Analysis of the flow of non-Newtonian viscoelastic fluids in fractal reservoir with the fractional derivative

In this paper, fractional order derivative, fractal dimension and spectral dimension are introduced into the seepage flow mechanics to establish the relaxation models of non-Newtonian viscoelastic fluids with the fractional derivative in fractal reservoirs. A new type integral transform is introduced, and the flow characteristics of non-Newtonian viscoelastic fluids with the fractional order derivative through a fractal reservoir are studied by using the integral transform, the discrete Laplace transform of sequential fractional derivatives and the generalized Mittag-Leffler functions. Exact solutions are obtained for arbitrary fractional order derivative. The long-time and short-time asymptotic solutions for an infinite formation are also obtained. The pressure transient behavior of non-Newtonian viscoelastic fluids flow through an infinite fractal reservoir is studied by using the Stehfest's inversion method of the numerical Laplace transform. It is shown that the clearer the viscoelastic characteristics of the fluid, the more the fluid is sensitive to the order of the fractional derivative. The new type integral transform provides a new analytical tool for studying the seepage mechanics of fluid in fractal porous media.

Effect of bubble morphology and behavior on power consumption in non-Newtonian fluids’aeration process

Due to a prolonged operation time and low mass transfer efficiency, the primary challenge in the aeration process of non-Newtonian fluids is the high energy consumption, which is closely related to the form and rate of impeller, ventilation, rheological properties and bubble morphology in the reactor. In this perspective, through optimal computational fluid dynamics models and experiments, the relationship between power consumption, volumetric mass transfer rate(kLa) and initial bubble size(d0) was constructed to establish an efficient operation mode for the aeration process of non-Newtonian fluids. It was found that reducing the d0could significantly increase the oxygen mass transfer rate, resulting in an obvious decrease in the ventilation volume and impeller speed. When d0was regulated within 2-5 mm,an optimal kLa could be achieved, and 21% of power consumption could be saved, compared to the case of bubbles with a diameter of 10 mm.

A Novel Blood Coagulation Measuring Method Based on the Viscoelasticity of Non-Newtonian Fluid

There are complex and perfect coagulation, anticoagulation and fibrinolysis systems in the human body and their fine regulatory mechanisms. Once the coagulation system and its regulatory mechanisms are destroyed, bleeding or thrombosis will occur very soon. In the blood coagulation, the blood viscoelasticity changes. Therefore, the thrombus elasticity measurement technology can be used to continuously monitor the changing blood viscoelasticity in order to study the process of coagulation. The results of the interaction among the various components of the blood can be obtained from coagulation to fibrinolysis by bedside detection. The traditional electromagnetic induction sensors, based on conventional coil inductance, are manufactured complexly, high cost and non-linear. Therefore, this paper proposes a non-Newtonian fluid viscoelasticity measurement method based on the piezoelectric effect. We use the piezoelectric bimorphs with the diameter of 21 mm and the total thickness of 0.38 mm and DSM coupling probes with the length of 3 mm, 5 mm and 7 mm to design the piezoelectric bimorphs driver. The viscoelasticity of different non-Newtonian fluids is tested. The vibration amplitudes of the piezoelectric bimorphs and liquid surfaces range from 0.43 μm to 3.52 μm. Consequently, the feasibility of in vitro detection of thrombus is confirmed in principle and the above scheme is validated theoretically and experimentally, which provides the basis for the measurement of blood viscoelasticity, the in vitro detection of thrombus and the manufacture of blood coagulation instrument.

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.

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.

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.

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.

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.

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.

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.

ON THE EXISTENCE OF LOCAL CLASSICAL SOLUTION FOR A CLASS OF ONE-DIMENSIONAL COMPRESSIBLE NON-NEWTONIAN FLUIDS

In this paper, the aim is to establish the local existence of classical solutions for a class of compressible non-Newtonian fluids with vacuum in one-dimensional bounded intervals, under the assumption that the data satisfies a natural compatibility condition. For the results, the initial density does not need to be bounded below away from zero.

