© Hindawi Publishing Corp. NONLINEAR PERISTALTIC TRANSPORT OF MHD FLOW THROUGH A POROUS MEDIUM

In order to determine the characteristics of peristaltic transport of magnetohydrodynamic flow through a porous medium, the motion of a hydromagnetic (electrically conducting), viscous, and incompressible fluid in planer channel filled with a homogeneous porous medium and having electrically insulated walls that are transversely displaced by an infinite, harmonic travelling wave of large wavelength was analyzed using a perturbation expansion in terms of a variant wave number. We obtain an explicit form for the velocity field, a relation between the pressure rise and flow rate, in terms of Reynolds number, wave number, Hartmann number, permeability parameter, and the occlusion. The effects of all parameters of the problem are numerically discussed and graphically explained. 2000 Mathematics Subject Classification: 76S05. 1. Introduction. Peristalsis is now well known to the physiologists as one of the major mechanisms for fluid transport in many biological systems. In particular, peristaltic mechanism may be involved in swallowing food through the oesophagus, urine transport from kidney to bladder through the ureter, movement of chyme in the gastrointestinal tract, transport of spermatozoa in the ductus efferents of the male reproductive tracts and in the cervical canal, movement of the ovum in the fallopian tubes, and in the vasomotion of small blood vessels as well as blood flow in arteries. In addition, peristaltic pumping occurs in many practical applications involving biomechanical systems. Also, finger and roller pumps are frequently used for pumps corrosive or very pure materials so as to prevent direct contact of the fluid with the pump’s internal surfaces. A number of analytical [3, 5, 7, 9, 11, 12, 16, 23], numerical, and experimental [2, 10, 19, 20, 21] studies of peristaltic flow of different fluids have been reported. Several review articles have been written [8, 14]. Also a summary of analytical papers up to 1984 has been presented in [18]. Most of the analytical studies use perturbation series in a small parameter such as Reynolds number or a dimensionless wave number, which, unfortunately, limits the range of validity of the results. However, a perturbation method does provide explicit information about the physical effects of that parameter. Also, the analytical results can be used to check the calculations of wider-range numerical methods.


Introduction.
Peristalsis is now well known to the physiologists as one of the major mechanisms for fluid transport in many biological systems.In particular, peristaltic mechanism may be involved in swallowing food through the oesophagus, urine transport from kidney to bladder through the ureter, movement of chyme in the gastrointestinal tract, transport of spermatozoa in the ductus efferents of the male reproductive tracts and in the cervical canal, movement of the ovum in the fallopian tubes, and in the vasomotion of small blood vessels as well as blood flow in arteries.In addition, peristaltic pumping occurs in many practical applications involving biomechanical systems.Also, finger and roller pumps are frequently used for pumps corrosive or very pure materials so as to prevent direct contact of the fluid with the pump's internal surfaces.A number of analytical [3,5,7,9,11,12,16,23], numerical, and experimental [2,10,19,20,21] studies of peristaltic flow of different fluids have been reported.Several review articles have been written [8,14].Also a summary of analytical papers up to 1984 has been presented in [18].Most of the analytical studies use perturbation series in a small parameter such as Reynolds number or a dimensionless wave number, which, unfortunately, limits the range of validity of the results.However, a perturbation method does provide explicit information about the physical effects of that parameter.Also, the analytical results can be used to check the calculations of wider-range numerical methods.
It has been established that the biological systems, in general, are greatly addicted by the application of the external magnetic field.Moreover, the MHD flow of a fluid in a channel with elastic rhythmically contracting walls (peristaltic flow) is of interest in connection with certain problems of the movement of conductive physiological fluids, and with the need for theoretical research on the operation of a peristaltic MHD compressor, also the principle of magnetic field may be used in clinical application (magnetic resonance imaging MRI).The effect of moving magnetic field on blood flow was studied by Agrawal and Anwaruddin [1], they observed, for the flow of blood in arteries with arterial disease like arterial stenosis or arteriosclerosis, that the influence of magnetic field may be utilized as a blood pump in carrying out cardiac operations.Also, the magnetohydrodynamic flow with suspension has been studied by Parsad and Ramacharyulu [6], and Srivastava and Agrawal [17] considered the blood as an electrically conducting fluid and that it constitutes a suspension of red cells in plasma.
Flow through a porous medium has been studied by a number of workers employing Darcy's law [15].Some studies about this point have been made by Varshney [22], Raptis and Perdikis [13], and El-Dabe and El-Mohendis [4].
Here, we are interested in the nonlinear peristaltic pumping of MHD flow through a porous medium, and due to the complexity of the nonlinear equations of motion, we only consider the case: a symmetric, harmonic, infinite wave train having a wavelength that is large relative to the gap between the walls; transverse displacement only; and electrically conducting fluid.This problem may be considered as a mathematical representation to the case of gall bladder and bile duct with stones under a uniform magnetic field.The gall stones cause fibrosis of the gall bladder, thus when a stone is later impacted in the common bile duct, jaundice results and gall bladder cannot dilate as it fibrosed as a result of the cholecystitis due to stones.
A regular perturbation series is used to solve the problem; variables are expanded in a power series of the parameter δ, which is defined as the ratio of half width of the channel to the wavelength of the peristaltic wave.Closed form solutions up to order δ 2 are presented.

