Oldroyd-B Fluid Research Articles

Steady-State Solutions for Two Mixed Initial-Boundary Value Problems Which Describe Isothermal Motions of Burgers’ Fluids: Application

Steady-state solutions of two mixed initial-boundary value problems are presented in equivalent forms. They describe isothermal permanent motions of incompressible Burgers’ fluids over an infinite flat plate that applies time-dependent shear stresses to the fluid. More exactly, they are the first exact solutions for motions of Burgers’ fluids with differential expressions of the shear stress or velocity on the boundary. The obtained results are designed to make equivalent solutions for motions caused by an infinite plate moving in its plane at velocities that seem to be similar to previous shear stresses. It is simple to limit all results for the purpose of providing efficient results for incompressible Oldroyd-B, Maxwell, second grade and Newtonian fluids undergoing comparable motions. They may also be used to estimate how long it will take to get to a steady or permanent state.

Thermophoretic particle deposition impact in the Oldroyd-B fluid flow influenced by a magnetic dipole with an exponential thermal heat source

The role of non-Newtonian fluids can’t be denied owing to their immense industrial and engineering applications. Taking this point in mind, the motivation of the current investigation is to examine the influence of a magnetic dipole on two-dimensional nonelectrical conducting Oldroyd-B fluid flow in the existence of thermophoretic particle deposition along a linearly stretching surface. The exponential space-based thermal source and the temperature-dependent heat source/sink are incorporated into the thermal equation. The fluid concentration is strengthened with binary activation energy and Arrhenius chemical reaction. The set of ordinary differential equations that are attained after applying the similarity transformations is resolved numerically by the bvp4c approach of MATLAB software. The impact of diverse parameters on velocity, temperature, and concentration are graphically examined. The quantities of physical importance are also computed and analyzed numerically. The results revealed that fluid velocity and temperature have opposing trends for the ferromagnetic interaction parameter. It is also witnessed that the fluid concentration arguments for the chemical reaction parameter. The validation of the model in the limiting case is also given.

Spatiotemporal linear stability of viscoelastic Saffman–Taylor flows

The temporal and spatiotemporal linear stability analyses of a displaced Oldroyd-B fluid with the mean flow in a horizontally aligned, square, Hele–Shaw cell are reported to identify the regions of topological transition of the advancing interface. The viscosity of the displacing fluid is negligible in comparison to the displaced fluid. While all the inertial terms in the model are retained, the interface is assumed to evolve on a slow timescale compared with the timescale of the perturbation (or the so-called “quasi-stationary” approximation). The parameters governing stability are the Reynolds number Re=b2ρU012η2L, the elasticity number E=12λ(1−ν)η2ρb2, and the ratio of the solvent to the polymer solution viscosity ν=ηsη2, where b,L,U0,ρ,λ are the cell gap, the cell length (or width), the mean flow velocity, the density of the driven fluid, and the polymer relaxation time, respectively. Reasonably good agreement on the relative finger width data computed with our model and the experimental data in the Stokes and the inertial Newtonian regime is found. In the asymptotic limit E(1−ν)≪1, the critical Reynolds number, Rec, diverges as Rec∼[E(1−ν)]−5/3 and the critical wavenumber, αc, increases as αc∼[E(1−ν)]−2/3. In a confined domain, the temporal stability analysis indicates (a) the destabilizing influence of the inertial terms, (b) the destabilizing impact of the finite boundaries near the wall, and (c) the stabilizing impact of elasticity until a critical Reynolds number. The Briggs idea of analytic continuation is deployed to classify regions of absolute and convective instabilities as well as the evanescent modes. The phase diagram reveals the presence of an absolutely unstable region at high values of Reynolds and elasticity number, confirming the role of fluid inertia in triggering a pinch-off.

A Fully Implicit Log-Conformation Tensor Coupled Algorithm for the Solution of Incompressible Non-Isothermal Viscoelastic Flows.

