Numerical solution of a casson fluid flow in the entrance of annular tubes

The solution of the momentum equation for a Casson fluid flowing in the entrance region of an annular tube has been obtained. The results have been presented for a large range of radii ratio and dimensionless yield stress. The mathematical accuracy of the numerical procedure is demonstrated by comparing the asymptotic velocity profiles at large axial distance with fully developed solution [1]. In addition, the results of the numerical solution for the case of yield stress equal to zero are compared with the entrance flow solution for a Newtonian fluid [2].     

Peristaltic transport of a Herschel-Bulkley fluid in an inclined tube

Peristaltic flow of Herschel-Bulkley fluid in an inclined tube is analyzed. The velocity distribution, the stream function and the volume flow rate are obtained. Also, when the yield stress ratio τ→0, and when the inclination parameter α=0 and the fluid parameter n=1, the results agree with those of Jaffrin and Shapiro (Ann. Rev. Fluid Mech. 3 (1971) 13) for peristaltic transport of a Newtonian fluid in a horizontal tube. The effects of τ and n on the pressure drop and the mean flow are discussed through graphs. Furthermore, the results for the peristaltic transport of Bingham and power law fluids through a flexible tube are obtained and discussed. The results obtained for the flow characteristics reveal many interesting behaviors that warrant further study of the effects of Herschel-Bulkley fluid on the flow characteristics.     

Numerical analysis of the transient conjugated heat transfer in a circular duct with a power-law fluid

We treat numerically in this paper, the transient analysis of a conjugated heat transfer process in the thermal entrance region of a circular tube with a fully developed laminar power-law fluid flow. We apply the quasi-steady approximation for the power-law fluid, identifying the suitable time scales of the process. Thus, the energy equation in the fluids is solved analytically using the well-known integral boundary layer technique. This solution is coupled to the transient energy equation for the solid where the transverse and longitudinal heat conduction effects are taken into account. The numerical results for the temporal evolution of the average temperature of the tube wall, _av, is plotted for different nondimensional parameters such as conduction parameter, , the aspect ratios of the tube, and ₀ and the index of power-law fluid, n.     

Injection and suction of an upper-convected maxwell fluid through a porous-walled tube

The steady-state, similarity solutions of the flow of an upper-convected Maxwell fluid through a tube with a porous wall are constructed by asymptotic and numerical analyses as functions of the direction of flow through the tube, the amount of elasticity in the fluid, as measured by the Deborah number De, and the degree of fluid slip along the tube wall. Fluid slip is assumed to be proportional to the local shear stress and is measured by a slip parameter β that ranges between no-slip (β = 1) and perfect slip (β = 0). The most interesting results are for fluid injection into the tube. For β = 1, the family of flows emanating from the Newtonian limit (De = 0) has a limit point where it turns back to lower values of De. These solutions become asymptotic to De = 0) and develop an O(De) boundary layer near the tube wall with singularly high stresses matched to homogeneous elongational flow in the core. This solution structure persists for all nonzero values of the slip parameter. For β ≠ 1, a family of exact solutions is found with extensional kinematics, but nonzero shear stress convected into the tube through the wall. These flows differ for low De from the Newtonian asymptote only by the absence of the boundary layer at the tube wall. Finite difference calculations evolve smoothly between the Newtonian-like and extensional solutions because of approximation error due to under-resolution of the boundary layer. The radial gradient of the axial normal stress of the extensional flow is infinite at the centerline of the tube for De > 1; this singularity causes failure of the finite difference approximations for these Deborah numbers unless the variables are rescaled to take the asymptotic behavior into account.     

Entrance flow of a yield-power law fluid

In the present work we have obtained the numerical solution of the momentum equation for a Yield-Pseudoplastic power-law fluid flowing in the entrance region of a tube. The accuracy of the numerical results is checked by comparing the asymptotic values of friction coefficients and velocity profiles with the corresponding results from the analytical solutions for the fully-developed region. The results of the entrance flow solution for the power-law exponent equal to unity (Bingham fluid) are also in agreement with the numerical solution for a Bingham fluid. Detailed results are presented for wide ranges of yield numbers and power law exponents.

