STUDY OF PERISTALTIC MOTION OF NON–NEWTONIAN FLUIDS THROUGH POROUS MEDIUM

Abstract

The word peristaltic comes from a Greek word ''Peristaltikos'' which means clasping and compressing. Thus ‘Peristalsis’ is the rhythmic sequence of smooth muscle contractions that progressively squeeze one small section of the tract and then the next, to push the content along the tract. It induces propulsion and mixing movements and pumps the content against pressure rise, thus finding enormous physiological, biomedical and industrial applications. This is an important mechanism for the mixing and transporting of fluids. This kind of fluid transport appears in many biological systems which have smooth muscle tubes such as, the swallowing of food through the esophagus, movement of chyme in the lower intestinal and gastrointestinal tract, transport of urine from kidney to the bladder, cervical canal and fallopian tube in females, lymphatic vessels and small blood vessels, etc. are some of the physiological applications. Mechanical devices like finger pumps, roller pumps are also based on the peristaltic mechanism. Nuclear industries also use peristaltic transport to curb the flow of toxic liquid to reduce the contamination of the environment. Biomedical instrument such as heart lung machine works on this principle. Even worms like earthworms use peristalsis for their movement. It is also speculated that tall trees may depend on peristalsis for the translocation of water. The translocation of water involves its motion through the porous matrix of the trees. Though peristalsis is a well known mechanism in biological system, it is only four decades ago that it was given its theoretical and experimental analysis of its fluid dynamic aspects and has recently become the object of scientific research for both mechanical and physiological situations. ii A porous medium is a material containing pores or spaces between solid material through which liquid or gas can pass. Examples of natural porous media are beach sand, sandstone, limestone, the human lung, bile duct, gall bladder with stones in small blood vessels. Flow through a porous medium has several practical applications especially in geophysical fluid dynamics. Also, as most of the tissues in the body are deformable porous media Peristaltic transport of a bio fluid through a channel with permeable walls is of considerable importance in biology and medicine. Flow through porous media has been of significant interest in the recent years, to understand the complexity of disease like intestinal cystitis, bladder stones, bacterial stones, bacterial infections of kidneys, the various medical conditions (viz., tumor growth) and treatments (injections) and so on. Though Newtonian and several non-Newtonian models have been used to study the motion of blood, it is realized that Herschel-Bulkley model describes the behavior of blood very closely. Herschel-Bulkley fluids are a class of non-Newtonian fluids that require a finite stress, known as yield stress, in order to deform. Therefore, these materials behave like rigid solids when the local shear is below the yield stress. Once the yield stress is exceeded, the material flows with a non-linear stress-strain relationship either as a shear-thickening fluid, or a shear-thinning one. Few examples of fluids behaving in this manner include paints, food products, plastics, slurries, pharmaceutical products etc. Further, in small diameter tubes blood behaves like Herschel-Bulkley fluid rather than power law and Bingham fluids. Herschel-Bulkley fluid is considered to be the more general non-Newtonian fluid as it contains two parameters, the yield stress and the power law index. In addition Herschel-Bulkley fluid’s constitutive equation can be reduced to the constitutive equations of Newtonian, Power law, and Bingham fluid models, by suitable choice of the parameters. The iii same model can be used for larger arteries where the effect of yield stress can be ignored. Hence it is appropriate to model blood as a Herschel–Bulkley fluid rather than Casson fluid. In view of all the studies mentioned above, the study of peristaltic motion of a non-Newtonian fluid, in particular the Herschel Bulkley fluid has been considered to analyze the flow of blood in small vessels. The flow analysis is worked out to explore the impacts of various parameters of concern under uniform, non-uniform, inclined, heat transfer and magnetic effects in the thesis. MATHEMATICA software is used to analyze the results graphically. The plots of both the pressure difference and the frictional force against flow rate show that the flow rate is more in convergent channel when compared to divergent channel. The frictional force behaves reversely in comparison with the pressure rise in accordance with the results of many prior research establishments. The porous lining of the wall increases the Mechanical efficiency. The absolute value of the Nusselt number increases with rise in Brinkmann number, and the yield stress. The trapped bolus shrinks with rise in the magnetic field.