M.S. Abu Zaytoon et al./ Elixir Appl. Math. 96 (2016) 41336-41340 41336 1.Introduction Dependence of fluid viscosity on pressure is not a recent knowledge or phenomenon; rather, it has been known, investigated and modelled since the days of Stokes and Barus, (cf. [1], [2] and the references therein). However, studies of flow of pressure-dependent fluids through porous structures and its applications is relatively more recent and is motivated by both industrial applications and theoretical fundamentals of the nature of flow. Many applications both in industry and nature involve processes with high pressures that warrant consideration of pressure-dependent viscosity variations of fluids flowing in free space or in porous media, ( cf. [3-6] and the references therein]. These applications involve chemical and process technologies, such as pharmaceutical tablet production, ground water and crude oil pumping, food processing technologies, lubrication theory, and microfluidics [1-9]. From a theoretical point of view, concern arises in how high pressures affect fluid viscosity, and what parameters in porous media are important in modelling the phenomena. The pioneering work of Rajagopal and co-workers (cf. [5- 9] and the references therein), among others, provided detailed analysis, models and many answers to questions that arise when this type of flow is considered. They developed a number of flow models based on mixture theory and reported on various functional forms of dependence of viscosity on pressure, in addition to introducing the limitations of the Darcy drag force in the modelling process. Others have considered modelling the flow through porous media using averaging theorems (which have worked well in other flow problems, [10-13]) and ended with a Darcy-Lapwood- Brinkman model. An advantage of using averaging theorems is the development of models that are valid for variable medium porosity, hence permeability. The Darcy-Lapwood-Brinkman model developed in [11] remains silent about Forchheimer effects that arise due in part to the porous microstructure. This is the subject matter of the current work where intrinsic volume averaging will be applied to the Navier-Stokes equations with pressure-dependent viscosity. The effects of the porous microstructure will be accounted for using mathematical idealizations of the pore structure, based on the concept of Representative Unit Cell (RUC) that was introduced by Du Plessis and Masliyah, [14,15 ] and the geometric factors of Du Plessis , [16] and of Du Plessis and Diedericks, [17]. Both granular and consolidated media microstructures are considered. 2. Model Equations The unsteady, Navier-Stokes flow of an incompressible fluid with pressure-dependent viscosity is governed by the equations of continuity and momentum, expressed as: 0    v  …(1)   g T p v v v t                 …(2) where   T v v T ) (         …(3) v  is the velocity vector field, p is the pressure,  is the fluid density, ) ( p    is the pressure-dependent viscosity of the fluid. When the flow domain is a porous structure, the above governing equations are valid locally (microscopically) in the pore space. However, due to complexity of the pore structure and its boundary (that is, porous matrix) it is customary to seek a macroscopic form of the equations, obtained by averaging the governing equations over a control volume, V , referred to as a Representative Elementary Volume (REV), [14,15]. An REV is composed of a fluid-phase contained in the pore space,  V , and a (stationary) solid-phase contained in the porous matrix solid of volume s V . Tele: E-mail address: hamdan@unb.ca © 2016 Elixir All rights reserved ARTICLE INFO Article history: Received: 4 May 2016; Received in revised form: 25 June 2016; Accepted: 30 June 2016; Keywords Pressure-dependent Viscosity, Intrinsic volume averaging. Averaged Equations of Flow of Fluid with Pressure-Dependent Viscosity through Porous Media M.S. Abu Zaytoon 1 , F.M. Allan 2 , T.L. Alderson 1 and M.H. Hamdan 1 1 Department of Mathematics and Statistics, University of New Brunswick, P.O. Box 5050,Saint John, New Brunswick, Canada E2L 4L5. 2 Department of Mathematical Sciences, United Arab Emirates University, P.O. Box 15551,Al-Ain, UAE. ABSTRACT Equations governing the flow of a fluid with variable viscosity through an isotropic porous structure are derived using the method of intrinsic volume averaging. Viscosity of the fluid is assumed to be a variable function of pressure, and the effects of the porous microstructure are modelled in terms of Darcy resistance, Brinkman shear term, and Forchheimer effects. © 2016 Elixir All rights reserved. Elixir Appl. Math. 96 (2016) 41336-41340 Applied Mathematics Available online at www.elixirpublishers.com (Elixir International Journal)