In this work, a block-coupled algorithm is presented, which can compute laminar, incompressible, non-isothermal, viscoelastic flow problems based on the log-conformation tensor approach. The inter-equation coupling of the incompressible Cauchy linear momentum and mass conservation equations is obtained in a procedure based on the Rhie–Chow interpolation. The divergence of the log-conformation tensor term in the linear momentum equations is implicitly discretized in this work. In addition, the velocity field is considered implicitly in the log-conformation tensor constitutive equations by expanding the advection, rotation and the rate of deformation terms with a Taylor series expansion truncated at the second-order error term. Finally, the advection and diffusion terms in the energy equation are also implicitly discretized. The mass, linear momentum, log-conformation tensor constitutive model and energy-discretized linear equations are joined into a block-matrix following a monolithic framework. Validation of the newly developed algorithm is performed for the non-isothermal viscoelastic matrix-based Oldroyd-B fluid flow in the axisymmetric 4:1 planar sudden contraction benchmark problem.

CONSTRUCTION OF THE EXACT SOLUTION OF BLOOD FLOW OF OLDROYD-B FLUIDS THROUGH ARTERIES WITH EFFECTS OF FRACTIONAL DERIVATIVE MAGNETIC FIELD AND HEAT TRANSFER

In this paper, we construct the exact solution of blood flow of Oldroyd-B fluids to represent the non-Newtonian characteristics of biofluids and to study the unsteady flow of blood and heat transfer through arterial segment with external magnetic field applied normal to the flow direction in the presence of thermal radiation and body acceleration. In the study, we investigate the influence of Caputo time-fractional parameter, heat transfer, external body acceleration, and magnetic field on the blood velocity and temperature distribution in a straight circular cylindrical vessel. The constitutive partial differential equation and temperature equation governing blood flow in the arterial wall are solved using Laplace and finite Hankel transforms. In addition, Gaver Stehfest’s algorithm is used for the inverse Laplace transform. The results obtained are interpreted graphically and discussed from the physiological point of view. The graphs show that the rate of heat transfer and blood velocity become less with the increase in the external magnetic field strength which is very good in regulating blood flow as well as temperature distribution during treatment. Based on the graphical representations, blood velocity and temperature distribution both decrease in ascending order of the fractional parameter as memory effect. Further, owing to the dissipation of energy caused by blood viscoelasticity and magnetic field effect, during pulsatile flow of blood, the heat transfer rate at the wall of the artery is enhanced. The significance of this study can be found in the application fields such as biomedical engineering, medicine and pathology.

Line contact lubricated by a fluid described by non-Newtonian Giesekus model

Abstract Modelling hydrodynamic lubrication helps engineers and material scientists designing lubricants that minimize viscous energy losses without compromising the durability of engine components. This type of modelling requires selecting lubricant rheological constitutive equations that provide realistic information on fluid behaviour within tribological contact zones. Due to complexity of differential constitutive equations the modelling effort was limited to generalized Newtonian or Oldroyd-B fluids. In this paper, we present more realistic description of hydrodynamic lubrication using the Giesekus fluid capturing both viscous shear thinning and the development of elastic stresses at high Weissenberg numbers. The problem analysis is done using a modified regular perturbation method with respect to the small mobility parameter $\alpha $. The perturbation analysis modification is caused by the fact that the problem contains several a priori unknown boundaries. The actual perturbation solutions for functions of pressure and gap, as well as the coordinate of exit from the contact and the exit film thickness, are obtained as linear functions of $\alpha $. The effects of the model parameters such as the mobility factor, relaxation time and the polymer contribution to viscosity on pressure, shear and extensional viscosities within the contact zone are analysed. It is shown that the presence of polymers helps to increase the gap and reduce the viscous losses associated with hydrodynamic friction. This result is consistent with empirical observations of fuel economy enhancement for polymer-containing lubricants.

Chemical Reaction and Radiation Effects on MHD Flow of Oldroyd-B Fluid through Porous Medium Past an Exponentially Stretching Sheet with Heat Sink

In this article, the effect of electromagnetic force with the chemical and thermal radiation effect on the Oldroyd-B fluid past an exponentially stretched sheet with a heat sink and porous medium was studied. The governing system of nonlinear partial differential equations has transformed into a system of ordinary differential equations using similarity transformations. The system is then solved numerically using a successive linearization method. The numerical results of velocity, temperature, and concentration profiles are represented graphically. Several parameters’ effects are investigated and examined. Local Nusselt number, skin friction coefficient, porosity, Deborah numbers, and local Sherwood number numerical values are listed and analyzed. The results reveal that many parameters have a significant impact on the fluid flow profiles. The concentration profiles were considerably affected by the reaction rate parameter, and the concentration thickness of the boundary layer decreased as the reaction rate parameter increased. The results of the analysis were compared to the results of existing works and found to be in excellent agreement.

