1.1
Motivation for the study
Tribology is the science of interacting surfaces in relative motion. It is
specifically concerned with the friction, wear and lubrication of these surfaces.
Although tribology has conventionally been associated with the surface interaction
of mechanical systems, concepts of tribology have also been important in the study
of biological systems. Biotribology is one of the newest fields to emerge in the
discipline of tribology. It can be described as the study of friction, wear and
lubrication of biological systems, mainly synovial joints such as the human hip and
knee. Synovial joint is a perfect tribological creation of the nature with low friction
and wear resistance acting without reparation during service. Hence the application
of principles of hydrodynamic lubrication theory in medicine and biology is a
rapidly expanding field. The human joint is a self acting and dynamic load bearing
structure, which uses a porous and elastic biomaterial as well as highly nonNewtonian lubricants for its functioning.
Leonardo da vinci (1452-1519) named as the father of modern tribology.
Later, Nikolai Pavlovich petro and Osborne Reynolds around 1880, who recognized
the hydrodynamic nature of lubrication, and introduced a theory of fluid-film
lubrication. Till today, Reynolds steady state equation of fluid film lubrication
F
v
is valid for hydrodynamic lubrication of thick films where the frictional
D
force F, is proportional to both the sliding velocity v, and the bulk fluid viscosity  ,
and inversely proportional to the film lubricant thickness D.
The scientific study of lubrication began with Rayleigh together with Stokes
discussed the feasibility of theoretical treatment of film lubrication. The
1
understanding of hydrodynamic lubrication began with the classic experiments of
Tower (1884), in which the existence of a film was detected from measurements of
pressure within the lubricant, and of Petrov (1883), who reached the same
conclusion from friction measurements. Reynolds (1886) closely followed this
work, in his analytical paper he used a reduced form of the Navier-Stoke’s
equations in association with the continuity equation to generate a second-order
partial differential equation for the pressure in the narrow, conversing gap between
bearing surfaces. This pressure enables a load to be transmitted between the
surfaces with extremely low friction, since the surfaces are completely separated by
a fluid film.
1.1.1 Biological Bearings
Human skeleton is a living framework inside the body and is made up of
bones that are strong and light. The skeleton supports us but also lets us move. It
also surrounds the softer parts of the body and helps to stop them from being
injured. The human skeleton is made up of bones of different shapes and sizes.
These bones are link together to form the skeleton, which shapes the body (Walker,
2005). Mobility in humans is made possible by bone and muscles acting together.
However, mobility also requires joints. Joints are the structures that connect the
bones of the skeleton. They can be classified by the degree of movement they
permit. Those that allow little movement are slightly moveable. Those that allow no
movement are immovable joints (Chiras, 2011). There are three main basic types of
joints, fibrous (immoveable), cartilaginous (partially moveable), and the synovial
(freely moveable) joint. These joints can be explained as follows
2
 Fibrous Joints (Immoveable joints): These joints allow no movement, and
are composed of fibrous (dense) connective tissue. The skull sutures
and syndesmoses such as the connection between the tibia and fibula are
fibrous joints.
 Cartilaginous joints (Partially moveable joints): These joints allow very
little movement, and are characterized by a connection between adjoining
bones made of cartilage. The pubic symphysis, intervertebral joints and
connection between the first rib and sternum are slightly movable
cartilaginous joints.
 Synovial joints (freely moveable joints): The synovial joints are by far the
most common classification of joints within the human body. They are
highly moveable, and all have a membrane (the inner layer of capsule)
which secretes synovial fluid (a lubricating liquid) and cartilage known as
hyaline cartilage which pads the ends of the articulating bones. The hip joint
is example of synovial joint.
Figure 1.1 is a simple schematic diagram of a synovial joint. The two bones
that make up the synovial joint are covered by cartilage referred to as articular
cartilage. The synovial membrane covers the non-cartilaginous surfaces of the joint
capsule or cavity. The synovial membrane secretes synovial fluid in to the joint
capsule. The synovial fluid provides nutrients for the articular cartilage it also acts
as a lubricant. The synovial joints allow for articulation of the surfaces with a
minimal amount of friction or wear. The joint is made up of several complex
components some of these will be discussed in the subsequent sections.
3
Figure 1.1: A schematic diagram of a synovial joint
There are of 6 types of synovial joints, which can be distinguished by their function
and range of movement
1. Hinge joint
These joints occur when the convex surface of bone fits into the
concave surface of the other. It allows for a flexion and extension movement
in one plane only. Examples of these types of joints are the elbow and
phalanges. The representation of a hinge joint is shown in Fig.1.2.
Figure 1.2: A representation of a hinge joint
4
2. Ball and Socket joint
In these types of joints, the head of one bone fits into the cup-like
socket of another bone. This allows for freedom of movement in all
directions, from the single common centre between the bones. Examples of
ball and socket joints are the hip and shoulder. The representation of ball
and socket joint is shown in Fig. 1.3.
Figure 1.3: A representation of ball and socket joint
3. Condyloid joint
This joint is similar to a ball and socket joint the head of one bone
fits into an oddly shaped depression of the other bone. It allows movement
in different planes, but not rotational. An ellipsoidal joint is a form of a
condyloid joint. Examples of condyloid joints are the wrist (ellipsoidal) and
the femoral-tibial joint. The representation of a condyloid joint is shown in
Fig. 1.4.
5
Figure 1.4: A representation of condyloid joint
4. Pivot joint
In this joint, the end of one bone rotates around the axis of another
bone. Generally, the cylindrical surface of one bone rotates within the ring
of the other. Examples of these types of joints are between the atlas and axis
in the neck and between the radius and the ulna in the forearm. The
representation of pivot joint is shown in Fig. 1.5.
Figure 1.5: A representation of pivot joint
6
5. Saddle joint
In this joint, both bones have concave and convex regions. The
opposing surfaces complement each other. The movement of a saddle joint
is similar to that of a condyloid joint and allows movement in two planes
but not rotation. An example of this type of joint is the thumb. The
representation of a saddle joint is shown in Fig. 1.6.
Figure 1.6: A representation of saddle joint
6. Plane or Gliding joint
This joint allows for sliding or gliding movement of flat or nearly
curved bones as one bone moves over the other. It allows for both back and
forth and twisting movements. The movement of such joints is limited, in
many cases by the surrounding ligaments. Examples of these types of joints
are those between the carpal bones it also occurs between vertebrae and
between the tarsal bones. The representation of a plane joint is shown in Fig.
1.7.
7
Figure 1.7: A representation of a plane joint
In this thesis two types of synovial joints have been investigated namely human
knee joints and hip joints.
1.1.2 Synovial Knee Joint
The human knee joint is considered to be one of the most complicated
articulations within the human body. It is a synovial joint between the condyles of
the femora and tibia, which consists of the patella and femur with the trochlear on
the surface (Harris, 1985). The knee joint plays an essential role over a lifetime,
which is potentially more than 70 years under healthy circumstances.
Figure 1.8 shows an anatomical model of the human knee joint. The knee
joint is the pivotal hinge joint that permits flexion and extension as well as a slight
medial and lateral rotation. The knee joint is very vulnerable since it supports nearly
the whole weight of the body. A knee joint consists of the following components.
The thin anterior and posterior capsule is enhanced on the femoral condyles side by
strong lateral ligaments. Extracapsular and intracapsular ligaments are located on
both sides, which reinforce the attached capsules. The synovial membrane is not
located on the articulating surfaces, as is commonly understood, but covers the
8
cruciate ligaments, the infrapatellar fat pad and the popliteus tendon. Bursae around
the knee together with the synovial membrane excrete synovial fluid which
lubricates the cavum articulate to maintain the knee joints normal function (Harris,
1985). Therefore, the synovial membrane plays an important role in sealing in the
synovial fluid and generating ample glucose to nourish the cartilage. The articular
surfaces of femur, tibia and patella are all covered with hyaline cartilage. The
capsule of knee joint is attached to the margins of articular surfaces. It surrounds
the sides and the posterior aspect of the joint.
Figure 1.8: Schematic diagram of synovial knee joint
In addition, the hyaline cartilage that covers the articular surfaces is a
protective and active material. The femur and tibia are separated by articular
cartilage and lubricated through synovial fluid. Articular cartilage contains a large
proportion of water, extracellular collagen matrix and cells. The thickness of the
9
articular cartilage varies within different joints, and from area to area in the same
joint. Articular cartilage is supposed to remain stable under physiological activities
