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Peristaltic transport is a form of fluid transport that occurs when a progressive wave of area contraction or expansion propagates along the length of an extensible tube containing a liquid. In humans, peristalsis is found in the contraction of smooth muscles to propel contents through the digestive tract. Earthworms use a similar mechanism to drive their locomotion. The word is derived from New Latin and comes from the Greek peristallein, "to wrap around," from peri-, "around" + stallein, "to place". Peristaltic pumping also has engineering applications, e.g. in situations where it is desirable to prevent the mechanical parts of the pump from coming into contact with a corrosive fluid. Technical roller and finger pumps also operate according to this rule.
1.2. Biological Systems Associated With Peristalsis
After food is chewed into a bolus, it is swallowed and moved through the esophagus. Smooth muscles contract behind the bolus to prevent it from being squeezed back into the mouth, and then rhythmic, unidirectional waves of contractions will work to rapidly force the food into the stomach. This process works in one direction only and its sole purpose is to move food from the mouth into the stomach.
Fig: 1.1. Peristalsis in digestive track
1.2.2 Small Intestine
Once processed and digested by the stomach, the milky chyme is squeezed through the pyloric sphincter into the small intestine. Once past the stomach a typical peristaltic wave will only last for a few seconds, travelling at only a few centimeters per second. Its primary purpose is to mix the chyme in the intestine rather than to move it forward in the intestine. Through this process of mixing and continued digestion and absorption of nutrients, the chyme gradually works its way through the small intestine to the large intestine.
During vomiting the propulsion of food up the esophagus and out the mouth comes from contraction of the abdominal muscles; peristalsis does not reverse in the esophagus.
As opposed to the more continuous peristalsis of the small intestines, faecal contents are propelled into the large intestine by periodic mass movements. These mass movements occur one to three times per day in the large intestines and colon, and help propel the contents from the large intestine through the colon to the rectum.
1.2.3 Large Intestine
Large intestines normally exhibit four types of motions: 1. Rhythmic variations of tone, 2. Peristalsis, 3. Mass peristalsis and 4. Anti-peristalsis.
Fig.1.2. Peristalsis in Ureter
Peristalsis is not equivalent as rush peristalsis seen in the small intestine. It is a weak peristalsis alternately shortening and elongating in the transverse colon.
A movement is a modified type of peristalsis characterized by the following sequence of events: First, a constrictive ring occurs in response to a distended or irritated point in the colon, usually in the transverse colon. Then, rapidly thereafter the 20 or more centimeters colon distal to the constriction lose their haustrations and instead contract as a unit, forcing the fecal material in this segment en masse further down the colon. The contraction develops progressively more force for about 30 seconds, and relaxation and then occurs during the next 2-3 minutes. Then, another mass movement occurs, this time perhaps farther along the colon.
In the early stages of excessive gastrointestinal irritation, anti-peristalsis begins to occur often many minutes before vomiting appears. The anti-peristalsis may begin as far down in the intestinal tract as the ileum, and the anti-peristaltic wave travels backward up the intestine at a rate of 2-3 cm/sec; this process can actually push a large share of the intestinal contents all the way back to the duodenum and stomach within 3-5 minutes. Then, as these upper portions of the gastrointestinal tract, especially the duodenum, become overly distended, this destination becomes the exiting factor that initiates the actual vomiting act. In man it is rarely seen but is well marked in animals such as cat.
1.2.4 Renal System
The ureter propels the urine from the kidneys into the bladder by peristaltic contraction of smooth muscle layer. This is an intrinsic property of the smooth muscle and is not under autonomic nerve control. The waves of contraction originate in a pacemaker in the minor calyces. Peristaltic waves occur several times per minute, increasing in frequencies with the volume of urine produced, and send little spurts of urine into the bladder.
1.3. Classification of Fluids
1. 3.1 Newtonian Fluid
If shear stress is linearly proportional to the rate of strain, the fluid is called as a Newtonian fluid. Newtonian behavior has been observed in all gases, in liquids or solutions of materials of low molecular weight.
The constitute equation for Newtonian fluid is
Where is the shear stress, is the shear rate and is the viscosity of the fluid.