Thermal Convection for an Oldroyd-B Fluid in an Anisotropic Porous Medium Underlying a Fluid Layer

Abstract We investigate the thermal convection in a fluid layer overlying an anisotropic porous medium with Oldroyd-B fluids in the paper. The linear stability analysis and the Chebyshev pseudo-spectral method are used to obtain the numerical results. We found that the neutral curves are bimodal at some depth ratios. The proper values of the permeability ratio and the thermal diffusivity ratio are found to stabilize the stationary convection and the oscillatory convection. Furthermore, the variation of permeability ratio does not affect the bimodal nature of the neutral curves. In addition, the increase in the stress relaxation time has an unstable effect on the oscillatory convection, but the strain retardation time has the opposite effect.

Birefringent strands drive the flow of viscoelastic fluids past obstacles

The flow of polymer solutions past solid obstacles or through porous media gives rise to rich physical phenomena over a wide range of spatial and temporal scales. Viscoelasticity, in particular, can induce a strong nonlinear response with an increase of flow resistance even for a solution whose viscosity decreases in simple shear flow. Various hypotheses have been proposed to explain this phenomenon but a clear picture of the pore-scale mechanisms involved and their impact upon larger scales is still lacking. Here, we show that localized zones of large polymeric stress, known as birefringent strands, drive the flow of an Oldroyd-B fluid through two-dimensional arrays of cylinders. Combining a recently developed numerical scheme with high performance computing, we find that these strands generate a complete reorganization of the flow with an increase of stagnation zones, a reinforcement of preferential paths and a splitting of flow channels. Furthermore, we show that this reorganization is the source of an increase in the viscous dissipation of the solvent and also that the stretching of polymer molecules in the strands is associated with entropy production. Both these phenomena yield a global increase in dissipation that can be directly linked to the increase of flow resistance. Our results demonstrate that the birefringent strands – not the elongational viscosity – drive the flow of viscoelastic fluids through porous media and that the increase of flow resistance can occur even at steady state, before the transition to elastic turbulence.

Magneto-Oldroyd-B fluid flow over an exponentially accelerated vertical porous plate with heat source and reactive agents

ABSTRACT The present research work describes about flowing magnetohydrodynamic generalized Oldroyd-B liquid model in an exponentially accelerated vertical porous boundless device with heat source and reacting species and heat source. The fully developed Oldroyd-B fluid model is stimulated by heat source/sink and species reaction and vertical device under the influence of gravity. The physical flow equation is modelled in the form of partial derivative which is mutated into the dimensionless partial derivative equations by invoking suitable non-dimensional variables. The partial system of derivative equations is tackled by employing perturbation technique. The outcomes for the velocity flow field, thermal and species distributions are exhibited graphically. Computed outcomes for the plate friction, heat gradient and Sherwood number are provided in table. Furthermore, correlation in terms of validation has been tabulated. Some obtained results show that an increase in heat source and time reduces velocity plots but increase the rate of particle temperature. Owing to the Lorentz force produced by the imposed magnetic field, the velocity alongside velocity degenerates. Time parameter compliments the fluid temperature, particle concentration and fluid transport. The heat transfer rate accelerates with higher Prandtl number and heat source, but mass transfer rate retards with higher Schmidt number.

Effect of resistive and radiative heats on enhanced heat transfer of parabolic trough solar collector

The total amount of existing solar energy on earth is more than the World's current and estimated energy requirements. The radiative heat from the sun is capable of producing electricity. This study deals with hydrogen energy storage by growing the heat transmission enactment of PTSC (Parabolic Trough Solar Collector) in the presence of thermic heat, resistive heat, buoyancy, magnetic field, and uneven heat sink/source. The Oldroyd-B and Maxwell fluids are the shear-thinning fluids with rheological differences. This study investigates the best heat transfer performer among these two, which helps to enhance the heat transfer in PTSC. A mathematical model was established and resolved numerically to examine this. The study outcomes are deliberated with the assistance of pictorial and numerical evaluations. Thermal radiation with Oldroyd-B fluid combination helps to enhance the heat transmission rate of PTSC to another level.