Nomenclature

Nomenclature a constant - D diameter - F dimensionless pressure gradient in (4.3) - f _x friction factor in (5.1) - f _app total friction factor in (5.2) - K entrance pressure drop coefficient - n power law exponent - p pressure - r radial co-ordinate - R radius of a tube - Re Reynolds number (5.3) - s rate of shear, u/r - u axial velocity - average velocity - v velocity in radius direction - x axial co-ordinate - y normal co-ordinate - Y yield number in (4.4) - z dimensionless axial distance =(x/D)/Re - z ₁ 1/z Greek Symbols plug flow radius in (4.6) - _eff effective viscosity - density - shear stress - _y yield stress - dimensionless stream function     

Modeling of the phase lag causing fluidelastic instability in a parallel triangular tube array

Fluidelastic instability is considered a critical flow induced vibration mechanism in tube and shell heat exchangers. It is believed that a finite time lag between tube vibration and fluid response is essential to predict the phenomenon. However, the physical nature of this time lag is not fully understood. This paper presents a fundamental study of this time delay using a parallel triangular tube array with a pitch ratio of 1.54. A computational fluid dynamics (CFD) model was developed and validated experimentally in an attempt to investigate the interaction between tube vibrations and flow perturbations at lower reduced velocities U_r=1–6 and Reynolds numbers Re=2000–12 000. The numerical predictions of the phase lag are in reasonable agreement with the experimental measurements for the range of reduced velocities U_g/fd=6–7. It was found that there are two propagation mechanisms; the first is associated with the acoustic wave propagation at low reduced velocities, U_r<2, and the second mechanism for higher reduced velocities is associated with the vorticity shedding and convection. An empirical model of the two mechanisms is developed and the phase lag predictions are in reasonable agreement with the experimental and numerical measurements. The developed phase lag model is then coupled with the semi-analytical model of Lever and Weaver to predict the fluidelastic stability threshold. Improved predictions of the stability boundaries for the parallel triangular array were achieved. In addition, the present study has explained why fluidelastic instability does not occur below some threshold reduced velocity.     

Thermal convection for a thermodependent herschel-bulkley fluid in an annular duct

This article presents a numerical and experimental investigation of the thermal convection for a thermodependent Herschel-Bulkley fluid in an annular duct under the conditions of a uniform heat flux density on the outer wall and an insulated inner wall. In the numerical analysis, it is assumed that: (i) the rheological behavior of the fluid can be expressed through the Herschel-Bulkley law: $\tau = \tau _s + K\dot \gamma ^n $ ; (ii) the flow is fully developed at the inlet; (iii) all fluid properties except consistency indexK are constant. TheK?T relation used isK=K ₀exp(?bT). The results obtained enable us to characterize completely the dynamical and thermal fields. The numerical solution is in good agreement with the experimental data, showing the reasonableness of the computed results.     

Effect of yield stress on the flow of a Casson fluid in a homogeneous porous medium bounded by a circular tube

The effect of yield stress on the flow characteristics of a Casson fluid in a homogeneous porous medium bounded by a circular tube is investigated by employing the Brinkman model to account for the Darcy resistance offered by the porous medium. The non-linear coupled implicit system of differential equations governing the flow is first transformed into suitable integral equations and are solved numerically. Analytical solution is obtained for a Newtonian fluid in the case of constant permeability, and the numerical solution is verified with that of the analytic solution. The effect of yield stress of the fluid and permeability of the porous medium on shear stress and velocity distributions, plug flow radius and flow rate are examined. The minimum pressure gradient required to start the flow is found to be independent of the permeability of the porous medium and is equal to the yield stress of the fluid.     