1.3.2. Non-Newtonian Fluid
Non-Newtonian fluids generally exhibit a nonlinear relationship between the shear stress and the rate of strain. Foodstuffs (like banana juice, apple juice, and ketch up), blood, slurries, sperm, intra uterine fluid, etc. behave like non-Newtonian fluids.
In this thesis an attempt is made to study the following non-Newtonian fluids:
a. Carreau Fluid
The constitutive equation for a Carreau fluid is
where ï´ is the extra stress tensor, is the infinite shear - rate viscosity, is the zero shear - rate viscosity, ï‡ is the time constant, n is the dimensionless power law index and is defined as
Here is the second invariant of strain - rate tensor.
Note that the above model reduces to Newtonian model for or.
b) Third Order Fluid
The constitutive equation for in a third order fluid is
= - ,
Where is the pressure, - porosity of the porous medium, - the identity tensor and is the extra stress tensor
c) Jeffrey Fluid
The Jeffrey model is relatively simpler linear model using time derivatives instead of convected derivatives, for example Oldroyd-B model does; it represents a rheology different from the Newtonian.
The constitute equation for the Jeffrey fluid is
Where is the dynamic viscosity of the fluid, is the shear rate, is the ratio of relaxation time to retardation time and is the retardation time and dots over the quantities denote differentiation with respect to time. The Jeffrey fluid model helps to treat both the MHD Newtonian and non-Newtonian problems analytically under long wavelength and low Reynolds number considerations.
d) Prandtl Fluid
The constitutive equation for in a third order fluid is
= - ,
Where is the pressure, - the identity tensor and the extra stress tensor is given by
in which and are material constants of Prandtl fluid model.
e) Williamson Fluid
The constitutive equation for a Williamson fluid
Where is the extra stress tensor, is the infinite shear rate, viscosity is the zero shear rate viscosity, is the time constant and is defined as
Where is the second invariant stress tensor. We consider in the constitutive equation the case for which and so we can write.
The above model reduces to Newtonian for
1.4 Mathematical Modeling
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences (such as physics, biology, earth science, meteorology) and engineering disciplines (e.g. computer science, artificial intelligence), but also in the social sciences (such as economics, psychology, sociology and political science); physicists, engineers, statisticians, operations research analysts and economists use mathematical models most extensively. A model may help to explain a system and to study the effects of different components, and to make predictions about behavior.
A mathematical model usually describes a system by a set of variables and a set of equations that establish relationships between the variables. Variables may be of many types; real or integer numbers, boolean values or strings, for example. The variables represent some properties of the system. The actual model is the set of functions that describe the relations between the different variables.
1.5. Literature Survey
In past five decades, many mathematical and computational models were developed to describe fluid flow in a tube undergoing peristalsis with prescribed wall motions. In earlier analytical studies, simplifying assumptions were made, including zero Reynolds number, small-amplitude oscillations, infinite wavelength, as well as symmetry of the channel (Jaffrin and Shapiro . Subsequently, non-uniform channel geometry, and effects of finite length channels (Eytan and Elad , Fauci , Li and Brasseur , Pozrikidis , Takabatake et al. ).
In the literature, several works pertaining to peristaltic motion have been done for Newtonian fluid. Such approach is true in ureter but it fails to give an adequate understanding of peristalsis in blood vessels, chyme moment in intestine, semen transport in ducts efferentus of male reproductive tract, in transport of spermatozoa and in cervical canal. In these body organs, the fluid viscosity varies across the thickness of the duct. Also, the assumption that most of the physiological fluid behave like Newtonian fluid is not true in reality. With all these facts in mind, it is clear that viscoelastic rheology is the correct way of properly describing the peristaltic flow. Peristaltic transport of blood in small vessels was investigated using the viscoelastic fluid by Bohme and Fredrich , power-law fluid by Radhakrishnamacharya , micro polar fluid by Srinivasacharya et al. , casson fluid by Srivastava and Srivastava . Peristaltic flow of a second -order fluid in a planar channel and in an axisymmetric tube has been studied by Siddiqui et al. , Siddiqui and Schwarz  under long- wave length assumption. The power-law model was used to study the fluid transport in the male reproductive tract by Srivastava and Srivastava , small intestine and esophagus by Srivastava and Srivastava . Peristaltic flow of third order fluid has been investigated by Siddiqui Schwarz  for planar channel and by Hayat et al . Hayat et al.  have discussed the effect of endoscope on the peristaltic motion of a Jeffrey fluid in a tube. The non- Newtonian fluids are Bingham and Herschel- Bulkley fluids. Vajravelu et al., ,  made a detailed study on the effect of yield stress on peristaltic pumping of a Herschel - Bulkley fluid in an inclined tube and a channel. All these investigations are confined to hydrodynamic study of a physiological fluid obeying some yield stress model.
