Joseph used the Darcy law to describe **flow** in the **porous** **medium** [4]. They circumvented the formal mathematical treatment of the fluid momentum transport across the interface by constructing an empirical so-called hydrodynamic boundary condition [4]. Further steps were made when non- Darcian effects in the composite **medium** were accounted for by using the Brinkman-Forchheimer- extended Darcy equation in the **porous** **medium** [5]. The boundary condition at the interface with clear fluid was studied by Ochoa-Tapia and Whitaker [6,7], who were able to match the Brinkman-extended Darcy law to the Stokes equations by requiring discontinuity in the stress but retaining continuity in velocity using a sophisticated averaging procedure [6,7]. The boundary condition was termed the stress jump boundary condition and was further utilized by Kuznetzov [8] for derivation of the governing **flow** equations for fluid **flow** in the channel bound to a cylindrical **porous** **medium**.

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A lot of investigations have been done in order to understand the mechanisms of the transport of particulate suspension **flow** through **porous** **medium**. In general, Deep Bed Filtration studies have been conducted to analyse the mechanism involved in the processes of capturing and retaining particles occurs throughout the entire depth of the filter and not just on the filter surface. In this study, the deep bed filtration mechanism and the several mechanisms for the capture of sus- pended particles are explained then the size exclusion mechanism has been focused (particle capture from the suspen- sion by the rock by the size exclusion). The effects of particle flux reduction and pore space inaccessibility due to selec- tive **flow** of different size particles will be included in the model for deep bed filtration. The equations for particle and pore size distributions have been derived. The model proposed is a generalization of stochastic Sharma-Yortsos equa- tions. Analytical solution for low concentration is obtained for any particle and pore size distributions. As we will see, the averaged macro scale solutions significantly differ from the classical deep bed filtration model.

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Two-dimensional, nonlinear, steady, MHD laminar boundary layer **flow** with heat and mass transfer of a viscous, incompressible and electrically conducting fluid over a **porous** surface embedded in a **porous** **medium** in the presence of a transverse magnetic field including viscous and Joules dissipation is considered for investigation. An uniform transverse magnetic field of strength B 0 is applied parallel to y-axis. Consider a polymer sheet emerging out of a slit at x = 0 , y = 0 and subsequently being stretched, as in a polymer extrusion process. Let us assume that the speed at a point in the plate is proportional to the power of its distance from the slit and the boundary layer approximations are applicable. In writing the following equations, it is assumed that the induced magnetic field, the external electric field and the electric field due to the polarization of charges are negligible. Under these conditions, the governing boundary layer equations of momentum, energy and diffusion with visc- ous and Joules dissipation are

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6. Tasawar H, Meraj M, Ioan P. Heat and mass transfer for Dufour and Soret effect on mixed convection boundary layer **flow** over a stretching vertical surface in a **porous** **medium** filled with a viscoelastic fluid. Commun Nonlinear Sci Numer Simulat 2010; 15:1183-1196. 7. Nazibuddin A, Dhruba PB, Himanshu K. Unsteady MHD free convective **flow** past a vertical

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Abstract: The main theme of the present examined the influence of heat transfer on magnetohydrodynamics (MHD) for the oscillatory **flow** of Williamson fluid with variable viscosity model for two kinds of geometries "Poiseuille **flow** and Couette **flow**" through a **porous** **medium** channel. The momentum equation for the problem, is a non-linear differential equations, has been found by using "perturbation technique" and intend to calculate the solution for the small number of Weissenberg (We <<1) to get clear forms for the velocity field by assisting the (MATHEMATICA) program to obtain the numerical results and illustrations. The physical features of Darcy number, Reynolds number, Peclet number, magnetic parameter, Grashof number and radiation parameter are discussed simultaneously through presenting graphical discussion. Investigated through graphs the variation of a velocity profile for various pertinent parameters. While the velocity behaves strangely under the influence of the Brownian motion parameter and local nanoparticle Grashof number effect. On the basis of this study, it is found that the velocity directly with Grashof number, Darcy number, radiation parameter, Reynolds number and Peclet number, and reverse variation with magnetic parameter and frequency of the oscillation and discussed the solving problems through graphs.

