ENERGY CONSERVATION FOR SOLUTIONS OF INCOMPRESSIBLE VISCOELASTIC FLUIDS

Some sufficient conditions of the energy conservation for weak solutions of incompressible viscoelastic flows are given in this paper.First,for a periodic domain in R^(3),and the coefficient of viscosity μ=0,energy conservation is proved for u and F in certain Besovs paces.Furthermore,in the whole space R^(3),it is shown that the conditions on the velocity u and the deformation tensor F can be relaxed,that is,u∈B_(3,c(N))^(1/3),and F∈B_(3,∞)^(1/3).Finally,when μ>0,in a periodic domain in R^(d) again,a result independent of the spacial dimension is established.More precisely,it is shown that the energy is conserved for u∈L^(T)(0,T;L^(n)(Ω))for any 1/r+1/s≤1/2,with s≥4,and F∈L^(m)(0,T;L^(n)(Ω))for any 1/m+1/n≤1/2,with n≥4.

Electromagnetohydrodynamic (EMHD) flow of fractional viscoelastic fluids in a microchannel

This study investigates the electromagnetohydrodynamic(EMHD)flow of fractional viscoelastic fluids through a microchannel under the Navier slip boundary condition.The flow is driven by the pressure gradient and electromagnetic force where the electric field is applied horizontally,and the magnetic field is vertically(upward or downward).When the electric field direction is consistent with the pressure gradient direction,the changes of the steady flow rate and velocity with the Hartmann number Ha are irrelevant to the direction of the magnetic field(upward or downward).The steady flow rate decreases monotonically to zero with the increase in Ha.In contrast,when the direction of the electric field differs from the pressure gradient direction,the flow behavior depends on the direction of the magnetic field,i.e.,symmetry breaking occurs.Specifically,when the magnetic field is vertically upward,the steady flow rate increases first and then decreases with Ha.When the magnetic field is reversed,the steady flow rate first reduces to zero as Ha increases from zero.As Ha continues to increase,the steady flow rate(velocity)increases in the opposite direction and then decreases,and finally drops to zero for larger Ha.The increase in the fractional calculus parameterαor Deborah number De makes it take longer for the flow rate(velocity)to reach the steady state.In addition,the increase in the strength of the magnetic field or electric field,or in the pressure gradient tends to accelerate the slip velocity at the walls.On the other hand,the increase in the thickness of the electric double-layer tends to reduce it.

A new mixed subgrid-scale model for large eddy simulation of turbulent drag-reducing flows of viscoelastic fluids

A mixed subgrid-scale（SGS） model based on coherent structures and temporal approximate deconvolution（MCT） is proposed for turbulent drag-reducing flows of viscoelastic fluids. The main idea of the MCT SGS model is to perform spatial filtering for the momentum equation and temporal filtering for the conformation tensor transport equation of turbulent flow of viscoelastic fluid, respectively. The MCT model is suitable for large eddy simulation（LES） of turbulent dragreducing flows of viscoelastic fluids in engineering applications since the model parameters can be easily obtained. The LES of forced homogeneous isotropic turbulence（FHIT） with polymer additives and turbulent channel flow with surfactant additives based on MCT SGS model shows excellent agreements with direct numerical simulation（DNS） results. Compared with the LES results using the temporal approximate deconvolution model（TADM） for FHIT with polymer additives, this mixed SGS model MCT behaves better, regarding the enhancement of calculating parameters such as the Reynolds number.For scientific and engineering research, turbulent flows at high Reynolds numbers are expected, so the MCT model can be a more suitable model for the LES of turbulent drag-reducing flows of viscoelastic fluid with polymer or surfactant additives.

REGULARITY FOR A GENERALIZED JEFFREY'S INTEGRAL MODEL FOR VISCOELASTIC FLUIDS

We prove a local existence of a strong solution v ：Ω×T→R^3 for a system of nonlinear integrodifferential equations describing motion of an incompressible viscoelastic fluid using standard mathematical tools. The problem is considered in a bounded, smooth domain ΩСR^3 with a Dirichlet boundary condition and a standard initial condition.