A Numerical Study of Laminar Reciprocating Flow in a Pipe of Finite Length

A numerical investigation has been carried out for a laminar incompressible reciprocating flow in a circular pipe with a finite length. An examination of the governing equations and boundary conditions indicates that a sinusoidally reciprocating flow is governed by three similarity parameters: the kinetic Reynolds number Re, the dimensionless oscillation amplitude A_o, and the length to diameter ratio L/D. The numerical solution for the velocity profiles of a developing reciprocating flow shows that at any instant of time, there exist three flow regimes in the pipe, namely, an entrance regime, a fully developed regime and an exit regime. The numerical results for the fully developed region are shown to be in excellent agreement with the analytical solution. Based on the numerical results, a correlation equation of the space-cycle averaged friction coefficient for a laminar developing reciprocating pipe flow has been obtained in terms of the three similarity parameters.     

The inlet effect on the drag factor of a sphere in a tube

The creeping motion Ground a sphere situated axisymmetrically near the entrance of asemi-infinite circular cylindrical tube is analyzed using infinite series solutions for thevelocity components. pressure and the stream function. Truncating the infinite series. thecorresponding coefficients in the series are determined by a collocation technique. The dragfactor and the stress distribution on the surface of the sphere are calculated for the sphere inmotion in quiescent fluid and for the flow with uniform velocity at the entrance past a rigidlyheld sphere. The results indicate that a sphere near the entrance which has a uniformentrance velocity profile will suffer larger drag than that in infinite tube.Theconvergence of the collocation technique is tested by numerical calculation. It is shown thatthe technique has good convergence properties.     

Structure and evolution of the airflow generated by a slender body entry near a tube

When a slender body moving forward in open air enters into a confined region, two important unsteady aerodynamic phenomena are generated. An exiting flow is created with a direction opposite to the body movement and inside the confined region, a compression wave is formed. Generation mechanism of compression wave have been extensively studied but so far, no detailed investigation of the exiting flow has ever been reported. The experimental study presented in this paper was undertaken to gain insight into the structure and the evolution of the exit-flow. Experiments were conducted with an axisymmetric apparatus and the explored range of the moving body speed was 5–50 m/s. The study focused on the influence of the body speed and the body nose geometry on the flow. It was shown that the air ejected from the tube entrance generates an annulus jet accompanied by a vortex ring. The vortex development was clarified using laser sheet visualizations associated with unsteady pressure and velocity measurements at the tube entrance. It is constituted by four phases, the pre-vortex phase, the vortex development phase, the vortex convection phase and the vortex breakdown phase. The duration of each of these steps was found to be independent of both the studied parameters in a non-dimensional time scale. Furthermore, neither the body speed nor the nose geometry induced significant changes on the vortex ring evolution, except for extreme conditions (low body speed, V_M.B.<15 m/s, and/or very long nose geometry, L_nose/D_M.B.>6). The evolution of the vortex ring was compared to that of ‘classical’ vortex ring generated at a tube exit by a piston motion with large non-dimensional stroke length. Main similarities and differences were discussed in the paper. In particular, the formation number of vortex ring observed in our experiments was found to be significantly smaller.     

Heat transfer in an inner loop reactor

General expressions for evaluating heat transfer in reactors with a draft tube have been derived. Theoretical results show that heat transfer can be enhanced for flow pattern (A) compared with that in an open duct whenK ₁ is not departed from 0.5. The competition of the residence time and the volumetric flow rate of the fluids in the annulus and in the draft tube may be used to explain the fluid behaviors. For flow patterns (B) and (C) as in a loop reactor, introduce of recycling can augment the heat transfer rate for large Graetz number, especially whenK ₁,K ₂ orR increases. The competition between the premixing and the residence time effects of the fluid may be used to describe the fluid behaviors. Moreover, asymptotic solutions for all flow patterns were also derived.     

Nanofluids thermal behavior analysis using a new dispersion model along with single-phase

A numerical study has been performed to analyze nanofluids convective heat transfer. Laminar α-Al₂O₃-water nanofluid flows in an entrance region of a horizontal circular tube with constant surface temperature. Numerical analysis has been carried out using two different single-phase models (homogenous and dispersion) and two-phase models (Eulerian–Lagrangian and mixture). A new model is developed to consider the nanoparticles dispersion. The transport equations for the tube with constant surface temperature were solved numerically using a control volume approach. The effects of nanoparticles volume fraction (0.5, 1 %) and Reynolds number (650 ≤ Re ≤ 2300) on nanofluid convective heat transfer coefficient were studied. The results are compared with the experimental data and it is shown that the homogenous single-phase model is underestimated and the mixture model is overestimated. Although the Eulerian–Lagrangian model gives a reasonable prediction for the thermal behavior of nanofluids, the dispersion single-phase model gives more accurate prediction despite its simplicity.     

