Buoyancy-assisted flow of yield stress fluids past a cylinder: Effect of shape and channel confinement

The aiding-buoyancy mixed convection heat transfer in Bingham plastic fluids from an isothermal cylinder of elliptical and circular shape in a vertical adiabatic channel is numerically investigated. For a fixed shape of the elliptical cylinder E = 2 (ratio of major to minor axes), the effect of confinement is studied for three values of blockage ratio, B, defined as the ratio of the channel width to the circumference of the cylinder/π, as 6.5, 2.17 and 1.3. In order to delineate the role of cross-section of the cylinder, results are also presented here for a circular cylinder of the same heat transfer area as the elliptical cylinder. The results presented herein span the range of conditions as: Bingham number, 0 ≤ Bn ≤ 100, Reynolds number, 1 ≤ Re ≤ 40, and Prandtl number, 1 ≤ Pr ≤ 100 over the range of Richardson number Ri = 0 (pure forced convection) to Ri = 10. Extensive results on drag coefficient, local and surface averaged values of the Nusselt number and yield surfaces are presented herein to elucidate the combined effects of buoyancy, blockage ratio and fluid yield stress. The morphology of the yield surfaces shows that the unyielded plug regions formed upstream and downstream of the cylinder grow faster at low Reynolds numbers with the increasing yield stress effects under the weak buoyancy forces, i.e., small values of Grashof or Richardson number. The heat transfer enhancement is observed with the increasing channel-confinement due to the sharpening of the temperature gradients near the surface of the cylinder. The average Nusselt number shows a positive dependence on the Reynolds number, Prandtl number and Richardson number irrespective of the shape of the cylinder or the type of fluid. By employing the modified definitions of the dimensionless parameters (based on the two choices of the overall effective fluid velocity), predictive correlations have been established for estimating the value of the average Nusselt number in a new application.     

Pulsatile flow of Herschel–Bulkley fluid through catheterized arteries – A mathematical model

The pulsatile flow of blood through catheterized artery has been studied in this paper by modeling blood as Herschel–Bulkley fluid and the catheter and artery as rigid coaxial circular cylinders. The Herschel–Bulkley fluid has two parameters, the yield stress θ and the power index n. Perturbation method is used to solve the resulting quasi-steady nonlinear coupled implicit system of differential equations. The effects of catheterization and non-Newtonian nature of blood on yield plane locations, velocity, flow rate, wall shear stress and longitudinal impedance of the artery are discussed. The existence of two yield plane locations is investigated and their dependence on yield stress θ, amplitude A, and time t are analyzed. The width of the plug core region increases with increasing value of yield stress at any time. The velocity and flow rate decrease, whereas wall shear stress and longitudinal impedance increase for increasing value of yield stress with other parameters held fixed. On the other hand, the velocity, flow rate and wall shear stress decrease but resistance to flow increases as the catheter radius ratio (ratio of catheter radius to vessel radius) increases with other parameters fixed. The results for power law fluid, Newtonian fluid and Bingham fluid are obtained as special cases from this model.     

Helical flows of second grade fluid due to constantly accelerated shear stresses

The helical flows of second grade fluid between two infinite coaxial circular cylinders is considered. The motion is produced by the inner cylinder that at the initial moment applies torsional and longitudinal constantly accelerated shear stresses to the fluid. The exact analytic solutions, obtained by employing the Laplace and finite Hankel transforms and presented in series form in term of usual Bessel functions of first and second kind, satisfy both the governing equations and all imposed initial and boundary conditions. In the limiting case when α → 0, the solutions for Newtonian fluid are obtained for the same motion. The large-time solutions and transient solutions for second grade fluid are also obtained, and effect of material parameter α and kinematic viscosity ν is discussed. In the last, the effects of various parameters of interest on fluid motion as well as the comparison between second grade and Newtonian fluids are analyzed by graphical illustrations.     