The magneto hydrodynamic flow of blood in a channel having walls that execute peristaltic waves using long wavelength approximation has been discussed by Agrawal and Anwaruddin . Peristaltic flow of blood under the effect of a magnetic field in a non-uniform channel has investigated by Mekheimer . The effect of magnetic field on the peristaltic flow of a Johnson-Segalman fluid in a planar channel has investigated by Elshahad and Haroun . Hayat et al.  have analyzed peristaltic motion of a third order fluid under the effect of magnetic field in tube. Hayat et al.  have studied the effects of endoscope and magnetic field on the peristaltic flow of a Jeffrey fluid in a tube. Peristaltic motion of a Jeffrey fluid under the effect of magnetic field in a tube was discussed by Hayat and Ali . Ali et al.  have investigated peristaltic flow of MHD fluid in a channel with variable viscosity under the effect of slip condition.
Flow through porous medium occurs in filtration of fluids and seepage of water in river beds. Movement of underground water and oils, limestone, rye bread, wood, the human lung, bile duct, gallbladder with stones, and small blood vessels are some important examples of flow through porous medium. A model of a combined porous peristaltic pumping system was discussed by Reese and Rath . The effects of porous boundaries on peristaltic pumping through a porous medium have been investigated by El Shehawey and Husseny . El Shehawey and Sebaei  discussed the peristaltic transport in cylindrical tube through a porous medium. Mekheimer and AL-Arabi  made a detailed study on the peristaltic pumping of a conducting fluid through porous medium. Non-linear peristaltic transport through a porous medium in an inclined planar channel has studied by Mekheimer  taking the gravity effect on pumping characteristics. Peristaltic flow of a Maxwell fluid through a porous medium in a channel with Hall effects has investigated by Hayat et al. . Peristaltic transport of a Newtonian fluid through a porous medium in an asymmetric channel has analyzed by El Shehawey et al. .
Much attention has been confined to symmetric channels or tubes, but there exist also flow situations where the channel flow may not be symmetric. Mishra and Rao  studied the peristaltic flow of a Newtonian fluid in an asymmetric channel in a recent research. In another attempt, Rao and Mishra  discussed the non-linear and curvature effects on peristaltic flow of a Newtonian fluid in an asymmetric channel when the ratio of channel width to the wavelength is small. Very recently, Haroun  extended the analysis of reference Mishra and Rao for a third order fluid. An example for peristaltic type motion is the intra-uterine fluid flow due to myometrial contractions, where the myometrial contractions may occur in both symmetric and asymmetric directions. An interesting study was made by Eytan and Elad  whose results have been used to analyze the fluid flow pattern in a non-pregnant uterus. In another paper, Eytan et al.  discussed the characterization of non-pregnant women uterine contractions as they are composed of variable amplitude and arrange of different wavelength. Haroun  have investigated the peristaltic flow of a fourth grade fluid in an inclined asymmetric channel. Ali and Hayat  have investigated the peristaltic motion of a Carreau fluid in an asymmetric channel. Peristaltic transport of a micropolar fluid in a asymmetric channel as investigated by Ali and Hayat .Peristaltic transport of a Johnson-Segalman fluid in an asymmetric channel has studied by Hayat et al .
It should be noted that the nomenclatures of esch chapter is independent of other.
The introduction of each of the other chapters of the thesis contains some more related literature.