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CHAPTER 2

PRELIMINARIES

Discussion of Biological Terms:

2.1 Human Biliary System

The human biliary system consists of the organs and ducts i.e. common bile duct, cystic

duct and common hepatic duct, which are involved in the production and transportation

of bile. The transport of bile from the liver hepatocytes (bile is produced in hepatocytes)

to the small intestine is accomplished via a system of bile ducts that collectively are

called the biliary system. The biliary system consists of canaliculi; spaces between

adjacent hepatocyte cells. Further on towards the small intestine each canaliculus feeds

into a true bile duct. Small bile ducts join together to form a large and common bile duct,

which dumps bile into the duodenum (a part of the small intestine). A powerful muscle

known as a sphincter is located at the entry point to the small intestine, and functions to

control the flow rate of the bile [15].

The gallbladder and the ducts that carry bile and other digestive enzymes from the liver,

gallbladder, and pancreas to the small intestine are called the biliary system. The anatomy

of the human biliary system consists of-

1. The Liver

2. The Gallbladder

3. The Biliary Tract

Functions of the biliary system

The biliary system's main function includes the following:

· to drain waste products from the liver into the duodenum

· to help in digestion with the controlled release of bile.

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Composition of bile

Bile or gall is a bitter-tasting, dark green to yellowish brown fluid, produced by the liver

that aids the process of digestion of lipids in the small intestine. Bile is stored in the

gallbladder and upon eating is discharged into the duodenum. Bile is a composition of the

following materials: water, bile salts , bilirubin and cholesterol. Bile is the greenish-

yellow fluid (consisting of waste products, cholesterol, and bile salts) that is secreted by

the liver cells to perform two primary functions, including the following:

· to carry away waste

· to break down fats during digestion

Bile is produced in hepatocytes, which are the main functional cells of the liver. The bile

flows into the bile ducts. As bile flows through the bile ducts, it is modified by the

addition of secretions from the epithelial cells that line the ducts. The main route for

eliminating cholesterol from the body is through the bile. Cholesterol, being fatty in

composition, cannot dissolve in the water-based fluids that are present in the body Bile salt is

the actual component which helps in the breakdown and absorption of fats.The bile acids

and lecithin in bile allow cholesterol to dissolve into solution. If this mechanism goes wrong,

cholesterol can precipitate out of the bile solution, forming gallstones. Aside from its

function as waste disposal fluid, bile also helps neutralize stomach acid in the small intestine,

providing a more hospitable environment for enzymes that break down food. Bile acids

function to promote the complete digestion of food, by facilitating the uptake of fat-soluble

vitamins through the wall of the small intestine.

2.2 Liver

The liver is a vital organ present in vertebrates and some other animals. This organ plays

a vital role in metabolism and has a number of functions including glycogen storage,

decomposition of red blood cells, hormone production and detoxification. It produces

bile, an alkaline compound which helps in the digestion of lipids.

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2.3 The Biliary Tract

The biliary tract (or biliary tree) is the common term for the path by which bile is

secreted by the liver on its way to the duodenum.It is referred to as a tree because it

begins with many small branches which end in the common bile duct. Pressure inside the

biliary tree can give rise to gallstones.

Mechanism of bile flow in biliary tract

1. When the liver cells secrete bile, it is collected by a system of ducts that flow from the

liver through the right and left hepatic ducts. These ducts ultimately drain into the

common hepatic duct.

2. The common hepatic duct then joins with the cystic duct from the gallbladder to form

the common bile duct, which runs from the liver to the duodenum.

3.Then, when food is eaten, the gallbladder contracts and releases stored bile into the

duodenum to help break down the fats

The biliary tract comprises of the following three ducts:

· Cystic duct

· Common hepatic duct

· Common bile duct

Cystic Duct

The cystic duct is the short duct that joins the gallbladder to the common bile duct. The

presence of the “Valves of heister” in the lumen complicates the cystic duct geometry.

These valves consist of several semi lunar folds. The number of folds in the cystic duct

varies from 2 to 14.The cystic duct diameter ranges from 2 to 5 mm and its length from

60 mm.