Analytical solution of unsteady three-dimensional MHD boundary layer flow and heat transfer due to impulsively stretched plane surface

The unsteady magnetohydrodynamic viscous flow and heat transfer of Newtonian fluids induced by an impulsively stretched plane surface in two lateral directions are studied by using an analytic technique, namely, the homotopy method. The analytic series solution presented here is highly accurate and uniformly valid for all time in the entire region. The effects of the stretching ratio and the magnetic field on the surface shear stresses and heat transfer are studied. The surface shear stresses in x- and y-directions and the surface heat transfer are enchanced by increasing stretching ratio for a fixed value of the magnetic parameter. For a fixed stretching ratio, the surface shear stresses increase with the magnetic parameter, but the heat transfer decreases. The Nusselt number takes longer time to reach the steady state than the skin friction coefficients. There is a smooth transition from the initial unsteady state to the steady state.     

Axisymmetric flow over a backward-facing step in an annular pipe

The incompressible flow of a Newtonian fluid over a backward-facing step is investigated numerically. The geometry is an annular pipe in which the radius of the inner cylinder decreases suddenly. Keeping the radial expansion ratio fixed axisymmetric flows are computed for outlet radius ratios from 0.1 to 1 (ratio of the inner to the outer outlet radius). The Reynolds number at which the flow separates from the outer cylinder decreases as the outlet radius ratio decreases for constant inlet geometry. The growth with Reynolds number of the recirculation zone on the inner outlet cylinder just behind the step is strongly reduced when the recirculation zone on the outer cylinder is established. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Viscoelastic flow in a curved channel: A similarity solution for the Oldroyd-B fluid

A similarity solution is used to analyse the flow of the Oldroyd fluid B, which includes the Newtonian and Maxwell fluids, in a curved channel modelled by the narrow annular region between two circular concentric cylinders of large radius. The solution is exact, including inertial forces. It is found that the non-Netonian kinematics are very similar to the Newtonian ones, although some stress components can become very large. At high Reynolds number a boundary layer is developed at the inner cylinder. The structure of this boundary layer is asymptotically analysed for the Newtonian fluid. Non-Newtonian stress boundary layers are also developed at the inner cylinder at large Reynolds numbers.     

Steady flows of Cosserat–Bingham fluids

The equations describing the steady flow of Cosserat–Bingham fluids are considered, and existence of weak solution is proved for the three‐dimensional boundary‐value problem with the use of the Lipschitz truncation argument. In contrast to the classical Bingham fluid, the micropolar Bingham fluid supports local micro‐rotations and two types of plug zones. Our approach is based on an approximation of the constitutive relation by a generalized Newtonian constitutive relation and a subsequent limiting process. Copyright © 2016 John Wiley & Sons, Ltd.     

Analysis of viscoelastic fluid flow with temperature dependent properties in plane Couette flow and thin annuli

The steady state flow in very thin annuli has been studied analytically for the case where the annular gap is much smaller than the radius of the inner cylinder and for the outer cylinder rotating at constant angular speed and the inner cylinder at rest. The cylinders were subjected to two different thermal boundary conditions. The exponential effect of temperature on the relaxation time and the viscosity coefficient was accounted into the governing differential equations using Nahme’s law. Effects of viscous dissipation as well as εDe² (viscoelastic index for SPTT constitutive equation) on the dimensionless velocity and temperature profiles have been investigated. Results show that while the properties of the fluid depend on temperature, the velocity and temperature profiles are different compared to those obtained with constant physical properties. The Nahme–Griffith number increases whereas εDe² as a viscoelastic index decreases when temperature dependent physical properties are considered. In addition, the results indicate that the viscous dissipation has a sensible effect on heat transfer and the Nusselt number decreases with an increase in the Nahme–Griffith number.     

Accelerated slip flow past a cylinder

This study deals with boundary layer flow along the entire length of a stationary semi-infinite cylinder under a steady, accelerated free-stream. Considering flow at reduced dimensions, the no-slip boundary condition is replaced with a Navier boundary condition. Asymptotic series solutions are obtained for the shear stress coefficient in terms of the Bingham number that corresponds to prescribed values of both the slip coefficient and the index of acceleration. By investigating motion at small and large axial distances, the series solutions are presented. For flow in the intermediate distances, exact and interpolated numerical solutions are obtained. Using these results, the shear stress along the entire cylinder wall is evaluated in terms of the parameters of acceleration and slip.     

