**Nonlinear Effects in Flow in Porous Duct**

**ABSTRACT**

In general, it is assumed in some non viscous flows that the flow velocity is constant at a cross-section. In this thesis, If we impose more realistic boundary conditions by, for example, introducing viscosity, and suction at walls, the net mass flow will change since the continuity equation must hold. The convective acceleration terms will be products of variables such that a nonlinear behavior will take place in the flow. The work will consist of deriving all the equations and parameters needed to described this kind of flow. An approximate analytic solution for the case of small Reynold number Re is discussed.Expression for the velocity components and pressure are obtained.The governing nonlinear differential equation that cannot be solved analytically is solved numerically using Runge-Kutta Program and the graph of axial and radial velocity profiles have drawn.

**TABLE OF CONTENTS**

Dedication iii

Acknowledgement iv

Abstract v

Nomenclature vi

1 Introduction 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Previous work on wall suction and porous duct . . . . . . . . . . . . . . . . . 2

1.3 Study of fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.1 Computational fluid dynamic CFD . . . . . . . . . . . . . . . . . . . 3

1.4 Boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Scope of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Governing equations of fluid dynamics 5

2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 The Continuity equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 The Momentum equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 Coutte flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 Plane Poiseuille Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.6 Hagen poiseuille flow(or pipe flow) . . . . . . . . . . . . . . . . . . . . . . . 10

3 Flow over porous wall 12

3.1 Uniform Suction on a plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Flow between plates with bottom injection and top suction . . . . . . . . . . 13

3.3 Flow in a porous duct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Results 19

4.1 Approximate analytic solution(perturbation) . . . . . . . . . . . . . . . . . . 19

4.2 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

vii

5 Discussion and Conclusion 22

5.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.3 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Hafeez

**CHAPTER ONE**

**Introduction**

**1.1 Introduction**

In the previous years, the problems of fluid flow through porous duct have arouse the interest of Engineers and Mathematicians, the problems have been studied for their possible applications in cases of membrane filtration, transpiration cooling, gaseous diffusions and drinking water treatment as well as biomedical engineerings. Such flows are very sensitive to the Reynold number and change from laminar to transitional to turbulent flow as the Reynold number increases. To evaluate the pressure changes that result from incompressible flow in pipes, ducts, and flow systems. Assume for a moment frictionless flow.This is unrealistic for duct flow, but nevertheless useful foe seeing what factors effect the pressure. The Bernoulli equation indicates that the pressure will only change if we have a change in potential or velocity. For example in a horizontal duct of constant cross-section (constant potential and velocity) the pressure would be constant. If this duct had an increase in area (decrease in velocity), the pressure would increase; for a constant area duct sloping upwards (increase potential), the pressure would decrease.Diffusers are added to ducts to increase the pressure recovery at the exit of the system.The cases where an exact solution for the Navier-Stokes equations can be obtained are of particular importance in order to describe the fluid motion of viscous flows. However, since the Navier-Stokes equations are non-linear, there cannot be a general method to solve analytically full equations [1]. Exact solution on the other hand are

very important for many reasons. The provide a reference solution to verify the accuracies of many approximate methods, such as numerical and/or empirical. Although, nowadays,

computer techniques make the complete integration of the Navier-Stokes equations feasible, the accuracy of numerical results can be established only by the comparison with an exact solution [2]. The Navier-Stokes equations were extensively studied in the literature. Exact solution already known are one dimensional or parallel shear flows, rectangular motion flows, or duct flows [2, 3, 4]. The flow of fluids over boundaries of porous materials has many applications in practice, such as a boundary layer control and transpiration processes. Exact solutions are generally easy to find when suction or injection is applied to a fluid. In the case of flows through porous media, a simple solution of the Navier-Stokes equations can be obtained for the flow over a porous plane boundary at which there is a uniform suction velocity [5].

A porous medium is typically composed by solid particles (grains) and interstitial spaces (pores) that are connected, permitting a fluid to flow.Such a definition is quite generic, since it includes coarse materials, like granules, and soils with extremely fine texture, like clays, with a great range of intermediate possibilities. In a natural porous medium the distribution of pores with respect to shape and size is irregular.Examples of natural porous media are beach sand, sandstone, limestone, rye bread, wood and the human lung.On the pore scale (the microscopic scale) the flow quantities (velocity, pressure e.t.c) will be clearly irregular.