Formulation of the problem.
Consider the two-dimensional unsteady hydromagnetic flow of a viscous, incompressible, and electrically conducting fluid in an infinite channel having width 2a and filled with a homogeneous porous medium.A uniform magnetic filed B 0 is acting along the Ȳ -axis and the induced magnetic field is assumed to be negligible.We assume an infinite wave train travelling with velocity c along an electrically insulated walls.We choose a rectangular coordinate system for the channel with X along the center line in the direction of wave propagation, and Ȳ transverse to it.The geometry of the wall surface is defined as where b is the wave amplitude and λ is the wavelength.We carry out this investigation in a coordinate system moving with the wave speed, in which the boundary shape is stationary.The coordinates and velocities in the laboratory frame ( X, Ȳ ) and the wave frame ( x, ȳ) are related by where Ū , V and ū, v are the velocities components in the corresponding coordinate systems.Using these transformations and introducing the following dimensionless variables: we find that the equation which governs the MHD flow in terms of the stream function ψ(x, y) after eliminating the pressure gradient is where and the dimensionless parameters: Reynolds number Re = caρ/µ, wave number δ = 2πa/λ, Hartmann number M = σ /µB o a (suitably greater than √ 2), and permeability parameter K = k/a 2 , where ρ is the density, µ the viscosity of the fluid, σ electrical conductivity of the fluid, and k the permeability of the porous medium.

Rate of volume flow and boundary conditions. The instantaneous volume flow rate in the fixed frame is given by
where h is a function of X and t.
The rate of volume flow in the wave frame is given by where h is a function of x alone.If we substitute (2.2) into (3.1) and make use of (3.2), we find that the two rates of volume flow are related through 3) The time-mean flow over a period T at a fixed position X is defined as Substituting (3.3) into (3.4) and integrating, we get On defining the dimensionless time-mean flows θ and F , respectively, in the fixed and wave frame as one finds that (3.5) may be written as where We note that h represents the dimensionless form of the surface of the peristaltic wall: where is the amplitude ratio or the occlusion.
If we select the zero value of the streamline at the streamline (y = 0): then the wall (y = h) is a streamline of value The boundary conditions for the dimensionless stream function in the wave frame are

Perturbation solution.
In order to solve the present problem, we expand the flow quantities in a power series of the small parameter δ as follows: On substituting (4.1) into (2.4) and (3.13), collecting terms of equal powers of δ, and then equating the coefficients of like powers on both sides of the equations, we obtain the following set of problems.

Zero-order problem.
We need to solve subject to the boundary conditions: where The solution of (4.2) in terms of stream function is given by We point out that this problem is essentially the classical Poiseuille MHD flow.
It can be easily shown that [9] lim

First-order problem.
On substituting the zero-order solution (4.4) into the equation of motion obtained for first order in δ, one finds that the latter equation reduces to The solution of (4.6) subject to the first-order boundary conditions: is given as where the constants c 11 , c 12 , c 13 , and D are listed in Appendix A.

