Some Analytical Solutions of Laminar and Incompressible Flows of Viscid Fluids

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2017-10-11 Some Analytical Solutions of Laminar and Incompressible Flows of Viscid Fluids

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Some Analytical Solutions of Laminar and Incompressible Flows of Viscid Fluids

Tadesse Zenebe Mesfin

Department of Mathematics Collage of Science Bahir Dar University

September, 2017 Bahirdar, Ethiopia

Some Analytical Solutions of Laminar and Incompressible Flows of Viscous Fluids

A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in Mathematics.

Tadesse Zenebe Mesfin

Adivisor: Dr. Eshetu Haile

Department of Mathematics Collage of Science Bahir Dar University

September, 2017 Bahirdar, Ethiopia

The Dissertation Entitled “Some Analytical Solutions of Laminar and Incompressible Flows of Viscous Fluids” by Tadesse Zenebe is approved for the Degree of Master of Science in Mathematics.

Board of Examiners

Name Signature

Adviser: Dr. Eshetu Haile ------

Examiner 1:------

Examiner 2:------

Date------

Acknowledgements

First, I would like to thank Dr. Eshetu Haile for his continuous support by giving source of reading material, unreserved advice and guidance by reading my project and giving constructive comments which helped me to arrange the report. I am grateful to Tewodros higher preparatory and secondary school teachers who gave me helpful advice and comments. Finally, I want to thank information technology professional of our school for helping computer related work.

Abstract

This project work is concerned about some simple analytical solutions of laminar and incompressible viscous fluid flows. Particularly it deals about the velocity distributions of some simple fluid flow problems (Couette and Poisueille) flows. Solving non-linear Navier- Stokes equations are complex and difficult. To reduce this complexity and difficulty we have considered one-dimensional, laminar and incompressible viscous fluid flow problems. we have been used mat lab codes to solve those fluid flow problems analytically.

Table of Contents page List of Figures iv CHAPTER ONE:INTRODUCTION AND PRELIMINARIES 1 1.1 Introduction 1 1.2 Preliminaries 3 1.2.1 Definition of Basic Terminologies. 3 1.2.2 Types of Fluid Flow 5 1.2.3 The Concept of Continuum 8 1.2.4 Governing Equations 9 1.2.5 Streamlines, path lines, stream tubes and filaments 11 CHAPTER TWO:CONSERVATION LAWS 12 2.1 Lagrangian and Eulerian Description of Motion 12 2.2 Fluid Element, Fluid Particle and Their Properties 13 2.3 Mass Balance 15 2.4 Momentum Balance 16 2.5 Energy Equation 17 2.6 Viscous Stress Tensor 20 2.7 Navier-Stokes Equations 20 2.8 Summary of Equations in Conservation Form 21 CHAPTER THREE:SOLUTIONS FOR LAMINAR FLOW OF INCOMPRESSIBLE VISCOUS FLUIDS 3.1 Steady-Laminar Flow Between Two Parallel Plates In Relative Motion (Couette Flow) 22 3.2 steady State, Laminar Flow between Stationary parallel plates(poiseuille flow) 24 3.3 Axially Moving Concentric Cylinders (Couette Flow) 26 3.4 Flows between Rotating Concentric Cylinder (Couette Flow) 28 3.5 Steady Laminar Flow Through A Cylindrical Pipe. 30 3.6 Combined Couette - Poiseuille Flow between Parallel Plates 33 CONCLUSION 35 References 36 Appendix 37

iii

List of figures Contents pages

Fig.1 Lagrangian description of fluid motion 12

Fig.2 Eulerian description of fluid motion 13

Fig.3 Fluid element for conservation laws 13

Fig.4 Mass balance 15

Fig.5 Viscous stress aligned with the direction of coordinate axis 16

Fig.6 Forces aligned in the x- coordinate direction 17

Fig.7 Work done by surface stress in x- direction 18

Fig.8 Energy flux due to heat conduction 19

Fig.9 Fluid flow between two parallel plates in relative motion of the above plate 23

Fig.10 Velocity profile of Couette flow between parallel plates 23

Fig.11 Fluid flow between two fixed parallel plates 24

Fig.12 The velocity profile of fluid flow between two fixed parallel plates 25

Fig.13 Axially moving concentric cylinders 26

Fig.14 Velocity profile of a fluid flow between axially moving concentric cylinders 27

Fig.15 Velocity distribution between stationary outer and rotating inner cylinders 29

Fig.16 Fluid flow through a cylindrical pipe 30

Fig.17 Velocity profile for steady-laminar flow through a cylindrical pipe 32

Fig.18 Combined couette-p0iseuille flow between parallel plates 33

Fig.19 Velocity profile of combined Couette – Poiseuille flow between parallel plates 34

CHAPTER ONE

INTRODUCTION AND PRELIMINARIES

1.1 Introduction When a fluid is flowing through a closed channel such as a pipe or between two flat plates, either laminar flow or turbulent flow may occur depending on the velocity and viscosity of the fluid. Laminar flow tends to occur at lower velocities and large fluid viscosity, below a threshold at which it becomes turbulent. Turbulent flow is a less orderly flow regime that is characterized by eddies or small packets of fluid particles which result in lateral mixing [5]. In non-scientific terms, laminar flow is smooth while turbulent flow is rough.

In fluid dynamics the dimensionless Reynolds number is an important parameter in the equations that describe whether fully developed flow conditions lead to laminar or turbulent flow. In fully developed flow the velocity profile and momentum does not change in the direction of fluid flow. The phenomenon of laminar, transitional and turbulent flow regime was first investigated in an experiment by injecting some dye streaks into the flow in a glass pipe by the British engineer Osbourne Reynolds (1842–1912) did over a century ago. From the experiment he observes that the dye streak forms a straight and smooth line at low velocities when the flow is laminar, has bursts of fluctuations in the transitional regime and zigzags rapidly and randomly when the flow becomes fully turbulent. These shows in laminar flow the viscous force are much more important than the inertial forces and are usually counterbalanced by pressure or gravitational effects. If the Reynolds number is very small, much less than 1, then the fluid will exhibit Stokes or creeping flow, where the viscous forces of the fluid dominate the inertial forces.

The specific calculation of the Reynolds number and the values where laminar flow occurs will depend on the geometry of the flow system and flow pattern. The common example is flow through a pipe. For such systems, laminar flow occurs when the Reynolds number is below a critical value of approximately 2,040, though the transition range is typically between 1,800 and 2,100 [2].