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From the technological point of view, MHD free- convection flows have also great significance for the applications in the fields of stellar and planetary magnetospheres, aeronautics, chemical engineering, and electronics. The effects of magnetic field on free convection **flow** of electrically conducting fluids past through a **porous** **medium** has been studied many authors. Ahmed [14] looked the effects of unsteady free convective MHD **flow** through a **porous** **medium** bounded by an infinite vertical **porous** plate. Chaudhary and Arpita Jain [15] have discussed the MHD heat and mass diffusion **flow** by natural convection past a surface embedded in a **porous** **medium**. Kim [16] studied unsteady MHD convection **flow** of polar fluids past a semi-infinite vertical-moving **porous** plate in a **porous** **medium**. Soundalgekar [17] obtained approximate solutions for the two-dimensional **flow** of an incompressible, viscous fluid past an infinite **porous** vertical plate with constant suction velocity normal to the plate, the difference between the temperature of the plate and the free stream is moderately large causing the free convection currents. Raptis [18] studied mathematically the case of unsteady two-dimensional natural convective heat transfer of an incompressible, electrically conducting viscous fluid via a highly **porous** **medium** bound by an infinite vertical **porous** plate. The effects of magnetic field on free convection **flow** of electrically conducting fluids past a plate has been studied by many authors such as Soundalgekar [19], Singh et al.[20].

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Magneto hydrodynamic free convection **flow** past an infinite vertical plate oscillating in its own plane was first studied by Soundalgekar [1] in case of an isothermal plate. Mansour [2] has studied the interaction of free convection with thermal radiation of the oscillatory **flow** past a vertical plate. Soundalgekar and Takhar [3] have considered radiation effects on free convection **flow** past a semi-infinite vertical plate. Helmy [4] has investi- gated MHD unsteady free convective **flow** past a vertical **porous** plate. Hossain et al. [5] have analyzed the heat transfer response of MHD free convective **flow** along a vertical plate to surface temperature oscillations. Radia- tion and free convection **flow** past a moving plate was considered by Raptis and Perdikis [6]. Hossain et al. [7] have described the effect of radiation on free convection from a **porous** vertical plate. Kim [8] has founded an unsteady MHD convective heat transfer past a semi-infinite vertical **porous** moving plate with variable suction. Radiation effects on the free convection over a vertical flat plate embedded in **porous** **medium** with high porosity have been studied by Hossain and Pop [9]. El-Arabawy [10] studied the effect of suction/injection on a micro polar fluid past a continuously moving plate in the presence of radiation. The effects of thermal radiation on the **flow** past an oscillating plate with variable temperature have been studied by Pathak et al. [11]. Chandrakala and Raj [12] have studied the effects of thermal radiation on the **flow** past a semi infinite vertical isothermal plate with uniform heat flux in the presence of transversely applied magnetic field. Das [13] has analyzed the exact solution of MHD free convection **flow** and mass transfer near a moving vertical plate in the presence of thermal radiation. Chandrakala [14] have studied the radiation effects on **flow** past an impulsively started vertical oscil- lating plate with uniform heat flux. Ibrahim and Makinde [15] have studied the radiation effect on chemically reacting MHD boundary layer **flow** of heat and mass transfer through a **porous** vertical flat plate. Radiation ef- fect on natural convection near a vertical plate embedded in **porous** **medium** with ramped wall temperature has been studied by Das et al. [16]. Chandrakala [17] studied the effects of thermal radiation on the **flow** past an in- finite vertical oscillating plate with uniform heat flux. Janaand Manna [18] studied the effects of radiation on unsteady MHD free convective **flow** past an oscillating vertical **porous** plate embedded in a **porous** **medium** with oscillatory heat flux. Vidyasagar and Ramana [19] studied the radiation effect on MHD free convection **flow** of Kuvshinshiki fluid with mass transfer past a vertical **porous** plate through **porous** **medium**. Radiation effects on mass transfer **flow** through a highly **porous** **medium** with heat generation and chemical reaction has studied by Mohammed Ibrahim [20].

