Finite Element Analysis of Two-Dimensional Peristaltic Flows

Abstract

The finite element method is an advance numerical technique which is validly used to solve complicated problem in structural engineering and physics. Finite element technique can easily apply on complex problem and geometry, therefore it gains equal attention to other field of engineering, especially in fluid mechanics and heat flow problem. Many fluid models based on Navier-Stokes equation in which higher order nonlinearity appears, therefore it is complicated to find an accurate numerical solution. Finite element method helps us to solve these types of complex phenomena. Peristaltic motion is one of the important phenomena in fluid mechanics, which gain significant attention from many scientists and engineers in last four decades. Peristaltic flow has numerous applications in industrial science, physiological flow and bioscience. Typical examples of industrial systems and physiological, where peristaltic mechanism is involved are, flow of urine from kidney to the bladder, flow of chyme in small intestine, the small blood vessels as well as blood flow in arteries, spermatic fluid transport in female reproductive tract, swallowing of nutriment through oesophagus, blood flow through capillaries, etc. The transport in corrosive fluid in the nuclear industry, diabetes pumps, roller pumps and pharmacological delivery systems involve peristaltic mechanism. During the literature survey, it is noted that many authors find the analytical solution of peristaltic motion for Newtonian and non-Newtonian fluid by neglecting the inertial effects and using many assumptions like long wave length, small time mean flow rate, small amplitude ratio etc. The main purpose of this study is that the find accurate numerical results without neglecting the inertial effects and using any assumption. Finite element method is applied to find the numerical results of peristaltic motion for Newtonian and non-Newtonian fluid under different physical situations. The present study is valid for moderate Reynolds number and any wavelength. Moreover, the present study helps for further investigation in peristaltic motion. This thesis consists of eight chapters. Chapter 1 contains a background of finite element method and basic definition of fluid mechanics. To understand the basic produces of finite element method, solve examples are also given in this chapter which can help the beginner. The rest of the chapter contains two parts. Chapter 2 to 4 contains computational study of peristaltic motion in two dimensional channel flow problem and Chapter 5 to 7 contains computational study of peristaltic motion in axisymmetric tube problem. Chapter 8 gives the concluding remarks and future work. In chapter 2, finite element solution obtained for two dimensional MHD peristaltic flow of Newtonian fluid in an inclined channel against moderate Reynolds number and wave number at different wave shapes. The results are compared with the existing analytical result of Jaffrin (1973), numerical results of Dennis-Chang (1969) and Takabatake et al. (1989) and experimental results of Weinberg et al. (1971) in presence of Reynolds number and wave number. It is found that the results obtained without imposing the assumptions of long wavelength and low Reynolds number are significantly different from their counterparts based on long wavelength and low Reynolds number assumptions. It concludes that the present study obtained gives more accurate results as compare to old FEM results of Takabatake et al. (1989). It is also noted that the present results are well matched with experimental result of Weinberg et al. (1971) and theoretical result of Jaffrin (1973) against high Reynolds number and wave number. It is observed that the longitudinal velocity decreases near the channel centre with increasing Reynolds number and wave number. However, it increases near the channel centre with increasing Hartmann number. Moreover, the longitudinal velocity is less sensitive to the values of Reynolds number and wave number in the range and 1, respectively. It is also noticed that, the flow behaviour is not significant effects at different wave shapes. These results are published in Journal of the Korean Physical Society,71(12) 950-962. In chapter 3, heat transfer effect is observed on peristaltic flow problem against moderate Reynolds number and wave number in channel numerically. The finite element technique is used to find the numerical solution. Here again, It is found that the obtained results are significantly different from previous results without imposing any assumptions. The results of velocity, pressure rise, streamline and isothermal line are presented graphically. The obtained solution upto Reynold number 100 by using time mean flow rate Q = 1.4, wave number 0.1 and amplitude rate at 0.5. It is concluded that thermal effects are more for water based fluid as compared