Second-order problem.
As before, we insert the zero-and first-order solutions into the equations of motion obtained for δ 2 and find that Using the zero-order and the first-order solutions in (4.9) and then applying the boundary conditions: we find that the stream function ψ 2 turns out to be (4.12) 5. Pressure gradient.When the flow is steady in the wave frame, one can characterize the pumping performance by means of the pressure rise per wavelength.On substituting (4.1) into the dimensionless equation of motion and equating the coefficient of like powers of δ on both sides of the equation, we obtain a set of partial differential equations for ∂p 0 /∂x, ∂p 1 /∂x, and ∂p 2 /∂x.
The nondimensional pressure rise and the nondimensional friction force per wavelength in the wave frame are defined, respectively, as Since ∂p/∂x is periodic in x, the pressure rise and the friction force per wavelength in the longitudinal direction are independent of y [11], and the integrals in (5.1) can be evaluated numerically on the axis at y = 0. Now, and we compute the pressure rise and friction force per wavelength at each order for a wall shape of the sinusoidal form defined by (3.9).Using the zero-, first-, and second-order terms for the pressure gradient in (5.1), we obtain where p (2) λ , and F (2) are the pressure rise, the friction force, and the flow rate, respectively, in the wave frame to the second order in δ, p (2) (5.4) We note that the relation between the dimensionless flow rate in the frame F (2)  and that in the time-mean flow rate in the lab frame θ (2) is given by θ (2) = F (2) + 1. (5.5) We have here the second order of the pressure rise as a nonlinear function of the time-mean flow.For completeness, we give the second-order expression of the stream function ψ (2) in terms of F (2) , ψ (2) = F (2) + (1/γ) tanh γh y h − (1/γ) tanh γh − F (2) + h sinh γy γ cosh γh h − (1/γ) tanh γh (5.6) 6. Discussion of results.The results of our analysis are presented as follows: (1) pumping characteristics; (2) the streamlines and trapping regions for the parameters {Re,ϕ,δ,γ, θ (2) }.
We define the maximum pressure rise ( p (2) λ ) max , which is obtained by putting θ (2) = 0 and would occur in a channel, whose exit was shut off, as ( p (2) θ=0 ). Figure 6.4(a) is a graph of ( p (2) λ ) max versus K for varying values of M at δ = 0.06, ϕ = 0.2, and Re = 1, which shows the effects of M and K on the pumping rate, and we observe that for the same ( p (2) λ ) max the affect of magnetic field increases as permeability parameter increases.Figure 6.4(b) is a graph of the maximum pressure rise ( p (2) λ ) max versus the amplitude ratio ϕ for different values of K at δ = 0.06, M = 2.5, and Re = 1.It is clear that the maximum pressure rise increases as ϕ and M increase and decrease as K increases.

Streamlines and fluid trapping.
The phenomenon of trapping, whereby a bolus (defined as a volume of fluid bounded by a closed streamlines in the wave frame) is transported at the wave speed, has been studied by several investigators (Shapiro et al. [9], Jaffrin [7], and Siddiqui and Schwarz [16]).Figures 6.5(a), 6.5(b), and 6.5(c) are graphs of the streamlines for the conditions {ϕ = 0.4,δ = 0.06,M = 1.5,K = 1, Re = 0}, and θ (2) = 0.4, 1.2, 1.8, respectively.In Figure 6.5(a) the case of typical Stokes nontrapping region for peristaltic pumping, the adverse pressure gradient ( p (2) λ ) > 0 is almost large enough to cause zero pumping.While Figure 6.5(b) shows the center line trapped eddy, which was described by Jaffrin [7] and Siddiqui [16], the width of the eddy (measured on the centerline) increases and the eddy lifts off the center line and forms a torus-shaped eddy with flow through the center, in case of increasing θ (2) .We also observe that a smaller trapped bolus in the crest of the peristaltic wave (see Figure 6.5(c)), and as θ 2 becomes large, the flow has the same form as the flow through a two-dimensional symmetric channel, having an axial sinusoidal variation of its width.
The effect of the Hartmann number M on the peristaltic velocity is demonstrated in Figures 6.6(a), 6.6(b), and 6.6(c) for {ϕ = 0.4,δ = 0.06, Re = 1,θ (2) = 0.8,K = 1,M = 2, 5, 7}, which shows that as the M increases the center line trapped eddy disappears and the fluid moves like a block, and as M becomes larger the rigidity of the fluid appears.The effect of the permeability parameter on the peristaltic velocity is demonstrated in Figures 6.7(a), 6.7(b), and 6.7(c) for {ϕ = 0.4,δ = 0.06, Re = 1.0,θ (2) = 0.8,M = 2,K = 0.02, 0.08, 0.15}, which shows that as the permeability parameter increases the center line trapped eddy appears.Small values of K shows some sort of rigidity of fluid.

Appendices
A. Constants of the first-order solution in δ in terms of F (2)   c 01 = F (2)

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning δ c 11 y 2 sinh γy + c 12 y cosh γy + c 13 sinh γy + Dy + δ 2 c 34 y 4 sinh γy + c 35 y 3 cosh γy + c 36 y 2 sinh γy + c 37 y cosh γy + c 38 sinh γy + c 39 y cosh 2γy + c 40 sinh 2γy − c 29 γ 2 y 3 + 6y γ 2 − A γ 2 y + E sinh γy .

Figure 6 .
2(a) is a graph of the pressure change per wavelength ( p(2)

Figure 6 .
Figure 6.3(a) is a graph of the pressure change per wavelength ( p (2)

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation