A mathematical model for the peristaltic flow of chyme movement in small intestine

A mathematical model based on viscoelastic fluid (fractional Oldroyd-B model) flow is considered for the peristaltic flow of chyme in small intestine, which is assumed to be in the form of an inclined cylindrical tube. The peristaltic flow of chyme is modeled more realistically by assuming that the peristaltic rush wave is a sinusoidal wave, which propagates along the tube. The governing equations are simplified by making the assumptions of long wavelength and low Reynolds number. Analytical approximate solutions of problem are obtained by using homotopy analysis method and convergence of the obtained series solution is properly checked. For the realistic values of the emerging parameters such as fractional parameters, relaxation time, retardation time, Reynolds number, Froude number and inclination of tube, the numerical results for the pressure difference and the frictional force across one wavelength are computed and discussed the roles played by these parameters during the peristaltic flow. On the basis of this study, it is found that the first fractional parameter, relaxation time and Froude number resist the movement of chyme, while, the second fractional parameter, retardation time, Reynolds number and inclination of tube favour the movement of chyme through the small intestine during pumping. It is further revealed that size of trapped bolus reduces with increasing the amplitude ratio whereas it is unaltered with other parameters.     

Interaction of peristaltic flow with pulsatile flow in a circular cylindrical tube

The effect of pulsatile flow on peristaltic transport in a circular cylindrical tube is analysed. The flow of a Newtonian viscous incompressible fluid in a flexible circular cylindrical tube on which an axisymmetric travelling sinusoidal wave is imposed, is considered. The initial flow in the tube is induced by an arbitrary periodic pressure gradient. A perturbation solution with amplitude ratio (wave amplitude/tube radius) as a parameter is obtained when the frequency of the travelling wave and that of the imposed pressure gradient are equal. The interaction effects of periodic wall induced flow and periodic pressure imposed flow are visualized through the presence of substantially different components of steady and higher harmonic oscillating flow in the first order flow solution. Numerical results show a strong variation of steady state velocity profiles with boundary wave number and Reynolds number and a strong phase shift behaviour of the flow in the radial direction.     

Mathematica numerical simulation of peristaltic biophysical transport of a fractional viscoelastic fluid through an inclined cylindrical tube

This paper studies the peristaltic transport of a viscoelastic fluid (with the fractional second-grade model) through an inclined cylindrical tube. The wall of the tube is modelled as a sinusoidal wave. The flow analysis is presented under the assumptions of long wave length and low Reynolds number. Caputo's definition of fractional derivative is used to formulate the fractional differentiation. Analytical solutions are developed for the normalized momentum equations. Expressions are also derived for the pressure, frictional force, and the relationship between the flow rate and pressure gradient. Mathematica numerical computations are then performed. The results are plotted and analysed for different values of fractional parameter, material constant, inclination angle, Reynolds number, Froude number and peristaltic wave amplitude. It is found that fractional parameter and Froude number resist the flow pattern while material constant, Reynolds number, inclination of angle and amplitude aid the peristaltic flow. Furthermore, frictional force and pressure demonstrate the opposite behaviour under the influence of the relevant parameters emerging in the equations of motion. The study has applications in uretral biophysics, and also potential use in peristaltic pumping of petroleum viscoelastic bio-surfactants in chemical engineering and astronautical applications involving conveyance of non-Newtonian fluids (e.g. lubricants) against gravity and in conduits with deformable walls.     

Peristaltic Transport of Carreau-Yasuda Fluid in a Curved Channel with Slip Effects

The wide occurrence of peristaltic pumping should not be surprising at all since it results physiologically from neuro-muscular properties of any tubular smooth muscle. Of special concern here is to predict the rheological effects on the peristaltic motion in a curved channel. Attention is focused to develop and simulate a nonlinear mathematical model for Carreau-Yasuda fluid. The progressive wave front of peristaltic flow is taken sinusoidal (expansion/contraction type). The governing problem is challenge since it has nonlinear differential equation and nonlinear boundary conditions even in the long wavelength and low Reynolds number regime. Numerical solutions for various flow quantities of interest are presented. Comparison for different flow situations is also made. Results of physical quantities are interpreted with particular emphasis to rheological characteristics.     

Mixed Convection Peristaltic Flow of Third Order Nanofluid with an Induced Magnetic Field

This research is concerned with the peristaltic flow of third order nanofluid in an asymmetric channel. The governing equations of third order nanofluid are modelled in wave frame of reference. Effect of induced magnetic field is considered. Long wavelength and low Reynolds number situation is tackled. Numerical solutions of the governing problem are computed and analyzed. The effects of Brownian motion and thermophoretic diffusion of nano particles are particularly emphasized. Physical quantities such as velocity, pressure rise, temperature, induced magnetic field and concentration distributions are discussed.     