to gases. It is also observed that the bolus appear in the whole region at small time mean flow rate and move to carets region when time flow increase. It is noted that more thermal effect observed against high time mean flow rate. It is also noted that the positive pumping region appears at time mean flow rate Q < 0.45, free pumping at Q = 0.45 and co-pumping region appear at Q > 0.45. It is observed that the longitudinal velocity reduces near the wall with increasing Reynolds number, but enhance by increasing values of wavelength. It is noted that the temperature profile increases sharply due to increase in all the parameters accept wavelength. These results are submitted in Journal of Theoretical and Applied Mechanics. Chapter 4 described the numerical solution of peristaltic motion for Non-Newtonian fluid against high Reynolds number and wave number in a channel. The micropolar fluid is considered as a non-Newtonian fluid. The obtained governing partial differential equations converted into stream-vorticity form and then use Galerkin’s finite element technique to obtained numerical solution. The obtained solution is well convergent even high Reynolds number and wave number. It is concluded that the velocity decreases near the peristaltic wall and increases in the centre of the channel by increasing micropolar. It is observed that the velocity decreases near the centre of channel by increasing the values of Reynolds number whereas reversing near the boundaries. It is noted that the streamline are not disturbed by taking the large value of Re. It is also seen that there is no restriction by choosing the value of parameter in channel flow problem. These results are published in Journal of the Brazilian Society of Mechanical Sciences and Engineering (39) 4421-4430. Chapter 5 discussed MHD peristaltic motion through an inclined tube at high Reynolds numbers and wave number. The governing equation obtained in axisymmetric form and then converted in stream-vorticity form without imposing any assumption. The obtained results against higher value of Reynolds number in tube are significant different of those result obtain in channel flow problem. Most of studies available in literature are carried out low Reynolds numbers assumption which makes the simple nonlinear problem. It is noted that the velocity of the fluid is maximum at Re = 15 at inlet part of the wave for large magnetic effect and Re = 18 against small magnetic effect at centre of the tube. After increasing the value of Reynolds numbers, the velocity filed decreases and remains stable at higher Reynolds numbers. For higher value of Hartman number, the trapping bolus and the size of boluses increase due to increase in velocity of fluid. The pressure rise against time mean flow for different value of Reynold numbers, magnetic field and the amplitude ratio increases by increasing the values of these parameters. These results are published in Biophysical Reviews 11,139-147,(2019). In chapter 6, the effect of heat transfer of peristaltic motion in a tube against the high Reynolds number and wave number is observed. The finite element technique is used to solve the governing partial differential equation and obtained the numerical results graphically. The present results are valid for arbitrary Reynolds number, wave number and amplitude ratio. The streamline and isothermal line is plotted at different value of parameters. It is noted that the heat effect increases by increasing Reynolds number and wave. It is observed that more heat effects are more for water based fluid as compared to gases. The pressure rise per wave length for time mean flow at different value of Reynolds number, magnetic number and the amplitude ratio also presented through graphs. These results are published in Journal of the Korean Physical Society, 73(9) 1290-1302. Chapter 7 discussed peristaltic motion in a tube for non-Newtonian fluid at high Reynolds number. The micropolar fluid is considered as a non-Newtonian fluid. The obtained governing partial differential equations converted into stream-vorticity. The domain is discretize into non-uniform mesh using quadratic triangular element and then use Galerkin’s finite element technique to obtained numerical solution. The obtained solution is well convergent even high Reynolds number and wave number. The obtained numerical results of velocity, pressure rise, streamline, vorticity and microrotation are presented graphically. It is noted that the number of bolus and the size of bolus increases by increasing Reynolds number and decreases by increasing time mean flow rate. The micropolar parameter and coupling number do not have much effect on trapping bolus. It is also noticed the rotation of the fluid particle is faster for small coupling number, micropolar parameter and time mean flow rate. These results are published in Journal of the Brazilian Society of Mechanical Sciences and Engineering 41:104, (2019)