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The human cystic duct system functions as a single conduit for the transport of bile

during both filling and emptying of the gall bladder.

Common Hepatic Duct

The common hepatic duct is the duct formed by the convergence of the right hepatic duct

and the left hepatic duct. The common hepatic then joins the cystic duct coming from the

duct to form the common bile duct.

Common Bile Duct

This duct carries bile from the gall bladder and liver into the duodenum (the upper part of

the intestine).

2.4. The Gallbladder

The gallbladder is a pear-shaped organ that stores and concentrates bile. It is

approximately 3 to 4 inches long and about 1 inch wide. The gallbladder stores about 50

mL of bile, which is released when food containing fat enters the digestive tract. The bile

emulsifies fats in partly digestive food. Humans have an organ close to the liver, the

gallbladder that can store and concentrate bile. Food stimulates the production of a hormone

called cholecystokinen, which acts to trigger the secretion of bile.

Function

The function of the gallbladder is to store and concentrates bile. The bile emulsifies fats

and neutralizes acids in partly digested food.

Conditions and Diseases of gallbladder

Sometimes the substances contained in bile (like cholesterol) crystallizes in the

gallbladder leading to the formation of stones called gallstones. They can cause

inflammation of gallbladder. Most common biliary diseases are:

· Cholelithiasis - the presence of gallstones and

· Cholecystitis – the inflammation of gallbladder

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2.5 Gallstones

Gallstones are small, pebble-like substances that develop in the gallbladder. The

gallbladder is a small, pear-shaped sac located below your liver in the right upper

abdomen. Gallstones form when liquid stored in the gallbladder hardens into pieces of

stone-like material. The liquid—called bile—helps the body digest fats. Bile is made in

the liver, then stored in the gallbladder until the body needs it. The gallbladder contracts

and pushes the bile into a tube—called the common bile duct—that carries it to the small

intestine, where it helps with digestion[31,32].

Bile contains water, cholesterol, fats, bile salts, proteins, and bilirubin—a waste product.

Bile salts break up fat, and bilirubin gives bile and stool a yellowish-brown color. If the

liquid bile contains too much cholesterol, bile salts, or bilirubin, it can harden into

gallstones.

The two types of gallstones are cholesterol stones and pigment stones. Cholesterol stones

are usually yellow-green and are made primarily of hardened cholesterol. They account

for about 80 percent of gallstones. Pigment stones are small, dark stones made of

bilirubin. Gallstones can be as small as a grain of sand or as large as a golf ball. The

gallbladder can develop just one large stone, hundreds of tiny stones, or a combination of

the two.

Gallstones can block the normal flow of bile if they move from the gallbladder and lodge

in any of the ducts that carry bile from the liver to the small intestine. The ducts include

the hepatic ducts, which carry bile out of the liver, cystic duct which takes bile to and

from the gallbladder, common bile duct, which takes bile from the cystic and hepatic

ducts to the small intestine.

Bile trapped in these ducts can cause inflammation in the gallbladder or in the ducts.

Other ducts open into the common bile duct, including the pancreatic duct, which carries

digestive enzymes out of the pancreas. Sometimes gallstones passing through the

common bile duct provoke inflammation in the pancreas—called gallstone pancreatitis—

an extremely painful and potentially dangerous condition.

If any of the bile ducts remain blocked for a significant period of time, severe damage or

infection can occur in the gallbladder, liver, or pancreas. Left untreated, the condition can

be fatal. Warning signs of a serious problem are fever, jaundice, and persistent pain.

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Causes of gallstones

Scientists believe cholesterol stones form when bile contains too much cholesterol, too

much bilirubin, or not enough bile salts, or when the gallbladder does not empty

completely or often enough. The reason these imbalances occur is not known. The stones

tend to develop in people who have liver cirrhosis, biliary tract infections, or hereditary

blood disorders— such as sickle cell anemia—in which the liver makes too much

bilirubin.

The mere presence of gallstones may cause more gallstones to develop. Other factors that

contribute to the formation of gallstones, particularly cholesterol stones, include-

1. Sex. Women are twice as likely as men to develop gallstones. Excess estrogen

from pregnancy, hormone replacement therapy, and birth control pills appears to

increase cholesterol levels in bile and decrease gallbladder movement, which can

lead to gallstones.