Molecular dynamics simulation of the flow mechanism of shear-thinning fluids in a microchannel

Shear-thinning fluids have been widely used in microfluidic systems,but their internal flow mechanism is still unclear.Therefore,in this paper,molecular dynamics simulations are used to study the laminar flow of shear-thinning fluid in a microchannel.We validated the feasibility of our simulation method by evaluating the mean square displacement and Reynolds number of the solution layers.The results show that the change rule of the fluid system's velocity profile and interaction energy can reflect the shear-thinning characteristics of the fluids.The velocity profile resembles a top-hat shape,intensifying as the fluid's power law index decreases.The interaction energy between the wall and the fluid decreases gradually with increasing velocity,and a high concentration of non-Newtonian fluid reaches a plateau sooner.Moreover,the velocity profile of the fluid is related to the molecule number density distribution and their values are inversely proportional.By analyzing the radial distribution function,we found that the hydrogen bonds between solute and water molecules weaken with the increase in velocity.This observation offers an explanation for the shear-thinning phenomenon of the non-Newtonian flow from a micro perspective.

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.

GLOBAL EXISTENCE OF STRONG SOLUTIONS OF NAVIER-STOKES EQUATIONS WITH NON-NEWTONIAN POTENTIAL FOR ONE-DIMENSIONAL ISENTROPIC COMPRESSIBLE FLUIDS

The aims of this paper are to discuss global existence and uniqueness of strong solution for a class of isentropic compressible navier-Stokes equations with non-Newtonian in one-dimensional bounded intervals. We prove two global existence results on strong solutions of isentropic compressible Navier-Stokes equations. The first result shows only the existence. And the second one shows the existence and uniqueness result based on the first result, but the uniqueness requires some compatibility condition.

Pressure transient characteristics of non-uniform conductivity fractured wells in viscoelasticity polymer flooding based on oil-water two-phase flow

Polymer flooding in fractured wells has been extensively applied in oilfields to enhance oil recovery.In contrast to water,polymer solution exhibits non-Newtonian and nonlinear behavior such as effects of shear thinning and shear thickening,polymer convection,diffusion,adsorption retention,inaccessible pore volume and reduced effective permeability.Meanwhile,the flux density and fracture conductivity along the hydraulic fracture are generally non-uniform due to the effects of pressure distribution,formation damage,and proppant breakage.In this paper,we present an oil-water two-phase flow model that captures these complex non-Newtonian and nonlinear behavior,and non-uniform fracture characteristics in fractured polymer flooding.The hydraulic fracture is firstly divided into two parts:high-conductivity fracture near the wellbore and low-conductivity fracture in the far-wellbore section.A hybrid grid system,including perpendicular bisection(PEBI)and Cartesian grid,is applied to discrete the partial differential flow equations,and the local grid refinement method is applied in the near-wellbore region to accurately calculate the pressure distribution and shear rate of polymer solution.The combination of polymer behavior characterizations and numerical flow simulations are applied,resulting in the calculation for the distribution of water saturation,polymer concentration and reservoir pressure.Compared with the polymer flooding well with uniform fracture conductivity,this non-uniform fracture conductivity model exhibits the larger pressure difference,and the shorter bilinear flow period due to the decrease of fracture flow ability in the far-wellbore section.The field case of the fall-off test demonstrates that the proposed method characterizes fracture characteristics more accurately,and yields fracture half-lengths that better match engineering reality,enabling a quantitative segmented characterization of the near-wellbore section with high fracture conductivity and the far-wellbore section with low fracture conductivity.The novelty of this paper is the analysis of pressure performances caused by the fracture dynamics and polymer rheology,as well as an analysis method that derives formation and fracture parameters based on the pressure and its derivative curves.

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.

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.

Slip flow and variable properties of viscoelastic fluid past a stretching surface embedded in a porous medium with heat generation

This study examines theoretically and computationally the non-Newtonian boundary layer flow and heat transfer for a viscoelastic fluid over a stretching continuous sheet embedded in a porous medium with variable fluid properties, slip velocity, and internal heat generation/absorption. The flow in boundary layer is considered to be generated solely by the stretching of the sheet adjacent to porous medium with boundary wall slip condition. Highly nonlinear momentum and thermal boundary layer equations governing the flow and heat transfer are reduced to set of nonlinear ordinary differential equations by appropriate transformation. The resulting ODEs are successfully solved numerically with the help of shooting method. Graphical results are shown for non-dimensional velocities and temperature. The effects of heat generation/absorption parameter, the porous parameter, the viscoelastic parameter, velocity slip parameter, variable thermal conductivity and the Prandtl number on the flow and temperature profiles are presented. Moreover, the local skin-friction coefficient and Nusselt number are presented. Comparison of numerical results is made with the earlier published results under limiting cases.

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.