Oldroyd-B Fluid Research Articles

Transient computations using the natural stress formulation for solving sharp corner flows

In this short communication, we analyse the potential of the natural stress formulation (NSF) (i.e. aligning the stress basis along streamlines) for computing planar flows of an Oldroyd-B fluid around sharp corners. This is the first attempt to combine the NSF into a numerical strategy for solving a transient fluid flow problem considering the momentum equation in Navier–Stokes form (the elastic stress entering as a source term) and using the constitutive equations for natural stress variables. Preliminary results of the NSF are motivating in the sense that accuracy of the numerical solution for the extra stress tensor is improved near to the sharp corner. Comparison studies among the NSF and the Cartesian stress formulation (CSF) (i.e. using a fixed Cartesian stress basis) are conducted in a typical benchmark viscoelastic fluid flow involving a sharp corner: the 4 : 1 contraction. The CSF needs a mesh approximately 10 times smaller to capture similar near singularity results to the NSF.

On the performance of heat absorption/generation and thermal stratification in mixed convective flow of an Oldroyd-B fluid

Abstract This investigation explores the thermally stratified stretchable flow of an Oldroyd-B material bounded by a linear stretched surface. Heat transfer characteristics are addressed through thermal stratification and heat generation/absorption. Formulation is arranged for mixed convection. Application of suitable transformations provides ordinary differential systems through partial differential systems. The homotopy concept is adopted for the solution of nonlinear differential systems. The influence of several arising variables on velocity and temperature is addressed. Besides this, the rate of heat transfer is calculated and presented in tabular form. It is noticed that velocity and Nusselt number increase when the thermal buoyancy parameter is enhanced. Moreover, temperature is found to decrease for larger values of Prandtl number and heat absorption parameter. Comparative analysis for limiting study is performed and excellent agreement is found.

Numerical methods and analysis for simulating the flow of a generalized Oldroyd-B fluid between two infinite parallel rigid plates

In recent years, non-Newtonian fluids have been widely applied in a number of engineering applications. One particular subclass of non-Newtonian fluids is the generalized Oldroyd-B fluids with fractional derivative constitutive equations, which can be used to approximate the response of many dilute polymeric liquids. Different to the general time fractional diffusion equation, the constitutive equation not only has a multi-term time derivative but also possesses a special time fractional operator on the spatial derivative, which is challenging to approximate. The literature reported on the numerical solution of this model is extremely sparse. In this paper, we will consider the finite difference method for its discretisation and propose a new scheme to approximate the time fractional derivative. Then we derive an implicit finite difference scheme and establish some new theoretical analysis of the stability and convergence. Furthermore, we present a numerical scheme to improve the time order. Finally, we present two numerical examples to show the effectiveness of our method and apply it to solve the generalized Oldroyd-B fluid model.

Oscillating Motion of an Oldroyd-B Fluid with Fractional Derivatives in a Circular Cylinder

Oscillating Motion of an Oldroyd-B Fluid with Fractional Derivatives in a Circular Cylinder

Some exact solutions for the rotational flow of Oldroyd-B fluid between two circular cylinders

In this article, an effort has been made to compute and inspect the rotational flow of Oldroyd-B fluid between two co-axial circular cylinders. As novelty, the rotation of both cylinders produces motion of the fluid. The velocity field and tangential stress corresponding to the motion of an Oldroyd-B fluid with fractional derivatives have been obtained using Laplace and Hankel integral transforms. The velocity profiles of the respective model are graphically presented and deliberated for the pertinent parameters. Corresponding solutions are also obtained for ordinary Oldroyd-B fluid, ordinary and fractional Maxwell fluids, ordinary and fractional second-grade fluids, and Newtonian fluid as limiting cases of our general solutions.