A linear route to elasto-inertial turbulence

Pipe flow of Newtonian fluids is well known to exhibit a transition from the laminar to turbulent regime usually at a Reynolds number ~ 2000, despite being linearly stable at all Reynolds numbers. In stark contrast, we show that pressure-driven pipe flow of a viscoelastic (Oldroyd-B) fluid is linearly unstable to axisymmetric perturbations. The dimensionless groups that govern stability are the Reynolds number Re = ρUmaxR/η, the elasticity number E = λη/(ρR2) and the ratio of solvent to solution viscosity β = ηs/η; here, R is the pipe radius, Umax is the maximum velocity of the base flow, ρ is the fluid density, and λ is the polymer relaxation time. The unstable mode has a phase speed close to Umax over the entire unstable region in the Re-E-β space. In parameter regimes accessible to experiments (e.g. β > 0.6 and E > 0.08), the critical Reynolds number Rec ~ 400, with the associated eigenfunctions spread out across the pipe cross section. In the asymptotic limit of E(1-β) << 1, but fixed E, the critical Reynolds number for instability and the critical wavenumber diverge as Rec ~ (E (1-β))-3/2 and kc ~ (E(1-β))-1/2 respectively. The unstable eigenfunction in this limit is localized near the centerline, implying that the unstable mode belongs to a class of viscoelastic ‘center modes'.The instability identified in this study comprehensively dispels the prevailing notion of pipe flow of viscoelastic fluids being linearly stable in the Re-W plane (W = Re E being the Weissenberg number). The prediction of a linear instability is consistent with several experimental studies on pipe flow of polymer solutions, ranging from reports of ‘early turbulence’ in the 1970's to the more recent discovery of ‘elasto-inertial turbulence’ (B. Hof and coworkers, Proc. Natl. Acad. Sci., 110, 10557–10562 (2013)), wherein transition is observed at Re much lower than 2000. An analogous center-mode instability is predicted for viscoelastic channel flows over a similar range of parameters. Thus, there is the suggestion of a universal linear mechanism that underlies the onset of turbulence in rectilinear viscoelastic shearing flows for sufficiently elastic dilute polymer solutions, marking a possible paradigm shift in our understanding of transition in such flows. We will end with a discussion of the possible transition scenarios in the Re-W-β space for viscoelastic shearing flows.

Magnetic Dipole and Thermophoretic Particle Deposition Impact on Bioconvective Oldroyd-B Fluid Flow over a Stretching Surface with Cattaneo-Christov Heat Flux.

This study emphasizes the performance of two-dimensional electrically non-conducting Oldroyd-B fluid flowing across a stretching sheet with thermophoretic particle deposition. The heat and mass transfer mechanisms are elaborated in the presence of a magnetic dipole, which acts as an external magnetic field. The fluid possesses magnetic characteristics due to the presence of ferrite particles. The gyrotactic microorganisms are considered to keep the suspended ferromagnetic particles stable. Cattaneo–Christov heat flux is cogitated instead of the conventional Fourier law. Further, to strengthen the heat transfer and mass transfer processes, thermal stratification and chemical reaction are employed. Appropriate similarity transformations are applied to convert highly nonlinear coupled partial differential equations into non-linear ordinary differential equations (ODEs). To numerically solve these ODEs, an excellent MATLAB bvp4c approach is used. The physical behavior of important parameters and their graphical representations are thoroughly examined. The tables are presented to address the thermophoretic particle velocity deposition, rate of heat flux, and motile microorganisms’ density number. The results show that the rate of heat transfer decreases as the value of the thermal relaxation time parameter surges. Furthermore, when the thermophoretic coefficient increases, the velocity of thermophoretic deposition decreases.

The Oscillatory Flow of Oldroyd-B Fluid with Magnetic Disturbance

The magnetic field intensity will be nondeterminacy with the flow of charged particles thrown out by solar activities, the overlap of adjacent magnetic islands or non-axisymmetric magnetic interference in tokamaks and so on. The model of a generalized Oldroyd-B fluid with fractional derivative under oscillating pressure gradient and magnetic field with some disturbance will be considered in this paper. The disturbance is regarded as the background noise of the system, and the model is described by a fractional stochastic differential equation. Time and space are discretized by L1, L2 schemes based on piecewise linear interpolation and the central difference quotient method. We demonstrate the effects of the amplitude and period of the oscillating pressure gradient, magnetic parameter, fractional parameters and noise on the velocity field, and two special cases are given.