A common application of laminar flow is in the smooth flow of a viscous liquid through a tube or pipe. In that case, the velocity of flow varies from zero at the walls to a maximum along the

1 cross-sectional center of the vessel. The flow profile of laminar flow in a tube can be calculated by dividing the flow into thin cylindrical elements and applying the viscous force to them [4].

If the fluid is assumed to be of constant density, vanishes from the continuity equation and the second viscosity coefficient disappears, from Newton's law, the Navier- Stokes equation are not greatly simplified through, if the first viscosity µ allowed to vary with temperature and pressure (and hence with position). However, if we assume that µ is constant, many terms vanish and leaving us with a much simpler Navier-Stokes equation for constant viscosity. This is an excellent point of departure in the theory of incompressible viscous flow. In a compressible fluid, accelerations are important, and continuity is not trivial. There for density is an important variable in case of compressible fluid.

A mathematical theory of fluid motion had been worked out by Leonhard Euler, Joseph Louis Lagrange, Alfred George Greenhill, C.V. Coates, Gröbli and others during the eighteenth and early nineteenth centuries. Fluid flows are governed by partial differential equations which represent conservation laws for the mass, momentum, energy and the interrelation ship between the flow variables and their evolution in time and space. The basic differential equations describing the flow of Newtonian fluids are commonly called the Navier–Stokes equations, named in honor of the French mathematician L. M. H. Navier (1785–1836) and the English Mechanician Sir G. G. Stokes (1819–1903), who were responsible for their formulation. These equations of motion provide a complete mathematical description of the flow of incompressible Newtonian fluids.

In Parallel flows only one velocity component is different from zero, two-dimensional, incompressible fluid have this characteristic. Examples for which analytical solutions exist are parallel flow through a straight channel Couette flow and Hagen-poiseuille flow i.e. flow in a cylindrical pipe.

1.2 Preliminaries

1.2.1 Definition of Basic Terminologies. Fluid:-is any material that unable to prevent the deformation caused by a shear stress (or a it is any substance that deforms continuously when subjected to a shear stress, no matter how small). This continuing deformation in shear is the characteristics of all fluids. In a fluid at rest all shear stresses must be absent. In the Eulerian description, a fluid is an indivisible continuous material that moves through space under the action of external and internal forces.

Fluid Mechanics:-is concerned with understanding, predicting, and controlling the behavior of a fluid. The field of fluid mechanics has historically been divided into two branches, fluid statics (fluid at rest) and fluid dynamics (fluid in motion). Fluid statics or hydrostatics, is concerned with the behavior of a fluid at rest or nearly so.

Fluid Dynamics:-is "the branch of applied science that is concerned with the movement of liquids and gases," according to the American Heritage Dictionary. Fluid dynamics is one of the two branches of fluid mechanics, which is the study of fluids and how forces affect them.

Fluid dynamics provides methods for studying the evolution of stars, ocean currents, weather patterns, plate tectonics and even blood circulation. Some important technological applications of fluid dynamics include rocket engines, wind turbines, oil pipelines and air conditioning systems.

Reynolds’s Number:-It is the most common dimensionless group in fluid mechanics. It can be VD interpreted as the ratio of inertial forces to viscous forces for example Re = for fluids in a  pipe, where D is the hydraulic diameter of the pipe, V is the mean velocity of the fluid, μ is the dynamic viscosity of the fluid, ρ is the density of the fluid.

If the Re is small, viscous forces dominate the flow and inertial forces can be neglected. Conversely, if Re is large, inertial forces dominate outside of boundary layers.

Viscosity of a Fluid:-denoted by, µ, is defined as the constant of proportionality between shear du stress and the transverse velocity gradient, i.e.    . µ is also called absolute or dynamic dy

3 viscosity, but a more descriptive name is shear viscosity. In a fluid flow viscosity is a tendency to convert the useful energy content of the fluid into heat. The useful energy lost appears as an increase in the internal energy of the fluid to a rise in temperature. The shear viscosity of a fluid is strongly dependent on temperature but not strongly dependent on pressure. In most cases shear viscosity increases with temperature for gases but decreases with temperature for liquids.

Pressure of a Fluid:-is the compressive normal component of the force applied by a fluid at rest or in motion to a surface, divided by the area of that surface. Pressure differences can cause a fluid to flow or be caused by a fluid flow. F Mathematically P= which is the absolute pressure. P is zero in a perfect vacuum. A

Density of a Fluid:-denoted by ρ, is defined as the mass of the fluid per unit volume. The density of a fluid varies in general with temperature and pressure. The relation between these properties is called an equation of state.

M Density (ρ) = , where V is the Volume of the fluid and M is the mass of the fluid. V

Temperature of a Fluid:-is a thermodynamic state variable that provides a measure of the internal energy stored in the fluid. In a fluid in equilibrium the temperature is proportional to the mean kinetic energy of the random motion of the molecules. Temperature differences may create a fluid to flow, or they may be a consequence of a fluid flow. Temperature differences in a fluid are always accompanied by the flow of heat by molecular conduction. The relationship between heat flux, temperature gradient and thermal conductivity in horizontal flow is given by Fourier’s law of heat.

Body Forces:-are long-range forces that act on a volume of fluid in such a way that the magnitude of the body force is proportional to the mass or volume of the fluid element. Body forces act on a fluid but are not applied by a fluid. They exert their influence on fluids at rest and in motion without the need for physical contact between the fluid and the external source of the body force. Gravitational, centrifugal, electromagnetic forces, etc. are examples of body forces.

Surface Forces:-are short-range forces that act on an element of fluid through physical contact between the molecules of a fluid and the molecules of a bounding material. Surface forces not

4 only act on a fluid but are applied by a fluid to its surroundings. These forces exist at every interface involving a fluid both in a fluid at rest and in a fluid in motion.

In developing a model for the surface force, we may distinguish three different situations: the surface force exerted by a fluid on a structure, the surface force applied to a fluid by a structure, and the state of stress in a fluid. The orientation of any surface is specified by the outward unit normal of the surface such as those exerted by pressure or shear stress.

In a fluid in motion, the surface force per unit area acting on an infinitesimal surface generally has both normal and tangential components. The total surface force acting on any surface in contact with fluid is found by using a surface integral to sum the individual contributions from each infinitesimal surface element. The total surface force is a vector quantity whose components may be resolved in any desired direction.