Parameters identification for the unsteady helical flows of a generalized Oldroyd-B fluid model based on its numerical solution

In this paper, we mainly consider the problem of parameter identification for the unsteady helical flows of a generalized Oldroyd-B fluid between two infinitely long coaxial circular cylinders. The implicit finite difference method is employed to obtain the numerical solution for the direct problem. By means of the Levenberg–Marquardt method, the numerical inversion for identifying the three parameters of the model is implemented simultaneously, that is, the Riemann–Liouville time-fractional derivative α, relaxation time λ, and retardation time λr. To test the validity of the proposed numerical method, varied initial guesses and the observed data, whether or not they involve random error, have been used to verify the reliability of the estimation. This paper provides an efficient method to obtain the estimated value of the unknown parameters for the generalized non-Newtonian fluids model.

The role of body flexibility in stroke enhancements for finite-length undulatory swimmers in viscoelastic fluids

The role of passive body dynamics on the kinematics of swimming micro-organisms in complex fluids is investigated. Asymptotic analysis of small-amplitude motions of a finite-length undulatory swimmer in a Stokes–Oldroyd-B fluid is used to predict shape changes that result as body elasticity and fluid elasticity are varied. Results from the analysis are compared with numerical simulations and the numerically simulated shape changes agree with the analysis at both small and large amplitudes, even for strongly elastic flows. We compute a stroke-induced swimming speed that accounts for the shape changes, but not additional effects of fluid elasticity. Elasticity-induced shape changes lead to larger-amplitude strokes for sufficiently soft swimmers in a viscoelastic fluid, and these stroke boosts can lead to swimming speed-ups. However, for the strokes we examine, we find that additional effects of fluid elasticity generically result in a slow-down. Our high amplitude strokes in strongly elastic flows lead to a qualitatively different regime in which highly concentrated elastic stresses accumulate near swimmer bodies and dramatic slow-downs are seen.

A non-Fourier heat flux approach to model MHD Oldroyd-B fluid flow due to bidirectional stretching surface

Prime focus of this research is to address the MHD Oldroyd-B fluid flow and heat transfer due to stretching of plane elastic surface in two lateral directions. The phenomenon of heat transfer is modeled under a non-Fourier heat flux theory featuring thermal relaxation effects. Flow field is subjected to uniform vertical magnetic field. Furthermore, fluid with variable thermal conductivity is considered. Appropriate transformations are applied to yield the similar form of governing equations . A powerful analytical approach namely Homotopy Analysis Method (HAM) is opted to develop series solutions involving exponentially decaying functions. The results show that flow fields are appreciably affected with the variation in embedded parameters, especially with fluid relaxation and retardation times. It is found that the structure of solutions depends on a parameter measuring the ratio of stretching rates. Thermal penetration depth decreases for increasing values of thermal relaxation time. Thermal relaxation time has an effective role in improving the cooling process of the stretching surface which is vital in some industrial processes.

Stability of plane Couette flow of a power-law fluid past a neo-Hookean solid at arbitrary Reynolds number