2. Family history. Gallstones often run in families, pointing to a possible genetic

link.

3. Weight. A large clinical study showed that being even moderately overweight

increases the risk for developing gallstones. The most likely reason is that the

amount of bile salts in bile is reduced, resulting in more cholesterol. Increased

cholesterol reduces gallbladder emptying. Obesity is a major risk factor for

gallstones, especially in women.

4. Diet. Diets high in fat and cholesterol and low in fiber increase the risk of

gallstones due to increased cholesterol in the bile and reduced gallbladder

emptying.

5. Rapid weight loss. As the body metabolizes fat during prolonged fasting and

rapid weight loss—such as “crash diets”—the liver secretes extra cholesterol into

bile, which can cause gallstones. In addition, the gallbladder does not empty

properly.

6. Age. People older than age 60 are more likely to develop gallstones than younger

people. As people age, the body tends to secrete more cholesterol into bile.

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7. Ethnicity. American Indians have a genetic predisposition to secrete high levels

of cholesterol in bile. In fact, they have the highest rate of gallstones in the United

States. The majority of American Indian men have gallstones by age 60.

8. Cholesterol-lowering drugs. Drugs that lower cholesterol levels in the blood

actually increase the amount of cholesterol secreted into bile. In turn, the risk of

gallstones increases.

9. Diabetes. People with diabetes generally have high levels of fatty acids called

triglycerides. These fatty acids may increase the risk of gallstones

Symptoms of gallstones

As gallstones move into the bile ducts and create blockage, pressure increases in the

gallbladder and one or more symptoms may occur. Symptoms of blocked bile ducts are

often called a gallbladder “attack” because they occur suddenly. Gallbladder attacks often

follow fatty meals, and they may occur during the night. A typical attack can cause:

· Steady pain in the right upper abdomen that increases rapidly and lasts from 30

minutes to several hours

· Pain in the back between the shoulder blades

· Nausea

· Vomiting

· Sweating

· Restlessness

Diagnoses of gallstones

Frequently, gallstones are discovered during tests for other health conditions. When

gallstones are suspected to be the cause of symptoms, the doctor is likely to do an

ultrasound exam—the most sensitive and specific test for gallstones. A handheld device,

which a technician glides over the abdomen, sends sound waves toward the gallbladder.

The sound waves bounce off the gallbladder, liver, and other organs, and their echoes

make electrical impulses that create a picture of the gallbladder on a video monitor. If

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gallstones are present, the sound waves will bounce off them, too, showing their location.

Other tests may also be performed like:

· Computerized tomography (CT) scans. The CT scan is a noninvasive x ray

that produces cross-section images of the body. The test may show the gallstones

or complications.

· Cholescintigraphy (HIDA scan). The patient is injected with a small amount of

non harmful radioactive material that is absorbed by the gallbladder, which is

then stimulated to contract. The test is used to diagnose abnormal contraction of

the gallbladder or obstruction of the bile ducts.

· Endoscopic retrograde cholangiopancreatography (ERCP). ERCP is used to

locate and remove stones in the bile ducts.

· Blood tests. Blood tests may be performed to look for signs of infection or

obstruction.

2.6 Treatment of Gallstones

Surgical

Surgery to remove the gallbladder is the most common way to treat symptomatic

gallstones. (Asymptomatic gallstones usually do not need treatment.) Each year more

than 500,000 Americans have gallbladder surgery. The surgery is called

cholecystectomy.

Nearly all cholecystectomies are performed with laparoscopy. After giving medication to

sedate the patient, the surgeon makes several tiny incisions in the abdomen and inserts a

laparoscope and a miniature video camera. The camera sends a magnified image from

inside the body to a video monitor. While watching the monitor, the surgeon uses the

instruments to carefully separate the gallbladder from the liver, bile ducts, and other

structures. Then the surgeon cuts the cystic duct and removes the gallbladder through one

of the small incisions. Recovery after laparoscopic surgery usually involves only one

night in the hospital, and normal activity can be resumed after a few days at home.