The total surface force acting on an arbitrary volume of fluid may be expressed as a volume integral of the dot product of the deloperator and the stress tensor. The dot product, •σ, is the stress divergence. The stress divergence may be interpreted as the surface force per unit volume. To have a net surface force on a volume of fluid, there must be a spatial variation in one or more components of the stress tensor describing the state of stress in the fluid. If the stresses are uniform, the stress divergence is zero at every point in a fluid and the net surface force acting on a volume of fluid is also zero.

1.2.2 Types of Fluid Flow

1.2.2.1 Classification of Fluid Flow According to Dimensions A flow is characterized as having one, two or three dimensions depending on the corresponding number of components needed to describe its Eulerian velocity field.

One Dimensional Flow:-It is a unidirectional flow. The velocity of the x- component u is nonzero but the y-component v and z-component w are both zero at all points in the velocity field in the Cartesian coordinate system, then the flow is 1D. Similarly, in a cylindrical coordinate system, if the velocity of the z-component is nonzero and the velocity of the r-component and

θ-component and are zero, then the flow field is 1D.

Two Dimensional Flows:-If the velocity of the x-components u and the y-component v are nonzero but the z-component is zero at all points in the velocity field in Cartesian coordinate system, then the flow is 2D. Similarly in cylindrical coordinate system, if the velocity of the z- component vz and the r-component vr are nonzero but the θ-component Vθ is zero at all points in the velocity field, then the flow is 2D.

Three Dimensional Flows:- If the velocity of the x-component , the y-component and the z- component are each non zero at all points in the velocity field in the cartesian coordinate system, then the flow is 3D.similarly in cylindrical coordinate system, if all the velocity components vz , vr and vθ are nonzero, then the flow is 3D.

1.2.2.2 Classification of Fluid Flow According to Velocity Variation Steady Flow:-It is a flow in which the fluid properties and conditions such as velocity, pressure, temperature, density, etc. associated with the motion of the fluid are independent of time. In steady flow the flow pattern remains unchanged with time. It is important to realize that identifying a flow as steady does not imply the absence of acceleration. In a steady flow the local u acceleration, is zero but the convective acceleration (u• ) u, need not be zero. t

Mathematically = 0, etc.

Unsteady Flow:-It is a flow in which the fluid properties and conditions associated with the motion of the fluid depend with time. In unsteady flow the flow pattern varies with time.

T Mathematically  0 t

Uniform Flow:-It is a flow in which the fluid particles possess equal velocities at each section of the channel or pipe in the flow.

Non-Uniform Flow:-It is a flow in which the fluid particles possess different velocities at each section of the channel or pipe in the flow.

Laminar Flow:-In fluid dynamics, laminar flow (or streamline flow) occurs when a fluid flows in parallel layers, with no disruption between the layers [3]. At low velocities, the fluid tends to flow without lateral mixing and adjacent layers slide past one another like playing cards. There

6 are no cross-currents perpendicular to the direction of flow, nor eddies or swirls of fluids [5]. In laminar flow, the motion of the particles of the fluid is very orderly with particles close to a solid surface moving in straight lines parallel to that surface [9]. Laminar flow is a flow regime characterized by high momentum diffusion and low momentum convection. Generally laminar flow is amenable to observation, measurement and prediction.

Turbulent Flow:-is a less orderly (or highly disordered) flow regime that is characterized by velocity fluctuations, eddies or small packets of fluid particles which result in lateral mixing. Turbulent flow is encountered in almost all flows in nature and engineering practice. This type of flow consists of a chaotic, disordered and unsteady motion of fluid that is generally difficult to visualize measure and predict. There are no analytical solutions for turbulent flow and computational models of turbulence are limited in their applicability. Thus, experimental results are necessary for engineering designs involving turbulent flows.

1.2.2.3 Classification of fluid flow according to the type of fluid Ideal Fluid Flow:-is a fluid flow with zero viscosity and no surface tension. In this flow the velocity distribution is assumed uniform.

Real Fluid Flow:-is a fluid flow which considers viscosity, compressibility and surface tension. In this type of flow the velocity which causes the formation of shear stress is taken in to consideration.

Compressible Fluid Flow:-it is a flow at which the change in pressure or temperature in the flow results change in volume and density.

Gases are generally treated as compressible fluids.

Incompressible Fluid Flow:-it is a flow in which the flow density remains unchanged throughout the flow when the pressure or temperature changes in the flow.

Liquids are generally treated as incompressible fluids.

1.2.2.4 Classification of fluids According to Shear strain- Relation Newtonian Fluids:-In continuum mechanics, a Newtonian fluid is a fluid in which the viscous stresses arising from its flow, at every point, are linearly [10] proportional to the local strain rate (the rate of change of its deformation over time) [4] [6-7]. That is equivalent to saying that those

7 forces are proportional to the rates of change of the fluid's velocity vector as one move away from the point in various directions.

More precisely, a fluid is Newtonian only if the tensors that describe the viscous stress and the strain rate are related by a constant viscosity tensor that does not depend on the stress state and velocity of the flow.

dr Mathematically   . dt Water, air, milk, alcohol, glycerol, thin motor oil, etc. are examples of Newtonian fluids that have different density properties, which allow a different shear stress and rate flow graphs within a given interval. Non-Newtonian Fluids:-are fluids that have more complicated nonlinear relationships between the shear stress and strain rate.

Human blood, soup, printer ink, shampoo, saliva, honey, butter, paint, tooth paste, lubricants, polymer solution, etc. are examples of non- Newtonian fluids.

1.2.3 The Concept of Continuum Fluids are assumed to obey the continuum assumption. Fluids are composed of molecules that collide with one another and solid objects. However, the continuum assumption assumes that fluids are continuous rather than discrete. Consequently, it is assumed that properties such as density, pressure and temperature and flow velocity are well defined at infinitesimally small points in space and vary continuously from one point to another. The fact that the fluid is made up of discrete molecules is ignored. Mathematically speaking, the continuum hypothesis allows the use of differential calculus in the modeling and solution of fluid mechanics problems. Each fluid property is considered to be a continuous function of position and time. The continuum model generally assumes that all properties are described by continuous functions.

1.2.4 Governing Equations Continuity Equation

In rectangular coordinate: = 0.

 1  1   In cylindrical coordinate  rv  v  v = 0. t r r r r   z z

 1  1  1  In spherical coordinate:  r 2v  v sin  v  = 0, where ρ t r 2 r r rsin   rsin   is the fluid density, 혂= (u, v, w) in cartesian coordinate system, 혂= (vz, , in cylindrical coordinate system, 혂= , in spherical coordinate system are the flow velocity vector and t is time.