The linear stability of plane Couette flow of a power-law fluid past a deformable solid is analyzed at arbitrary Reynolds number (Re). For flow of a Newtonian fluid past a deformable solid, at high Re, there are two different modes of instability: (i) “wall modes” (Γ∝Re−1∕3) and (ii) “inviscid modes” (Γ∝Re−1) where Γ=VμfGR is the non-dimensional shear-rate in the fluid (V, μf, G, and R denote the top-plate velocity, fluid viscosity, shear modulus of the solid, and fluid thickness, respectively). In this work, we consider the power-law model for the fluid to elucidate the effect of shear-thickening/shear-thinning behaviour on the modes of instability present in the flow, especially at moderate and high Re. At high Re, our numerical results show that wall modes exhibit different scalings in Γ (VηfGR, where ηf is Newtonian-like constant viscosity) vs Re for different values of the power-law index (n), and the scaling exponents are different from that for a Newtonian fluid. This drastic modification in the scaling of wall modes is not observed in viscoelastic (modelled as upper-convected Maxwell or Oldroyd-B fluids) plane Couette flow past a deformable solid. We show that the difference in scaling exponents can be explained by postulating that the wall modes in a power-law fluid are determined by the actual viscosity corresponding to the shear rate of the laminar flow denoted by ηapp. A non-dimensional shear rate based on this viscosity Γapp=VηappGR can be defined, and we show that the postulate Γapp∼Re−1∕3 (motivated by the wall-mode scaling in a Newtonian fluid) captures all the numerically observed scalings for Γ vs Re for different values of n>0.3, which is found to be Γ∼Re−12n+1. Further, we numerically evaluated the wall layer thickness and this agreed with the theoretical scaling of δ∼Re−n2n+1. Interestingly, the theoretical and numerical prediction of wall modes is found to be valid for power-law index, n≥ 0.3. For n≤ 0.3, there is a marked departure from the wall-mode scalings, and our results show a scaling of Γ ∼ Re−1 corresponding to inviscid modes. The variation of the power-law index (n) can stabilise/destabilise the inviscid mode when compared with Newtonian fluid, and this result is observed only in the power-law model and is not seen in the flow of viscoelastic fluid past deformable surfaces. Thus, the present study shows that the shear-rate dependence of viscosity has a significant impact on both the qualitative and quantitative aspects of stability of non-Newtonian fluid flow past deformable surfaces.

An analytical treatment for MHD mixed convection boundary layer flow of Oldroyd-B fluid utilizing non-Fourier heat flux model

In current framework, an analytical treatment for mixed convection flow of an electrically conducting Oldroyd-B fluid adjacent to a vertical stretchable surface is provided. A non-Fourier heat flux approach is employed to formulate the energy balance relation. Using similarity approach, the governing equations are changed to a set of non-linear differential equations which are tackled by well-known analytical approach called homotopy analysis method (HAM). Appropriate range of auxiliary parameter is obtained by plotting the so called ℏ-curves. Velocity and temperature profiles are computed and elucidated in the existence of new physical mechanism, that is, thermal relaxation time in current research. The main implication of this research is that the relaxation and retardation times considerably alter the flow behavior near the surface. The results predict that heat penetration into the fluid reduces as the relaxation time of heat flux enlarges. Furthermore, the change in temperature gradient at the surface with increasing thermal relaxation time appears similar in magnitude at all considered Prandtl numbers. In assisting flow regime, we noticed a growth in velocity and temperature profiles for increasing strength of buoyancy force.

Flow resistance of viscoelastic flows in fibrous porous media

In this work, we present flow simulations of viscoelastic (Oldroyd-B and Leonov) fluids in the unidirectional fibrous porous media to investigate effects of the elasticity on the flow resistance. Extensive numerical simulations were carried out for the Oldroyd-B and Leonov fluids to investigate the flow resistance in some typical geometries of unidirectional fibrous porous media: square, hexagonal, as well as randomly aligned arrays of unidirectional fibers. To the best knowledge of the authors, we report for the first time the stiff increase of flow resistance through numerical simulations above a certain Weissenberg number. Considerable growth in the flow resistance has turned out to be closely associated with microstructure of porous media through the extensional stretch of polymers. For the hexagonal and randomly aligned fibers, where there are enough spaces behind fibers, significant amount of polymer stretch is observed along with substantial growth of the corresponding flow resistance. On the other hand, only minor change in flow resistance and negligible polymer stretch are observed in case of square-packing porous media.