Micro-macro models for viscoelastic fluids： modelling, mathematics and numerics Dedicated to the NSFC-CNRS Chinese-French summer institute on fluid mechanics in 2010

This paper is an introduction to the modelling of viscoelastic fluids, with an emphasis on micromacro （or multiscale） models. Some elements of mathematical and numerical analysis are provided. These notes closely follow the lectures delivered by the second author at the Chinese Academy of Science during the Workshop ＂Stress Tensor Effects on Fluid Mechanics＂ in January 2010.

Blow-up Criterion for Three-dimensional Compressible Viscoelastic Fluids with Vacuum

In this paper, we study a Cauchy problem for the equations of 3D compressible viscoelastic fluids with vacuum. We establish a blow-up criterion for the local strong solutions in terms of the upper bound of the density and deformation gradient.

Numerical simulation of thermal convection of viscoelastic fluids in an open-top porous medium with constant heat flux

This paper makes a numerical study of the buoyancy-driven convection of a viscoelastic fluid saturated in an open-top porous square box under the constant heat flux boundary condition. The effects of the relaxation and retardation times on the onset of the oscillatory convection, the convection heat transfer rate and the flow pattern transition are investigated. It is shown that a large relaxation time can destabilize the fluid flow leading to an early onset of the thermal convection and a high heat transfer rate, while a large retardation time tends to stabilize the flow and suppress the convection onset and the heat transfer. After the convection sets in, the flow bifurcation appears earlier with the increase of the relaxation time and the decrease of the retardation time, resulting in more complicated flow patterns in the porous medium. Furthermore, with the increase of the ratio of the relaxation time to the retardation time, the fluid may be blocked from flowing through the open-top boundary, which may be caused by the viscoelastic effect. Finally, the comparison of our results with those under isothermal heating boundary conditions reveals that the heat transfer rate correspo- nding to a constant heat flux boundary is always higher.

Thermal instability and heat transfer of viscoelastic fluids in bounded porous media with constant heat flux boundary

A numerical simulation is performed for thermal instability and heat transfer of viscoelastic fluids in bounded porous media under the bottom constant heat flux boundary condition. The results for six different combinations of relaxation and retardation times demonstrate the existence of the thermal instability induced flow bifurcation. It is found that the increase of the relaxation time can enhance the heat transfer efficiency by disturbing the fluid flow and facilitating the bifurcation. The increase of the retardation time can stabilize the flow and postpone the bifurcation, leading to simpler flow pattern and lower heat transfer rate.

On the some issues of particle motion in the flow of viscoelastic fluids

Particle motion in confined shear flow of viscoelastic fluids is very common in nature and has a wide range of applications.Understanding and mastering the motion characteristics of particles in viscoelastic fluids has important academic value and practical significance.In this paper,we first introduce the related equations and characteristic parameter,and then emphasize the following issues:the lateral equilibrium position of particle;interaction and aggregation of multiple particles;the chain structure formed by multiple particles;and the motion of non-spherical particle.Finally,some unresolved issues,challenges,and future research directions are highlighted.

Performance of EDAB-HCl Acid Blended System as Fracturing Fluids in Oil Fields

Due to the high price and formation damage of the guargum fracturing fluid, many oilfields are more and more interested in surfactant based fracturing fluids. The rheological properties of erucicamide dimethyl amidopropyl betaine （EDAB）-HC1 acid blended system and its suitability as fracturing fluid were investigated in this work. The effects of pH, concentration of EDAB, and temperature on the rheological properties of the blended system were studied. The results show that addition of EDAB improved the viscosity of the system from less than 10 mPa. s to about 400 mPa. s, which could retard the acid-rock reaction to about one half at 60℃ and one quarter at 90℃ comparing to straight HC1 acid, suggesting that there is sufficient time for the blended fluid to react with formation rock when it is used as fracturing fluid in an oil field. Core flow tests demonstrated that the EDAB-acid blended fluid could divert itself from high permeability formation core to low permeability one, thus ensuring proper acid placement in the target reservoirs.