during human life and plays the role of a safeguard. However, the thickness of
articular cartilage might be diminished and localized fissures and lesions may form
as a result of collagen fibrillation and synovial fluid lubrication transition. As a
result, various bone diseases including osteoarthritis can occur in this situation
(Freeman, 1976). Therefore, it is very important to study cartilage denaturalization
under collagen fibrillation and synovial fluid lubrication mechanisms.
1.1.3 Hip Joint
The hip joint is one of the most important joints in the human body. It
allows us to walk, run, and jump. It bears our body’s weight and the force of the
strong muscles of the hip and leg. Yet the hip joint is also one of our most flexible
joints and allows a greater range of motion than all other joints in the body except
for the shoulder. The hip joint is a ball and socket joint, formed by the head of the
femur and the acetabulum of the pelvis. The dome shaped head of the femur forms
the ball, which fits snuggly into the concave socket of the acetabulum. The surface
of the femur is covered with a thin layer of hyaline cartilage which acts to allow
smooth movement of the joint. Hyaline cartilage lines both the acetabulum and the
head of the femur, providing a smooth surface for the moving bones to glide past
each other. Hyaline cartilage also acts as a flexible shock absorber to prevent the
collision of the bones during movement. Between the layers of hyaline cartilage,
synovial membranes secrete watery synovial fluid to lubricate the joint capsule. The
joint is physically large, and it is capable of carrying remarkably high loads. Simple
models of the geometry and musculature of the joint indicate that the load carried
10
by the hip joint, as a person stands motionless on one leg, is on the order of 2.5
times his body weight (Inman and Verne, 1947). A much more detailed analysis of
the forces acting on the hip has shown that, for a typical young adult, the hip joint
can be high as 5 or 6 times his body weight during walking (Paul, 1967). The load
carrying surfaces of both ball and socket are covered by a thin layer of elastic
cartilage which is lubricated by a synovial fluid. The schematic diagram of human
hip joint is shown in Fig. 1.9.
Figure 1.9: A schematic diagram of hip joint
11
1.1.4 Articular Cartilage
Cartilage is flexible connective tissue found in many areas in the bodies of
humans, such as the nose, ear and joints. Articular cartilage is the hyaline type that
covers the ends of the long bones in the knee, shoulder and hip joint to facilitate
load- carriage and lubrication within the synovial environment. Cartilage is
composed of specialized cells called chondroblasts that produce a large amount
of extracellular matrix composed of collagen fibers, abundant ground substance rich
in proteoglycan, and elastin fibers. Cartilage is classified in three types, elastic
cartilage, hyaline cartilage and fibrocartilage, which differ in the relative amounts
of these three main components (Pratt Rebecca, 2012).
Cartilage is a white connective tissue which is synthesized and maintained
by cells called chondrocyte. In human joints, the thickness of the articular cartilage
layer varies from 0.5 to 1.5 mm in upper extremity joints, such as the hand and the
shoulder and from 1 to 6 mm in lower extremity joints, such as the hip, knee, and
ankle (Ateshian and Hung, 2006). Under normal conditions, articular cartilage
provides low friction and wear over a life span. It is a highly hydrated tissue, with a
porosity varying from 68 to 85 per cent in adult joints (Mow and Rik Huiskes,
2005).
Normal healthy cartilage is an avascular, alymphatic and aneural tissue
found at the end of articulating bones. It is responsible for providing a load-bearing,
low friction interface for diarthrodial joints. The cartilage tissue is classified into
four zones: superficial or tangential, middle or transitional, deep or radial and
calcified zone, where each of those zones is characterized by depth-varying cellular
12
density and extracellular matrix properties (Jadin et al. 2005). The cell density of
the superficial zone is high and chondrocytes are arranged parallel to the surface of
cartilage. Chondrocytes are randomly dispersed in the middle zone and a lower
cellular density can be observed. In the deep zone, chondrocytes have a columnar
organization (Hunziker et al. 2002). It has been demonstrated that a tight interaction
exists between those zones to regulate cell proliferation and secretion (Blewis 2007).
All zones are building by chondrocytes, which are sparsely distributed within the
negatively charged cartilaginous extracellular matrix (Laver-Rudich and Silbermann,
1985).
The superficial zone of the normal cartilage is the most distinct and reaches
up to 200 μm. Its upper most layer, only about 3 μm thick, is called the lamina
splendors, is made of non-banded, randomly arranged fine collagen filaments of 4–
12 nm in diameter. Below this layer there are sheets of a well identified period, each
sheet with predominantly parallel, banded collagen fibrils, their diameter increasing
with depth.
Articular cartilage is elastic, fluid-filled and backed by a relatively
impervious layer of calcified cartilage and bone. This means that load induced
compression of cartilage will force interstitial fluid to flow laterally within the
tissue and to surface through adjacent cartilage. As that area in turn becomes load
bearing, it is partially protected by the newly expressed fluid above it. One of the
important factors for the degeneration of the cartilage surfaces of human joints is a
breakdown of the mechanism of natural lubrication. Hence it is important to make
an attempt to understand the role of synovial fluid as a lubricant and the factors
which causes the fluid to lose its ability.
13
1.1.5 Synovial Fluid
Normal synovial fluid is clear, pale yellow, viscid and does not clot.
Studies of mammalian synovial fluid have found considerable similarities among
species, although notable differences do exist. The majority of investigative work
determining the composition of synovial fluid has been performed on bovine
synovial fluid mainly because large quantities of it are available.
Synovial fluid is a plasma dialysate modified by constituents secreted by
the joint tissues. The major difference between synovial fluid and other body fluids
derived from plasma is the high content of hyaluronic acid (mucin) in synovial
fluid. The exact source of the hyaluronic acid has been the subject of debate .It is
generally assumed, however, that both fibroblasts beneath the synovial membrane
intima and synovial membrane–living cells produce this mucopolysaccharide
constituent of synovial fluid. Hyaluronic acid is a nonsulfated polysaccharide
composed of equimolar quantities of D-glucuronic acid and N-acetyle-Dglucosamine residues (Curtiss, 1964). It was first identified in joint fluid by acetic
acid precipitation. The normal viscosity of synovial fluid is due to the hyaluronic
acid.
Synovial fluid is believed to have two main functions: to aid in the
mechanical functions of joints by lubrication of the articulating surface. Articular
cartilage has no blood, nerve, or lymphatic supply (Curtiss, 1964). Glucose for
articular cartilage chondrocyte energy is transported from the periarticular
vasculature to the cartilage by the synovial fluid. Understanding conditions, the
glucose concentrations of synovial fluids usually approximately equal to that of
14
blood (Curtiss, 1964). A decreased amount of synovial fluid glucose may be
associated with articular diseases, particularly septic and immune mediated arthritis
(Cohen et al.1975).
Distribution of synovial fluid electrolytes is in accord with the Donnan
theory of membrane equilibrium of solute, adding credence to the theory that
synovial fluid is indeed a plasma dialysate (Curtiss, 1964 and Perman, 1980). The
distribution ratio of most electrolytes between synovial fluid and serum is 1. The
average ratio of total anions of serum to total anions of synovial fluid is 1.99. Ionic
calcium ratios are approximately 1.18. This higher ratio is thought to be due to the
base combining power of mucin for calcium (Perman, 1980).
The normal volume of synovial fluid obviously varies from joint to joint .In
the dog , the average is 0.24ml (0.01ml-1.0ml) (Sawyer 1963). Experimental work
with dogs has shown that the pH is lowered by exercise and returns to a higher
value at rest. As noted above, the viscosity of synovial fluid is due to the hyaluronic
acid precipitation of synovial fluid mucin with weak acetic acid (mucin clot test)
which leaves a fluid with a viscosity similar to water. The lubricating ability of
synovial fluid is often equated with its normal viscosity. However, experimentally
induced trypticdigestion of hyaluronic acid will destroy the synovial fluid
lubricating abilities without altering its lubricating capacity (McCutchen, 1978).
Almost all of the protein constituents of synovial fluid are derived from plasma.
The passage of plasma proteins to synovial fluid is related to the size and shape of
the protein molecule. Most proteins with molecular weights less than 100,000
Daltons are readily transferred from one fluid space to another (Perman, 1980).
Vascular permeability is altered by inflammation, which accounts for protein
15
content changes in diseased synovial fluid. Immunoglobulin, immune complexes
and complement are produced by cells accumulating in the inflamed synovial
membrane and periarticular lymph nodes and find their way to the synovial fluid.
Normal synovial fluid complement levels in humans are approximately 10%
of the serum values. In the inflamed joint synovial fluid complement levels will
vary. The long term patterns of variation have some prognostic value in human
rheumatoid arthritis patients (Cohen et al, 1974). The proteins of coagulation are
not found in normal synovial fluid, while proteins of the plasmin system may be
