Exact and numerical simulation of peristaltic flow of a non‐Newtonian fluid with inclined magnetic field in an endoscope

In the present article, we have studied the effects of inclined magnetic field on the peristaltic flow of Jeffrey fluid through the gap between two coaxial inclined tubes. The inner tube is rigid, whereas the outer tube has sinusoidal wave traveling down its wall. The governing equations are simplified using long wave length and low Reynolds number approximations. Exact and numerical solutions have been derived for velocity profile. The expressions for pressure rise and friction force are calculated using numerical integration. Graphical results and trapping phenomenon is presented at the end of the article to see the physical behavior of different parameters. Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical studies of fluid flow through tubes with double constrictions

The flow fields in the neighbourhood of double constrictions in a circular cylindrical tube were studied numerically. The effects on the streamline, velocity and vorticity distributions as the flow passes through the constrictions in the tube were studied in the Reynolds number range 5–200. Double constrictions with dimensionless spacing ratios of 1, 2, 3 and ∞ were studied for a 50% constriction. It is noted that when the Reynolds number is below 10, no recirculation region is formed in the above constricted flow. For Reynolds numbers greater than 10, a recirculation region forms downstream of each of the constrictions. For constriction spacing ratios of 1, 2, and 3, when the Reynolds number is high, a recirculation region spreads between the valley of the constrictions. The recirculation region formed between the two constrictions has a diminishing effect on the generation of wall vorticity near the second constriction area. In general, the peak value of wall vorticity is found slightly upstream of each of the constrictions. When the Reynolds number is increased, the peak wall vorticity value increases and its location is moved upstream. Maximum wall vorticity generated by the first constriction is found to be always greater than the maximum wall vorticity generated by the second constriction. The extent of this spreading of the recirculation region from the first constriction and its effects on the second constriction depend on the constriction spacing ratio and the flow Reynolds number.     

The effect of flexible tube vibration on pressure drop and heat transfer in heat exchangers considering viscous dissipation effects

The pressure drop and heat transfer coefficient in tube bundle of shell and tube heat exchangers are investigated considering viscous dissipation effects. The governing equations are solved numerically. Because of temperature-dependent viscosity the equations should be solved simultaneously. The flexible tubes vibration is modeled in a quasi-static method by taking the first tube of the row to be in 20 asymmetric positions with respect to the rest of the tubes which are assumed to be fixed and time averaging the steady state solutions corresponding to each one of these positions .The results show that the eccentricity of the first tube increases pressure drop and heat transfer coefficients significantly comparing to the case of rigid tube bundles, symmetrically placed. In addition, these vibrations not only compensate the effect of viscous dissipations on heat transfer coefficient but also increase heat transfer coefficient. The constant viscosity results obtained from our numerical method have a good agreement with the available experimental data of constant viscosity for flexible tube heat exchangers.     

Quasi-similar solutions for blast waves with internal heat transfer effects

The point explosion problem with internal heat transfer effects taken into account is analysed. The classical inviscid solutions to this problem yield a non-physical phenomenon of infinite temperature and zero density at the center of explosion for all times. With heat transfer fluxes considered, the solution near the center of symmetry is improved and finite values for temperature are obtained. The non-self-similar solution of the problem is based on the quasi-similar approximate technique which reduces the non-linear partial differential equations governing the problem to ordinary differential ones. However, this formulation yields a two-point boundary-value problem. To facilitate the integration, the flow field is first divided into two regions: an outer inviscid region and an inner region where dissipation effects are manifested. This results in two sets of ordinary differential equations expressing the conservation equations in the inner and outer regions which are then solved and matched together to yield the composite solution. Secondly, the problem is then transformed into an initial-value one. Using the results of the composite solution, the governing equations can be integrated directly from the center until the shock front. The structure of the non-self-similar flow fields with internal heat transfer effects is then fully determined for specified values of the heat transfer parameters.     

