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 RESEARCH ON THE COHERENT STRUCTURE IN THE VISCOELASTIC SECOND-ORDER MIXING LAYERS

Numerical simulations have been performed in time-developing plane mixing layers of the viscoelastic second-order fluids with pseudo-spectral method. Roll-up, pairing and merging of large eddies were examined at high Reynolds numbers and low Deborah numbers. The effect of viscoelastics on the evolution of the large coherent structure was shown by making a comparison between the second-order and Newtonian fluids at the same Reynolds numbers.

ANALYTICAL SOLUTION OF FLOW OF SECOND-ORDER NON-NEWTONIAN FLUIDS THROUGH ANNULAR PIPES

This paper presents an analytical solution to the unsteady flow of the second-order non-Newtonian fluids by the use of intergral transformation method. Based on the numerical results, the effect of non-Newtonian coefficient Hc and other parameters on the flow are analysed. It is shown that the annular flow has a shorter characteristic time than the general pipe flow while the correspondent velocity, average velocity have a ... nailer value for a given Hc. Else, when radii ratio keeps unchanged, the shear stress of inner wall of annular flow will change with the inner radius -compared with the general pipe flow and is always smaller than that of the outer 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.

Second-order two-scale computational method for ageing linear viscoelastic problem in composite materials with periodic structure

The correspondence principle is an important mathematical technique to compute the non-ageing linear viscoelastic problem as it allows to take advantage of the computational methods originally developed for the elastic case. However, the correspon- dence principle becomes invalid when the materials exhibit ageing. To deal with this problem, a second-order two-scale （SOTS） computational method in the time domain is presented to predict the ageing linear viscoelastic performance of composite materials with a periodic structure. First, in the time domain, the SOTS formulation for calcu- lating the effective relaxation modulus and displacement approximate solutions of the ageing viscoelastic problem is formally derived. Error estimates of the displacement ap- proximate solutions for SOTS method are then given. Numerical results obtained by the SOTS method are shown and compared with those by the finite element method in a very fine mesh. Both the analytical and numerical results show that the SOTS computational method is feasible and efficient to predict the ageing linear viscoelastic performance of composite materials with a periodic structure.

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.

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.

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.

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.

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.

Second-order random wave solutions for interfacial internal waves in N-layer density-stratified fluid

This paper studies the random internal wave equations describing the density interface displacements and the velocity potentials of N-layer stratified fluid contained between two rigid walls at the top and bottom. The density interface displacements and the velocity potentials were solved to the second-order by an expansion approach used by Longuet-Higgins （1963） and Dean （1979） in the study of random surface waves and by Song （2004） in the study of second- order random wave solutions for internal waves in a two-layer fluid. The obtained results indicate that the first-order solutions are a linear superposition of many wave components with different amplitudes, wave numbers and frequencies, and that the amplitudes of first-order wave components with the same wave numbers and frequencies between the adjacent density interfaces are modulated by each other. They also show that the second-order solutions consist of two parts： the first one is the first-order solutions, and the second one is the solutions of the second-order asymptotic equations, which describe the second-order nonlinear modification and the second-order wave-wave interactions not only among the wave components on same density interfaces but also among the wave components between the adjacent density interfaces. Both the first-order and second-order solutions depend on the density and depth of each layer. It is also deduced that the results of the present work include those derived by Song （2004） for second-order random wave solutions for internal waves in a two-layer fluid as a particular case.