found in variable quantities (Perman, 1980). Normal synovial fluid does not clot but
may exhibit thixotropy, (McCutchen, 1978) the property of certain gels to become
fluid when shaken. On standing at room temperature, normal synovial fluid may
assume a gelatin–like appearance. When shaken it will resume its normal fluid
nature.
Many enzymes have been found in the normal synovial fluid nature of
domestic animals and humans. Alkaline phosphatase, acid phosphatase lactic
dehydrogenase, and other enzymes are present in detectable quantities (Perman,
1980). Synovial fluid to serum ratios of these and other enzymes vary with the
species studied and with the presence of articular disease. Enzymes enter the
synovial fluid directly from the plasma or may be produced locally by the synovial
membrane or released by synovial fluid macrophages.
16
1.2
Types of Lubrication
Lubrication is defined as the process of adding a solid, liquid, or gas to
reduce friction and/or wear at the interface between two surfaces which are in
relative motion. In the synovial joint, lubrication is provided by the synovial fluid,
which has been previously described. Joint lubrication has been studied extensively,
with over two dozen proposed theories. Several types of the lubrication mechanism
are believed to occur in the functioning of human joints e.g., Hydrodynamic,
Boundary, Elastohydrodynamic and Weeping.
1.2.1 Hydrodynamic Lubrication
Hydrodynamic lubrication occurs when non-parallel rigid bearing surfaces
lubricated by a fluid film move tangentially with respect to each other (i.e. slide on
each other), forming a converging wedge of fluid. A lifting pressure is generated in
this wedge by the fluid viscosity as the bearing motion drags the fluid into the gap
between the surfaces.
In 1932 MacConaill first proposed the hydrodynamic lubrication for
synovial joints. MacConaill hypothesized that a wedge shaped film would be
formed between the articulating joint surfaces by the synovial fluid, hence bearing
the load and preventing friction and wear of the joint surfaces. However, there was
no explanation for the low friction of joints starting from rest and he had not taken
into account the incongruent nature of the opposing surfaces. Also, it seemed
unlikely since hydrodynamic lubrication requires low loads and high surface
velocities which are not found in the human synovial joints. Hydrodynamic
lubrication is a poor model for synovial joints; however, it may occur during the
17
high speed non-accelerating rotatary motion of the femur during the swing phase of
gait.
Hydrodynamic lubrication was one of the earliest modes of lubrication
implicated within the joint capsule. This mechanism requires uninterrupted motion
in the same direction to maintain the integrity of the wedge and is especially
efficient in bearings. These requirements indicate that pure hydrodynamic
lubrication is unlikely to be solely applicable to the synovial joint which requires
frequent changes of direction or efficient functioning at slow speeds, up to 0.1 m/s
(Dowson 1973; 1990) or at rest. Further problems with this mechanism are
highlighted when one examines the fluid film thickness: minimum fluid thickness is
in the order of 10-7- 10-3 µm during the peak loading period of a normal walking
cycle (Mow and Mak, 1987). When compared to the average height of the healthy
cartilage asperities (approximately 2-6 µm) (Dowson, 1990) it is obvious that
hydrodynamic lubrication alone would rapidly result in considerable wear of the
cartilage and is inadequate under the conditions within the synovial joint.
Hydrodynamic lubrication is very effective in machinery in which shafts
generally rotate continuously in one direction. It cannot work so well in human
joints which never keep turning in one direction for more than a small fraction of a
revolution and are constantly stopping and starting. Boundary lubrication could
work in such circumstances but human and animal joints have much lower
coefficients of friction than boundary lubrication can be expected to give squeeze
film lubrication is probably important in activities such as walking in which the
load is taken off the joints once in each step, allowing the cartilage surfaces to
separate and synovial fluid to flow in between them.
18
1.2.2 Boundary Lubrication
In boundary lubrication, the bearing surfaces are not separated by the
lubricant and the load is carried by the asperity contacts. Thin surface films may
form on the bearing surfaces through absorption or chemical reaction and therefore
the contact lubrication mechanism is dependent on the physical and chemical
properties of these thin surface films. Although the friction in this case is less than
for the unlubricated condition, it can still be approximately two orders of magnitude
higher than for full fluid film lubrication. Due to the contacting of the asperities, the
wear rate for boundary lubrication is also expected to be higher than for full fluid
film lubrication.
1.2.3 Elastohydrodynamic Lubrication
In this type of lubrication, the bearing surfaces are separated by a highly
pressurized fluid film that causes elastic deformation of the solid surfaces. Hence,
the pressure and thickness of the hydrodynamic film depend on both the rheological
properties of the fluid and the elastic deformation of the bearing surfaces. The high
film pressure increases the fluid viscosity, producing a tendency for both the
thickness and the shear resistance of the fluid film to increase due to the exponential
dependence of viscosity on pressure.
1.2.4 Weeping Lubrication
Weeping lubrication is a form of hydrostatic lubrication in which interstitial
fluid of articular cartilage flows out onto its surface when a load is applied to it. The
cartilage act as a self-pressurizing sponge when the pressure is released the fluid
flows back into the cartilage.
19
In this thesis the hydrodynamic lubrication characteristics of squeeze film
bearings in general and synovial joints in particular are presented in the subsequent
chapters.
1.3
Lubricants
Lubricant is any substance that reduces friction and wear, providing smooth
running and satisfactory life for machine elements. Most lubricants are liquids such
as mineral oils, synthetic esters, silicon fluids and water. But the lubricants may
also be solids that are used in dry bearings, greases for use in rolling element
bearings, or gases for use in gas bearings. Another approach to reducing friction
and wear is to use bearings such as ball bearings, roller bearings or air bearings,
which in turn require internal lubrication themselves, or to use sound, in the case
of lubrication. In addition to industrial applications, lubricants are used for many
other purposes. Other uses include cooking (oils and fats in use in frying pans, in
baking to prevent food sticking), bio-medical applications on humans (e.g.
lubricants for artificial joints), ultrasound examination etc.
Solid lubricants are a thin film of solid interposed between two rubbing
surfaces. They provide effective lubrication by shearing easily to prevent wear and
maintaining low coefficients of friction. Most solid lubricants can shear easily due
to their lamellar crystalline structure allowing ease of sliding between layers. They
contain close, strongly bonded layers connected by only weak forces between each
layer. It is thought that the strong bonding within the layer helps reduce wear
damage. Common examples of lamellar solid lubricants are graphite and
molybdenum disulfide (Erdemir, 2001). Evidence has been presented which
20
suggests that surfactant found in synovial fluid acts in the same way as lamellar
solid lubricants, such as graphite in joint prosthesis (Purbach et al. 2002; Hills
2002).
The natural lubricant, called synovial fluid, is a clear, viscous fluid which
serves three purposes: it lubricates the articulating surfaces, carries nutrients to the
cartilage cells, or chondrocytes, and transports waste products away from the
cartilage. Lubrication of synovial surfaces by synovial fluid requires hyaluronate
and is due to boundary lubrication occurs when each bearing surface is coated or
impregnated with a thin layer of lubricant that keeps the sliding surface apart,
allowing case of motion with a low coefficient of friction between the sliding
surfaces. Hyaluronate sticks to the synovial surfaces. The lubricating properties of
synovial fluid in a soft tissue system are directly related to the concentration and
molecular weight of the hyaluronate, which is also determined by viscosity.
However, it is not the viscosity of synovial fluid that is responsible for lubrication
of this system but the stickiness or boundary phenomena exhibited by the fluid.
Viscous solutions containing no hyaluronate do not lubricate a soft tissue system
nearly as well as solutions containing hyaluronate of equal or even lower viscosity.
The lubricating properties of synovial fluid on articular cartilage were
originally attributed to its viscosity, which in turn is due to the presence of
hyaluronate or mucin. However, viscosity or the resistance of a fluid to shearing
forces is not the same as lubricating effectiveness. Digestion of synovial fluid
hyaluronate by hyaluronidase, which totally destroys the viscous nature of the fluid,
does not decrease the lubricating properties of synovial fluid on articular cartilage
when compared with a non viscous buffer. This is in contrast to the finding that
21
proteolytic digestion of synovial fluid on decreases its lubricating abilities. A
glycoprotein has been isolated from the fluid deprives of its lubricating properties.
1.3.1 Classification of Lubricants
In general the lubricants are classified in to two types viz; Newtonian and
non-Newtonian.
a) Newtonian fluid
The viscosity  is measured by the slope of stress-shearing rate curve. For
natural fluids like water, air, oil and so on viscosity does not vary with rate of
strain. That is fluids with constant viscosity are known as Newtonian fluids. For
Newtonian fluids shear is linearly proportional to the rate of strain.