But in typical experiments the quantities of interest are measured over areas that cross many pores and such spaces-averaged (macroscopic) quantities change in regular manner

with respect to space and time, ans hence are amenable to theoretical treatment.

How we treat a flow through a porous structure is largely a question of distance, the distance between the problem solver and the actual flow structure (Bejan et.al 2004). When the distance is short , the observer sees only one or more channels, or one or two open or closed cavities. In this case it is possible to use conventional fluid mechanics and convective heat transfer to describe what happen at every points of the fluid- and-solid-filled spaces. When the distance is large so that there are many channels and cavities in the problem solver’s field vision, the compilation of the flow path rule out the convectional approach. In this limit, volume-averaging and global measurements (e.g, permittivity, conductivity) are useful in describing the flow and in simplifying the description. As engineers focus more on and more on designed porous media at decreasing pores scales, the problems tend to fall between the extreme noted above. In this intermediate range, the challenge is not only to describe the coarse porous structures, but also to optimize flow elements and assembled them. The resulting flow structures are designed porous media (see Bejan et.al 2004; Bejan 2004b).

The usual way of deriving the laws governing the macroscopic variable is to begin with standard equations obeyed by the fluid and obtain the the macroscopic equations by averaging over volume or areas containing many pores. In normal case a porous medium can be modeled as continuum by considering a representative volume element on which we are define averaged quantities next extend to the whole domain occupied by the system as regular functions of the spaces co-ordinates. The volume element must be large enough to contain a sufficiently high number of grains and pores, but much smaller than the typical size os the system. In this way we define a basic geometric property, the porosity , ie the volume fraction occupied by the pores. When the following fluid is incompressible we may likewise introduce the saturation S ie the fraction of volume available to the flow (ie the pore volume) which is occupied by the fluid.

These are both numbers between 0 and 1. We say that the medium is saturated when S = 1, and unsaturated if 0 < S < 1. When S = 0 the medium is dry. It may be useful to expose
some fundamental fact, about the incompressible flows through porous media.
The fundamental experimental law, very well known and must frequently used to describe the flow of liquid through porous media is Darcy’s law which dates back to 1856 [1].
To the authors’ knowledge, there has not been an exhaustive review of the literature from numerous disciplines that have contributed to this subject.In the present study the full steady two dimensional Navier-Stokes equations are considered for the case of incompressible porous flow. An exact non-linear effects is shown by employing perturbation techniques and the Runge-Kutta program for the case of two-dimensional unsteady flows in a porous duct.
**1.2 Previous work on wall suction and porous duct**

Previous reviews of flow in porous duct tended to focus only on one specific aspects of subject at a time such as membrane filtration (Belfort and Nagat,1985), the description of boundary Hafeez Yusuf Hafeez 2 [email protected] conditions (Sahraoui and Kaviany, 1992) or the existence of exact solutions (Wang,1991). [7]

Also wall suction was early recognized to stabilize the boundary layer and critical Reynold number for natural transition of 46130 was obtained by Hughes and Reid (for reference see Drazin and Reid 1981). Hughes and Reid did however not take into account the advective term introduced in the modified Orr-sommerfield equation and consequently the critical

Reynold number was corrected to 54370 by Hocking (1975). The stabilization effect of wall suction is mainly due to the change of mean velocity profile in this case.In the review of Joslin (1908), it is also noticed that the uniform wall suction is not only a tool for laminar flow control but can also used to damp out already existence turbulence. Rioual et al. (1996) investigated the power balance of flat plate used as an airfoil. Uniform suction was found to reduce the work drag and an optimum for the suction velocity was obtained, leading to a reduction of power consumption. In order to obtain the stabilizing effects on the boundary layer, one must assure that the chosen material is able to provide continuous suction [8].

**1.3 Study of fluid**

For most of the the 19th and 20th centuries there were two approaches to study of fluid motion: Theoretical and Experimental approach. Many contribution to our understanding

of fluid behaviour was made through the years by both of these methods. But today because of power of modern digital computers there is yet third way to study fluid dynamic:

Computational fluid dynamic or in short CFD. There are three main approaches to study of fluid dynamics i) Theoretical ii) Experimental and iii) Computational, we note (and justify) that out of these computational will be emphasized [10].