Peristaltic flow of a couple stress fluid in an asymmetric channel

This paper presents an analysis of the peristaltic flow of a couple stress fluid in an asymmetric channel. The asymmetric nature of the flow is introduced through the peristaltic waves of different amplitudes and phases on the channel walls. Mathematical modelling corresponding to a two-dimensional flow has been carried out. The flow analysis is presented under long wavelength and low Reynolds number approximations. Closed form solutions for the axial velocity, stream function and the axial pressure gradient are given. Numerical computations have been carried out for the pressure rise per wavelength, friction forces and trapping. It is noted that there is a decrease in the pressure when the couple stress fluid parameter increases. The variation of the couple stress fluid parameter with the size of the trapped bolus is also similar to that of pressure. Furthermore, the friction force on the lower channel wall is greater than that on the upper channel wall.     

Peristaltic transport of a couple stress fluid in a uniform and non-uniform channels

The problem of peristaltic transport of a couple stress fluid in uniform and non-uniform two-dimensional channels has been investigated under zero Reynolds number with long wavelength approximation. Blood is represented by a couple stress fluid (a fluid which its particles size are taken into account, a special case of a non-Newtonian fluid). It is found that the pressure rise decreases as the couple stress fluid parameter gamma increases (i.e., small size fluid particle). So the pressure rise for a couple stress fluid (as a blood model) is greater than that for a Newtonian fluid. Also the pressure rise increases as the amplitude ratio phi increases for different values of gamma. Further, the pressure rise in the case of non-uniform geometry is found to be much smaller than the corresponding value in the case of uniform geometry. Finally, the maximum pressure rise when the mean flow rate over one period of the wave, Q = 0, increases as phi increases and gamma decreases.     

Interaction of Peristaltic Flow with Pulsatile Couple-Stress Fluid

1IntroductionPeristalsisIsnow－wellknowntothephyslologlststobeoneofthem8JormechanismforfluidtransportInmanybiologicalsystems．Inpatlcular，peristaltlcmechanismmaybeInvolvedInswal－lowing恤throughtheesophagus，urinetransPOrtfromkidneytobladderthoughuner．Inaddl－tion，perlstaltlcpumpingoccursInmanypracticalapplicationsInvolvingbio－mechanicalsystems．Thestudyofthemechanismofperistalsis，Inbothmechanicalandphysiologicalsituations，hasre－centlybecometheoNectofs。；ent；f；crese。roh．S；nce…     

Interaction of peristaltic motion with Poiseuille flow

The effect of Poiseuille flow on peristaltic transport has been investigated in a two-dimensional mathematical model of peristalsis for the case when the wall of the channel executes a sinusoidal motion of small amplitude. Closed-form solutions have been obtained for limiting values of Reynolds number and the Poiseuille flow parameter, while the method of Frobenius series solution has been used for the general case. It is found that the mean flow reversal is strongly dependent on the Poiseuille flow. The position of flow reversal may change drastically from the center of the channel to the boundaries. Numberical results are reported for various values of the physical parameters of interest.     

Peristaltic transport of blood: Casson model--II

The problem of peristaltic transport of blood in a uniform and non-uniform tube has been investigated, under zero Reynolds number and long wavelength approximation. Blood is represented by a two-layered fluid model consisting of a central layer of suspension of all erythrocytes, etc., assumed to be a Casson fluid, and a peripheral layer of plasma as a Newtonian fluid. A comparison of results with those without peripheral layer shows that the magnitude of the pressure rise, under a given set of conditions is smaller in the case of model with peripheral layer. It is found that, for a given flow rate, the pressure rise decreases as the viscosity of the peripheral layer decreases, and for a given non zero pressure drop, the flow rate increases as the viscosity of the peripheral layer decreases. However, the flow is independent of the presence of the peripheral layer, for zero pressure rise. Further, the pressure rise in the case of non-uniform geometry is found much smaller than the corresponding value in the uniform geometry.     

Asymptotic method for peristaltic transport

The purpose of this paper is to justify an asymptotic method developed for the study of peristaltic transport in a tube of arbitrary cross section. Within the framework of long wave approximation, the three-dimensional nonlinear Navier-Stokes equations are reduced to a sequence of two-dimensional linear boundary value problems of Laplace and biharmonic operators. It is shown that, if a Reynolds number is less than some constant, the solution of the approximate equations is indeed an asymptotic approximation to the exact solution of the problem as the ratio of the maximum radius of the tube to the wave length of the peristaltic motion of the wall tends to zero, and the error estimates are expressed inL ₂ norms. Furthermore, under the same condition the exact solution is shown to be unique and stable under arbitrary perturbation of spatially periodic disturbance. Application of the stability condition to peristaltic transport in a tube of circular cross section is given.     