Heat transfer to laminar flow of nongray gases through a circular tube

Analyses are presented for infrared radiative energy transfer in gases when other modes of energy transfer simultaneously occur. Fully developed laminar flow of an absorbing emitting gas in a circular tube is considered under the conditions of uniform wall heat flux. Nongray as well as gray formulations are presented, and results are obtained for illustrative cases. Appropriate limiting solutions of the governing equations are obtained and conduction-radiation interaction parameters are evaluated. The influence of variable wall emittance (gray and nongray) upon radiative energy transfer in nongray gases is investigated. In particular, nongray results are obtained, in the large path length limit, for the flow of CO₂ through stainless steel tubes of various compositions. Finally, a correlation is presented which can be utilized to extend all nongray results for the parallel plate geometry, already available in literature, to yield results for the corresponding case of a circular tube. This work was supported by the National Science Foundation through Grant No. GK-16755.     

Radiation effects on Marangoni convection boundary layer over a permeable surface

The problem of Marangoni convection boundary layer flow that can be formed along the interface of two immiscible fluids when the wall is permeable, where there is suction or injection effect, is considered. Similarity equations are obtained through the application of similarity transformation techniques. The effects of suction/injection and radiation parameters on the heat transfer characteristics are numerically studied using the shooting method for a fixed value of the Prandtl number (Pr=0.7). Numerical results are obtained for the surface temperature gradient or the heat transfer rate as well as the temperature profiles for some values of the governing parameters. Comparisons with known results from the open literature show very good agreements. The results indicate that the heat transfer rate at the surface decreases as the radiation parameter increases. Further, results show that multiple (dual) solutions exist for a certain range of the governing parameters.     

Fluid flow of incompressible viscous fluid through a non-linear elastic tube

The study of viscous flow in tubes with deformable walls is of specific interest in industry and biomedical technology and in understanding various phenomena in medicine and biology (atherosclerosis, artery replacement by a graft, etc) as well. The present work describes numerically the behavior of a viscous incompressible fluid through a tube with a non-linear elastic membrane insertion. The membrane insertion in the solid tube is composed by non-linear elastic material, following Fung’s (Biomechanics: mechanical properties of living tissue, 2nd edn. Springer, New York, 1993) type strain–energy density function. The fluid is described through a Navier–Stokes code coupled with a system of non linear equations, governing the interaction with the membrane deformation. The objective of this work is the study of the deformation of a non-linear elastic membrane insertion interacting with the fluid flow. The case of the linear elastic material of the membrane is also considered. These two cases are compared and the results are evaluated. The advantages of considering membrane nonlinear elastic material are well established. Finally, the case of an axisymmetric elastic tube with variable stiffness along the tube and membrane sections is studied, trying to substitute the solid tube with a membrane of high stiffness, exhibiting more realistic response.     

Numerical simulation of complex flows of non-Newtonian fluids using the stream tube method and memory integral constitutive equations

In this paper a memory integral viscoelastic equation is considered for simulating complex flows of non-Newtonian fluids by stream tube analysis. A formalism is developed to take into account co-deformational memory equations in a mapped computational domain where the transformed streamlines are parallel and straight. The particle-tracking problem is avoided. Evolution in time and related kinematic quantities involved with a K-BKZ integral constitutive model are easily taken into account in evaluating the stresses. Successive subdomains, the stream tubes, may be considered for computing the main flow in abrupt axisymmetric contractions from the wall to the central flow region. The ‘peripheral stream tube’ close to the duct wall is determined by developing a non-conventional modified Hermite element. A mixed formulation is adopted and the relevant non-linear equations are solved numerically by the Levenberg-Marquardt algorithm. Although the singularity at the section of contraction is not involved explicitly, the results obtained for the peripheral stream tube clearly show the singularity effects and the extent of the recirculating zone near the salient corner. The algorithm is stable even at high flow rates and provides satisfactory solutions when compared with similar calculations in the literature.     

Nonlinear axisymmetric deformations of an elastic tube under external pressure

The problem of the finite axisymmetric deformation of a thick-walled circular cylindrical elastic tube subject to pressure on its external lateral boundaries and zero displacement on its ends is formulated for an incompressible isotropic neo-Hookean material. The formulation is fully nonlinear and can accommodate large strains and large displacements. The governing system of nonlinear partial differential equations is derived and then solved numerically using the C++ based object-oriented finite element library Libmesh. The weighted residual-Galerkin method and the Newton-Krylov nonlinear solver are adopted for solving the governing equations. Since the nonlinear problem is highly sensitive to small changes in the numerical scheme, convergence was obtained only when the analytical Jacobian matrix was used. A Lagrangian mesh is used to discretize the governing partial differential equations. Results are presented for different parameters, such as wall thickness and aspect ratio, and comparison is made with the corresponding linear elasticity formulation of the problem, the results of which agree with those of the nonlinear formulation only for small external pressure. Not surprisingly, the nonlinear results depart significantly from the linear ones for larger values of the pressure and when the strains in the tube wall become large. Typical nonlinear characteristics exhibited are the “corner bulging” of short tubes, and multiple modes of deformation for longer tubes.     