Lagrangian Mesh Model with Regridding for Planar Poiseuille Flow

AbstractMany biological settings involve complex fluids that have non-Newtonian mechanical responses that arise from suspended microstructures. In contrast, Newtonian fluids are liquids or mixtures of a simple molecular structure that exhibit a linear relationship between the shear stress and the rate of deformation. In modeling complex fluids, the extra stress from the non-Newtonian contribution must be included in the governing equations.In this study we compare Lagrangian mesh and Oldroyd-B formulations of fluid-structure interaction in an immersed boundary framework. The start-up phase of planar Poiseuille flow between two parallel plates is used as a test case for the fluid models. For Newtonian and Oldroyd-B fluids there exist analytical solutions which are used in the comparison of simulation and theoretical results. The Lagrangian mesh results are compared with Oldroyd-B using comparable parameters. A regridding algorithm is introduced for the Lagrangian mesh model. We show that the Lagrangian mesh model simulations with regridding produce results in close agreement with the Oldfoyd-B model.

Time-dependent viscoelastic properties of Oldroyd-B fluid studied by advection-diffusion lattice Boltzmann method

Time-dependent viscoelastic properties of Oldroyd-B fluid were investigated by lattice Boltzmann method (LBM) coupled with advection-diffusion model. To investigate the viscoelastic properties of Oldroyd-B fluid, realistic rheometries including step shear and oscillatory shear tests were implemented in wide ranges of Weissenberg number (Wi) and Deborah number (De). First, transient behavior of Oldroyd-B fluid was studied in both start up shear and cessation of shear. Stress relaxation was correctly captured, and calculated shear and normal stresses agreed well with analytical solutions. Second, the oscillatory shear test was implemented. Dynamic moduli were obtained for various De regime, and they showed a good agreement with analytical solutions. Complex viscosity derived from dynamic moduli showed two plateau regions at both low and high De limits, and it was confirmed that the polymer contribution becomes weakened as De increases. Finally, the viscoelastic properties related to the first normal stress difference were carefully investigated, and their validity was confirmed by comparison with the analytical solutions. From this study, we conclude that the LBM with advection-diffusion model can accurately predict time-dependent viscoelastic properties of Oldroyd-B fluid.

Nonlinear three-dimensional stretched flow of an Oldroyd-B fluid with convective condition, thermal radiation, and mixed convection

The effect of non-linear convection in a laminar three-dimensional Oldroyd-B fluid flow is addressed. The heat transfer phenomenon is explored by considering the non-linear thermal radiation and heat generation/absorption. The boundary layer assumptions are taken into account to govern the mathematical model of the flow analysis. Some suitable similarity variables are introduced to transform the partial differential equations into ordinary differential systems. The Runge-Kutta-Fehlberg fourth- and fifth-order techniques with the shooting method are used to obtain the solutions of the dimensionless velocities and temperature. The effects of various physical parameters on the fluid velocities and temperature are plotted and examined. A comparison with the exact and homotopy perturbation solutions is made for the viscous fluid case, and an excellent match is noted. The numerical values of the wall shear stresses and the heat transfer rate at the wall are tabulated and investigated. The enhancement in the values of the Deborah number shows a reverse behavior on the liquid velocities. The results show that the temperature and the thermal boundary layer are reduced when the non-linear convection parameter increases. The values of the Nusselt number are higher in the non-linear radiation situation than those in the linear radiation situation.

Melting heat and thermal radiation effects in stretched flow of an Oldroyd-B fluid

The incompressible flow of a non-Newtonian fluid with mixed convection along a stretching sheet is analyzed. The heat transfer phenomenon is discussed through thermal radiation. The effects of the melting heat transfer and heat generation/absorption are also taken. Suitable transformations are utilized to attain the nonlinear ordinary differential expressions. The convergent series solutions are presented. The fluid flow, temperature, and surface heat transfer rate are examined graphically. It is observed that the velocity decreases when the relaxation time increases while increases when the retardation time is constant. The results also reveal that the temperature distribution reduces when the radiation parameter increases.