A Quantitative Investigation of a Peristaltic Model for Phloem Translocation

An analysis is made of a peristaltic model of phloem translocationIt is postulated that the periodic action of contractile orbending organelles drive a longitudinal flow of solution withintubules which connect sieve pores in successive sieve plates Plausible values are assumed for the velocity of propagationof the contraction wave, the frequency, the amplitude, and theviscosity of the solution and its concentration. Using relationswhich describe peristaltic flow, predictions are made for thevalues of parameters such as the velocity of solution, the drivingpressure, and rate of energy dissipation. These predicted valuesare seen to be reasonable when compared with the known propertiesof other biological contractile systems. Thus the model is quantitativelyacceptable.     

Transient magneto-peristaltic flow of couple stress biofluids: A magneto-hydro-dynamical study on digestive transport phenomena

Magnetic fields are increasingly being utilized in endoscopy and gastric transport control. In this regard, the present study investigates the influence of a transverse magnetic field in the transient peristaltic rheological transport. An electrically-conducting couple stress non-Newtonian model is employed to accurately simulate physiological fluids in peristaltic flow through a sinusoidally contracting channel of finite length. This model is designed for computing the intra-bolus oesophageal and intestinal pressures during the movement of food bolus in the digestive system under magneto-hydro-dynamic effects. Long wavelength and low Reynolds number approximations have been employed to reduce the governing equations from nonlinear to linear form, this being a valid approach for creeping flows which characterizes physiological dynamics. Analytical approximate solutions for axial velocity, transverse velocity, pressure gradient, local wall shear stress and volumetric flow rate are obtained for the non-dimensional conservation equations subject to appropriate boundary conditions. The effects of couple stress parameter and transverse magnetic field on the velocity profile, pressure distribution, local wall shear stress and the averaged flow rate are discussed with the aid of computational results. The comparative study of non-integral and integral number of waves propagating along the finite length channel is also presented. Magnetic field and non-Newtonian properties are found to strongly influence peristaltic transport.     

Peristaltic transport in a channel with a porous peripheral layer: model of a flow in gastrointestinal tract

Peristaltic transport in a two dimensional channel, filled with a porous medium in the peripheral region and a Newtonian fluid in the core region, is studied under the assumptions of long wavelength and low Reynolds number. The fluid flow is investigated in the waveframe of reference moving with the velocity of the peristaltic wave. Brinkman extended Darcy equation is utilized to model the flow in the porous layer. The interface is determined as a part of the solution using the conservation of mass in both the porous and fluid regions independently. A shear-stress jump boundary condition is used at the interface. The physical quantities of importance in peristaltic transport like pumping, trapping, reflux and axial velocity are discussed for various parameters of interest governing the flow like Darcy number, porosity, permeability, effective viscosity etc. It is observed that the peristalsis works as a pump against greater pressure in two-layered model with a porous medium compared with a viscous fluid in the peripheral layer. Increasing Darcy number Da decreases the pumping and increasing shear stress jump constant beta results in increasing the pumping. The limits on the time averaged flux Q for trapping in the core layer are obtained. The discussion on pumping, trapping and reflux may be helpful in understanding some of the fluid dynamic aspects of the transport of chyme in gastrointestinal tract.     

A long wavelength solution for a microorganism swimming in a channel

The hydrodynamics of a microorganism swimming in a channel is investigated. The microorganism is modeled as a two-dimensional sheet swimming at low Reynolds numbers between two rigid walls. The wavelengths of the propulsive waves passing down the sheet are assummed to be very large compared to the channel spacing, but the amplitude of the propulsive waves is arbitrary. Explicit analytical solutions for the propulsive velocity and the rate of energy dissipated in terms of the wave amplitude, channel spacing, wave number, and wave speeds are given.     

Influence of Hall Current and Viscous Dissipation on Pressure Driven Flow of Pseudoplastic Fluid with Heat Generation: A Mathematical Study

In this paper, we study the influence of heat sink (or source) on the peristaltic motion of pseudoplastic fluid in the presence of Hall current, where channel walls are non-conducting in nature. Flow analysis has been carried out under the approximations of a low Reynolds number and long wavelength. Coupled equations are solved using shooting method for numerical solution for the axial velocity function, temperature and pressure gradient distributions. We analyze the influence of various interesting parameters on flow quantities. The present study can be considered as a mathematical presentation of the dynamics of physiological organs with stones.     