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The temperature conditions near the wall play an important role in several industrial applications and it was extensively studied by a number of researchers using different sets of thermal conditions at the boundary plate. Chandra et al. [10] studied the effect of ramped wall temperature on unsteady free natural convective **flow** of incompressible viscous fluid near a vertical plate. The dimensionless governing equations have been solved by using Laplace transform method. Then, Rajesh [11] also studied the unsteady free convection incompressible viscous fluid in vertical plate with ramped wall temperature but in presence of thermal radiation and MHD effects. The author used Laplace transform method to find the analytical solution. Deka et al. [12] has produced the exact solution for unsteady natural convection **flow** past an infinite vertical plate with ramped wall temperature passing through a **porous** **medium**. Laplace transform also has been used to solve the dimensionless governing equations of the fluid flows.

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Transport of suspensions and emulsions in **porous** media occurs in numerous processes of environmental, chemical, petroleum and civil engineering. In this work, a mass balance particle transport equation which includes filtration has been developed. The steady-state transport equation is presented and the solution to the complete advective-dispersion equation for particulate suspension **flow** has been derived for the case of a constant filter coefficient. This model in- cludes transport parameters which are particle advective velocity, particle longitudinal dispersion coefficient and filter coefficient. This work recommends to be investigated by particle longitudinal dispersion calculation from experimental data, directly. Besides, the numerical model needs to be developed for general case of a transition filter coefficient. Keywords: Particulate Suspension **Flow**; Filtration Theory; Filtration Coefficient; Transport; **Porous** **Medium**;

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The study of **flow** of Non-Newtonian fluid has gained considerable importance in various disciplines in recent years, several researchers like Mishra and Roy (1967), Mishra and Achary (1972), Wang (1971), Radhakrishnamacharya and Maiti (1977), Bhatnagar (1979), have studied visco-elastic **flow** problems. Rockwell et al. assumed that blood is visco-elastic fluid of small elasticity whose circulations in artery can be well explained by suitable visco-elastic fluid model. Esmond and Clark (1966) have shown that the pulsatile visco- elastic **flow** in **porous** channel is important in the dialysis of blood in artificial kidneys Roy et l (1981), studied the **flow** of Walter liquid of short memory co-efficient. Rahim, Ramanamurthy and Thiagarajan (2002) studied numerical solution for unsteady **flow** of a visco-elastic fluid through a **porous** **medium** bounded by two **porous** plates.

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“Dharmaiah and veeraKrishna [1] studied finite difference analysis on mhd free convection **flow** through a **porous** **medium** along a vertical wall. Veerakrishna and dharmaiah [2] analyzed mhd **flow** of a rivlin-ericson fluid through a **porous** **medium** in a parallel plate channel under externally applied boundary acceleration.in [2],[3],[4]. Dharmaiah performed effects of radiation, chemical reaction and soret on unsteady and Ramprasad et al., reported unsteady mhd convective heat and mass transfer **flow** past an inclined moving surface with heat absorption in [5],[6],[7].Analyzed chemical response, radiation and also dufour consequences on casson magneto hydro dynamics fluid **flow** on a vertical plate using Heating source/sink [8],[9],[10]over a vertical permeable plate Dharmaiah studied An unsteady magnetohydro lively heat transport **flow** in a rotating parallel plate station via a solid **medium** with radiation influence in [11],[12],[13],[14],[15].

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The **flow** between two parallel plates is a classical prob- lem that has many applications in accelerators, aerody- namic heating, electrostatic precipitation, polymer tech- nology, petroleum industry, purification of crude oil, fluid droplets and sprays. Such a **flow** model is of great interest, not only for its theoretical significance, but also for its wide applications to geophysics and engineering. A lot of research work concerning the **flow** between two parallel plates studied in a rotating system have appeared, for example, Batchelor [1], Ganapathy [3], Gupta [4] and Mazumder [5]. The flows through **porous** **medium** are very much prevalent in nature and therefore, the study of such flows has become of principal interest in many sci- entific and engineering applications. This type of flows has shown their great importance in petroleum engineer- ing to study the movements of natural gas, oil and water through the oil reservoirs; in chemical engineering for the filtration and water purification processes. Further, to study the underground water resources and seepage of water in river beds one need the knowledge of the fluid **flow** through **porous** **medium**. Therefore, there are num- ber of practical uses of the fluid **flow** through **porous** media. Rotation has an immense importance in various phenomena such as in cosmical fluid dynamics, meteor-