Theoretical and experimental studies on the coefficient of discharge and spray cone angle of a swirl spray pressure nozzle using a power-law non-Newtonian fluid

An extension of earlier work is made in the present paper to determine both theoretically and experimentally the coefficient of discharge and spray cone angle of a swirl nozzle using a time-independent purely viscous power-law non-Newtonian fluid. The theoretical predictions are made through an approximate analytical solution of the hydrodynamics of flow inside the nozzle. Experiments are carried out with aqueous solutions of CMC (carboxymethyl cellulose sodium salt) powder of various concentrations as the working fluids. The rheological properties of the working fluids are established by a capillary tube viscometer. From both the theoretical and experimental analyses, the pertinent independent input parameters are recognised as the generalised Reynolds number at inlet to the nozzle Re_Gi, the flow behaviour index of the fluid n, length-to-diameter ratio of the swirl chamber L₁/D₁, spin chamber angle 2α and the orifice-to-swirl-chamber-diameter ratio D₁/D₁. Although the theory predicts the correct qualitative trend in all cases, it does not agree well with the experimental results. Therefore, on the basis of the theoretical results, emperical relationships between nozzle characteristics and input parameters heve been established. Finally it is recognised that, regarding the injection conditions and fluid properties, the generalised Reynolds number at nozzle inlet Re_Gi and the flow behaviour index n have inverse and direct effects, respectively, on the coefficient of discharge, but have a negligible influence on the spray cone angle. Amongst the nozzle geometries, an increase in the values of D₂/D₁ and 2α or a decrease in the value of L₁/D₁ decrease the coefficient of discharge and increase the spray cone angle.     

Numerical investigation of flow and heat transfer in a gas-droplet separated turbulent stream using the Reynolds stress transfer model

The results of the numerical modeling of flow structure, turbulence, and heat transfer in a gas-droplet stream after sudden tube expansion on the basis of the Eulerian approach are presented. The gas phase turbulence was modeled using the Reynolds stress transfer model modified to allow for the presence of particles. The results are compared with those obtained using the two-equation k-ε model. The latter results overestimate the heat transfer in the separation flow as compared with the Reynolds stress transfer model. The heat transfer is shown to considerably increase, when evaporating droplets are incorporated in the separation flow (by a factor of more than 1.5 compared with the case of a single-phase flow at a small mass concentration of the droplets M _L1 ≤ 0.05). The addition of the disperse phase in the turbulent gas flow leads a slight increase in the recirculation zone length. Good agreement with the experimental data indicates the adequacy of the numerical model developed.     

Purely elastic flow asymmetries in flow-focusing devices

The flow of a viscoelastic fluid through a microfluidic flow-focusing device is investigated numerically with a finite-volume code using the upper-convected Maxwell (UCM) and Phan-Thien–Tanner (PTT) models. The conceived device is shaped much like a conventional planar “cross-slot” except for comprising three inlets and one exit arm. Strong viscoelastic effects are observed as a consequence of the high deformation rates. In fact, purely elastic instabilities that are entirely absent in the corresponding Newtonian fluid flow are seen to occur as the Deborah number (De) is increased above a critical threshold. From two-dimensional numerical simulations we are able to distinguish two types of instability, one in which the flow becomes asymmetric but remains steady, and a subsequent instability at higher De in which the flow becomes unsteady, oscillating in time. For the UCM model, the effects of the geometric parameters of the device (e.g. the relative width of the entrance branches, WR) and of the ratio of inlet average velocities (VR) on the onset of asymmetry are systematically examined. We observe that for high velocity ratios, the critical Deborah number is independent of VR (e.g. De_c ≈ 0.33 for WR = 1), but depends non-monotonically on the relative width of the entrance branches. Using the PTT model we are able to demonstrate that the extensional viscosity and the corresponding very large stresses are decisive for the onset of the steady-flow asymmetry.     