Navier-Stoke Equation

In Rectangular Coordinate:

 u u u u   p   2u  2u  2u  X-momentum:   u  v  w         S MX .    2 2 2   t x y z  x  x y z 

 v v v v   p  2v 2v 2v  Y-momentum:   u  v  w         S .    2 2 2  my  t x y z  y  x y z 

 w w w w   p   2w  2w  2w    u  v  w         S .    2 2 2  mz Z- momentum:  t x y z  z  x y z 

In cylindrical coordinate:

R-momentum:

2 2 2 2  u u u u u u    p   u 1 u u 1  u  u 2 u   r  u r   r  r      r  r  r  r  r     S .  r   2 2 2 2 2 2  mr  t r r  z r  r  r r r r r  z r  

θ-Momentum:

 u u u u u u u  1 p  2u 1 u u 1 2u 2u 2 u     u   r      u             r     r z   2 2 2 2 2 2   t r r r  z  r   r r r r r  z r  

 Sm .

z- Momentum:

 u u u u u   p  2 z 1 u 1 2u 2u   z  u z   z  u z     z  z  z   S .  r z   2 2 2 2  mz  t r r  z  z  r r r r  z 

In Spherical Coordinate

R-momentum:

 u u u u u u  2  2   r r  r  r    ur        t r r  r sin  r r   p   2u 2 u 2u 1  2u cot u 1  2u 2 u 2u cot 2 u    r  r  r  r  r   r         S .  2 2 2 2 2 2 2 2 2 2 2  m r  r r r r r  r  r sin   r  r r sin  

θ- Momentum:

 u u u u u u u u u 2 cot 2     r         ur       t r r r  r sin  r 

1 p   2u 2 u u 1  2u cot u 1  2u 2 u 2cot u                  r     S .  2 2 2 2 2 2 2 2 2 2 2  m r   r r r r sin  r  r  r sin   r u r sin   ф- Momentum:

 u u u u u u u u cot u u     r          ur        t r r r  r r sin   1 p   2u 2 u u 1  2u cot u 1  2u 2 u 2cot u                 r     S ,  2 2 2 2 2 2 2 2 2 2 2  m r sin   r r r r sin  r  r  r sin   r sin  r sin   where µ is the viscosity of the fluid, Smi's are body forces in their respective directions, ρ is the fluid density, 혂= (u, v, w) in cartesian coordinate system, 혂= ( , , in cylindrical

10 coordinate system, 혂= , in spherical coordinate system are the flow velocity vector, P is Pressure(or compressive force) and t is time.

Energy Equation:

In rectangular coordinates:

In cylindrical coordinates: ( ) ( ) ( )

In spherical coordinates:

( ) ( ) ( ) , where is enthalpy,

is the thermal conductivity of the fluid, is temperature and is the viscous dissipation function:

  2 2 2 2 2 2    u   v   w   u v   u w   v w   2 2    2                       (div혂) .  x  y  z  y x  z x  z y 3         

The viscous dissipation function governs the rate at which mechanical energy of the flow is converted to heat. The second law of thermodynamics requires that the dissipation term is always positive.

1.2.5 Streamlines, path lines, stream tubes and filaments Streamlines:-are lines which show how each particle is moving at a given instant.

Path lines:-are lines which show how a given particle is moving at each instant.

Stream tubes:-is a tube obtained by drawing the streamlines through each point of a closed curve.

A stream filament:-is a stream tube whose cross-section is a curve of infinitesimal dimensions. Note:-When the motion is steady, the path lines coincide with the streamlines.

CHAPTER TWO

CONSERVATION LAWS

Three conservation laws are used to solve fluid dynamics problems and may be written in integral or differential form. The conservation laws may be applied to a region of the flow called a control volume. A control volume is a discrete volume in space through which fluid is assumed to flow. The integral formulations of the conservation laws are used to describe the change of mass, momentum or energy within the control volume. Differential formulations of the conservation laws apply Stokes' theorem to yield an expression which may be interpreted as the integral form of the law applied to an infinitesimally small volume (at a point) within the flow.

In order to get a complete and fundamental understanding of the fluid dynamic problems the governing equations should be first explained.

2.1 Lagrangian and Eulerian Description of Motion The general equations of fluid motion in differential form may be derived by applying Lagrangian and Eulerian description.

Lagrangian Description

A fluid flow field can be thought of as being comprised of a large number of finite sized fluid particles which have mass, momentum, internal energy and other properties. Mathematical laws can then be written for each fluid particle.

Fig.1 Lagrangian description of fluid motion

Eulerian Description

Another view of fluid motion is the Eulerian description. In the Eulerian description of fluid motion, we consider how flow properties change at a fluid element that is fixed in space and time (x, y, z, t), rather than following individual fluid particles.

Fig.2 Eulerian description of fluid motion

2.2 Fluid Element, Fluid Particle and Their Properties The behavior of the fluid is described in terms of macroscopic fluid properties such as velocity u, pressure p, density ρ, temperature T, energy E. Those Properties are averages of sufficiently large number of molecules.

A fluid element is the smallest volume for which the continuum assumption is valid.

Fig.3 Fluid element for conservation laws

Faces are labeled North, East, West, South, Top and Bottom

Properties at faces are expressed as first two terms of a Taylor series expansion

p 1 1 E.g. P: Pw= p - δx and PE= P+ δx x 2 2

Rate of Change for a Stationary Fluid Element

In most cases we are interested in the changes of a flow property for a fluid element or fluid volume that is stationary in space. However, some equations are derived for fluid particles. The total derivative per unit volume of an arbitrary property ф is given by:

For a moving fluid particle of an arbitrary property ф,

( ) (2.1)

For a fluid element of this arbitrary conserved property ф:

  div(u)  0, Continuity equation. (2.2) t

()  div(u)  0 , Arbitrary property. (2.3) t

Rate of Change for a Fluid Particle

A fluid particle is a volume of fluid moving with the flow and experiences two rates of changes.

Change due to changes in the fluid as a function of time.

Change due to the fluid particle moves to a different location in the fluid with different conditions.

The sum of these two rates of changes for a property per unit mass ф is called the total or substantial derivative .

dx dy dz , where  u,  v,  w. dt dt dt

These results in: 혂.grad ф. (2.4)

Fluid Particle and Fluid Element

We can derive the relationship between the equations for a fluid particle (Lagrangian) and a fluid element (Eulerian) as follows:

  div (ρф혂) = ρ* + * += . t

Since * + is zero because of continuity, then the above relation becomes:

(ρф혂) = , i.e.

