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.

Flow field simulation and establishment for mathematical models of flow area of spool valve with sloping U-shape notch machined by different methods

Precise function expression of the flow area for the sloping U-shape notch orifice versus the spool stroke was derived. The computational fluid dynamics was used to analyze the flow features of the sloping U-shape notch on the spool, such as mass flow rates, flow coefficients, effiux angles and steady state flow forces under different operating conditions. At last, the reliability of the mathematical model of the flow area for the sloping U-shape notch orifice on the spool was demonstrated by the comparison between the orifice area curve derived and the corresponding experimental data provided by the test. It is presented that the bottom arc of sloping U-shape notch （ABU） should not be omitted when it is required to accurately calculate the orifice area of ABU. Although the theoretical flow area of plain bottom sloping U-shape notch （PBU） is larger than that of ABU at the same opening, the simulated mass flow and experimental flow area of ABU are both larger than these of PBU at the same opening, while the simulated flow force of PBU is larger than that of ABU at the same opening. Therefore, it should be prior to adapt the ABU when designing the spool with proportional character.

Development of Mathematical Model of Slag Flow Field and Temperature Field in ESR System

The heat transfer phenomena and driving forces of slag bath flow in ESR process were analyzed, and the mathematical models in ESR system were reviewed and evaluated. The electromagnetic force is the main driving force for the flow formation in the ESR slag pool, and the temperature difference in the pool creates a convective flow in the system. The shape of the electrode tip has an effect on electromagnetic field distribution in slag pool, thus affects the flow pattern. Finally an improved mathematical model of slag pool flow was proposed.

Mathematical Modeling,Field Calibration and Numerical Simulation of Low-Speed Mixed Traffic Flow in Cities

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Modeling Tracer Flow Characteristics in Different Types of Pores: Visualization and Mathematical Modeling

Structure of porous media and fluid distribution in rocks can significantly affect the transport characteristics during the process of microscale tracer flow.To clarify the effect of micro heterogeneity on aqueous tracer transport,this paper demonstrates microscopic experiments at pore level and proposes an improved mathematical model for tracer transport.The visualization results show a faster tracer movement into movable water than it into bound water,and quicker occupancy in flowing pores than in storage pores caused by the difference of tracer velocity.Moreover,the proposed mathematical model includes the effects of bound water and flowing porosity by applying interstitial flow velocity expression.The new model also distinguishes flowing and storage pores,accounting for different tracer transport mechanisms(dispersion,diffusion and adsorption)in different types of pores.The resulting analytical solution better matches with tracer production data than the standard model.The residual sum of squares(RSS)from the new model is 0.0005,which is 100 times smaller than the RSS from the standard model.The sensitivity analysis indicates that the dispersion coefficient and flowing porosity shows a negative correlation with the tracer breakthrough time and the increasing slope,whereas the superficial velocity and bound water saturation show a positive correlation.

Application of the Sediment Mathematical Modelling on Planned Project of Lingdingyang Bay

Based on the combined hydraulic calculation for the eastern network region at the Pearl River estuary and several outlets to the Lingdingyang Bay, the sediment calculation modelling was introduced in the establishment of the sediment mathematical model for Lingdingyang Bay and the eastern region with one and two dimensional flow calculation. Model adjustment and verification were performed in conjunction with field data. The simulated results coincide well with measured data.In addition the model is applied to predict the shore-line planning scheme of Lingdingyang Bay.The theoretical criterion is provided for the shore line plan in the model.And a new mathematical simulated method is put out to research the planning engineering concerned with one-dimensional net rivers and two-dimensional estuary.