Because the abdominal muscles are not cut during laparoscopic surgery, patients have

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less pain and fewer complications than after “open” surgery, which requires a 5- to 8-inch

incision across the abdomen.

If tests show the gallbladder has severe inflammation, infection, or scarring from other

operations, the surgeon may perform open surgery to remove the gallbladder. In some

cases, open surgery is planned. Recovery from open surgery usually requires 3 to 5 days

in the hospital and several weeks at home. Open surgery is necessary in about 5 percent

of gallbladder operations. The most common complication in gallbladder surgery is

injury to the bile ducts. An injured common bile duct can leak bile and cause a painful

and potentially dangerous infection. Mild injuries can sometimes be treated non

surgically. Major injury, however, is more serious and requires additional surgery.

Nonsurgical Treatment

Nonsurgical approaches are used only in special situations—such as when a patient has a serious

medical condition preventing surgery—and only for cholesterol stones. Stones commonly recur

within 5 years in patients treated non surgically.

· Oral dissolution therapy. Drugs made from bile acid are used to dissolve gallstones.

The drugs ursodiol (Actigall) and chenodiol (Chenix) work best for small cholesterol

stones. Months or years of treatment may be necessary before all the stones dissolve.

Both drugs may cause mild diarrhea, and chenodiol may temporarily raise levels of blood

cholesterol and the liver enzyme.

· Contact dissolution therapy. This experimental procedure involves injecting a drug

directly into the gallbladder to dissolve cholesterol stones. The drug—methyl tert-butyl

ether— can dissolve some stones in 1 to 3 days, but it causes irritation and some

complications have been reported. The procedure is being tested in symptomatic patients

with small stones.

Prevention of Gallstones

Such factors that increase the risk of developing gallstones, such as age, sex and ethnic

origin cannot be altered. However, it is possible that having a vegetarian diet may reduce

the risk of developing gallstones. Vegetarians have a significantly lower risk of

developing gallstones, compared to people who eat meat. Many experts say that a diet

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low in fat and high in fruit and vegetables, including plenty of dietary fiber may help

protect people from developing gallstones. Controlling the bodyweight may also help

prevent the formation of gallstones. However, crash dieting and rapid weight loss are risk

factors in the development of gallstones.

Life without Gallbladder

Fortunately, we can live without our gallbladder. The liver produces enough bile to digest

a normal diet. If a person's gallbladder is removed the bile reaches the small intestine

from the liver through the hepatic ducts, rather than being stored in the gallbladder. A

small proportion of patients who have had their gallbladder removed will experience

softer and more frequent stools for a while because their bile flows into the small

intestine more often.

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Preliminaries of Fluid Dynamics

2.7 Fluid Dynamics

Fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow-the

science of fluids (liquids and gases).

Fluid

The defining property of the fluid lies in the ease with which they can be deformed. The

fluid consists of liquids and gases. The most important difference between them is there

compressibility [19]. Gases can be compressed much more easily. Fluids are assumed to

obey the continuous assumption. Fluids are composed of molecules that collide with one

another and solid objects. However, the continuum assumption considers fluids to be

continuous, rather than discrete. Consequently, properties such as density, pressure,

temperature, and velocity are taken to be well-defined at infinitesimally small points, and

are assumed to vary continuously from one point to another.

2.8 Compressible and Incompressible Fluid

All fluids are compressible to some extent that is changes in pressure or temperature will

result in changes in density. However, in many situations the changes in pressure and

temperature are sufficiently small that the changes in density are negligible.

· Compressible fluid: its density can change (eg, gases)

· Incompressible fluid: its density is constant (eg, liquids)

· 2 2.9 Viscous and Inviscid fluid

An infinitesimal fluid element is acted upon by two types of forces, namely,body forces

and surface forces. The body force is proportional to the mass of body on which it acts

while the surface force is proportional to the surface area on which it acts. The surface

force is resolved into two types of components, one normal and the other tangential to the

surface element on which surface force acts.The normal force per unit area is called

normal stress or pressure while the tangential force per unit area is said to be the shearing

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stress.A fluid is said to be viscous when normal as well as shearing stresses exist. On the

other hand, a fluid is said to be inviscid when it does not exert any shearing stress.