First general solutions for unsteady unidirectional motions of rate type fluids in cylindrical domains

The dimensionless velocity field corresponding to the motion of an Oldroyd-B fluid between two infinite coaxial circular cylinders induced by the outer cylinder that rotates around its symmetry axis with an arbitrary time-dependent velocity is determined using the integral transforms technique. The corresponding solution for the motion through an infinite circular cylinder is obtained as a limiting case of the previous solution. It is used to generate exact solutions for the motion due to the cylinder that applies an arbitrary longitudinal shear stress to the fluid. Finally, for a check of results that have been obtained as well as to correct some results from the literature and to get some physical insight for oscillating motions, three special cases are considered and some known results are recovered as limiting cases. The required time after which the fluid moves according to the steady-state solutions is graphically determined for sine and cosine oscillating motions.

Analytical and Numerical Solutions of Oscillating Flow at a Trapezoidal Pressurewaveform

This study aimed to develop a mathematical model of an unsteady Burgers’ fluid in a circular cylinder with a trapezoidal pressure waveform described by an infinite Fourier series. An analytical solution was obtained for the governing equation using the Bessel transform method together with similarity arguments. The validity of the solution was verified using a numerical inversion method based on Stehfest’s method. Limiting cases were considered to examine the fluid flow performance of different fluids. Our results show that the Newtonian and Oldroyd-B fluids performed similar velocity time variation for the trapezoidal waveform of oscillating motion, whereas the velocity time variation was different for Maxwell and Burger’s fluids. Moreover, it is evident that the material constant of a Burgers’ fluid is another important factor that affects flow performance in an oscillating flow.

Flow of variable thermal conductivity Oldroyd-B fluid with generalized Fourier's and Fick's laws

Here two-dimensional Oldroyd-B fluid flow has been studied in the presence of generalized Fourier's and Fick's laws. The flow is generated due to an exponentially stretchable surface. The process of heat and mass transfer through theory of Cattaneo-Christov double diffusion characterizing the features of thermal and concentration relaxation factors is studied. Temperature dependent thermal conductivity of fluid is taken into consideration. The set of partial differential equations governing the flow of an Oldroyd-B fluid and heat and mass transfer through the Cattaneo-Christov double diffusion theory is developed. The appropriate transformations correspond to the strong nonlinear ordinary differential system. Optimal homotopy scheme is employed for the solutions of governing system. The optimal values of nonzero auxiliary parameters are computed. The solution expressions of temperature and concentration are examined through the plots. The contributions of several pertinent parameters are studied in detail. Further the heat and mass transfer rates are characterized by numerical data. Our computations reveal that the heat and mass transfer rates are higher for larger values of thermal and concentration relaxation parameters.

Slip effects on MHD boundary layer flow of Oldroyd-B fluid past a stretching sheet: An analytic solution

Present analysis is made for the laminar flow of Oldroyd-B fluid induced by a deforming sheet in the existence of transverse magnetic field. Flow model is constructed in the presence of slip boundary condition. The governing problem even after utilizing boundary layer approximations comprises of non-linear differential equation with the non-linear boundary condition. Such boundary condition for the no slip case is linear. Highly accurate analytic solutions for velocity distribution are derived by powerful homotopy analysis approach. Permissible values of the convergence control parameter are obtained by the so-called h-curves. The behavior of parameters on the solutions is shown graphically. In the light of numerical results, we predict that slip effect gives opposition to the momentum transport phenomenon. Moreover, the behaviors of relaxation and retardation time are qualitatively opposite. A comparative study of slip and no-slip cases is also discussed.

Three-dimensional simulation of two viscoelastic droplets impacting onto a rigid plate using smoothed particle hydrodynamics

In this article, a smoothed particle hydrodynamics method is developed to simulate the dynamic process of the impact of two viscoelastic droplets onto a rigid plate. The Oldroyd-B fluid is considered as the rheological model to describe the viscoelastic characteristics. An artificial stress is added into the momentum equation to remove the tensile instability. The solution of the problem of two successive impacts of droplets are demonstrated to be in good agreement with the literature data. The problem of two droplets impacting simultaneously onto a rigid plate is investigated.