Oldroyd-B Fluid Research Articles

Transient Electro-osmotic Slip Flow of an Oldroyd-B Fluid with Time-fractional Caputo-Fabrizio Derivative

In this article, the electro-osmotic flow of Oldroyd-B fluid in a circular micro-channel with slip boundary condition is considered. The corresponding fractional system is represented by using a newly defined time-fractional Caputo-Fabrizio derivative without singular kernel. Closed form solutions for the velocity field are acquired by means of Laplace and finite Hankel transforms. Additionally, Stehfest’s algorithm is used for inverse Laplace transform. The solutions for fractional Maxwell, ordinary Maxwell and ordinary Newtonian fluids are obtained as limiting cases of the obtained solution. Finally, the influence of fractional and some important physical parameters on the fluid flow are spotlighted graphically.

Numerical simulation of Oldroyd-B fluid with application to hemodynamics

Oldroyd-B viscoelastic fluid is numerically simulated using the stabilized Galerkin least squares finite element method. The instabilities due to the connective nature of the Oldroyd-B model is treated using the discrete viscous elastic split stress method. The model is used to study the behavior of the flow of blood through an abdominal aortic segment. The results show that the viscoelastic nature of the blood should be considered specially at low shear rates.

Weak nonlinear analysis of Darcy-Brinkman convection in Oldroyd-B fluid saturated porous media under temperature modulation

The analysis of chaotic convection of Oldroyd-B fluid in porous medium under temperature modulation is presented on the basis of Darcy-Brinkman model. A four-dimensional nonlinear system resulting from a truncated Galerkin expansion of the conservation and constitutive equations is set up to deal with the stability of convection with different Darcy number and Darcy-Rayleigh number. The chaotic behavior is controlled by Darcy-Rayleigh number while Darcy number has a great influence on the stability of system. With the lower boundary which is heated in a time-periodic manner, the system will change over to a state of periodic or chaotic motion from steady convection.

Equilibrium positions of the elasto-inertial particle migration in rectangular channel flow of Oldroyd-B viscoelastic fluids

In this paper, the lateral migration of a neutrally buoyant spherical particle in the pressure-driven rectangular channel flow of an Oldroyd-B fluid is numerically investigated with a fictitious domain method. The aspect ratio of the channel cross-section considered is 1 and 2, respectively. The particle lateral motion trajectories are shown for the bulk Reynolds number ranging from 1 to 100, the ratio of the solvent viscosity to the total viscosity being 0.5, and a Weissenberg number up to 1.5. Our results indicate that the lateral equilibrium positions located on the cross-section midline, diagonal line, corner and channel centreline occur successively as the fluid elasticity is increased, for particle migration in square channel flow with finite fluid inertia. The transition of the equilibrium position depends strongly on the elasticity number (the ratio of the Weissenberg number to the Reynolds number) and weakly on the Reynolds number. The diagonal-line equilibrium position occurs at an elasticity number ranging from roughly 0.001 to 0.02, and can coexist with the midline and corner equilibrium positions. When the fluid inertia is negligibly small, particles migrate towards the channel centreline, or the closest corner, depending on their initial positions and the Weissenberg number, and the corner attractive area first increases and then decreases as the Weissenberg number increases. For particle migration in a rectangular channel with an aspect ratio of 2, the transition of the equilibrium position from the midline, ‘diagonal line’ (the line where two lateral shear rates are equal to each other), off-centre long midline and channel centreline takes place as the Weissenberg number increases at moderate Reynolds numbers. An off-centre equilibrium position on the long midline is observed for a large blockage ratio of 0.3 (i.e. the ratio of the particle diameter to the channel height is 0.3) at a low Reynolds number. This off-centre migration is driven by shear forces, unlike the elasticity-induced rapid inward migration, which is driven by the normal force (pressure or first normal stress difference).

Analytical Solution of a Fractional Differential Equation in the Theory of Viscoelastic Fluids

The aim of this paper is to present analytical solutions of fractional delay differential equations of an incompressible generalized Oldroyd-B fluid with fractional derivatives of Caputo type. Using a modification of the method of separation of variables the main equation with non-homogeneous boundary conditions is transformed into an equation with homogeneous boundary conditions, and the resulting solutions are then expressed in terms of Green functions via Laplace transforms. Different situations for the unsteady flows of a generalized Oldroyd-B fluid, including a flow with a moving plate, are considered via examples.

Retraction Note: Velocity and shear stress for an Oldroyd-B fluid within two cylinders

The Editors-in-Chief are retracting this article [1] because it overlaps with previously published articles [2,3]. None of the authors have responded to correspondence about this retraction.