**1.3.1 Computational fluid dynamic CFD**

Computational fluid dynamics constitute a new “third approach” in the philosophical study and development of the whole discipline of fluid dynamic. It is today an equal partner with theory and experiment in the analysis and solution of fluid dynamic problems. It nicely and synergistically complements the other two approaches of theory and experiment, but it will never replace either of these approaches (as sometime suggested). There will always be need

for theory and experiment.

Computational fluid dynamic, the field of study devoted to solution of the equation of fluid flow through use of computer (or more recently, several computers working in parallel).

Modern engineers applied both experimental and computational fluid dynamic analyses and the two complement each other, In addition experimental data are often use to validate

computational fluid dynamic solution by matching the computationally and experimentally determined global quantities. Computational fluid dynamic is the art of replacing integral or partial derivatives (as the case may be) in these equation with discretized algebraic forms, which in turn are solved to obtain numbers for the flow field value at discrete points in time and/or space. The end product of Computational fluid dynamic is indeed a collection of numbers, in contract to a closed-form analytical solution.

Computational fluid dynamic is widely used both in everyday activities and in design of modern engineering system, is also playing a strong role as a design tool and research tool such as analysis of aircraft, boat, cooling of electronic components and transportation of water, oil and natural gas e.t.c. It also used in the design of heating and air-conditioning system, hydraulic brake. It can also be considered in design of building and bridges.

Computational fluid dynamic is a major tool in solving hydrodynamic problems associated Hafeez Yusuf Hafeez 3 [email protected] with ships submarines, torpedoes e.t.c, it also used to improve the performance of modern cars and tracks (environment quality, fuels economy e.t.c).

**1.4 Boundary condition**

The boundary conditions and sometimes called initial conditions, dictate the particular solutions to be obtained from the governing equations. First, let us review the physical boundary conditions for a viscous flow. Here, the boundary condition on the surface assumes zero relative velocity between the surface and the gas immediately at the surface.

This is called the no-slip condition. If the surface is stationary with the flow moving past , then V(u,v,w)=0 i.e u = v = w = 0 at the surface (for a viscous flow) .At wall no-slip condition is enforced: the tangential fluid velocity is equal to the wall velocity, and normal velocity components is set to zero. This no-slip condition prevents the shear stress at wall to become infinite. Finally, we note that the only physical boundary conditions along a wall for continuum viscous flow are the no-slip conditions discussed above; These boundary conditions are associated with velocity at the wall. Other flow properties such as pressure and density at the wall, fall out

as part of the solution [9].

**1.5 Scope of the Thesis**

In this thesis, mainly the computational aspects of fluid dynamics will be studied. Moreover, the approach to be taken in these thesis will be to emphasize the importance and utility of the “equation of motion” (continuity and Navier-stokes equations) that cannot be solved analytically. This thesis consists of all derivation of the equations and parameter needed to described the flow with the flow velocity is not constant at cross-section.

The assumptions which the work is based upon are:

**The continuum hypothesis:** The discrete molecular structure of fluid is replaced by a continuum distribution.

**Incompressibility:** The fluid is considered incompressible, i.e the density does not vary.This restricts the results to fluid at relatively low speed.

**Newtonian Fluid:** The relation between shear stress and deformation is linear.

**Viscous flow:** Is the one where the transport phenomena of friction, thermal conductions, and/or mass diffusions are included.

**Laminar flow:** For not too large Reynold numbers the flow look smooth. The streamlines are nicely parallel to each other, as in thin layers. We call this laminar flow.

**Kinematic viscosity ():** can be derived from shear viscosity () by dividing shear viscosity by the fluid density.

The chapters of this thesis contain the following materials. Chapter two is devoted to the background of Navier-stokes equations (i.e continuity and momentum equations )

their derivations and mainly poiseuille and coutte flow for the case of pipe and plate is considered. Chapter three is mainly about the Asymptotic suction flow (i.e flow in porous media and duct with suction and injection between the plates),while chapter four deals with the results (i.e analytical solution using perturbation techniques and the numerical solutions using Runge-Kutta program) and chapter five comprises Discussion, Conclusion and future works.

Hafeez Yusuf Hafeez