( ) + ( ) = ( ) (2.5)

2.3 Mass Balance Conservation of Mass:-The rate of change of fluid mass inside a control volume must be equal to the net rate of fluid flow into the volume. This statement shows that mass is neither created nor destroyed in the control volume.

Rate of change is: (ρ δx δy δz) = (δx δy δz).

The inflows (positive) and outflows (negative) are shown here:

Fig.4 Mass balance of a fluid element

Continuity Equation

Summing all the inflow and outflow terms of the above fluid element and dividing by the volume δxδyδz results in:

= 0. (2.6)

In vector notation: where, is change in density and div (ρ혂) is the net flow of mass across boundaries (convective term).

For incompressible fluids, = 0.

And the equation becomes: divu= = 0. (2.7)

2.4 Momentum Balance Conservation of Momentum:- Newton’s second law of motion states that any change in momentum of the fluid within the control volume will be due to the net flow of momentum into the volume and the action of external forces(or Surface forces and body forces) acting on the fluid within the volume.

Rate of increase of x-, y-, and z-momentum are respectively:

Viscous Stress

Viscous stresses denoted by τ, are forces per unit area. Its Unit is N/m2or Pa.

The nine stress components are . Suffix notation τij is used to indicate direction.

are normal stresses e.g. is the stress in the z-direction on a z-plane.

– Other stresses are shear stresses e.g. τzy is the stress in the y-direction on a z-plane.

Forces aligned with the direction of a coordinate axis are positive and opposite direction is negative.

Fig.5 Viscous stress aligned with the direction of a coordinate axis.

Forces in the x-direction

Fig.6 Forces aligned in the x- coordinate direction.

Net force in the x-direction is the sum of all the force components in that direction. So the rate of change of x-momentum for a fluid particle equal to the sum of the forces due to surface stresses shown in Fig.6, plus the body forces which are usually lumped together into a source term SMi and dividing by the volume δxδyδz results in:

( p  xx )  yx  zx ρ +  +SMx. (2.8) x y z

In equation (2.8) p is a compressive stress and is a tensile stress [1].

Similarly, for y- and z-momentum:

Dv  xy ( p  yy )  zy  +   SMY . Dt x y z (2.9) Dw   ( p  )  xz  yz  zz  S . Dt x y z MZ

2.5 Energy Equation

Conservation of Energy:-In the first law of thermodynamics is the heat added plus the total work done must be equal to the change in total energy of the system.

The work term includes the work on the boundaries due to normal and tangential stresses plus any direct work added to the system.

w = w normal stress + w shear stress + w direct.

Energy can be converted from one form to another but the total energy in a closed system remains constant.

Energy E = i + ½ (u2+v2+w2) = internal (thermal) energy + kinetic energy.

Potential energy (gravitation) is usually treated separately and included as a source term. we will derive the energy equation by setting the total derivative equal to the change in energy as a result of work done by viscous stresses and the net heat conduction.

Work Done by Surface Stresses in x-direction

Fig.7 Work done by surface stress in x-direction.

The total rate of work done by surface stresses for x-components is the sum of all terms in Fig.7 above and in similar fashion we can drive the y- and z-components.

Add all components of work done by surface stresses and divide by δxδyδz to get the work done per unit volume by the surface stresses:

u  u yx  u  v xy  v yy  v zy  w  w yz  w  ( 혂) + xx   zx     xz   zz . x y z x y z x y z

Energy Flux due to heat conduction

Fig.8 Energy flux due to heat conduction

The heat flux vector q has three components, qx, qy, and qz.

Summing all terms and dividing by δxδyδz gives the net rate of heat transfer to the fluid particle per unit volume.

Thus, energy flux due to conduction is given by:

q q q  x  y  z  qdiv. x y z (2.10)

Fourier’s law of heat conduction relates the heat flux to the local temperature gradient:

(2.11)

In vector form: q = grad (2.12)

Thus, energy flux due to conduction is: (κgradΤ). (2.13)

This is the final form used in the energy equation.

Setting the total derivative for the energy in a fluid particle equal to the previously derived work and energy flux terms, results the following energy equation:

(ρ혂) + ( grad ) + SE+

u xx  u yx  u zx  v xy  v yy  v zy  w xz  w yz  w zz            . (2.14)  x y z x y z x y z 

Note that we also added a source term SE that includes sources (potential energy, sources due to heat production from chemical reactions), etc.

2.6 Viscous Stress Tensor We will express the viscous stresses as functions of the local deformation rate (strain rate) tensor. There are two types of deformation:

Linear deformation rates due to velocity gradient

Volumetric deformation rates due to expansion or compression.

Using an isotropic (first) dynamic viscosity µ for the linear deformations and a second viscosity λ=-2/3 µ for the volumetric deformations results in.

( ) ( )

[ ] ( ) ( )

( ) ( ) [ ]

Note: div u = 0 for incompressible fluid.

By substituting viscous dissipation and the stresses in the internal energy equation and rearranging results:

(i) Internal energy: +div (ρ I 혂) = -p +div혂 + div (k grad T) + +Si. t

2.7 Navier-Stokes Equations Including the viscous stress terms in the momentum balance and rearranging, results in the Navier-Stokes equations:

u p X-Momentum:  + div (ρ u 혂) =  + div (µ grad u) +SMX. t x

v p Y-Momentum: + div (ρ v 혂) =   div (µ grad v) + SMY. t y

w p z- Momentum: + div (ρ w 혂) =  +div (µ grad w) +SMZ. t z

2.8 Summary of Equations in Conservation Form  Mass:  div (ρ혂) = 0. t

u  p x- Momentum:  div(u 혂) =  div (µ grad u) + S . t t mx

v  p y - Momentum:  div (ρ v 혂) =  div (µ grad v) + S . t y my

w)  p z- Momentum: + div (ρ w 혂) =  div (µ grad w) + S . t z mz

i Internal energy:  div (ρ I 혂) = -p div혂 + div (k grad T) +   S . t i

CHAPTER THREE

SOLUTIONS FOR LAMINAR FLOW OF INCOMPRESSIBLE VISCOUS FLUIDS Because of the great complexity of the full compressible Navare-Stoke equations, no known general analytical solution exists. Hence, it is necessary to simplify the equations either by making assumptions about the fluid, about the flow or about the geometry of the problem such as the flow is laminar, steady, two-dimensional, the fluid is incompressible with constant properties and the flow is between parallel plates in order to obtain analytical solutions. By so doing it is possible to obtain analytical, exact and approximate Solutions to the Navier-stokes equations. An analytical solution is obtained when the governing boundary value problem is integrated using the methods of classical differential equations. The result is an algebraic expression giving the dependent variable(s) as a function(s) of the independent variable(s). An exact solution is obtained by integrating the governing boundary value problem numerically. The result is a tabulation of the dependent variable(s) as a function(s) of the independent variable(s). An approximate solution results when methods such as series expansion technique are used to solve the governing boundary value problem.