u
y
 
u
y
where  is the shear stress,  is the proportionality constant known as Newtonian
viscosity. A fluid that behaves according to Newton’s law of viscosity, with a
viscosity  that is independent of the stress, is said to be Newtonian fluid. For
example water, air, benzene, ethyl alcohol etc.
b) Non-Newtonian fluid
A fluid in which shear stress is not proportional to the rate of shear strain is
known as non-Newtonian fluid. Generally non-Newtonian fluids are those in which
22
the value of  , the Newtonian viscosity is not constant and are represented by a
non-linear flow curve. Non-Newtonian fluids are complex mixtures such as
ketchup, blood, pastes, gels, gravy, pie fillings, mud, printer ink etc. The nonNewtonian fluids are mainly classified into three categories.
i) Time-independent fluids
The time independent fluids are those fluids which are characterized by the
fact that the shear rate depends only on the shear stress, provided the temperature of
the fluid remains constant, and is a single valued function of it. Fluids with these
properties may be described by a relation of the form
u
 f  
y
Depending upon the above equation, three possibilities exist:
1. Shear thinning or Pseudoplastic fluids
Viscosity decreases with increasing velocity gradient. e.g.
polymer solutions and blood. At low shear rates u y the shear
thinning fluid is more viscous than the Newtonian fluid, and at high
shear rates it is less viscous. Good examples of shear thinning fluids are
paints.
2. Shear thickening or Dilatants fluids
In shear thickening fluids viscosity increases with an increase in
shear rate. Examples of shear thickening fluids are a mixture of cornstarch and water.
23
3. Viscoplastic fluids
The shear stress must exceed a certain yield value, yield stress,
before the fluid deforms and flows.
ii) Time dependent fluids
The viscosity of the fluid varies with the time of shear as well as the shear
rate. Such fluids are known as time dependent fluids. These are the fluids whose
strain rate is a function of time and are very complex in nature. A practical example
of time dependent fluid is the waxy crude oil which not only exhibit time-dependent
behavior but also exhibits a yield stress.
iii) Viscoelastic fluids
Fluids which exhibit both viscous and elastic characteristics when
undergoing deformation are known as viscoelastic fluids. These fluids are similar to
Newtonian but if there is a sudden large change in shear they behave like plastic.
Some fluids have elastic properties, which allow them to spring back when a shear
force is released. e.g. egg white.
In this thesis the non-Newtonian synovial fluid is modeled as a couplestress
fluids and micropolar fluids that have been considered to analyze the performance
characteristics of squeeze film bearings.
1.4
Couplestress Fluid
The long chain polysaccharide hyaluronic acid molecules present in the
synovial fluid motivates for modeling of the synovial fluid as a Stokes (1966)
24
couple-stress fluid in particular and the lubricants with additives in general. The
increasing use of fluids containing microstructures such as those containing
additives, suspensions, granular matter or long-chained polymer has been
emphasized due to the development of modern machine apparatus. Couple stresses
are found to appear in noticeable magnitude in liquids with very large molecules.
The additives stabilize the flow properties and minimize the sensitivity of the
lubricant for changes in the shear rate. The simplest generalization of the classical
theory of fluids by Stokes (1966) provides the macroscopic description of the
behavior of fluids containing a substructure such as lubricants with polymer
additives. This couple stress fluid model is intended to account the particle size
effects and is important for applications of pumping fluids such as animal bloods,
liquid crystals, polymer thickened oils and complex fluids.
The Stokes (1966) microcontinuum theory has been extensively used to
study the effect of couple stresses on the performance of various bearing systems
for the last few decades. Ramanaiah and Sarkar (1978) studied the optimum load
capacity of a slider bearing lubricated by a fluid with couplestress. They showed
that the optimum load capacity increases with the couplestress parameter. The
squeeze films between finite plates lubricated by fluids with couplestress have been
theoretically studied by Ramanaiah (1979) and found that the squeeze time
increases if a fluid with couplestress is used as the lubricant. Bujurke and
Jayaraman (1982) presented the influence of couple stresses in squeeze films. They
found that the bearings with couplestress fluid as lubricant provide significant load
supporting capacities which results in longer bearing life. Gupta and Sharma (1988)
have analyzed the effect of couplestress in hydrostatic thrust bearing. Bujurke et al.
25
(1990) analyzed the effect of couple stresses in squeeze film poro-elastic bearings
with special reference to synovial joints. They reported that the influence of
elasticity is to increase the load capacity and yield a longer squeeze film time. The
theoretical analysis of squeeze film action in porous layered bearings is studied by
Bujurke and Patil (1991) and presented the appropriate mathematical model to
characterize the poroelastic nature of the porous matrix and the non-Newtonian
aspect of the lubricant and have found that, the load capacity is higher as compared
with that of classical viscous lubrication. The influence of elasticity is to increase
the load capacity and yield a longer squeeze film time. Bujurke et al. (1992)
analyzed the influence of couple stresses on the dynamic properties of a double
layered porous slider bearing. The squeeze film characteristics of long partial
journal bearings lubricated with couple stress fluids was analyzed by Lin (1996).
He pointed out that, the presence of couple stresses provides an enhancement in the
load carrying capacity, and lengthens the response time of the squeeze film action
of the system as compared to the Newtonian lubricant case. Lin (1966a) studied the
couple stress effects on the squeeze film characteristics of hemispherical bearings
with reference to synovial joints. Effects of couple stresses on the lubrication of
finite journal bearings were presented by Lin (1997). He concluded that the effects
of enhanced the load-carrying capacity, as well as reduced friction parameter and
the attitude angle. Lin (1998) analyzed squeeze film characteristics of finite journal
bearings: couplestress fluid model and reported that, the presence of couple stresses
improves the characteristics of finite journal bearings operating under pure squeeze
film motion. Static and dynamic characteristics of externally pressurized circular
step thrust bearings lubricated with couplestress fluids were presented by Lin
(1999). Squeeze film characteristics between a sphere and a flat plate: couplestress
26
fluid model was presented by Lin (2000) and found that, the couplestress parameter
produce an increase in the load-carrying capacity and the response time as
compared to the classical Newtonian lubricant case. The mathematical modeling of
some biological bearings with couplestress fluids is studied by Walicki and Walicka
(2000). Naduvinamani et al. (2001) studied squeeze film lubrication of a short
porous journal bearing with couplestress fluids. They observed that, the lubricants
which sustain the couple stresses yield an increase in the load carrying capacity.
Naduvinamani et al. (2002) studied the squeeze film lubrication of a narrow porous
journal bearing with a couplestress fluid. Lin et al. (2003) has made the derivation
of dynamic couplestress Reynolds equation of sliding squeezing surfaces and
numerical solution of plane inclined slider bearings. They observed that the effects
of couple stresses provide an improvement on both the steady-state performance
and the dynamic stiffness and damping characteristics especially for the bearing
with a higher value of aspect ratio. Ma et al. (2004) analyzed a study of
dynamically loaded journal bearings lubricated with non-Newtonian couplestress
fluids. They found that the couplestress fluids lubrication improves the bearing
performance under dynamic loads. Ahmad and Singh (2007) studied a model for
couple-stress fluid film mechanism with reference to human joints. Separation of a
sphere from flat plate in the presence of couplestress fluids was studied by
Elasharkawy and Fadhalah (2008). They showed that, the additive characteristic
length has negligible effect on the thickness of the lubricant film at the separation
point. Naduvinamani and Patil (2009) presented numerical solution of finite
modified Reynolds equation for couplestress squeeze film lubrication of porous
journal bearings. They observed that, under cyclic load, the effect of couplestress is
to reduce the velocity of the journal centre and to increase the permissible height of
27
the squeeze film. Lin et al. (2010) studied the non-Newtonian couple stress effects
on the frictional and flow rate performances of wide composite slider bearings.
They observed that, comparing with the Newtonian lubricant composite slider