Effects of slip on the non-linear flows of a third grade fluid

The present analysis considers the non-linear problems of steady flow of a third grade fluid between the concentric cylinders. A complete analysis of mathematical modeling is made when no-slip condition is no longer valid. Exact analytic solutions of the following two non-linear problems are derived: (i) when inner cylinder moves and outer cylinder remains stationary and (ii) for inner cylinder at rest and outer cylinder in motion. Graphical results are presented to illustrate the analytic solutions. The corresponding results of no-slip condition are deduced as the limiting cases when the slip parameter is equal to zero.     

Unsteady longitudinal flow of a generalized Oldroyd-B fluid in cylindrical domains

Considering a fractional derivative model the unsteady flow of an Oldroyd-B fluid between two infinite coaxial circular cylinders is studied by using finite Hankel and Laplace transforms. The motion is produced by the inner cylinder which is subject to a time dependent longitudinal shear stress at time t = 0⁺. The solution obtained under series form in terms of generalized G and R functions, satisfy all imposed initial and boundary conditions. The corresponding solutions for ordinary Oldroyd-B, generalized and ordinary Maxwell, and Newtonian fluids are obtained as limiting cases of our general solutions. The influence of pertinent parameters on the fluid motion as well as a comparison between models is illustrated graphically.     

Exact solutions for the flow of a viscoelastic fluid induced by a circular cylinder subject to a time dependent shear stress

The velocity field and the adequate shear stress corresponding to the flow of a Maxwell fluid with fractional derivative model, between two infinite coaxial cylinders, are determined by means of the Laplace and finite Hankel transforms. The motion is due to the inner cylinder that applies a longitudinal time dependent shear to the fluid. The solutions that have been obtained, presented under integral and series form in terms of the generalized G and R functions, satisfy all imposed initial and boundary conditions. They can be easy particularizes to give the similar solutions for ordinary Maxwell and Newtonian fluids. Finally, the influence of the relaxation time and the fractional parameter, as well as a comparison between models, is shown by graphical illustrations.     

A numerical algorithm for modeling laminar flows in an annular channel with eccentricity

We propose a numerical algorithm for simulating steady laminar flows of an incompressible fluid in annular channels with eccentricity and rotation of the inner cylinder. This algorithm enables us to describe this class of flows for wide ranges of the annular channel and flow parameters. We present the implementation details and the results of testing this numerical method. For a series of flows in an annular clearance, we compare the numerical results to available analytic solutions and experimental data. In all cases under consideration the simulated data agree well with the available experimental, analytical, and numerical solutions.     

A note on the flow of a fluid with pressure-dependent viscosity in the annulus of two infinitely long coaxial cylinders

The dependence of the viscosity of fluids on pressure has been well established by experiments and it needs to be taken into consideration in problems where there is a large variation of pressure in the flow domain. In this paper we consider the flow of a fluid in the annulus between two cylinders whose viscosity depends on the pressure. First we consider the steady flow in the annulus due to the rotation of one cylinder with respect to the other. Then we study the problem of flow in the annular region due to torsional and longitudinal oscillations of one cylinder with respect to the other. In both the problems considered the flow is found to be markedly different from that for the incompressible Navier–Stokes fluid with constant viscosity.     

Helical flow of an Oldroyd-B fluid due to a circular cylinder subject to time-dependent shear stresses

Exact solutions corresponding to the unsteady helical flow of an Oldroyd-B fluid due to an infinite circular cylinder subject to torsional and longitudinal time-dependent shear stresses are established using Hankel transforms. These solutions, presented under series form in terms of Bessel functions J ₀(·), J ₁(·) and J ₂(·), can be easily specialized to give the similar solutions for Maxwell, Second grade and Newtonian fluids performing the same motion. Some characteristics of the motion, as well as the influence of pertinent parameters on the velocity profiles, are underlined by graphical illustrations.     