Low Reynolds number equi-bifurcation flow in a two-dimensional channel

The fluid flow in some physiological vessels such as the blood flow in blood vessels and the air flow through bronchi and bronchioles in the lungs undergoes a large number of bifurcations. The understanding of the bifurcation flow is of importance for a better comprehension of its effect in the blood and the air circulatory systems of the living body. The Reynolds number of flow in large blood vessels and bronchi is high and fluid inertia plays a dominant role in the bifurcation flow in such vessels. In small caliber blood vessels such as arterioles and capillaries, and bronchioles, the Reynolds number of flow is quite low and the effect of fluid inertia is negligible compared to the pressure and shear forces. In order to have a quantitative understanding of the bifurcation flow at low Reynolds numbers, the low Reynolds number equi-bifurcation flow in a two-dimensional channel at zero bifurcation angle is studied based on the Stokes approximation. The solution of the problem is posed as an infinite series, where the truncated version is used in numerical calculations. The results of this analysis is discussed in connection with the bifurcation flow of blood in small caliber blood vessels and that of the air in bronchioles in the lung.     

A mathematical simulation of the ureter: effects of the model parameters on ureteral pressure/flow relations

Ureteral peristaltic mechanism facilitates urine transport from the kidney to the bladder. Numerical analysis of the peristaltic flow in the ureter aims to further our understanding of the reflux phenomenon and other ureteral abnormalities. Fluid-structure interaction (FSI) plays an important role in accuracy of this approach and the arbitrary Lagrangian-Eulerian (ALE) formulation is a strong method to analyze the coupled fluid-structure interaction between the compliant wall and the surrounding fluid. This formulation, however, was not used in previous studies of peristalsis in living organisms. In the present investigation, a numerical simulation is introduced and solved through ALE formulation to perform the ureteral flow and stress analysis. The incompressible Navier-Stokes equations are used as the governing equations for the fluid, and a linear elastic model is utilized for the compliant wall. The wall stimulation is modeled by nonlinear contact analysis using a rigid contact surface since an appropriate model for simulation of ureteral peristalsis needs to contain cell-to-cell wall stimulation. In contrast to previous studies, the wall displacements are not predetermined in the presented model of this finite-length compliant tube, neither the peristalsis needs to be periodic. Moreover, the temporal changes of ureteral wall intraluminal shear stress during peristalsis are included in our study. Iterative computing of two-way coupling is used to solve the governing equations. Two phases of nonperistaltic and peristaltic transport of urine in the ureter are discussed. Results are obtained following an analysis of the effects of the ureteral wall compliance, the pressure difference between the ureteral inlet and outlet, the maximum height of the contraction wave, the contraction wave velocity, and the number of contraction waves on the ureteral outlet flow. The results indicate that the proximal part of the ureter is prone to a higher shear stress during peristalsis compared with its middle and distal parts. It is also shown that the peristalsis is more efficient as the maximum height of the contraction wave increases. Finally, it is concluded that improper function of ureteropelvic junction results in the passage of part of urine back flow even in the case of slow start-up of the peristaltic contraction wave.     

Flow of urine through the ureter: a collapsible, muscular tube undergoing peristalsis

In steady flow through nonuniform collapsible tubes a key concept is the compressive zone, at which flow limitation can occur at both high and low Reynolds numbers. Ureteral peristalsis can be considered as a series of compressive zones, corresponding to waves of active muscular contraction, that move at near-constant speed along the ureter towards the bladder. One-dimensional, lubrication-theory analysis shows that peristalsis can pump urine from kidney into the bladder only at relatively low mean rates of urine flow. Under these circumstances isolated boluses of urine are propelled steadily through the ureter (assumed uniform) by the contraction waves. At higher mean rates of flow the behavior depends on whether the frequency of peristalsis is higher or lower than a critical value. For frequencies above the critical value steady propagation of boluses that are in contact with contraction waves at both ends is possible. As the flow rate rises the urine begins to leak through the contraction waves and steady peristaltic flow breaks down. There is an upper limit to the mean flow rate that can be carried by steady peristalsis, which depends on the mechanical properties of the ureter. At high flow rates the peristaltic contractions do not pump but hinder the flow of urine through the ureter.     

Development of a general method for designing microvascular networks using distribution of wall shear stress

In the present study, theoretical formulations for calculation of optimal bifurcation angle and relationship between the diameters of mother and daughter vessels using the power law model for non-Newtonian fluids are developed. The method is based on the distribution of wall shear stress in the mother and daughter vessels. Also, the effect of distribution of wall shear stress on the minimization of energy loss and flow resistance is considered. It is shown that constant wall shear stress in the mother and daughter vessels provides the minimum flow resistance and energy loss of biological flows. Moreover, the effects of different wall shear stresses in the mother and daughter branches, different lengths of daughter branches in the asymmetric bifurcations and non-Newtonian effect of biological fluid flows on the bifurcation angle and the relationship between the diameters of mother and daughter branches are considered. Using numerical simulations for non-Newtonian models such as power law and Carreau models, the effects of optimal bifurcation angle on the pressure drop and flow resistance of blood flow in the symmetric bifurcation are investigated. Numerical simulations show that optimal bifurcation angle decreases the pressure drop and flow resistance especially for bifurcations at large Reynolds number.