Peristaltic flow and heat transfer in a vertical porous annulus, with long wave approximation

In this paper, we study the interaction of peristalsis with heat transfer for the flow of a viscous fluid in a vertical porous annular region between two concentric tubes. Long wavelength approximation (that is, the wavelength of the peristaltic wave is large in comparison with the radius of the tube) is used to linearise the governing equations. Using the perturbation method, the solutions are obtained for the velocity and the temperature fields. Also, the closed form expressions are derived for the pressure-flow relationship and the heat transfer at the wall. The effect of pressure drop on flux is observed to be almost negligible for peristaltic waves of large amplitude; however, the mean flux is found to increase by 10-12% as the free convection parameter increases from 1 to 2. Also, the heat transfer at the wall is affected significantly by the amplitude of the peristaltic wave. This warrants further study on the effects of peristalsis on the flow and heat transfer characteristics.     

Fluid flow through a partially filled cylinder

The laminar flow of viscous incompressible fluid through a partially filled circular cylinder is studied. The governing equations reduce to Laplace's equation, and the region of solution is the union of two intersecting circles. This region is mapped onto the unit circle by a conformal transformation, so that the problem may be solved by the Poisson integral formula. The integration is performed by a combination of residues and numerical integration. The results agree very well with exact analytic solutions at 50%, 85.4%, and 100% depth to diameter ratios. It is found that by not filling the cylinder completely, the flow rate can be increased by a significant 27.3%.     

Solution of boundary layer flow and heat transfer of an electrically conducting micropolar fluid in a non-Darcian porous medium

In this paper, we have studied the effects of radiation on the boundary layer flow and heat transfer of an electrically conducting micropolar fluid over a continuously moving stretching surface embedded in a non-Darcian porous medium with a uniform magnetic field has been analyzed analytically. The governing fundamental equations are approximated by a system of nonlinear locally similar ordinary differential equations which are solved analytically by applying homotopy analysis method (HAM). The effects of Darcy number, heat generation parameter and inertia coefficient parameter are determined on the flow. Convergence of the obtained series solution is discussed. The homotopy analysis method provides us with a new way to obtain series solutions of such problems. This method contains the auxiliary parameter which provides us with a simple way to adjust and control the convergence region of series solution. By suitable choice of the auxiliary parameter, we can obtain reasonable solutions for large modulus.     

Newtonian and non-Newtonian liquids rotating adjacent to a stationary surface

The interaction of a rotating flow and a stationary surface is discussed for a second-order non-Newtonian liquid. Similarity solutions of the governing partial differential equations are obtained for the case of the outer flow in solid-body rotation. The results for the Newtonian case are compared with Bödewadt's series solution of this problem. The non-Newtonian solutions indicate that for certain values of the parameters characterizing the non-linear viscous response and normal stress effects a larger secondary flow is induced in the boundary layer than in the Newtonian case.Also at North Carolina State University Raleigh (N.C.), U.S.A.     

轴对称横观各向同性热弹性圆柱的分解

为了得到精确的应力场、位移场、温度场,将扭转圆轴的精化理论研究方法推广到轴对称横观各向同性热弹性圆柱。利用Bessel函数以及轴对称横观各向同性热弹性圆柱的通解,给出了轴对称横观各向同性热弹性圆柱的分解定理。根据柱面齐次边界条件获得了精确的精化方程,精化方程可以分解为一阶方程、超越方程、温度方程,从而将横观各向同性热弹性圆柱的轴对称问题分解为轴向拉压问题、超越问题、热-应力耦合问题。超越部分对应端部自平衡情况,可以清晰地了解到端部应力分布对内部应力场的影响,热-应力耦合部分对应无外加应力场时圆柱内部因温度变化引起的热应力。     

Effects of heat and mass transfer peristaltic flow of?Williamson fluid in a vertical annulus

In the present investigation, we have studied the effects of mixed convection heat and mass transfer on peristaltic flow of Williamson fluid model in a vertical annulus. The governing equations of Williamson fluid model are simplified using the assumptions of long wavelength and low Reynold’s number. An approximated analytical and numerical solutions are found for the velocity field using (i) Perturbation method (ii) Shooting method. The comparisons of analytical and numerical solutions have been presented. The expressions for pressure rise, velocity against various physical parameter are discussed through graphs.     