精确计算Bingham流体圆管层流压降及速度分布规律的新方法

Bingham(宾汉)模型情况下,多采用通用公式进行圆管层流压降的解析计算,即将Bingham模型本构方程代入粘性流体圆管层流流动通用公式进行计算,仅能得到压降的解析解.新方法结合Bingham流体本构方程与运动方程,建立有关力学平衡方程,并运用代数方程的根式解理论对圆管层流流动时的非线性方程进行求解,可直接求得Bingham流体圆管层流压降及速度流核区半径的解析解,进一步可求得圆管层流速度解析解;Bingham流体圆管层流速度的直接影响因素为流量、塑性粘度和屈服值,研究发现速度流核宽度与屈服值成正比,与流量及塑性粘度成反比,且流核的宽度越大,流核区的速度越小.     

An experimental study of fixed points and chaos in the motion of spheres in a Stokes flow

We present the results of an investigation of a novel dynamicalsystem in which one, two or three solid spheres are free tomove in a horizontal rotating cylinder which is completely filledwith a highly viscous fluid. At low rotation rates, steady motionis found where the balls adopt stable equilibrium positionsrotating adjacent to the rising wall at a speed which is inagreement with available theory. At higher cylinder speeds time-dependentmotion sets in via Hopf bifurcations, and when one or two ballsare present the motion is strictly periodic. Perhaps surprisingly,it is found that three balls are required to produce low-dimensionalchaos.     

Numerical investigation of blood flow. Part I: In microvessel bifurcations

In some diseases there is a focal pattern of velocity in regions of bifurcation, and thus the dynamics of bifurcation has been investigated in this work. A computational model of blood flow through branching geometries has been used to investigate the influence of bifurcation on blood flow distribution. The flow analysis applies the time-dependent, three-dimensional, incompressible Navier–Stokes equations for Newtonian fluids. The governing equations of mass and momentum conservation were solved to calculate the pressure and velocity fields. Movement of blood flow from an arteriole to a venule via a capillary has been simulated using the volume of fluid (VOF) method. The proposed simulation method would be a useful tool in understanding the hydrodynamics of blood flow where the interaction between the RBC deformation and blood flow movement is important. Discrete particle simulation has been used to simulate the blood flow in a bifurcation with solid and fluid particles. The fluid particle method allows for modeling the plasma as a particle ensemble, where each particle represents a collective unit of fluid, which is defined by its mass, moment of inertia, and translational and angular momenta. These kinds of simulations open a new way for modeling the dynamics of complex, viscoelastic fluids at the micro-scale, where both liquid and solid phases are treated with discrete particles.     

Some new exact analytical solutions for helical flows of second grade fluids

The helical flow of a second grade fluid, between two infinite coaxial circular cylinders, is studied using Laplace and finite Hankel transforms. The motion of the fluid is due to the inner cylinder that, at time t = 0⁺ begins to rotate around its axis, and to slide along the same axis due to hyperbolic sine or cosine shear stresses. The components of the velocity field and the resulting shear stresses are presented in series form in terms of Bessel functions J₀(•), Y₀(•), J₁(•), Y₁(•), J₂(•) and Y₂(•). The solutions that have been obtained satisfy all imposed initial and boundary conditions and are presented as a sum of large-time and transient solutions. Furthermore, the solutions for Newtonian fluids performing the same motion are also obtained as special cases of general solutions. Finally, the solutions that have been obtained are compared and the influence of pertinent parameters on the fluid motion is discussed. A comparison between second grade and Newtonian fluids is analyzed by graphical illustrations.     

The asymptotics of neutral curve crossing in Taylor–Dean flow

The fluid flow between a pair of coaxial circular cylinders generated by the uniform rotation of the inner cylinder and an azimuthal pressure gradient is susceptible to both Taylor and Dean type instabilities. The flow can be characterised by two parameters: a measure of the relative magnitude of the rotation and pressure effects and a non-dimensional Taylor number. This work considers the small gap, large wavenumber limit for linear perturbations when the onset of the Taylor and Dean instabilities is concurrent. A consistent, matched asymptotic solution is found across the whole annular domain and identifies five regions of interest: two boundary adjustment regions and three internal critical points. Necessary conditions for the Taylor number and wavenumber are found and interpreted with reference to the suggestion of neutral curve kinks occurring at moderate wavelengths. Received: October 21, 2003; revised: November 11, 2004