Mathematical model of flow characteristic in multi-strand continuous casting tundishes

In this paper, the subject of mathematical model is a series of math expressions, which is used to calculate different regions＇ volume fraction and analyze flow characterization in multi-strand tundish. But research about mathematical model for multi-strand tundish is few, and so far, there has been no acknowledged math model for multi- strand tundish to describe its flow characteristic. If Sahai＇s model, which is originally proposed for the case of single-strand tundish （proposed in reference, and this model is widely used in the world）, is applied to describe flow feature in multi-strand tundish, the calculation results would be unreasonable. Based on the data of watermodel experiment results, the sum of each strand＇s dead region＇s volume fraction is bigger than 100%, and this obviously doesn＇T agree with reality; and the value of dead region＇s volume fraction is calculated to be minus according to mathematical simulation results data in another case. What＇s more, Sahai＇s model does not propose the standard of plotting the RTD-curve, and this makes scholars around the world can＇t achieve consensus of views about plotting RTD-curve. And the model doesn＇t consider the bypass flow and can＇t calculate its volume fraction, but bypass flow is critical to tundish metallurgy. And through Sahai＇s model, the calculation result of plug flow region＇s volume fraction is also not reasonable, because the model doesn＇t well describe the essence of plug flow. So these suggest that it is not reliable to apply Sahai＇s single-strand tundish model to multi-strand tundish case. Then a new model is attempted to propose in this paper for your discussion. In the new model, the standard of plotting RTD curve is definitely proposed, and relative calculation method is also proposed; and the feature of dead region is carefully studied and the model proposes a new method to calculate its volume fraction, and the calculation formula about its volume fraction can be adjustable according the actual demand; what＇s more, the new model considers the bypass flow and proposes a method to calculate its volume fraction for the first time, and then volume fraction of plug flow region, backmix flow region, dead region and bypass flow can be calculated and obtained at the same time; and this new model can better capture the deviation of reality flow pattern from ideal plug flow pattern, and reflects the feature of plug flow.

Mathematical Model and Theoretical Research of Flow Shear Constitutive Relation during Rheo-rolling of Semisolid Alloy

The mathematical model of flow shear constitutive relation during rheo-rolling process has been established. The distribution of velocity and shear stress in rolling cavity was investigated, and the effects of process parameters on shear stress of Sn-15 Pb alloy during rheo-rolling process were studied. In rolling cavity, the nearer the roll is, the bigger the velocity and shear stress are. The shear stress increases with the increment of the roll speed and the roll radius during rheo-rolling process, but deceases with the increment of the thickness of the strip. When the solid fraction of Sn-15 Pb alloy increases from 0.3 to 0.5, the shear stress increases slowly, but when the solid fraction increases from 0.5 to 0.6, the shear stress rapidly.

Simulation Analysis of New Energy Vehicle Engine Cooling System Based on K-E Turbulent Flow Mathematical Model

New energy vehicles have better clean and environmental protection characteristics than traditional fuel vehicles.The new energy engine cooling technology is critical in the design of new energy vehicles.This paper used oneand three-way joint simulation methods to simulate the refrigeration system of new energy vehicles.Firstly,a k-εturbulent flow model for the cooling pump flow field is established based on the principle of computational fluid dynamics.Then,the CFD commercial fluid analysis software FLUENT is used to simulate the flow field of the cooling pump under different inlet flow conditions.This paper proposes an optimization scheme for new energy vehicle engines’“boiling”phenomenon under high temperatures and long-time climbing conditions.The simulation results show that changing the radiator’s structure and adjusting the thermostat’s parameters can solve the problem of a“boiling pot.”The optimized new energy vehicle engine can maintain a better operating temperature range.The algorithm model can reference each cryogenic system component hardware selection and control strategy in the new energy vehicle’s engine.