2.10 Viscosity

Viscosity is a measure of the resistance of a fluid which is being deformed by either shear

or tensile stress. In everyday terms (and for fluids only), viscosity is "thickness" or

"internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick",

having a higher viscosity. Put simply, the less viscous the fluid is, the greater its ease of

movement (fluidity).

Viscosity describes a fluid's internal resistance to flow and may be thought of as a

measure of fluid friction [33]. With the exception of superfluids, all real fluids have some

resistance to stress and therefore are viscous, but a fluid which has no resistance to shear

stress is known as an ideal fluid or inviscid fluid. It can be said that: Internal friction

(viscosity) implies-

· Viscous fluid: there is internal friction

· Non-viscous fluid : internal friction is negligible

· Ideal fluid: incompressible and non-viscous

In general, in any flow, layers move at different velocities and the fluid's viscosity arises

from the shear stress between the layers that ultimately oppose any applied force. The

relationship between the shear stress and the velocity gradient can be obtained by

considering two plates closely spaced at a distance y, and separated by a homogeneous

substance. Assuming that the plates are very large, with a large area A, such that edge

effects may be ignored, and that the lower plate is fixed, let a force F be applied to the

upper plate. If this force causes the substance between the plates to undergo shear flow

with a velocity gradient u/y (as opposed to just shearing elastically until the shear stress

in the substance balances the applied force), the substance is called a fluid.

Figure 2.1

The applied force is proportional to the area and velocity gradient in the fluid:

where µ is the proportionality factor called

This equation can be expressed in terms of shear stress

differential form by Isaac Newton

between layers is proportional to the

the layers:

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Figure 2.1 Layers at different velocity

The applied force is proportional to the area and velocity gradient in the fluid:

is the proportionality factor called dynamic viscosity.

This equation can be expressed in terms of shear stress /F At = . Thus as expressed in

Isaac Newton for straight, parallel and uniform flow, the shear stress

ional to the velocity gradient in the direction perpendicular

The applied force is proportional to the area and velocity gradient in the fluid:

Thus as expressed in

and uniform flow, the shear stress

perpendicular to

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Types of viscosity

Newton's law of viscosity, given above, is a constitutive equation. It is not a fundamental

law of nature but an approximation that holds in some materials and fails in others. Non-

Newtonian fluids exhibit a more complicated relationship between shear stress and

velocity gradient than simple linearity. Thus there exist a number of forms of viscosity:

· Newtonian: fluids, such as water and most gases which have a constant viscosity.

· Shear thickening: viscosity increases with the rate of shear.

· Shear thinning: viscosity decreases with the rate of shear. Shear thinning liquids

are very commonly, but misleadingly, described as thixotropic.

· Thixotropic: materials which become less viscous over time when shaken,

agitated, or otherwise stressed.

· Rheopectic: materials which become more viscous over time when shaken,

agitated, or otherwise stressed.

· A Bingham plastic is a material that behaves as a solid at low stresses but flows as

a viscous fluid at high stresses.

· A magnetorheological fluid is a type of "smart fluid" which, when subjected to a

magnetic field, greatly increases its apparent viscosity, to the point of becoming a

viscoelastic solid.

Viscosity, the slope of each line, varies among materials.

Viscosity coefficients

Viscosity coefficients can be defined in two ways:

· Dynamic viscosity, also absolute viscosity (units Pa·s, Poise, P);

· Kinematic viscosity is the dynamic viscosity divided by the density (units cm2/s,

Stokes, St).

Figure 2.2

Units of viscosity

The usual symbol for dynamic viscosity used

mu (µ).The symbol η is also used

is the pascal-second (Pa·s), (equivalent to N·s/m

dynamic viscosity is the poise

1 P = 0.1 Pa·s and 1 cP = 1 mPa·s = 0.001 Pa·s

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Figure 2.2 Graph between strain and stress.

The usual symbol for dynamic viscosity used by fluid dynamicists — is the Greek letter

is also used by physicists. The SI physical unit of dynamic viscosity

(Pa·s), (equivalent to N·s/m2, or kg/ (m·s)). The cgs physical unit

poise (P).Also expressed as centipoise (cP).