bearing case, the effects of non-Newtonian couple stresses provide a reduction in
values of the friction parameter and the volume flow rate required. These
improvements on bearing characteristics are more emphasized with increasing
values of the couple stress parameter. Non-Newtonian couple stress poroelastic
squeeze film was analyzed by Nabhani et al. (2013). They found that, the numerical
results of the simulations show that all these effects have significant influences on
the porous squeeze film performance.
In the present thesis the lubrication characteristics of synovial joints with
couple stress fluid as lubricant is studied.
1.5
Micropolar Fluids
Fluids which exhibit certain microscopic effects arising from the micro
rotations of the fluid elements are called micro fluids. These fluids can support
stress moments and body moments and are influenced by the spin inertia. “A
subclass of micro fluids is called micropolar fluids which exhibit the microrotational effects and micro-rotational inertia also can support couple stress and
body couples only”. The behavior of synovial fluid is considered to be governed by
the micropolar fluid theory proposed by Eringen (1966). Physically they may
represent adequately the fluids consisting of bar like elements. Certain anisotropic
fluids, e.g. liquid crystals which are made up of dumbbell shaped molecules are of
this type. In fact, animal blood happens to fall into this category. Other polymeric
28
fluids and fluids containing certain additives may be represented by the
mathematical model underlying micropolar fluids. Recent experiments with fluids
containing extremely small amount of polymeric additives indicate that the skin
friction near a rigid body in such fluids are considerably lower than the same fluids
without additives. Thus the study of micropolar fluids has received considerable
attention due to their applications in a number of processes that occur in industries
such as the extrusion of polymer fluids, solidification of the liquid crystal, cooling
of metallic plate in a bath, exotic lubricants and colloidal and suspension solution.
Several investigators used the micropolar fluid for the study of many
different bearing systems. Agarwal et al. (1972) studied the squeeze film and
externally pressurized bearings lubricated with micropolar fluids and formed that
the time of approach is more for the micropolar fluids as compared to the
corresponding Newtonian fluids. The properties of micropolar lubricant were
examined by Maiti (1973) in reference to composite and stepped slider bearing. The
kinematic properties of potential flow and slow motions are investigated by
Indrasena (1973) by considering the motion of a steady, in viscid and
incompressible micropolar fluid of constant gyration parameter in the absence of
body forces and body couples. Another review article on the applications of
microcontinuum fluid mechanics was presented by Ariman et al. (1974). The
lubrication theory for micropolar fluids and its applications to a journal bearing is
presented by Prakash and Sinha (1975). In their study, the orders of magnitude
arguments are used to reduce the governing balance equations to a system of
coupled ordinary differential equations and are solved subject to appropriate
boundary conditions. The cyclic squeeze films in micropolar fluid lubricated
29
journal bearings is presented by Prakash and Sinha (1976) and have derived the
Reynolds equation for the general case of dynamic loading for fluid suspensions,
using the micropolar fluid theory. The optimum slider profile of a slider bearing
lubricated with a micropolar fluid is studied by Ramanaiah and Dubey (1977).
Zaheeruddin and Isa (1978) presented the study of micropolar fluid lubrication of
one dimensional journal bearings. The lubrication with micropolar liquids and its
application to short bearings is studied by Tipei (1979). Agarwal and Bhatt (1980)
made a theoretical study of a porous pivoted slider bearings lubricated with
micropolar fluid and proved that, their load capacity is greater than with Newtonian
fluid. Sinha and Singh (1981) applied the Eringen theory for the study of
lubrication of an inclined stepped composite bearing lubricated with lubricant
containing additives. Their results were in good agreement with experimental
observations. Nigam et al. (1982) studied the micropolar fluid film lubrication
between two parallel plates with reference to human joints. They studied that, the
effects of size and concentration variation of the hyaluronic acid molecules on the
mechanism of synovial joint. Prakash and Tiwari (1982) presented an analysis of
the squeeze film between porous rectangular plates including the surface roughness
effects. These authors shown that, the nominal geometry as characterized by the
aspect ratio of the plates has a profound effect on the system. Sinha et al. (1983)
studied the variation in viscosity with temperature in journal bearing lubricant with
additives using the micropolar theory. They established that, the additives in
lubricant increase the temperature in journal bearing. The analytical solution of the
problem of squeeze film lubrication of micropolar fluid between two parallel plates
is given by Bujurke et al. (1987). Singh et al. (1988) considered a model for
micropolar fluid film mechanism with reference to human joints. The performance
30
of finite journal bearings lubricated with micropolar fluids is analyzed by Khonsari
and Brewe (1989). The finite Reynolds equation for micropolar fluids is solved by
applying central finite difference scheme. It is shown that although the frictional
force associated with micropolar fluid is in general higher than that of a Newtonian
fluid, the friction co-efficient of micropolar fluids tends to be lower than that of
Newtonian. Huang and Wang (1990) studied the dynamic characteristics of finite
width journal bearings with micropolar fluid and have analyzed the dynamic
characteristics of finite width journal bearings lubricated with a micropolar fluid by
linear stability theory. The stiffness and damping coefficients and the critical
stability parameter for the micropolar fluid are obtained and it is reported that, the
normal stiffness coefficient is larger while the damping coefficient is smaller for
micropolar lubricants.
A new generalization of the Reynolds equation for a micropolar fluid and its
application to bearing theory is presented by Bessonov (1994). Khonsari (1994)
studied the effect of viscous dissipation on the lubrication characteristics of
micropolar fluids and shown that the heat generation due to viscous dissipation
plays an important role on the load carrying capacity of a journal bearing lubricated
with micropolar fluids. Lin (1996) analyzed the hydrodynamic lubrication of
journal bearings including micropolar lubricants and the three dimensional
irregularities. On the conical whirls instability of hydrodynamic journal bearing
lubricated with micropolar fluids is presented by Das et al. (2001). It is found that,
for any micropolar lubrication condition, the bearing is always stable as the ratio of
the moment of inertia approaches the value of conical whirl ratio. Static
characteristics of a circular journal bearing operating with micropolar lubricant
considering the effect of deformation of bearing liner were computed by Nair et al.
31
(2004). They have shown that static characteristics are greatly affected by the
volume concentration of additives in the lubricant. Das et al. (2005) have analyzed
the linear stability analysis of hydrodynamic journal bearing under micropolar
lubrication. Naduvinamani and Marali (2007) presented the dynamic Reynolds
equation for micropolar fluids and the analysis of plane inclined slider bearings
with squeezing effect and have shown that the micropolar fluids provide an
improved characteristics for both steady state and the dynamic stiffness and
damping characteristics. It is found that, the maximum steady load carrying
capacity is a function of the coupling parameter. Naduvinamani and Siddanagouda
(2008) studied the porous inclined stepped composite bearings with micropolar
fluid. It is observed that, the micropolar fluid lubricants provide an increased load
carrying capacity and decreased coefficient of friction as compared to the
corresponding Newtonian case. Naduvinamani and Santosh (2009) analyzed
theoretically the micropolar fluid squeeze film lubrication of short partial porous
journal bearing. Rahmatabadi et al. (2010) showed that the significant enhancement
in static performance for different configuration of non-circular bearings by
considering micropolar lubrication. Naduvinamani and Santosh (2011) studied the
micropolar fluid squeeze film lubrication of finite porous journal bearing. It is
observed that the micropolar fluid effect significantly increases the squeeze film
pressure and the load carrying capacity as compared to the corresponding
Newtonian case. Naduvinamani and Archana (2013) analyzed the effect of viscosity
variation on the micropolar fluid squeeze film lubrication of a short journal bearing.
In the present thesis the micropolar fluid squeeze film lubrication between rough