Second-order random interfacial wave solutions for two-layer fluid with a free surface

A previous study （Song. 2004. Geophys Res Lett, 31 （15）：L15302） of the second-order solutions for random interracial waves is extended in a constant depth, two-layer fluid system with a rigid lid is extended into a more general case of two-layer fluid with a top free surface. The rigid boundary condition on the upper surface is replaced by the kinematical and dynamical boundary conditions of a free surface, and the equations describing the random displacements of free surface, density-interface and the associated velocity potentials in the two-layer fluid are solved to the second order using the same expansion technology as that of Song （2004. Geophys Res Lett, 31 （15）：L15302）. The results show that the interface and the surface will oscillate synchronously, and the wave fields to the first-order both at the free surface and at the density-interface are made up of a linear superposition of many waves with different amplitudes, wave numbers and frequencies. The second-order solutions describe the second-order wave-wave interactions of the surface wave components, the interface wave components and among the surface and the interface wave components. The extended solutions also include special cases obtained by Thorpe for progressive interracial waves （Thorpe. 1968a.Trans R Soc London, 263A：563～614） and standing interracial waves （Thorpe. 1968b. J Fluid Mech, 32：489-528） for the two-layer fluid with a top free surface. Moreover, the solutions reduce to those derived for random surface waves by Sharma and Dean （1979.Ocean Engineering Rep 20） if the density of the upper layer is much smaller than that of the lower layer.

Nonlinear Dynamic Analysis of A Viscoelastic Pipe Conveying Fluid

Incremental harmonic balance method (IHBM) is applied to geometrically nonlinear vibration analysis of a simply supported pipe conveying fluid. the material of which is viscoelastic of the Kelvin- Voigt type. Some unstable phenomena - the appearance of the peak or jumps in the pipe's amplitude frequency curves, which are considered to be of importance to this kind of structure, are presented in the numerical results, and the influence of several parameters of the system on the dynamic characteristic of the pipe are also studied. It is believed that this is the first attempt to search for the periodic solution of the pipe and its intrinsic property with such a method.

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.

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.

Fluid discrimination incorporating viscoelasticity and frequency-dependent amplitude variation with offsets inversion

Frequency-dependent amplitude versus offset(FAVO)inversion is a popular method to estimate the frequency-dependent elastic parameters by using amplitude and frequency information of pre-stack seismic data to guide fluid identification.Current frequency-dependent AVO inversion methods are mainly based on elastic theory without the consideration of the viscoelasticity of oil/gas.A fluid discrimination approach is proposed in this study by incorporating the viscoelasticity and relevant FAVO inversion.Based on viscoelastic and rock physics theories,a frequency-dependent viscoelastic solid-liquid decoupling fluid factor is initially constructed,and its sensitivity in fluid discrimination is compared with other conventional fluid factors.Furthermore,a novel reflectivity equation is derived in terms of the viscoelastic solid-liquid decoupling fluid factor.Due to the introduction of viscoelastic theory,the proposed reflectivity is related to frequency,which is more widely applicable than the traditional elastic reflectivity equation directly derived from the elastic reflectivity equation on frequency.Finally,a pragmatic frequency-dependent inversion method is introduced to verify the feasibility of the equation for frequency-dependent viscoelastic solid-liquid decoupling fluid factor prediction.Synthetic and field data examples demonstrate the feasibility and stability of the proposed approach in fluid discrimination.

Effects of rotation and magnetic field on the nonlinear peristaltic flow of a second-order fluid in an asymmetric channel through a porous medium

In this paper, the effects of both rotation and magnetic field of the peristaltic transport of a second-order fluid through a porous medium in a channel are studied analytically and computed numerically. The material is represented by the constitutive equations for a second-order fluid. Closed-form solutions under the consideration of long wavelength and low Reynolds number is presented. The analytical expressions for the pressure gradient, pressure rise, friction force, stream function, shear stress, and velocity are obtained in the physical domain. The effects of the non-dimensional wave amplitude, porosity, magnetic field, rotation, and the dimensionless time-mean flow in the wave frame are analyzed theoretically and computed numerically. Numerical results are given and illustrated graphically in each case considered. Comparison was made with the results obtained in the presence and absence of rotation, magnetic field, and porosity. The results indicate that the effects of the non-dimensional wave amplitude, porosity, magnetic field, rotation, and the dimensionless time-mean flow are very pronounced in the phenomena.