Influence of viscous dissipation and double stratification on MHD Oldroyd-B fluid over a stretching sheet with uniform heat source

The present report explores the discussion on 3D flow of dual stratified Oldroyd-B fluid in the direction of thermal radiation, viscous dissipation and heat source/sink. Mathematical modeling is formulated with an applied magnetic field through a stretching surface. The present flow governing system has been transformed as nonlinear ODE via suitable transformations and then concluded by using bvp4c. The graphs are described and illustrated for various non-dimensional parameters. Thermal stratification and Prandtl number parameters reduce the temperature, whereas thermal radiation and heat source parameters show reverse behavior. Numerical results are used to obtain the values of frictional drag, rate of heat and mass transfers. Finally, numerical results are compared with previous established work in a limiting case.

DNS and LST stability analysis of Oldroyd-B fluid in a flow between two parallel plates

Several flows of practical interest are of viscoelastic fluids and it is desirable to know if these flows are in a laminar or turbulent state. In this paper, the laminar-turbulent transition is studied in which the convection of Tollmien–Schlichting waves is investigated for the incompressible two-dimensional flow between two parallel plates. The viscoelastic fluid adopted is modeled by the Oldroyd-B constitutive equation. Direct Numerical Simulation (DNS) and Linear Stability Theory (LST) were used to verify the stability of the viscoelastic fluid flow to unsteady disturbances. In the DNS formulation, the governing equations are discretized by high-order compact finite difference schemes for the spatial derivatives, while time integrations are carried out by a fourth order Runge–Kutta method. For the LST analysis, the Orr–Sommerfeld equation is modified to include viscoelastic effects and solved by a shooting method. In order to evaluate the flow stability when submitted to unsteady disturbances and to find the neutral stability curves, a range of numerical simulations was performed with different dimensionless parameters for the viscoelastic fluid flow and the results were compared with the Newtonian fluid flow. The influence of the elastic forces and the amount of the polymer concentration in the fluid were investigated. A wide range of polymer concentration was studied. The results show that the variation of the critical Reynolds number may be monotonic for certain conditions. The DNS simulations were carried out, and the amplification rates obtained were compared with the LST results, with a very good agreement. The DNS results allow also the visualization of the disturbances variation in streamwise and wall-normal directions, showing the behavior of the velocity components, vorticity and the non-Newtonian extra-stress tensor components. The DNS results show a significant change in flow structure for certain combinations of the polymer concentration and elastic forces parameters.

Analytical results on the unsteady rotational flow of fractional-order non-Newtonian fluids with shear stress on the boundary

The objective of this paper is to study the unsteady rotational flow of some non Newtonian fluids with Caputo fractional derivative through an infinite circular cylinder by means of the finite Hankel and Laplace transform. The novelty of the work is that motion is produced by applying tangential force not a specific but general function of time on the boundary. Initially the cylinder is at rest and after time $ t_{o} = 0^{+} $ it begins to rotate about its axis with an angular velocity $ \tau_{o} g(t) $. The obtained solutions of velocity field and shear stress have been presented under series form in terms of generalized $ G $-function, satisfying all imposed initial and boundary conditions. The corresponding solutions can be easily particularized to give similar solutions from existing literature for Oldroyd-B fluids, Maxwell fluids, Second grade fluids and Newtonian fluids with/without fractional derivatives performing similar motions.

Stagnation-point flow and heat transfer of upper-convected Oldroyd-B MHD nanofluid with Cattaneo–Christov double-diffusion model

Purpose The purpose of this paper is to investigate the two-dimensional stagnation-point flow, heat and mass transfer of an incompressible upper-convected Oldroyd-B MHD nanofluid over a stretching surface with convective heat transfer boundary condition in the presence of thermal radiation, Brownian motion, thermophoresis and chemical reaction. The process of heat and mass transfer based on Cattaneo–Christov double-diffusion model is studied, which can characterize the features of thermal and concentration relaxations factors. Design/methodology/approach The governing equations are developed and similarly transformed into a set of ordinary differential equations, which are solved by a newly approximate analytical method combining the double-parameter transformation expansion method with the base function method (DPTEM-BF). Findings An interesting phenomenon can be found that all the velocity profiles first enhance up to a maximal value and then gradually drop to the value of the stagnation parameter, which indicates the viscoelastic memory characteristic of Oldroyd-B fluid. Moreover, it is revealed that the thickness of the thermal and mass boundary layer is increasing with larger values of thermal and concentration relaxation parameters, which indicates that Cattaneo–Christov double-diffusion model restricts the heat and mass transfer comparing with classical Fourier’s law and Fick’s law. Originality/value This paper focuses on stagnation-point flow, heat and mass transfer combining the constitutive relation of upper-convected Oldroyd-B fluid and Cattaneo–Christov double diffusion model.