0.1 Pa·s and 1 cP = 1 mPa·s = 0.001 Pa·s

is the Greek letter

of dynamic viscosity

physical unit for

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2.11 Newtonian and Non-Newtonian fluid

Fluids that obey Newton’s law of viscosity are known as Newtonian Fluids. Newton’s

law states that the shear stress is proportional to velocity gradient.

i.e T=µ (du/dy) here µ is the viscosity.Common fluids like water, air and mercury

are all Newtonian Fluids.

Non -Newtonian Fluid are those that do not obey Newton’s law of viscosity. Fluids like

coal tar and paints are Non-Newtonian fluids.

2.12 Steady and Unsteady Flow

A steady flow is one in which the conditions (velocity, pressure and cross-section) may

differ from point to point but do not change with time.Whearas in unsteady flow, if at any

point in the fluid, the conditions change with time, the flow is described as unsteady(in

practice there is always slight variations in velocity and pressure, but if the average

values are constant, the flow is considered steady.

2.13 Laminar Flow and Turbulent flow

A flow, in which each fluid particle traces out a definite curve and the curve traced out by

any two different fluid particles do not intersect, is said to be laminar. On the other hand,

a flow, in which each fluid particle does not trace out a definite curve and the curve

traced out by fluid particles intersect, is said to be turbulent.

2.14 The Equation of Continuity

A basic principle of science and engineering is the conservation of mass. The continuity

equation is an expression of this basic principle in a particularly convenient form for the

analysis of materials processing operations.

The basic principle states that fluid can neither be created nor destroyed. In continuous

motion, the equation of continuity expresses the fact that the increase in the mass of the

fluid within any closed surface drawn in the fluid in any time must be equal to the excess

of the mass that flows in over the mass that flows out.

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Let u(x,y,z,t) , v(x,y,z,t) ,w(x,y,z,t) and p(x,y,z,t) denote respectively the three velocity

components and the pressure at the point (x,y,z) at time t in a fluid with constant density ρ and viscosity coefficient µ. Then ,the equation of continuity, which expresses the fact that the amount of fluid entering a unit volume per unit time is the same as the amount of the fluid leaving it per unit time, is given by in Cartesian co-ordinates as

0

u v wx y z¶ ¶ ¶

+ + =¶ ¶ ¶

Let us consider a fluid particle at M whose cylindrical coordinates are ),,( zr q where

.,20,0 ¥<<-¥££³ zr pq The equation of continuity in cylindrical coordinates is

given by-

z( v )( v ) ( v )1 1

0r r z

rrt r

qrr rrq

¶¶ ¶¶+ + + =

¶ ¶ ¶ ¶

Where ,rv vq and zv be the velocity components. For axi-symmetric case, we take vq =0,

and we also take ,r zv v and p to be independent of θ.In this case, the equation of continuity

is given by

z( v ) ( v )1

0r z

rrr

¶ ¶+ =

¶ ¶

2.15 The Equation of Motion

The equations of motion are obtained from Newton’s second law of motion which states

that the product of mass and acceleration of any fluid element is equal to the resultant of

all the external body forces acting on the element and to the surface forces acting on the

fluid volume due to the action of the remaining fluid on the same element. The equations

of motion, known as Navier –Stokes equations, for the flow of a Newtonian viscous

incompressible fluid are

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2 2 2

2 2 2( ) ( )

u u u u p u u uu v w X

t x y z x x y zr m¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶

+ + + = - + + +¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶

2 2 2

2 2 2( ) ( )

v v v v p v v vu v w Y

t x y z y x y zr m¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶

+ + + = - + + +¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶

2 2 2

2 2 2( ) ( )

w w w w p w w wu v w Z

t x y z z x y zr m¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶

+ + + = - + + +¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶

where u(x,y,z,t) , v(x,y,z,t) ,w(x,y,z,t) and p(x,y,z,t) are the three velocity components

and the pressure at the point (x,y,z) at time t in a fluid with constant density ρ and

viscosity coefficient µ. And X, Y and Z are the external body forces.