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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.