MHD free-convective flow of micropolar and Newtonian fluids through porous medium in a vertical channel

We have studied the fully-developed free-convective flow of an electrically conducting fluid in a vertical channel occupied by porous medium under the influence of transverse magnetic field. The internal prefecture of the channel is divided into two regions; one region filled with micropolar fluid and the other region with a Newtonian fluid or both the regions filled by Newtonian fluids. Analytical solutions of the governing equations of fluid flow are found to be in excellent agreement with analytical prediction. Analytical results for the details of the velocity, micro-rotation velocity and temperature fields are shown through graphs for various values of physical parameters. It is noticed that Newtonian fluids prop up the linear velocity of the fluid in contrast to micropolar fluid. Also the skin friction coefficient at both the walls is derived and its numerical values are offered through tables.     

Magnetohydrodynamic (MHD) flow of a second grade fluid in a channel with porous wall

An analysis has been carried out to study the effect of magnetic field on an electrically conducting fluid of second grade in a parallel channel. The coolant fluid is injected into the porous channel through one side of the channel wall into the other heated impermeable wall. The combined effect of inertia, viscous, viscoelastic and magnetic forces are studied. The basic equations governing the flow and heat transfer are reduced to a set of ordinary differential equations by using appropriate transformations for velocity and temperature. Numerical solutions of these equations are obtained with the help of Runge-Kutta fourth order method in association with quasi-linear shooting technique. Numerical results for velocity field, temperature field, skin friction and Nusselt number are presented in terms of elastic parameter, Hartmann number, Prandtl number and Reynolds number. Special case of our results is in good agreement with earlier published work.     

Bifurcation phenomena in two-phase natural circulation

Multiple steady-state solutions have been found theoretically for a two-phase natural circulation loop. This bifurcation phenomenon is attributed to the nonmonotonic behavior of the buoyancy and friction forces, and the discontinuities due to transitions between two-phase flow patterns. The paper outlines a general and consistent modeling method to describe two-phase natural circulation at steady state. The solution of the one-dimensional governing conservation equations is performed by a combination of analytical and numerical stages. The method is implicit, to treat the highly nonlinear problem by multinested iteration loops. Results are presented and discussed for the multiple solutions of the flow rate, temperature, quality and void distributions and two-phase elevations. Pressure effects are also studied. A comparison with available data shows that the theoretical model simulates the important characteristics of two-phase thermosyphons.     

An investigation on internal thermal flow of non-Newtonian fluid

In the present paper an unsteady thermal flow of non-Newtonian fluid is investigated which is of the fiow into axisymmetric mould cavity. In the second part an unsteady thermal flow of upper-convected Maxwell fluid is studied, For the flow into mould cavity the constitutive equation of power-law fluid is used as a rheological model of polymer fluid. The apparent viscosity is considered as a function of shear rate and temperature. A characteristic viscosity is introduced in order to avoid the nonlinearity due to the temperature dependence of the apparent viscosity. As the viscosity of the fluid is relatively high the flow of the thermal fluid can be considered as a flow of fully developed velocity field. However, the temperature field of the fluid fiow is considered as an unsteady one. The governing equations are constitutive equation, momentum equation of steady flow and energy conservation equation of non-steady form. The present system of equations has been solved numerically by the splitting differen     

The nonlinear dynamics of elastic tubes conveying a fluid

The Kirchhoff equations for elastic tubes are modified to include the effect of fluid flow. Using the techniques of linear and nonlinear analysis specially developed for the Kirchhoff equations, the effect of the fluid flow on the basic twist-to-writhe instability is investigated. The results suggest an intriguing modification of the bifurcation threshold due to the flow. Beyond threshold the buckled tube acquires a slight curvature which modifies the flow rate and results in a correction to nonlinearity of the amplitude equation governing the deformation dynamics.