anisotropic poroelastic rectangular plates with a special reference to synovial joint
lubrication is analyzed.
32
1.6
Surface Roughness
Roughness
is
one
of
the
most
important
surface
topographic
characterizations, which intuitively refers to the unevenness or irregularity of a
texture. It gives an idea of how smooth the surface is at a certain length scale.
Roughness is dependent on the vertical and horizontal resolution of the measuring
instruments. It is also a function of working length scale.
The study of the effects of surface roughness on hydrodynamic lubrication
of various bearing systems has been a subject of growing interest, mainly because,
in practice, most of the bearing surfaces are rough. The aspect ratio and the absolute
height of the asperities and valleys observed under microscope vary greatly,
depending on material properties and on the method of surface preparation. In
general, the height of roughness asperities is of the same order as the mean
separation in a lubricated contact.
The surface roughness of cartilage will be considered briefly from the point
of view of its relevance to lubrication: to the naked eye, the surface appears
glistening, smooth and free from noticeable unevenness, irregularities and
roughness. However, the issue of cartilage surface roughness has been the source of
considerable controversy. Most, if not all observations made with the scanning
electron microscope, whether a cartilage itself (Walker et al. 1968) or on cast
replicas (Dowson et al. 1968) show some surface irregularities. Ghadially (1983)
explains the presence of surface asperities as artifacts of tissue preparation;
however, there are a number of earlier studies which describe surface roughness.
Davies et al. (1962) suggests that the surface is very smooth; with irregularities in
33
the range of 0.02µm. Dowson et al. (1968), Jones and Walker (1968) found much
greater roughness, which increased with age. The values ranged from about 1 µm
for foetal cartilage to about 2.7 µm for adult cartilage. Further study by Sayles et al.
(1979) reported average roughness between 1 µm and 6 µm. The study of Dowson
(1990) indicated that, roughness asperities between 2-6 µm are commonly present is
generally accepted.
Stochastic models for hydrodynamic lubrication of rough surfaces were
studied by Christensen (1969-70). On the basis of stochastic theory, he developed
two different forms of Reynolds-type equation corresponding to two different types
of surface roughness. It is shown that the mathematical form of these equations is
similar but not identical to the form of the Reynolds equation governing the
behavior of smooth, deterministic bearing surfaces. It is shown that surface
roughness may considerably influence the operating characteristics of bearings and
that the direction of the influence depends upon the type of roughness assumed. The
effects are not, however, critically dependent upon the detailed form of the
distribution function of the roughness heights. Christensen (1971) presented some
aspects of the functional influence of surface roughness in lubrication. He displayed
that, surface roughness has a considerable effect on the functional characteristics of
a bearing operating in the hydrodynamic, and, especially, in the mixed lubrication
regime. Waviness and roughness in hydrodynamic lubrication was presented by
Tonder and Christensen (1972). They observed that the corrugation wavelength is a
major factor, pressure ripples vanishing with increasing corrugation density. It is
further shown that at the same time, the load-carrying capacity tends towards that
predicted by the authors' statistical roughness theory, the analysis thus constituting
34
a numerical proof of the mathematical soundness of that theory. An analysis of the
squeeze film between porous rectangular plates including the surface roughness
effects was studied by Prakash and Tiwari (1982). They observed that the nominal
geometry as characterized by the aspect ratio of the plates has a profound effect on
the system. Guha (1993) presented the analysis of dynamic characteristics of
hydrodynamic journal bearings with isotropic roughness effects. The effect of
roughness on the behavior of squeeze film in a spherical bearing is studied by
Gupta and Deheri (1996). A note on squeeze film between rough anisotropic porous
rectangular plates was analyzed by Bujurke and Naduvinamani (1998). They found
that the loci of maximum load are more sensitive to the anisotropic permeability
than the roughness parameter. Gururajan and Prakash (1999) presented surface
roughness effects in infinitely long porous journal bearings. Effect of surface
roughness in a narrow porous journal bearing was studied by Gururajan and
Prakash (2000). It is shown that the results are significantly different than those for
the case of an infinitely long journal bearing. The surface roughness effects on the
oscillating squeeze film behavior of long partial journal bearings are studied by Lin
et.al. (2002) on the basis of Christensen Stochastic model and found that the effect
of circumferential roughness provides a reduction in the mean bearing eccentricity
ratio as compared to smooth bearing case where as the longitudinal roughness
structure results in a reverse trend. Naduvinamani et al. (2003) analyzed effect of
surface roughness on characteristics of couplestress squeeze film between
anisotropic porous rectangular plates. They found that the surface roughness effects
are more pronounced for couplestress fluids as compared to the Newtonian fluids.
Chiang et al. (2004) presented linear stability analysis of a rough short journal
bearing lubricated with non-Newtonian fluids. They observed that when compared
35
to the smooth bearing lubricated with couple stress fluid, a decrease in the threshold
speed is found in the case of transverse roughness. Bujurke and Kudenatti (2006)
presented surface roughness effects on squeeze film poroelastic bearings. Finitedifference-based multigrid method is used for the solution of Reynolds equation.
The method has greatest advantage of minimizing the errors using correction
schemes in obtaining accurate solution as grid size tends to zero. The influences of
roughness and elasticity on bearing characteristics are discussed. Bujurke et al.
(2007) studied the effect of surface roughness on squeeze film poroelastic bearings
with special reference to synovial joints. Bujurke et al. (2007) investigated the
wavelet-multigrid analysis of squeeze film characteristics of poroelastic bearings.
They found that the poroelastic bearings with couple-stress fluid as lubricant
provide enhancement in pressure and ensure the increased load carrying capacity
compared with viscous fluids. This may be one of the reasons in the efficient
lubrication and proper functioning of synovial joints. Naduvinamani and Biradar
(2008) analyzed the surface roughness effects on the static and dynamic behavior of
squeeze film lubrication of short journal bearing with micropolar fluids. Numerical
solution of finite modified Reynolds equation for couple stress squeeze film
lubrication of porous journal bearings was analyzed by Naduvinamani and Patil
(2009). They found that the applied load is considered as a sinusoidal function of
time to simulate the bearings operating under cyclic loads. Under a cyclic load, the
effect of couple stress is to reduce the velocity of the journal centre and to increase
the minimum permissible height of the squeeze film. Naduvinamani and Kashinath
(2010) analyzed hydrodynamic analysis of rough curved pivoted porous slider
bearings with couple stress fluid. They found that the improved performance due to
the couple stresses and the presence of negatively skewed surface roughness.
36
However, the presence of porous facing and positively skewed surface roughness
affects the performance of the pivoted porous slider bearing. Elsharkawy and
Fadhalah (2011) studied squeeze film characteristics between a sphere and a rough
porous flat plate with micropolar fluids. They observed that, excessive permeability
of the porous layer causes a significant drop in the squeeze film characteristics and
minimizes the effect of surface roughness. For the case of limited or no
permeability, the azimuthal roughness is found to increase the load-carrying
capacity and squeeze time, whereas the reverse results are obtained for the case of
radial roughness. Walicka (2012) studied that, inertia effects in porous squeeze film
biobearing with rough surfaces lubricated by a power-law fluid. It is shown that the
inertia effects, power-law exponents, and surface roughness influence the
biobearing performance considerably part of this thesis is devoted for the study of
squeeze film lubrication between rough poroelastic rectangular plates with
micropolar fluid: A special reference to the study of synovial joint lubrication and
observed that the effect of surface roughness has considerable effects on lubrication
mechanism of synovial joints.
1.7
Basic Equations
1.7.1 Couplestress Fluids
The constitutive equations for force and couple stresses proposed by Stokes
(1966) are


