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 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.

Thermal instability of a viscoelastic fluid in a fluid-porous system with a plane Poiseuille flow

The thermal convection of a Jeffreys fluid subjected to a plane Poiseuille flow in a fluid-porous system composed of a fluid layer and a porous layer is studied in the paper.A linear stability analysis and a Chebyshevτ-QZ algorithm are employed to solve the thermal mixed convection.Unlike the case in a single layer,the neutral curves of the two-layer system may be bi-modal in the proper depth ratio of the two layers.We find that the longitudinal rolls(LRs)only depend on the depth ratio.With the existence of the shear flow,the effects of the depth ratio,the Reynolds number,the Prandtl number,the stress relaxation,and strain retardation times on the transverse rolls(TRs)are also studied.Additionally,the thermal instability of the viscoelastic fluid is found to be more unstable than that of the Newtonian fluid in a two-layer system.In contrast to the case for Newtonian fluids,the TRs rather than the LRs may be the preferred mode for the viscoelastic fluids in some cases.

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.

AN ANALYTICAL SOLUTION AND ANALYSIS OF CHARACTERS FOR VISCOELASTIC FLUID FLOW IN ANNULAR PIPE

In this paper an analytical solution to flow of second order and Maxwell fluids in annular pipe by using Hankel integral transform is presented. A derived formula can be used to analyze the behavior of rotatory velocity and shear stress; since the parameters of material and the gap size of annular pipe explicitly appear in the analytical formula one can easily analyze their effection on the flow behavior. This solution can provide a theoretical base to drilling engineering and polymer shaping techniques. In addition, it can be used to analyze the flow characters in concentric cylinder rheometer and obtain material constants with curve fitting procedure. By investigation it is found that when outer cylinder makes uniform rotatory the history curve of velocity and stress of Maxwell fluid exhibit obliquerectangle wave and raw-wave oscillation respectively. The wave period and amplitude increase with material constant Ha. This conclusion may be of significance in practice.

Differential Quadrature Method for Steady Flow of an Incompressible Second-Order Viscoelastic Fluid and Heat Transfer Model

The two-dimensional steady flow of an incompressible second-order viscoelastic fluid between two parallel plates was studied in terms of vorticity, the stream function and temperature equations. The governing equations were expanded with respect to a snmll parameter to get the zeroth- and first-order approximate equations. By using the differenl2al quadrature method with only a few grid points, the high-accurate numerical results were obtained.

Numerical study on polymer nanofibers with electrically charged jet of viscoelastic fluid in electrospinning process

Electrospinning is a useful and efficient technique to produce polymeric nanofibers. Nanofibers of polymers are electrospun by creating an electrically charged jet of polymer solution. Numerical study on non-Newtonian and viscoelastic jets of polymer nanofibers in electrospinning process is presented in this work. In particular, the effect of non-Newtonian rheology on the jet profile during the electrospinning process is examined. The governing equations of the problem are solved numerically using the Keller-Box method. The effects of yield stress and power-law index on the elongation, velocity, stress and total force are presented and discussed in detail. The results show that by increasing the values of yield stress, the fluid elongation is reduced significantly.

Stretching surface in rotating viscoelastic fluid

The boundary layer flow over a stretching surface in a rotating viscoelastic fluid is considered. By applying a similarity transformation, the governing partial differ- ential equations are converted into a system of nonlinear ordinary differential equations before being solved numerically by the Keller-box method. The effects of the viscoelastic and rotation parameters on the skin friction coefficients and the velocity profiles are thor- oughly examined. The analysis reveals that the skin friction coefficients and the velocity in the x-direction increase as the viscoelastic parameter and the rotation parameter in- crease. Moreover, the velocity in the y-direction decreases as the viscoelastic parameter and the rotation parameter increase.

Newtonian heating effects in three-dimensional flow of viscoelastic fluid

A mathematical model is constructed to investigate the three-dimensional flow of a non-Newtonian fluid. An in-compressible viscoelastic fluid is used in mathematical formulation. The conjugate convective process （in which heat the transfer rate from the bounding surface with a finite capacity is proportional to the local surface temperature） in three-dimensional flow of a differential type of non-Newtonian fluid is analyzed for the first time. Series solutions for the nonlinear differential system are computed. Plots are presented for the description of emerging parameters entering into the problem. It is observed that the conjugate heating phenomenon causes an appreciable increase in the temperature at the stretching wall.