Corrigendum to “The Oldroyd-B fluid in elastic instabilities, turbulence and particle suspensions”;

Corrigendum to “The Oldroyd-B fluid in elastic instabilities, turbulence and particle suspensions”;

HEAT TRANSFER OF MHD OLDROYD-B FLUID WITH RAMPED WALL VELOCITY AND TEMPERATURE IN VIEW OF LOCAL AND NONLOCAL DIFFERENTIAL OPERATORS

The theoretical study focuses on the examination of the convective flow of Oldroyd-B fluid with ramped wall velocity and temperature. The fluid is confined on an extended, unbounded vertical plate saturated within the permeable medium. To depict the fluid flow, the coupled partial differential equations are settled by using the Caputo (C) and Caputo Fabrizio (CF) differential time derivatives. The mathematical analysis of the fractionalized models of fluid flow is performed by Laplace transform (LT). The complexity of temperature and velocity profile is explored by numerical inversion algorithms of Stehfest and Tzou. The fractionalized solutions of the temperature and velocity profile have been traced out under fractional and other different parameters considered. The physical impacts of associated parameters are elucidated with the assistance of the graph using the software MATHCAD 15. We noticed the significant influence of the fractional parameter (memory effects) and other parameters on the dynamics of the fluid flow. Shear stress at the wall and Nusselt number also are considered. It’s brought into notice the fractional-order model (CF) is the best fit in describing the memory effects in comparison to the C model. An analysis of the comparison between the solution of velocity and temperature profile for ramped wall temperature and velocity and constant wall temperature and velocity is also performed.

Mathematical Simulation of Heat Transfer in Thermally Magnetised Oldroyd-B Fluid in Sakiadis Rheology with a Heat Reservoir

Sakiadis rheology of a generalised polymeric material, as well as a heat source or sink and a magnetic field, are all part of this study. Thermal radiations have been introduced into the convective heating process. The translation of a physical situation into a set of nonlinear equations was achieved through mathematical modelling. To convert the resulting partial differential equation into a set of nonlinear ordinary differential equations, appropriate transformations have been used. The velocity and temperature profiles are generated both analytically by HAM and numerically by the Runge–Kutta method (RK-4). In order to analyse the behaviour of the physical quantities involved, numerical and graphical depictions have been offered. To show that the acquired findings are correct, a nonlinear system error analysis has been offered. The heat flux study has been shown using bar charts. For the essential factors involved, the local Nusselt number and local Skin friction are calculated in tabular form. The fluid particles’ molecular mobility was slowed due to the magnetic field and porosity, and the heat transfer rates were demonstrated to be lowered when magnetic and porosity effects are present. This magnetic field and porosity effects regulating property has applications in MHD ion propulsion and power production, the electromagnetic casting of metals, etc. Furthermore, internal heat absorption and generation have diametrically opposed impacts on fluid temperature. The novelty of the present study is that no one has investigated the Sakiadis flow of thermal convection magnetised Oldroyd-B fluid in terms of a heat reservoir across a porous sheet. In limited circumstances, a satisfactory match is revealed when the collected values are compared to the existing work published corroborating the current attempt. The findings of this study are expected to be applicable to a wide range of technical and industrial processes, including steel extrusion, wire protective layers, fiber rolling, fabrication, polythene stuff such as broadsheet, fiber, and stainless steel sheets, and even the process of depositing a thin layer where the sheet is squeezed.

Variable heat source in stagnation-point unsteady flow of magnetized Oldroyd-B fluid with cubic autocatalysis chemical reaction

Understanding the flow behavior of non-Newtonian fluids required more research in terms of various aspects. Because the role of heat transfer in non-Newtonian flows is dominant due to its importance in technology. For these benefits, a theoretical investigation is carried out to analyze the time-dependent flow of non-Newtonian fluid with a focus on thermal and solutal transport. In the present study, we formulate a mathematical model for Oldroyd-B fluid by taking into account the effect of magnetic field. The governing flow field equations are constructed with the help of the rheological expression of Oldroyd-B fluid model. For physical relevance, the effects of non-uniform heat source/sink, Joule heating and thermal radiation with homogeneous-heterogeneous chemical reactions have been incorporated in stagnation point flow towards a stretching cylinder to modify the energy and concentration distributions. The highly non-linear ordinary differential equations are tackled analytically through the optimal homotopy analysis method (OHAM). Study reveals that, the strength of homogeneous and heterogeneous reactions leads to enhanced the concentration of the catalyst at the surface. Furthermore, the non-uniform heat source/sink parameter acts like a major controlling parameter for the heat transportation rate, and heat transport is also influenced by the surface-dependent heat source/sink.