T(i j )   p   Dk k  i j  2 Di j ,
T[i j ]   2 Wi j , k k 

2
 i j s Gs ,
(1.7.1.1)
(1.7.1.2)
37
and
Mi j  4  j , i  4 ' i , j ,
(1.7.1.3)
where




Di j 
1
Vi , j  V j , i ,
2
Wi j 
1
Vi , j  V j , i ,
2
i 
(1.7.1.4)
(1.7.1.5)
1
 i j k Vk , j ,
2
(1.7.1.6)
Where T( i j ) is the symmetric part and T[ i j ] is the antisymmetric part of Ti j ,
M i j the couple stress tensor, Di j the deformation rate tensor, Wi j the vorticity
tensor, Vi the velocity vector,  i
the vorticity vector , Gs the body couple,  i j
the Kronecker delta,  the density, p the pressure,  ijs the alternating unit tensor,
 and  are the material constants of the dimension of viscosity,  and   are the
   has the
material constants having the dimension of momentum. The ratio 
length squared and it characterizes the size of microstructure. In any flow of fluid
with microstructure, the departure from the classical theory depends on the relative
size of substructure compared with the linear dimensions of the flow. In the case of
lubrication problems, flow takes place in narrow recesses so that; the presence of
substructure in the lubricant may have a significant effect on the performance
characteristics of various bearing systems. In the analysis of the bearing problems
considered in this thesis, it is assumed that the body forces and body couples are
absent. Under this assumption, the governing equations of Stokes (1966)
38
couplestress fluid given in the equations (1.7.1.1)-(1.7.1.3) take the form in
Cartesian co-ordinate system as:
u v w
 
0,
x y z
(1.7.1.7)
2
 2 2 2 
u u
u
1 p   2u  2u  2u 
u v  w 
   2  2  2   kc  2  2  2  u ,
x y
z
 x  x y z 
 x y z 
(1.7.1.8)
2
  2v  2v  2v 
 2 2 2 
v v
v
1 p
u v  w 
 




k 
 v,
x y
z
 y  x 2 y 2 z 2  c  x 2 y 2 z 2 
(1.7.1.9)
and
2
 2 2 2 
w w
w
1 p   2 w  2 w  2 w 
u v  w 
   2  2  2   kc  2  2  2  w,
x
y
z
 z  x y z 
 x y z 
(1.7.1.10)
where u, v and w are the velocity components in the x, y and z directions,
respectively,       is the kinematic viscosity and kc     .
1.7.2 Micropolar Fluids
The field equations for micropolar fluids in vectorial form are (Eringen, 1966)
Conservation of Mass

    v  0
t
(1.7.2.1)
39
Conservation of linear momentum
   2   k      V      k      V  k   V     f