Analytical and computational solutions to laminar flow problems are both feasible and common, and the need for experiments is often minimal. However, laminar flows are relatively rare both in nature and in engineering practice because it undergoes a transition to turbulent as flow speeds increase.

3.1 Steady-Laminar Flow Between Two Parallel Plates In Relative Motion (Couette Flow).

Consider a two-dimensional incompressible, viscous, steady- laminar flow between two parallel plates separated by certain distance (2h) and assumed that pressure is constant.

The upper plate moves with constant velocity ( V =U) while the lower is fixed and assumed that the plates are very wide and long so that the flow is essentially axial (u  0 ; v =w =0). Further, the flow is considered far downstream from the entrance so that it can be treated as fully- developed.

Fig.9: Fluid flow between two parallel plates in relative motion of the top plate.

Body forces are normal to the direction of flow and they do not affect the flow. The boundary conditions are independent of x and z (infinite plates); so the function of velocity is u=u(y) only. The Navier-stokes equation of motion for a Newtonian fluid with constant ρ and µ may be written as: u Continuity = 0 (3.1) x

d 2u Momentum 0 =µ (3.2) dy 2

Integrating Eq. (3.2) twice gives u = c1y + c2

Using the no slip boundary conditions u (-h) = 0 and u (+h) = U; we obtain the value of U U c1 = and c2 = from u = c1y+c2. 2h 2

U  y  u 1  y  U the velocity distribution is u = 1    1 and    (3.3) 2  h  U 2  h  2h

Fig.10: Velocity profile of Cuoette flow between parallel plates.

3.2 steady State, Laminar Flow between Stationary parallel plates (poiseuille flow) In the figure below, both the upper and lower plates are fixed (the velocity at the boundaries are zero). Assume x1 = x and x2 = y

Fig.11: Fluid flow between two fixed parallel plates.

dp The flow is entirely driven by the presence of a pressure gradient and the pressure gradient is dx dp taken to have a constant value. If = 0, then u= 0 and there is no flow. For a more viscous dx fluid, a greater pressure gradient is required to achieve a given rate of flow, so pressure has linear profiles. Body forces are perpendicular to the direction of flow and so do not affect the flow and temperature is also constant. From the Cartesian version of the differential equation of motion for a Newtonian fluid with constant ρ and µ we have the equation:

dp d 2u dp 0 = - + µ ( )  = (3.4) dx dy 2 dx

Eq.(3.4) is the second order partial differential equation and the constant pressure gradient  dp p  p between the entrance and exit is given by = 2 1 where L is the length of the plate. dx L

p p Since v=w= 0, the y and z Navier-Stoke equation leads to  0and  0 which implies y z p=p(x) only. Thus the pressure gradient in (3.4) is the only total gradient.

The pressure must decrease in the flow direction in order to drive the flow against resisting wall shear stress.

By integrating Eq. (3.4) twice with respect to y, we get:

1 dp u= y 2  c y  c 2 dx 1 2

Using the no-slip boundary conditions u=0; at y=  h, the value of the constants are c1= 0 and

1 dp 2 c2 =  h and substituting the value of c1 and c2 in the above equation gives 2 dx

1 dp dp u= y 2  h2  ,  0 (3.5) 2 dx dx

du The maximum velocity (u max) occurs at the centerline y=0 and the velocity gradient = 0 at dy the midpoint of the gap between the plates.

 dp  h2  dp umax = -    ,  0 (3.6)  dx  2  dx

Therefore, the velocity profile is parabolic as shown in figure below.

fig.12: The velocity profile of fluid flow between two stationary parallel plates.

The volume flow rate passing between the plates (per unit depth) is calculated from the relation ship as follows:

3 h 1  dp  2h  dp  q = ∫   y 2  h2 dy =    h   (3.7)  2  dx  3  dx 

If ∆p represents the pressure-drop between two points at a distance L along x-direction, then Eq. (3.7) is expressed as,

2h3  p  dp q =   since  0 (3.8) 3  L  , dx

The average velocity (uavg) can be calculated as follows:

q  p  2 uavg    umax (3.9) 2h  L  3

The wall shear stress for this case can also be obtained from the definition of Newtonian fluid is

 u v    dp  h2  y 2   dp   dp  2u      y h     1    y   h   max (3.10)      2    yh    x y  y  dx  2  h  yh  dx   dx  h

3.3 Axially Moving Concentric Cylinders (Couette Flow). These flows are named in honor of M. Couette (1890), who performed experiments on the flow between a fixed and moving concentric cylinder.

Fig.13: Axially moving concentric cylinders.

Consider two long concentric cylinders with a viscous fluid between them and either the inner cylinder of radius (r = r0) move axially at u = u0 or the outer cylinder of radius (r = r1) move at u = u1. Assumed the fluid motion is steady, pressure and temperature are constant. The only non-zero velocity is uz but u'θ = ur= 0, so u= u(r) is the only velocity function and the axial momentum equation reduce to:

1   u  2u = r  = 0 (3.11) r r  r 

Integrating Eq. (3.11) twice provides that u = c1ln(r) + c2. (3.12)

If the inner cylinder moves the no-slip conditions are u (r0) = u0 and u (r1) = 0 and

c1 = , c2 = -c1 ln (r1) ⁄

Then by substituting the value of c1 and c2 in Eq. (3.12) we obtain

u = ⁄ and from the definition shear stress (τ) = µ (3.13) ⁄ ⁄

Similarly, if the outer cylinder is moving, the boundary conditions u (r0) = 0 and u (r1) = u1

⁄ Then the solution is u = and τ = µ (3.14) ⁄ ⁄

If the inner cylinder moves, the velocity distribution u(r) is concave where as it is convex if the outer cylinder moves. The velocity profile is highly dependent on the value of r and a moving cylinder, where r is the distance between the two cylinders.

Fig.14: Velocity profile of a fluid flow between axially moving concentric cylinders.