Impact of homogeneous–heterogeneous reactions and non-Fourier heat flux theory in Oldroyd-B fluid with variable conductivity

This article scrutinizes the influence of chemical reactions on flow of an Oldroyd-B fluid due to stretched cylinder. In vision of non-Fourier heat flux model, the heat transfer phenomenon is scrutinized. This enhanced constitutive model anticipates the time space upper-convected derivative which is recycled to depicting heat conduction mechanism. Additionally, heat transfer scrutiny is considered with the influence of thermal conductivity which is temperature dependent. Apposite conversions are engaged to acquire ODEs which are then deciphered analytically via homotopic approach. To highlight their physical consequences, the graphical portrayal of diverse considerations on velocity, temperature and concentration fields is depicted and conferred. It is scrutinized from this study that all the profiles are higher in the instance of the cylinder as equated to a flat plate. This scrutiny also reported that the thermal relaxation parameter decreases the temperature field while the Schmidt number and homogeneous response parameter display the conflicting performance on concentration field. In addition, an assessment in restrictive instance is also presented in this exploration, which ensures us that our outcomes are more precise.

A Lagrangian-Eulerian framework for simulation of transient viscoelastic fluid flow

A novel framework for simulation of transient viscoelastic fluid flow is proposed. The viscoelastic stresses are calculated at Lagrangian nodes which are distributed in the computational domain and convected by the fluid. The coupling between the constitutive equation and the fluid momentum equations is established through robust interpolation with radial basis functions.The framework is implemented in a finite volume based flow solver that combines an octree background grid with immersed boundary techniques. Since the distribution of the Lagrangian node set is performed entirely based on spatial information from the fluid solver, the ability to simulate flows in complex geometries is therefore as general as for the fluid solver itself.In the Lagrangian formulation the discretization of the convective terms in the constitutive equations is avoided. No re-formulation of the constitutive equation is required for stable solutions. Numerical experiments are performed of UCM and Oldroyd-B fluids in a channel flow and of a four mode PTT fluid in a confined cylinder flow. The computed flow quantities consistently converge and agree excellently with analytical and numerical data for fully developed and transient flow.

Unsteady flow of chemically reactive Oldroyd-B fluid over oscillatory moving surface with thermo-diffusion and heat absorption/generation effects

This theoretical investigation deals with the magnetohydrodynamic mass and heat transportation of oscillatory Oldroyd-B fluid flow under thermo-diffusion effects. Flow is produced due to periodic motion of sheet. Some interesting effects like heat absorption/generation and chemical reaction are superposed in the energy and mass species equations, respectively. By utilizing apposite variables, independent variables in the model equations are reduced. The set of these equations is solved with help of homotopy analysis method. Reliable results for different physical flow constraints are prepared for the velocity, temperature and concentration profiles. It is noted that the Deborah number in terms of relaxation time resists the motion of fluid particles at various time instants. Both temperature and concentration profiles increase by increasing Deborah number in terms of relaxation time. The presence of Dufour number may enhance the thermal boundary layer effectively. It is also concluded that the species concentration profile is promoted by increasing Hartmann number and Soret number. An excellent accuracy of obtained solution is observed with already reported numerical values as a special case.

Semi Analytical Solutions for Fractional Oldroyd-B Fluid Through Rotating Annulus

In this paper, semi analytical solutions for velocity field and tangential stress correspond to fractional Oldroyd-B fluid, in an annulus, are acquired by Laplace transforms and modified Bessel equation. In the beginning, cylinders are stationary, motion is produced after t = 0 when both cylinders start rotating about their common axis. The governing equations solved for velocity field and shear stress by using the Laplace transform technique. The inverse Laplace transform is alternately calculated by Stehfest's algorithm using “MATHCAD” numerically. The numerically obtained solutions are in the form of modified Bessel's equations of first and second kind and satisfying all the imposed physical conditions. Finally, there is a comparison between exact and obtained solutions. It is observed that semi analytical technique and exact technique are approximately the same and satisfy imposed boundary conditions. Through graphs, the impact of physical parameters (relaxation time, retardation time kinematic viscosity, and dynamic viscosity) and fractional parameters on both velocity and shear stress is observed.