Experimental Research and Mathematical Model of Drag Torque in Single-plate Wet Clutch

Reduction of drag torque is one of important potentials to improve transmission efficiency.Existing mathematical model of drag torque was not accurate to predict the decrease after oil film shrinking because of the difficulty in modeling the flow pattern between two plates.Flow pattern was considered as laminar flow and full oil film in the gap between two plates in traditional model.Subsequent equivalent circumferential degree model presented an improvement in oil film shrinking due to centrifugal force,but was also based on full oil film in the gap,which resulted difference between model prediction and experimental data.The objective of this paper is to develop an accurate mathematical model for the above problem by using experimental verification.An experimental apparatus was set up to test drag torque of disengaged wet clutch consisting of single friction and separate plate.A high speed camera was used to record the flow pattern through transparent quartz disk plate.The visualization of flow pattern in the clearance was investigated to evaluate the characteristics of oil film shrinking.Visual test results reveal that the oil film begins to shrink from outer radius to inner radius at the stationary plate and only flows along the rotating plate after shrinking.Meanwhile,drag torque decreases sharply due to little contact area between the stationary plate and the oil.A three-dimensional Navier-Stokes （N-S） equation based on laminar flow is presented to model the drag torque.Pressure distributions in radial and circumferential directions as well as speed distributions are deduced.The model analysis reveals that the acceleration of flow in radial direction caused by centrifugal force is the key reason for the shrinking at the constant feeding flow rate.An approach to descript flow pattern was presented on the basis of visual observation.The drag torque predicted by the model agrees well with test data for non-grooved wet clutch.The proposed model enhances the precision for predicting drag torque,and lays down a framework on which some subsequent models are developed.

Physical and Mathematical Modeling of the Argon-Oxygen Decarburization Refining Process of Stainless Steel

The available studies in the literature on physical and mathematical modeling of the argon oxygen decarburization (AOD) process of stainless steel have briefly been reviewed. The latest advances made by the author with his research group have been summarized. Water modeling was used to investigate the fluid flow and mixing characteristics in the bath of an 18 t AOD vessel, as well as the 'back attack' action of gas jets and its effects on the erosion and wear of the refractory lining, with sufficiently full kinematic similarity. The non rotating and rotating gas jets blown through two annular tuyeres, respectively of straight tube and spiral flat tube type, were employed in the experiments. The geometric similarity ratio between the model and its prototype (including the straight tube type tuyeres) was 1:3. The influences of the gas flow rate, the angle included between the two tuyeres and other operating parameters, and the suitability of the spiral tuyere as a practical application, were examined. These latest studies have clearly and successfully brought to light the fluid flow and mixing characteristics in the bath and the overall features of the back attack phenomena of gas jets during the blowing, and have offered a better understanding of the refining process. Besides, mathematical modeling for the refining process of stainless steel was carried out and a new mathematical model of the process was proposed and developed. The model performs the rate calculations of the refining and the mass and heat balances of the system. Also, the effects of the operating factors, including adding the slag materials, crop ends, and scrap, and alloy agents; the non isothermal conditions; the changes in the amounts of metal and slag during the refining; and other factors were all considered. The model was used to deal with and analyze the austenitic stainless steel making (including ultra low carbon steel) and was tested on data of 32 heats obtained in producing 304 grade steel in an 18 t AOD vessel. The changes in the bath composition and temperature during the refining process with time can be accurately predicted using this model. The model can provide some very useful information and a reliable basis for optimizing the process practice of the refining of stainless steel and control of the process in real time and online.

MATHEMATICAL MODELLING OF BUBBLE DRIVENFLOWS IN A NOVEL ALUMINUM ELECTROLYSIS CELL

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THE STUDY ON THREE-DIMENSIONAL MATHEMATICAL MODEL OF RIVER BED EROSION FOR WATER-SEDIMENT TWO-PHASE FLOW

Based on the tensor analysis of water-sediment two-phase how, the basic model equations for clear water flow and sediment-laden flow are deduced in the general curve coordinates for natural water variable-density turbulent how. Furthermore, corresponding boundary conditions are also presented in connection with the composition and movement of non-uniform bed material. The theoretical results are applied to the calculation of the float open caisson in the construction period and good results are obtained.