3.4 Flows between Rotating Concentric Cylinder (Couette Flow)

Consider the steady flow between two rotating concentric cylinders with stationary outer and moving inner cylinder and assume the inner cylinder has radius r0, angular velocity ω0 and the outer cylinder has r1, ω1 respectively. Assumed pressure and temperature are constant. From the geometry the only non-zero velocity component is uθ and the variables uθ must be functions of radius r. so the equations of motion in cylindrical coordinate system reduced to:

d 2u d  u  θ- Momentum      =0. (3.15) dr 2 dr  r  the boundary conditions are at r = r0, uθ = r0ω0 and at r = r1, uθ = r1ω1

c2 By integrating Eq. (3.15) we obtain the velocity distribution uθ = c1r + (3.16) r which is the sum of a solid body rotation and a “potential” vortex whose laplacian is zero,

c2 2 2 Now at r= r0 , u r0   c1r0   r00  c1r0  c2  r0 0 (a) r0

c2 2 2 At r= r1,u r1   c1r1   r11  c1r1  c2  r1 1 (b) r1

By solving Eq. (a) and (b) simultaneously we obtain c1 and c2 as follows

r 2  r 2  r 2  r 2  c  1 0 0 0 and c  r 2  r 2  1 1 0 0  1 2 2 2 0 0 0  2 2  r1  r0  r1  r0 

Substituting the value of c1 and c2 in Eq. (3.16) gives the solution of the θ-momentum equation:

2 2 2 2 2 2 2 2  r1 1  r0 0  r0 0 r1  r0  r0 r1 1  r0 0  uθ = r  after rearrangement the velocity  r 2  r 2  r r 2  r 2  1 0   1 0  distribution is given by:

⁄ ⁄ ⁄ u = r ω ( ) r ω ( ⁄ ) (3.17) θ 0 0 ⁄ + 1 1 ⁄ ⁄ ⁄

This is the sum of inner and outer-driven flows. If the limit of the inner cylinder vanishes (r0, ω0

= 0) then, u θ = ω1r = linear from Eq.(3.17) , i.e. steady outer rotation of a tube filled with fluid induces solid- body rotation. If the limit of the outer cylinder becomes very large and remains fixed (r1 = ∞, ω1 = 0), then

2 r0 0 U θ = (3.18) r

Which is a potential vortex driven in a viscous fluid by the no –slip condition. Since the total fluid momentum in Eq. (3.18) is infinite, it would take infinite time to generate this flow by steady rotation of the inner cylinder.

If there is a very small clearance between the cylinders i.e. r1 - r0 « r0 and let the outer cylinder r  r / r  r / r   r / r  r / r  0 0 1 1  0 0  remains fixed, then the velocity equation u θ =  r11  becomes r1 / r0  r0 / r1   r1 / r0  r0 / r1 

u  r  r0  = 1 –  (3.19) r00  r1  r0 

This is a linear Couette flow between effectively parallel plates. In general the velocity distribution is strongly dependence on r (or the graph is more parabolic if the value of r is large and linear if r is small), where r is the distance between the two concentric cylinders.

Fig.15: Velocity distribution between stationary outer and rotating inner cylinders.

3.5 Steady Laminar Flow Through A Cylindrical Pipe.

The circular pipe is our most celebrated viscous flow, first studied by Hagen (1839) and Poiseuille (1840). Couette flows are driven by moving walls where as poiseuille flows are generated by pressure gradients with application primarly to ductes with low speed flow named after J.L.M poiseuille(1840).

Fig.16: Fluid flow through a cylindrical pipe.

Since pipe is cylindrically symmetric it makes sense to choose cylindrical coordinates.

The velocity of the fluid is in the z- direction and it varies in r-direction. This implies that the velocity in the r and θ direction are zero and the velocity in the z-direction is only non-zero velocity i.e. vr=vθ=0 and vz  0.

The boundary conditions are no slip i.e. at r=a, vz = 0 and at r = 0, vz= finite.

Since the flow is steady-state with the velocityv  vr  0 and vz = constant, then the continuity and momentum equation is written as:

 Continuity equation: v   0 . z z

p r- Momentum:  0 . (3.20) r

p θ- Momentum:  0. 

p    v  z- Momentum:  r z . z r r  r 

p    v  Since is a function of z only and r z  is a function of r only, then we can write the z- z r r  r  momentum equation in ordinary differential equation form equivalently as follows dp  d  dv   r z  = constant (3.21) dz r dr  dr  dp dp p  p p  p p = constant implies pz=linear and  2 1  2 1   , P is pressure drop dz dz z2  z1 L L but, the flow is in the positive z-direction, we need have P2. Thus, we often define

 dp p  p p dp p  d  dv   1 2      r z . (3.22) dz L L dz L r dr  dr 

Integrating the second order differential equation in Eq. (3.22) gives

 r 2 p Vz=  c lnr c . (3.23) 4 L 1 2

Now we solve for the constants c1 and c2 using the boundaries

 a2 p a2 p At r=a, vz= 0 i.e. Vz (a) =  c ln(a)  c  0  c   c ln(a) and 4 L 1 2 2 4 L 1 at r=0, vz=finite i.e. vr (0) = 0+ c1 ln(0)  c2 = finite, but ln (0)  . This implies that ln (0) must a 2 p converge and c1  0, otherwise ln (0) diverges and vz is not finite. So c1  0 and c2  4 L

 r 2 p a2 p a2 p  r 2  There for v (r) =   1 . (3.24 z  2  4 L 4 L 4 L  a 

This shows that the velocity profile is parabolic and the maximum velocity is obtained at r=0 i.e. a 2 p vmax= . (3.25) 4 L

Fig.17: Velocity profile for steady laminar flow through a cylindrical pipe.

The total volume flow rate q for any duct is defined by q= (v.n)dA (3.26) s

From (3.26) n= êz, v =vz(r) êz and dA =2 rdr which is the crossectinal area of a pipe.

2 2 a a a p  r  q = (vz (r) êz. êz) 2 rdr= 1 2 rdr. 0 0  2  4 L  a 

2 3 2 2 4 2 2 2 a p a  r  a p r r  a p  a a    a   q= r  2 dr =   2  =  . 2 L 0  a  2 L 2 4a 2 L  2 4     0  

a 4 p q = . 8 L (3.27)

Equation (3.27) shows q is strongly dependence on the radius of the pipe. And the average

q 2 velocity is defined vavg = . since dA=2 rdr, then A= a by integrating dA on the interval A 0,a

2 a p vMax so vvag = = (3.28) 8 L 2

The wall shear stress is constant and is given by

 p  a  p  vavg  w = µ  =    4 (3.29)  L  2  L  a

3.6 Combined Couette - Poiseuille Flow between Parallel Plates

Another simple planner flow can be developed by imposing a pressure gradient between a fixed and moving plate as shown in Fig. 18. Let the upper plate moves with constant velocity (v=U) and a constant pressure gradient dp/dx is maintained along the direction of the flow.