Numerical implementation and error analysis of nonlinear coupled fractional viscoelastic fluid model with variable heat flux

The studies conducted in the recent past reveals that the non-Newtonian are extensively applied in several engineering applications. The atypical Oldroyd-B fluid, a specific sub-category of non-Newtonian fluids, with fractional derivative constitutive equations, is very expedient to determine the approximation of many dilute polymer liquids. Since very few studies regarding the numerical solution of the aforesaid model are conducted, so the literature that can be referred to in this regard is scarce. The current article investigates the unsteady magneto-hydrodynamic flow of irregular Oldroyd-B fluid with a modified thermal flux restricted between two infinite parallel plates. Here, it is assumed that the thermal conductivity of the fluid is variable, and the impact of body forces like gravity is accordingly analyzed in terms of non-linear convection. The linear acceleration of the lower permeable plate in its plane prompts the flow. Further, we have investigated the heat transfer in the flow field through a modified fractional thermal gradient and internal heat source. The temperature gradient accelerates with time and exists among the flow boundaries. In order to determine the approximate solution unified with finite difference approximation for Caputo fractional time derivatives, we have utilized the standard Galerkin finite element method. Moreover, we have conducted a convergence analysis of the suggested numerical scheme and offered some valid error approximations. In the non-integer viscoelastic model, the local number and coefficient of skin friction are also deliberated. We have also performed some upright numerical simulations to highlight the ascendency of characteristic flow parameters of velocity and temperature fields in the proposed scheme. Here, we have analyzed that the resistive force increases by 40%, corresponding to an increment in α ranging from 0 to 0.4. Further, the heat transfer rate is diminish by 10.94% against an increase in variable thermal conductivity while it is enhanced by 37.78% corresponding to increased in Pr.

Electroosmotic pressure-driven oscillatory flow and mass transport of Oldroyd-B fluid under high zeta potential and slippage conditions in microchannels

The appropriate analysis for microscale transport of complex fluids is essential in many scientific and technological devices to establish the accurate theoretical model. For example, the viscoelastic fluid plays a vital role in medical diagnostics to separate biomolecules using the concept of a microchannel flow under time-periodic oscillation effects. The available literature demonstrates no such analysis existed for the time-periodic oscillation of Oldroyd-B fluid subjected to the electroosmotic and slippage flow conditions. The present paper develops a new mathematical approach for an electroosmotic pressure-driven pulsatile-Oldroyd-B fluid flow between two parallel plates in microchannels under slipping actions on the wall surface. The flow takes place in an exponential electric field regime by imposing an exponential pressure gradient with time to simulate the elastic and viscous behaviors of the Oldroyd-B fluid. A finite difference method (MATLAB bvp4c) is employed to solve the strongly non-linear electrical potential and concentration equations. A time-periodic oscillation effect on the velocity profile, concentration distribution, and mass transport is analyzed under low or high zeta potentials at the wall. The results highlight that the elastic nature of fluid increases the oscillatory behavior of flow velocity compared to that of the viscous fluid. The zeta potential (ξ¯)in the flow field is responsible for increasing the flow velocity amplitude. Notably, the flow velocity for a high zeta potential (ξ¯=4) at the wall has increased approximately 142.75% at the peak point compared to the low zeta potential(ξ¯=0.1). The velocity amplitude for a higher elastic fluid (elasticity, E = 0.2) compared to a lower elastic fluid (E = 0.1) amplifies 21.01% at the peak point. Finally, it is noteworthy to demonstrate that for the Oldroyd-B fluid, a resonant behavior of the mass transfer rate is dependent on the combination of the parameters, namely angular Reynolds number and elasticity. In this context, it is evident that the interaction of the fluid elasticity and the oscillatory character of flow enhance the mass transfer rate. This innovative concept can be adapted to amplify the fluid mass transfer rate.