Finite element calculation of viscoelastic flow in a journal bearing: II. Moderate eccentricity

Finite element calculations of two-dimensional flows of viscoelastic fluids in a journal bearing geometry reported in an earlier paper (J. Non-Newt. Fluid Mech. 16 (1984) 141-172) are extended to higher eccentricity (ρ = 0.4); at this higher eccentricity flow separation occurs in the wide part of the gap for a Newtonian fluid. Calculations for the second-order fluid (SOF), upper-convected Maxwell (UCM), and the Giesekus models are continued in increasing Deborah number for each model until either a limit point is reached or oscillations in the solution make the numerical accuracy too poor to warrant proceeding. No steady solutions to the UCM model were found beyond a limit point De_c, as was the case for results at low eccentricities. The value of De_c was moderately stabel to mesh refinement. A limit point also terminated the calculations with a SOF model, in contradiction to the theorems for uniqueness and existence for this model. The critical value of De increased drastically with increasing refinement of the mesh, as expected for solution pathology caused by approximation error. Calculations for the Giesekus fluid with the mobility parameter α ≠ O showed no limit points, but failed when irregular oscillations destroyed the quality of the solution. The behavior of the recirculation region of the flow and the load on the inner cylinder were very sensitive to the value of α used in the Giesekus model. The recirculation disappeared at low values of De except when the mobility parameter α was so small that the viscosity was almost constant over the range of shear rates in the calculations. The recirculation persisted over the entire range of accessible De for the UCM fluid, the limit of α = O of the Giesekus model. The behavior of the recirculation is coupled directly to the viscosity by calculations with an inelastic fluid with the same viscosity predicted by the Giesekus model.     

Analysis of laminar flow forced convection heat transfer in the entrance region of a circular pipe

A numerical solution, for incompressible, steady-state, laminar flow heat transfer in the combined entrance region of a circular tube is presented for the case of constant wall heat flux and constant wall temperature. The development of velocity profile is obtained from Sparrow's entrance region solution. This velocity distribution is used in solving the energy equation numerically to obtain temperature profiles. Variation of the heat transfer coefficient for these two different boundary conditions for the early stages of boundary layer formation on the pipe wall is obtained. Local Nusselt numbers are calculated and the results are compared with those given byUlrichson andSchmitz. The effect of the thermal boundary conditions is studied by comparing the uniform wall heat flux results with uniform wall temperature.     

Peristaltic flow of a Johnson-Segalman fluid through a deformable tube

To understand theoretically the flow properties of physiological fluids we have considered as a model the peristaltic motion of a Johnson–Segalman fluid in a tube with a sinusoidal wave traveling down its wall. The perturbation solution for the stream function is obtained for large wavelength and small Weissenberg number. The expressions for the axial velocity, pressure gradient, and pressure rise per wavelength are also constructed. The general solution of the governing nonlinear partial differential equation is given using a transformation method. The numerical solution is also obtained and is compared with the perturbation solution. Numerical results are demonstrated for various values of the physical parameters of interest.      

Non-linear wave modulation in a prestressed viscoelastic thin tube filled with an inviscid fluid

In the present work, the propagation of weakly non-linear waves in a prestressed thin viscoelastic tube filled with an incompressible inviscid fluid is studied. Considering that the arteries are initially subjected to a large static transmural pressure P₀ and an axial stretch λ_z and, in the course of blood flow, a finite time-dependent displacement is added to this initial field, the governing non-linear equation of motion in the radial direction is obtained. Using the reductive perturbation technique, the propagation of weakly non-linear, dispersive and dissipative waves is examined and the evolution equations are obtained. Utilizing the same set of governing equations the amplitude modulation of weakly non-linear and dissipative but strongly dispersive waves is examined. The localized travelling wave solution to these field equations are also given.