluid) problem
where W = m0m1 - m2, α1 and α2 are the negative roots of Eq. (24).
5 Discussion of the magmatic fluid problem
This section deals with the graphics on the magmatic fluid. So, the interpretation of the relaxation time λ, Reynolds number R, Hartmann number M, Hall parameter m, the time parameter t, and the permeability parameter k have been studied on the pressure p, and the x and y components of the velocity distributions u and v.
Figures 1 and 2 show that as the permeability parameter k increases the horizontal velocity component u increases, while the vertical velocity component v decreases. Figures 3 and 4 illustrate the variation of the velocity components u and v with the Hall parameter m, which indicate that for small values of t (or at initial values of t) the curves are the same with no obvious different which for t > 2, the gap between the curves appears. Also, we can see that curves with small values of m (m = 0, 0.5) are vanishing rabidly than those for (m = 1,1.5) i.e., as the Hall parameter m increases the disturbance of the velocity components increase. (decreasing the number of density electrons or the electronic charges).
Figures 5 and 6 illustrate the variations of u and v with t for different values of the relaxation time λ, which show that for small values λ the disturbance in u and v will vanish rapidly than those as λ increases. Also, the figures show that the disturbance in u and v for a Newtonian fluid less than those for a Non-Newtonian fluid in the case of magma flow.
Figures 7 and 8 show that the variation with the Reynolds number R. As R increases the velocity components u and v increase. Figures 9 and 10 show that as the Hartmann number M increases the velocity components u and v decrease, i.e., the fluid moves as a block and takes a constant value for large values of M. Figures 11 and 12 illustrate the variation of u and v with t for different values of the y axis, which show that the velocity components take the initial values of the magma plate at y = 0 and the velocity components decreases as y increases. Figures 13 and 14 describe the variations of u and v with t for different values of U0 and V0 (velocities of the magma plate), the figures show that the gab between the curves decreases with time and finally vanishes and for certain values of t the velocity components u and v equal to zero. Also, the magnitudes of u and v increase with increasing U0 and V0.
Other cases of symmetry will be considered for other boundary value problems else where for other applications.
6 Concluding remarks
The significant features of Lie group analysis for hydro-magnetic Maxwell fluid through a porous medium have been presented. Similarities solutions are obtained and applied to an important phenomena in geology, which is the magmatic fluid. The main points have been summarized as follows:
As the Hall parameter m increases the disturbances of the velocity components are increase.
The disturbances in the fluid velocity components for a Newtonian fluid are less than those for a non-Newtonian fluid (magmatic fluid)
The magmatic fluid moves as a block for large values of the Hartmann number M.
The pressure near to the magma plate is higher for a magneto-magma flow than that for a magma flow without a magnetic field. Also, this pressure for a porous medium is less than that for a medium with high permeability.
The pressure increase near to the magma plate and take the constant value of the pressure deep in the magmatic fluid for large values of y.
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