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.

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.

NUMERICAL SI MULATION OF STEADY FLOWFOR VISCOELASTIC FLUID IN AN ECCENTRIC ANNULUS WITHINNER ROD MOVING AXIALLY

The governing equation about steady flow of viscoelastic fluids in an eccentric annulus with inner rod moving axially is established by using common conversion Maxwell constitutive model. The numerical solutions of the flow are obtained by control volume and ADI methods. The influence of the eccentricity, velocity of inner rod and elasticity of fluid to the velocity distribution, flow rate and radial force on the inner rod is obtained, and the reason for the severe eccentric wear of the sucker rod of polymer flood well is analyzed.

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.

NUMERICAL SIMULATION OF FLOWS FOR SECOND-ORDER VISCOELASTIC FLUID COUPLED WITH HEAT TRANSFER BY DIFFERENTIAL QUADRATURE METHOD

The problem of two-dimensional steady flow of an incompressible second-order viscoelastic fluid coupled with heat transfer between parallel plates was considered. A viscous dissipation function was included in the energy equation. When the elastic property of the fluid is weaker, the zeroth-order and first-order approximate governing equations were obtained by means of the perturbation method. To understand the behavior of flow near the tube wall, the half-domain was divided into two sub-domains, in which one is a thin layer near the wall called the inner domain and the remainder is called the outer domain. The governing equations in the inner domain and in the outer domain were discretized respectively by using the Differential Quadrature Method (DQM). The matching conditions at the interface between the inner and outer domains were presented. An iterative method for solving these discretized equations was given in this paper. The numerical results obtained agree with existing results.

SIMULATION OF VISCOELASTIC FLUID FLOWS IN EXPANSION GEOMETRY USING FINITE VOLUME APPROACH

The simulation results on viscoelastic fluid flows in sudden expansion geometry with different expansion ratios are presented. Oldroyd-B, linear Phan-Thien-Tanner （L-PTT） and Finitely Extensible Nonlinear Elastic （FENE-P） based constitutive equations were applied in two-dimensional Cartesian coordinates. The governing equations in transient and fully developed regions were solved using open source software called OpenFOAM. The flow patterns, including velocity profiles, shear stresses and first normal stress differences in some horizontal and vertical sections are illustrated. In addition, effects of the fluid type, flow dynamics and expansion ratio on the flow and vortex patterns in transient and fully developed regions are presented and discussed. The presented results show that existences of vortices cause the inverse velocity and negative stresses in expansion regions of the channel which increase with increment of expansion ratio and Weissenberg number （We）. Furthermore, some dead spaces can be observed at channel expansion regions close to the wall which are also increased. The results also show that at low We numbers all fluids show close behavior while at high We numbers the FENE-P fluid behavior shows high divergence from that of the two other fluids.

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 and scaling analysis of elasticity-induced lift force in a viscoelastic fluid between confining surfaces

How to accurately characterize the lift force on the particles near the solid surfaces is an ongoing challenge in fluid mechanics and microfluidic techniques, especially in a complex system with viscoelastic fluid or/and soft surface that is commonly encountered in a biological system. The motions of the particles in vicinity of a surface can be simplified to be a rigid cylinder surrounded by the viscoelastic fluid moving along a substrate which can be rigid or soft according to different cases. In such an inertial free system with a wide range of Weissenberg number (Wi < 5.00, representing the ratio of the elastic force to the viscous force), firstly we numerically evaluate the influence of the systematic parameters, including the polymer viscosity, the geometry and Wi, on the net normal force for a cylinder closely moving along a rigid substrate, and the elasticity-induced lift force in a scaled form. It is shown that a strong shear arises in the viscoelastic confinement between the moving cylinder and the rigid substrate, it leads to the asymmetry of the first normal stress distribution around the cylinder, and thus the lift force. Then, the influence of a soft substrate on the lift force is considered, and we find that the lift force induced by the viscoelastic fluid always dominates in magnitude over that induced by the soft substrate deformation. This work provides a reliable scaling that can be utilized to quantify the elasticity-induced lift force on the particles in a viscoelastic system, such as the micro- and nanofluidic systems in biological applications.