Conformable and Caputo’s Derivatives in Generalized Viscoelastic Models

We study two generalized versions of a system of equations which describe the time evolution of the hydrodynamic fluctuations of density and velocity in a linear viscoelastic fluid. In the first of these versions, the time derivatives are replaced by conformable derivatives, and in the second version left-handed Caputo’s derivatives are used. We show that the solutions obtained with these two types of derivatives exhibit significant similarities, which is an interesting (and somewhat surprising) result, taking into account that the conformable derivatives are local operators, while Caputo’s derivatives are nonlocal operators. We also show that the solutions of the generalized systems are similar to the solutions of the original system, if the order α of the new derivatives (conformable or Caputo) is less than one. On the other hand, when α is greater than one, the solutions of the generalized systems are qualitatively different from the solutions of the original system.

Stress Waves in Polymeric Fluids

This paper demonstrates the existence, propagation, transmission, reflection, and interaction of deviatoric stress waves in polymeric fluids for which the mathematical models are derived using conservation and balance laws (CBL) of Classical Continuum Mechanics (CCM) and the constitutive theories are based on the entropy inequality and representation theorem. The physical mechanisms of deformation in polymeric liquids that enable the stress wave physics are identified and are demonstrated to be valid using Maxwell, Oldroyd-B, and Giesekus polymeric fluids, and are illustrated using model problem studies. We assume polymeric fluids to be isotropic and homogeneous at the macro scale so that the CBL of the CCM can be used to derive their mathematical models. For simplicity, we assume the polymeric fluids to be incompressible in the present work.

PLANE SURFACE SUDDENLY SET IN MOTION IN A VISCOELASTIC FLUID WITH FRACTIONAL MAXWELL MODEL

The fractional calculus approach in the constitutive relationship model of viscoelastic fluid is introduced.The flow near a wall suddenly set in mo- tion is studied for a non-Newtonian viscoelastic fluid with the fractional Maxwell model.Exact solutions of velocity and stress are obtained by using the discrete in- verse Laplace transform of the sequential fractional derivatives.It is found that the effect of the fractional orders in the constitutive relationship on the flow field is signif- icant.The results show that for small times there are appreciable viscoelastic effects on the shear stress at the plate,for large times the viscoelastic effects become weak.

Unsteady rotating flows of a viscoelastic fluid with the fractional Maxwell model between coaxial cylinders

The fractional calculus is used in the constitutive relationship model of viscoelastic fluid. A generalized Maxwell model with fractional calculus is considered. Based on the flow conditions described, two flow cases are solved and the exact solutions are obtained by using the Weber transform and the Laplace transform for fractional calculus.

A revised Cattaneo-Christov micropolar viscoelastic nanofluid model with combined porosity and magnetic effects

The dynamics of non-Newtonian fluids along with nanoparticles is quite interesting with numerous industrial applications. The current predominately predictive modeling deals with the flow of the viscoelastic micropolar fluid in the presence of nanoparticles. A progressive amendment in the heat and concentration equations is made by exploiting the Cattaneo-Christov(C-C) heat and mass flux expressions. Besides, the thermal radiation effects are contributed in the energy equation and aspect of the radiation parameter, and the Prandtl number is specified by the one-parameter approach.The formulated expressions are converted to the dimensionless forms by relevant similarity functions. The analytical solutions to these expressions have been erected by the homotopy analysis method. The variations in physical quantities, including the velocity,the temperature, the effective local Nusselt number, the concentration of nanoparticles,and the local Sherwood number, have been observed under the influence of emerging parameters. The results have shown good accuracy compared with those of the existing literature.

THE DYNAMIC BEHAVIORS OF VISCOELASTIC PIPE CONVEYING FLUID WITH THE KELVIN MODEL

Based on the differential constitutive relationship of linearviscoelastic material, a solid-liquid coupling vibration equation forviscoelastic pipe conveying fluid is derived by the D'Alembert'sprinciple. The critical flow velocities and natural frequencies ofthe cantilever pipe conveying fluid with the Kelvin model (flutterinstability) are calculated with the modified finite differencemethod in the form of the recurrence for- mula. The curves betweenthe complex frequencies of the first, second and third mode and flowvelocity of the pipe are plotted. On the basis of the numericalcalculation results, the dynamic behaviors and stability of the pipeare discussed. It should be pointed out that the delay time ofviscoelastic material with the Kelvin model has a remarkable effecton the dynamic characteristics and stability behaviors of thecantilevered pipe conveying fluid, which is a gyroscopicnon-conservative system.