Fig. 18 Combined Couette-Poiseuille flow between parallel plates.

The Navier-Stokes combined equation and its solution will be the same as that of Poiseuille flow while the boundary conditions will change in this case;

du 2 dp 1  dp  y2     cons tant <0 and u =    c y + c (3.30) 2    5 6 dy dx   dx  2 

The constants can be found with two boundary conditions at the upper plate and lower plate;

At y= 0;u= 0 c6 = 0

U  b  dp  At y =b ;u=U c5 =    (3.31) b  2  dx 

After substitution of the constants, the general solution for Eq. (3.30) can be obtain

y 1  dp  u =U   y2  by b 2  dx 

u  y   b2  dp  y  or   1   1  (3.32) U  b   2U  dx  b 

The first part in the RHS of Eq. (3.32) is the solution for Couette wall-driven flow whereas the second part refers to the solution for Poiseuille pressure-driven flow. The actual velocity profile depends on the dimensionless parameter

 b2  dp  P=    (3.33)  2U  dx 

Several velocity profiles can be drawn for different values of P as shown in Fig. 19 With P= 0 , the simplest type of Couette flow is obtained with no pressure gradient. Negative values of P refers to back flow which means positive pressure gradient in the direction of flow.

Non dimentionalvelocity( )

Fig.19 Velocity profile of combined Couette- poiseuille flow between parallel plates.

CONCLUSION This short study gives an introduction to the analysis of steady- laminar and incompressible viscous fluids and defintions some basic terminologes related to fluid flow. The dimensionless Reynolds number is an important parameter of fluid flow equations to describe either the fully developed flow conditions lead to laminar or turbulent flow.

In this project work analytical solutions of fluid velocity have been discussed by using matlab codes. Particularilly we fined the solutions of simple Couette Flow and poiseuille Flow fluid problems.

The solutions for both the flows are exact solutions of Navier-Stokes equation. The velocity profile is linear for Couette flow with zero velocity at the lower plate with maximum velocity near to the upper plate and parabolic for Poiseuille flow with zero velocity at the top and bottom plate with maximum velocity in the central line.

It is not possible to treat many of the viscous phenomena which have been studied for future. Some topics omitted are mass transfer at the wall, temperature and pressure distributions,species diffusion due to dissociation, chemical reactions,three- dimentional flows,etc. for further studyof this interesting field of steady-laminar and incompressible viscous fluids.

References [1]André Bakker (2002 - 2006). Lecture – Notes on conservation Equations, Applied computational Fluid Dynamics.

[2] Avila K.; Moxey D.; de Lazar A.; Avila M.; Barkley D.; Hof B. (July 2011). The Onset of Turbulence in Pipe Flow Science. 333 (6039): 192–196. Bibcode: 2011Sci...333...192A

doi:10.1126/science.1203223.

[3] Batchelor, G. G. (2000). Introduction to Fluid Mechanics.

[4] Batchelor, G. K. (2000). An Introduction to Fluid Dynamics. Cambridge Mathematical Library series, Cambridge University Press. ISBN 0-521-66396-2.

[5] Geankoplis, Christie John (2003). Transport Processes and Separation Process Principles. Prentice Hall Professional Technical Reference. ISBN 978-0-13-101367-4.

[6]Kirby, B.J. (2010). Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices. Cambridge University Press. ISBN 978-0-521-11903-0.

[7] Kundu P. and Cohen I.(2010). Fluid Mechanics (Fourth ed.)

[8]Nave R.(2005).Laminar Flow. HyperPhysics. Georgia State University. Retrieved 23 November 2010.

[9] Noakes C.; Sleigh A. (January 2009). Real Fluids. An Introduction to Fluid Mechanics. University of Leeds. Retrieved 23 November 2010.

[10] Panton, Ronald L. (2013). Incompressible Flow (Fourth ed.). Hoboken: John Wiley & Sons. p. 114. ISBN 978-1-118-01343-4.

[11]. Rogers,D. F.(1992).Laminar flow analysis. Cambridge University Press. ISBN 0-521-41152-1.

Appendix Mat lab code for fig.10 clc; U=1; h=1; L=1; dx= 0.2; dy= 0.2; x=-1: dx: L; y=-1: dy: h; u=U/2*(1+y/h); plot (u, y,'-','color', 'r'); legend ('Analytical'); x label('Dimensionless velocity u/U'); y label('Dimensionless distance y/h');

Mat lab code for fig.12 clc; L=1; h=1; b=0.1; c=0.08; dy= 0.05; y=-1: dy: L; u= (-0.5*c/b)*(y.^2-h.^2); plot (u, y,'-'); legend ('Analytical'); x label(' velocity u'); y label('Dimensionless distance y/h');

Mat lab code for Fig.14 clc; h=2;

L=4; ro =1; r1=2; for Uo=2;

U1=0; dx=0.5; dy =0.5; x=0: dx: h; r=0: dy: L; a1=log (ro)-log (r1); a2= Uo*(log(r)-log (r1)); a3=a2/a1; u= Uo*log(r/r1)/log (ro/r1); end for Uo=0;

U1=2;

b1=U1*log(r/ro);

b2=log (r1/ro); b3=b1/b2; u2=U1*log(r/ro)/log (r1/ro); plot ( u, r,'-',u2,r,'-'); legend (' Analytical'); x label ('Dimensionless velocity u/U'); y label ('Dimensionless distance r');

Mat lab cod for fig.15 clc; m=1.5; w0=2; w1=0.1; r0=1; r1=2; dr= 0.05; r=1: dr: m; t1=r0*w0*(r1./r - r./r1)/(r1./r0-r0./r1); t2=r1*w1*(r./r0-r0./r)/(r1./r0-r0./r1); t3=t1./t2; u=t3; plot(u, r,'-'); legend ('Analytical'); y label('the gap distance r '); x label(' velocity u');

Mat lab code for fig.17 clc; L=1; a=1; b=0.01; c=0.05; dr= 0.2; r=-1: dr: L; u= (0.25*a^2*c/b)*(1-r.^2/a^2); plot(u, r,'-'); legend ('Analytical'); x label(' velocity u'); y label('Dimensionless distance r/a');