Double-Diffusive Convection in an Oldroyd-B fluid Layer-Stability of Bifurcating Equilibrium Solutions

Double-Diffusive Convection in an Oldroyd-B fluid Layer-Stability of Bifurcating Equilibrium Solutions

Multiple slip effects on mixed convection of Oldroyd-B fluid towards a stretchy surface with radiation and chemical reaction using Cattaneo-Christov heat flux

The steady two-dimensional MHD mixed convection flow and heat transfer characteristics of an Oldroyd-B fluid over a linear stretchy sheet is considered. The flow caused by a linear stretchy sheet with effects of velocity, temperature and concentration slips. The thermal radiation, internal heat generation effects are included in the study. The resulting governing PDEs rebuild into ODE model by applying a suitable similarity transformation and then are solved by homotopy analysis method. The effects of the various constants on the velocity, temperature and concentration fields are discussed. It is found that the velocity, temperature and concentration fields decrease when increasing the corresponding slip constants.

A material point method for simulation of viscoelastic flows

A novel variant of the Material Point Method is proposed to solve flow problems for incompressible viscoelastic fluids. As in the original spirit of the method, material points possess all necessary information about the material constitutive behavior and move with the flow. They are additionally employed as integration points, whose weights are computed as the system evolves in order to ensure exact integration of the discretized conservation equations. The method is shown to achieve quadratic convergence for transient start-up flow of the Oldroyd-B fluid. In addition, we apply the method to flow of a Giesekus fluid in concentric rotating cylinders, and in flow through an abrupt 4:1 contraction, comparing favorably to available analytical, experimental, and simulation results as appropriate.

Galerkin FEM for a time-fractional Oldroyd-B fluid problem

We consider the numerical approximation of a generalized fractional Oldroyd-B fluid problem involving two Riemann-Liouville fractional derivatives in time. We establish regularity results for the exact solution which play an important role in the error analysis. A semidiscrete scheme based on the piecewise linear Galerkin finite element method in space is analyzed, and optimal with respect to the data regularity error estimates are established. Further, two fully discrete schemes based on convolution quadrature in time generated by the backward Euler and the second-order backward difference methods are investigated and related error estimates for smooth and nonsmooth data are derived. Numerical experiments are performed with different values of the problem parameters to illustrate the efficiency of the method and confirm the theoretical results.

Numerical study of spheres settling in Oldroyd-B fluids

In order to simulate the motion of balls settling in Oldroyd-B fluids, we have generalized to three dimensions the distributed Lagrange multiplier based fictitious domain method we developed in the study by Hao et al. [“A fictitious domain/distributed Lagrange multiplier method for the particulate flow of Oldroyd-B fluids: A positive definiteness preserving approach,” J. Non-Newtonian Fluid Mech. 156, 95 (2009)] for viscoelastic particulate flow in two-dimensional channels. The effect of the fluid elastic number and that of the density ratio of the particle and fluid on the chaining of balls have been studied. For the cases of two balls released side-by-side at high elasticity numbers, the two balls attract each other first and then form a chain, and such a chain settles vertically. But at low elasticity numbers, the two balls either stay separate and interact periodically or attract each other, turn, and separate periodically. At high blockage ratios, a stronger wall effect enhances the formation of two ball chains. For the cases of three balls released side-by-side, the ball interaction is slightly more complicated due to its non-symmetrical initial configuration. At high elasticity numbers, either a three ball chain settles vertically or the leading two balls form a chain which leaves the third ball behind. But at low elasticity numbers considered in this article, only the leading two balls form a chain. For the vertical initial configuration, a three ball chain can be obtained at higher elasticity numbers and also the heavier balls can form a vertical chain easily.

Finite difference/finite element method for a novel 2D multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on convex domains

In this work, a novel two-dimensional (2D) multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on convex domains will be considered. Different from the general multi-term time-fractional diffusion-wave or sub-diffusion equation, the new equation not only possesses the diffusion-wave and sub-diffusion terms simultaneously but also has a special time-space coupled derivative. Although the analytic solution of this kind of equation can be derived using a multi-level sum of an infinite series and Fox H-functions, it is extremely complex and difficult to evaluate. Therefore, seeking the numerical solution of the equation is of great importance. In this paper, we will consider the finite element method for the novel 2D multi-term time fractional mixed diffusion equation. Firstly, we utilise the mixed L schemes to approximate the time fractional sub-diffusion term, diffusion-wave term and the coupled time-space derivative, respectively. Secondly, we establish the variational formulation and use the finite element method to discretise the equation. Then we adopt linear polynomial basis functions on triangular elements to derive the matrix form of the numerical scheme. Furthermore, we present the stability and convergence analysis of the numerical scheme. To show the effectiveness of our method, three examples are investigated, in which a 2D multi-term time fractional mixed diffusion equation on a circular domain and a 2D generalized Oldroyd-B fluid in a magnetic field are analysed.