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The thermo-physical properties of the fluid are assumed to be constant except for density dependency of the buoyancy term in the momentum equations. The **porous** **medium** is saturated with a fluid that is in local thermodynamic equilibrium with the solid matrix. The **medium** is assumed to be isotropic in permeability. In expressing the equations for the **flow** in the **porous** **medium**, it should be noted that the Darcy model presents a linear relationship between velocity of discharge and the pressure gradient. As the Darcy model does not hold when the **flow** velocity is not sufficiently small or when the permeability is high, extensions to this model known as Brinkman-extended or Forchheimer-extended models ex- ist. In short, the Brinkman term is found to be needed for satisfying a no-slip boundary condition at solid walls, whereas the Forchheimer term accounts for the form drag. Also in analogy with the Navier-Stokes equations, the Darcy model has been extended by including the material derivative. The necessity of the simultaneous inclusion of all or some of these extensions has been discussed in the literature [17]. Therefore, in order to cover extreme values of input parameters i.e., high permeability, high velocity and low thermal diffusivity; the governing equations for the **flow** and heat transfer in cylindrical polar coordinate system using following non-dimensional quantities:

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An exact solution of the governing equations has been obtained for the free and forced convection **flow** between infinitely long horizontal parallel plates embedded in a **porous** **medium**. It is found that the fluid velocity decreases with an increase in porosity parameter. It is also found that the critical Grashof number for which there is no **flow** reversal near the upper plate decreases with an increase in porosity parameter. Further, the fluid temperature increases with an increase in either porosity of the **porous** **medium** or the Grashof number.

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For better understanding the effect of nanofluid **flow** in a **porous** **medium**, a set of experiments were conducted on a horizontal glass micromodel for Al 2 O 3 -water nanofluids. To characterize the **flow** of nanofluids the same experiment was done by pure water. The glass micromodel was constructed by a photolithography technique. The Al 2 O 3 -water nanofluids were produced by a two-step method and no surfactant or PH changes were used. The nanofluids were made in different volume fractions of 0.1%, 0.5%, and 1%. The experimental results show that the pressure drop of nanofluids through the micromodel increases up to 43% at volume concentration of 0.01. Moreover , the Al 2 O 3 -water nanofluids behave like a Newtonian fluid and follow Darcy’s law at low Reynolds numbers. The permeability of the **porous** **medium** has been evaluated in different volume fraction of nanofluids and pure water. The results reveal that the assessed permeability doesn’t change significantly in various volume concentrations of nanoparticles in the constructed micromodel. A semi analytical correlation was proposed for calculating the permeability of such **porous** **medium**.