Numerical analysis for viscoelastic fluid flow with distributed/variable order time fractional Maxwell constitutive models

Fractional calculus has been widely used to study the flow of viscoelastic fluids recently,and fractional differential equations have attracted a lot of attention.However,the research has shown that the fractional equation with constant order operators has certain limitations in characterizing some physical phenomena.In this paper,the viscoelastic fluid flow of generalized Maxwell fluids in an infinite straight pipe driven by a periodic pressure gradient is investigated systematically.Consider the complexity of the material structure and multi-scale effects in the viscoelastic fluid flow.The modified time fractional Maxwell models and the corresponding governing equations with distributed/variable order time fractional derivatives are proposed.Based on the L1-approximation formula of Caputo fractional derivatives,the implicit finite difference schemes for the distributed/variable order time fractional governing equations are presented,and the numerical solutions are derived.In order to test the correctness and availability of numerical schemes,two numerical examples are established to give the exact solutions.The comparisons between the numerical solutions and the exact solutions have been made,and their high consistency indicates that the present numerical methods are effective.Then,this paper analyzes the velocity distributions of the distributed/variable order fractional Maxwell governing equations under specific conditions,and discusses the effects of the weight coefficient(α)in distributed order time fractional derivatives,the orderα(r,t)in variable fractional order derivatives,the relaxation timeλ,and the frequencyωof the periodic pressure gradient on the fluid flow velocity.Finally,the flow rates of the distributed/variable order fractional Maxwell governing equations are also studied.

Numerical Simulation of Viscoelastic Extrudate Swell Through Elliptical Ring Die

The numerical simulation of extrudate swell is significant in extrusion processing.Precise prediction of extrudate swell is propitious to the control of melt flow and the quality of final products.A mathematical model of three-dimensional（3D）viscoelastic flow through elliptical ring die for polymer extrusion was investigated.The penalty function formulation of viscoelastic incompressible fluid was introduced to the finite element model to analyze 3D extrusion problem.The discrete elastic viscous split stress（DEVSS）and streamline-upwind PetrovGalerkin（SUPG）technology were used to obtain stable simulation results.Free surface was updated by updating the streamlines which needs less memory space.According to numerical simulation results,the effect of zero-shear viscosity and elongation parameter on extrudate swell was slight,but with the increase of volumetric flow rate and relax time the extrudate swell ratio increased markedly.Finally,the numerical simulation of extrudate swell flow for low-density polyethylene（LDPE）melts was investigated and the results agreed well with others’work.These conclusions provided quantitative basis for the forecasting extrudate swell ratio and the controlling of extrusion productivity shape.

Accurate solutions for viscoelastic boundary layer flow and heat transfer over stretching sheet

In this article, we present accurate analytical solutions for boundary layer flow and heat transfer of an incompressible and electrically conducting viscoelastic fluid over a linearly stretching surface subject to a transverse uniform magnetic field using the homotopy analysis method （HAM） for two general types of non-isothermal boundary conditions. In addition, we demonstrate that the previously reported analytical solutions for the temperature field given in terms of Kummer＇s function do not converge at the boundary. We provide a graphical and numerical demonstration of the convergence of the HAM solutions and tabulate the effects of various parameters on the skin friction coefficient and wall heat transfer.

A corrected particle method with high-order Taylor expansion for solving the viscoelastic fluid flow

In this paper, a corrected particle method based on the smoothed particle hydrodynamics (SPH) method with high-order Taylor expansion (CSPH-HT) for solving the viscoelastic flow is proposed and investigated. The validity and merits of the CSPH-HT method are first tested by solving the nonlinear high order Kuramoto-Sivishinsky equation and simulating the drop stretching, respectively. Then the flow behaviors behind two stationary tangential cylinders of polymer melt, which have been received little attention, are investigated by the CSPH-HT method. Finally, the CSPH-HT method is extended to the simulation of the filling process of the viscoelastic fluid. The numerical results show that the CSPH-HT method possesses higher accuracy and stability than other corrected SPH methods and is more reliable than other corrected SPH methods.