1
 V

 V   V    V 2 
2
 t

 

(1.7.2.2)
Conservation of angular momentum

          v        v   k  V  2kv   l   jV
(1.7.2.3)
A superimposed dot in equation (1.7.2.3) indicates material differentiation.
For an incompressible fluid   const ,  v  0 and  is replaced by an unknown
pressure to be determined by the boundary conditions.
For the three dimensional steady motion of an incompressible micropolar
fluid under the assumption of negligible body forces and body couples, the field
equations (1.7.2.1)-(1.7.2.3) reduce to the following form
u v w
 
0
x y z
(1.7.2.4)
  2u  2u  2u 
 v v  p
 u
1
u
u 
2




 2  2  2     3  2     u  v  w 
2
y
z 
y
z 
 y z  x
 x
 x
(1.7.2.5)
  2v  2v  2v 
 v
v v
1
p
v
v 
 2     2  2  2     1  3      u  v  w 
2
y
z 
y
z 
 z x  y
 x
 x
(1.7.2.6)
40
 2w 2w 2w 
 v v  p
 w
1
w
w 
 2     2  2  2     2  1      u  v  w 
2
y
z 
y
z 
 x y  z
 x
 x
(1.7.2.7)
  2v1  2v1  2v1 
 w v 
 v
v
v 
  2  2  2        2  v1   j  u 1  v 1  w 1 
y
z 
y
z 
 y z 
 x
 x
(1.7.2.8)
  2v2  2v2  2v2 
 v
v
v 
 u w 
  2  2  2        2  v2   j  u 2  v 2  w 2 
y
z 
y
z 
 z x 
 x
 x
(1.7.2.9)
  2v3

2
 x

 v u 
 v3
 2v3  2v3 
v3
v3 





2

v


j
u

v

w





3
y
z 
y 2 z 2 
 x y 
 x
(1.7.2.10)
The constitutive equations for the stress tensor t kl
and the couple stress
tensor m kl are given in Cartesian co-ordinates as
tkl     vr ,r   kl    vk ,l  vl ,k   k Vt ,k klr vr 
(1.7.2.11)
mkl   vr ,r kl   vk ,l   vl ,k
(1.7.2.12)
where  kl is the Kronecker delta and klr is the alternating symbol. An index
followed by a comma represents partial differentiation with respect the space
variable xk .
The lubrication films to be analyzed are assumed to comply with usual
assumptions of hydrodynamic lubrication (Pinkus and Sternlicht, 1961) i.e., the
41
flow is laminar, body forces are neglected, no slip on the bearing surface and film is
sufficiently thin in comparison with the length and the span of the bearing equations
(1.7.2.1) to (1.7.2.12) are simplified by these assumptions, the velocity components
and micro-rotational components reduce to
Conservation of linear momentum:
v3 p
  2u

 0
   2 
2  y
y x

(1.7.2.13)
v1 p
  2w



 0 .

 2 
2  y
y z

(1.7.2.14)
Conservation of angular momentum:

 2 v1

 2 v3
y
y
2
2
 2  v1  
w
0 ,
y
(1.7.2.15)
 2  v3  
u
0
y
(1.7.2.16)
p
0
y
(1.7.2.17)
Conservation of mass:
u v w
 
0
x y z
(1.7.2.18)
The four differential equations (1.7.2.13)-(1.7.2.16) form two systems of
simultaneous equations. One consists of equations (1.7.2.13) and (1.7.2.16) and the
other of equations (1.7.2.14) and (1.7.2.15). Solution of these equations (1.7.2.13)
42
and (1.7.2.16) gives expressions for u and v3 and the solution of equations
(1.7.2.14) and (1.7.2.15) yields the expression for w and v1 . The second system
becomes identical with the first system if v3 is replaced by v1 . Hence it is necessary
to solve only the first system for u and v3 and the expression for w and v1 can be
written in a similar way.
The boundary conditions for the velocity and the micro rotation velocity are
(i) At the upper surface
 y  h ;
u  U11 , w  U12 , v  vh , v1  0 , v3  0
(ii) At the lower surface
(1.7.2.19)
 y  0 ;
u  U 21 , w  U 22 , v  v0 , v1  0 , v3  0
(1.7.2.20)
The solution of equations (1.7.2.13)-(1.7.2.16) subject to boundary conditions given
in (1.7.2.19) and (1.7.2.20) is obtained in
u
 2N2
1  y 2 p



D
y

11 
  m   D21 sinh  my   D31 cosh  my    D41
  2 x

(1.7.2.21)
w
1  y 2 p
 D12

  2 z
(1.7.2.22)
v1 
1  p

 y  D12    D22 cosh  my   D32 sinh  my  
2  z

 2N2
y  
  D22 sinh  my   D32 cosh  my    D42
m

v3  D21 cosh  my   D31 sinh  my  
1  p

 y  D11 
2   x

43
(1.7.2.23)
(1.7.2.24)
where
Dij (i  1, 2,3, 4 and j  1, 2) are constants and given by

 1
h p 
2N 2
1  cosh  mh   
Dij   U 2 j  U1 j sinh  mh  
h
sinh
mh

,
 

2 x j 
m
 D5


D2 j 
D3 j 

Dij
2
,
 1
1  cosh  mh   h p  h
1
N2
U

U

cosh
mh

1

h

sinh  mh    ,
  
 2 j 1j 
 

 
2
m

 2 x j  2
  D5

D4 j  Uij 
D5 

2N 2
 D3 j ,
m

h 
2N 2
sinh
mh

cosh  mh   1 ,





m

in which m 
N
, x1  x ,
l
x2  z .
The flow volume rates q x and q y along the x and z –axis respectively are given by
h
h
0
0
q x   udy and qz   wdy .
Substituting the expressions for u and w from Eqns. (1.7.2.21) and (1.7.2.22) into
the above integrals and simplifying, we get
qx 
f  N , l , h  p
h

,
U11  U 21  
2
12
x
(1.7.2.26)
44
qz 
f  N , l , h  p
h

.
U12  U 22  
2
12
z
(1.7.2.27)
 Nh 
Where f  N , h, l   h3  12l 2 h  6 Nlh coth 

 2l 
Integrating the equation of continuity (1.7.2.18) across the film and using
the equations (1.7.2.21) and (1.7.2.22) together with equations (1.7.2.19) and
(1.7.2.20), we get the generalized Reynolds equation for micropolar fluid in three
dimensional form as

p   
p 
 f  N , h, l     f  N , h , l   
x 
x  z 
z 
6
 U  U 

h
h
U11  U 21  h  12 h 12 22 12  vh  v0  U12  U 22 
x
z
x
z 

(1.7.2.28)
Where
 Nh 
f  N , h, l   h3  12l 2 h  6 Nlh coth 
,
 2l 
vh and v0 are normal velocity components of the surface at y = h and y = 0
respectively.
45