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in a **porous** media is expected to be slow. However, as shown by Munaf, et al. [5], inertial effects can become important in the **flow** of fluids through **porous** media under certain circumstances. In fact, in problems such as enhanced oil recovery where the oil is driven by steam at high pressure, when the pressure gradients are high or when the **flow** of dense fluids is considered, inertial ef- fects can become important, or at least significant enough to be not ignored. It might be necessary, in flows involving high pressures and high pressure gradients, to include the effect of the pressure on the viscosity as well as the “Drag” term that arises due to frictional effects at the pore. Recently, Subramaniam and Rajagopal [6] investigated **flow** of fluids at high pressures while the gradients of pressure is also high and allowed for the viscosity and the “Drag coefficient” to depend on the pressure. They found the results to be markedly different from the results for the constant viscosity and con- stant “Drag coefficient” in that the **flow** rates are very different and they also found the development of boundary layers (regions where the vorticity is much larger than the rest of the **flow** domain) wherein the high pressures are confined. Later, Kannan and Rajagopal [7] also studied the **flow** of fluids through an inclined **porous** media at high pressures and pressure gradients due to the effects of gravity and they also found results that show the devel- opment of boundary layers wherein the vorticity is concentrated. The flows considered by Subramaniam and Rajagopal [6] and Kannan and Rajagopal [7] are steady flows and due to the special form assumed for the **flow** field, the inertial term is identically zero. However, the **flow** field assumed in these and several other studies can only be viewed as approximations to flows that take place in a **porous** **medium** as they assume that the **flow** is unidirectional. It is important to recognize that flows through **porous** media are inherently unsteady and thus one has to include at the very least the unsteady part of the inertial term. Moreover, **flow** through **porous** media is never truly one-dimensional and thus one cannot neglect the non-linear term in the in- ertia on that basis. In fact, when very high pressure gradients are involved the **flow** will be turbulent. Here, we shall not consider turbulent flows. We shall however modify Brinkman’s equation to take into account the effects of inertia. A detailed discussion of the various assumptions that go into the development of Brinkman’s equation can be found in the recent article on a hierarchy for approximations for the **flow** of fluids through **porous** media by Rajagopal [8] 1 . Brinkman very astutely observed that “Equation (2) 2 ,

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Here, we are interested in the nonlinear peristaltic pumping of MHD ﬂow through a **porous** **medium**, and due to the complexity of the nonlinear equa- tions 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.

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From mathematical viewpoints, it is much more diffi- cult to solve a non-linear PDE than ODE. Generally speak- ing, it is difficult to solve nonlinear PDEs, especially by means of analytic method. Using the perturbation meth- ods or the traditional non-perturbation methods such as Lyapunov’s small parameter method [12], the δ-expan- sion method [13] and Adomian’s decomposition method [14], it is difficult to get analytic approximations con- vergent for all physical parameters in the infinite domain of the flows, because all of these techniques can not en- sure the convergence of approximation series. Currently, Cimpean et al. [15] applied the perturbation techniques, combined with numerical techniques, to solve a free con- vection non-similarity boundary-layer problem over a ver- tical flat sheet in a **porous** **medium**. Like most of pertur- bation solutions, their results are valid only for small and large x, which are regarded as perturbation quantities. Among analytic methods, the method of local similarity is most frequently used. Many researchers [16-19] have obtained the non-similarity solutions by using the method of local similarity. In some cases, the results given by this method agree with numerical solutions. However, the results given by this method are not very accurate and besides are valid only for small ξ in general.

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We have attempted to gain an understanding of instabil- ity of pressure gradient driven buoyancy assisted mixed convection in a vertical pipe ﬁlled with ﬂuid-saturated **porous** **medium**. To this end, we adopted Non-Darcy Brinkman-extended model. By means of linear theory, we were able to extract detailed information of transition of basic ﬂow through a **porous** **medium** for diﬀerent ﬂuids. The Spectral Collocation Method is used to solve the set of linear ordinary diﬀerential equations. The main objec- tive in this study was to investigate the dependency of the stability boundaries on the azimuthal wave numbers (n) for gas (Pr = 0.7), water (Pr = 7.0) and oil (Pr = 70). Four diﬀerent values (10 −1 , 10 −2 , 10 −3 and 10 −4 ) of Da were considered. Throughout the study, porosity (), viscosity ratio (Λ) and heat capacity ratio (σ) were given constant values of 0.9, 1 and 1, respectively. In

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A new dimension is added to the study of mixed convection **flow** past a stretching sheet embedded in a **porous** **medium** by considering the effect of thermal radiation. Thermal radiation effect plays a significant role in controlling heat transfer process in polymer processing industry. The quality of the final product depends to a certain extent on heat controlling factors. Also, the effect of thermal radiation on **flow** and heat transfer processes is of major important in the design of many advanced energy convection systems which operate at high temperature. Sasikumar and Govindarajan [1], discussed free convective MHD oscillating **flow** past parallel plates in a **porous** **medium**. Where the effects of various non dimensional parameters on velocity, temperature and concentration profiles have been analysed. The effect of magnetic field and convection on transient velocity, transient

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