A mathematical model capable of describing the liquid flow mainly in a blast furnace

The molten liquid flow inside a packed bed is a familiar momentum transportation phenomenon in a blast furnace. With regard to the reported mathematical models describing the liquid flow within a packed bed, there are some obstacles for their application in engineering design, or some limitations in the model itself. To overcome these problems, the forces from the packed bed to the liquid flow were divided into appropriate body and surface forces on the basis of three assumptions. Consequently, a new mathematical model was built to present the liquid flow inside the coke bed in a blast furnace. The mathematical model can predict the distribution of liquid flowrate and the liquid flowing range inside the packed bed at any time. The predicted results of this model accord well with the experimental data. The model will be applied considerably better in the simulation on the ironmaking process compared with the existent models.

Impact of pertinent parameters on foam behavior in the entrance region of porous media:mathematical modeling

Foam injection is a promising solution for control of mobility in oil and gas field exploration and development,including enhanced oil recovery,matrix-acidization treatments,contaminated-aquifer remediation and gas leakage prevention.This study presents a numerical investigation of foam behavior in a porous medium.Fractional flow method is applied to describe steady-state foam displacement in the entrance region.In this model,foam flow for the cases of excluding and including capillary pressure and for two types of gas,nitrogen(N2)and carbon dioxide(CO2)are investigated.Effects of pertinent parameters are also verified.Results indicate that the foam texture strongly governs foam flow in porous media.Required entrance region may be quite different for foam texture to accede local equilibrium,depending on the case and physical properties that are used.According to the fact that the aim of foaming of injected gas is to reduce gas mobility,results show that CO2 is a more proper injecting gas than N2.There are also some ideas presented here on improvement in foam displacement process.This study will provide an insight into future laboratory research and development of full-field foam flow in a porous medium.

Exact Solution of Unsteady Flow of Viscoelastic Fluid in a Pipe with Fractional Maxwell Model

The unsteady flow of viscoelastic fluid in a cylindrical pipe was investigated using the fractional Maxwell model. Two special cases of unsteady pipe flow were expressed. The first is start-up flow, and the second is oscillating flow. The exact solution of start-up flow under a constant pressure gradient was obtained by using the theories of Laplace transform and Fourier-Bessel series for fractional derivatives. The exact solution of oscillating flow was obtained by utilizing the separation of variables.

MATHEMATICAL MODELLING OF FLOW PHENOMENA IN METAL BATH COVERED WITH MOLTEN SLAG

By use of the two-component LDA and high speed camera,the water model of the flow field in metal bath under molten slag layer has been tested.On the basis of experimental results,the flow boundary conditions of liquid metal at the slag-metal interface of a gas injecting bath were deduced.The flow field and the turbulent parameters of the metal bath covered with slag were solved by the vorticity-stream function method.The results reveal that the flow velocity, turbulent energy and circulating rate of the melt under slag are lower than that of without slagcover.Another one “dead zone” of the lowest turbulent energy is formed in the top layer under cover of slag near the ladle linning.

Comparison on Mathematical Models for Description of Flow Curves of Stable Austenitic Steels

The flow curves were measured for the stable austenitic steels 304L and 304LN by means of tensile test at room temperature,which are described by the models σ=K1εn1 + exp(K2 + n2ε), σ=Kεn1+n2lnε and σ=σ0+Kεn (where, K1, K2, n1 andn2; K, n1 and n2; σ0, K and n are constant). The comparison of the maximum deviations and the consideration of thevariation of the work hardening rate with true strain show that the flow curves for the austenitic steels 304L and 304LN canbe described by the model σ=Kεn1+n2 lnε at higher precision.The derivatives of the models σ=K1εn1 + exp(K2 + n2ε) and σ=Kεn1+n2lnε with respect to true strain, exhibit theextreme at low true strain. This inherent character indicates that both models are unsuitable to describe the part of the workhardening rate curve at low true strain.

Mathematical Models for Flow of Chyme during Gastrointestinal Endoscopy

Intestinal infection has become a common disease in human and endoscopy can be a powerful means in diagnosis of intestinal illnesses. Mathematical models are developed for an inserted endoscope on the flow of chyme in the small intestine considering a Newtonian incompressible fluid flow, under an axisymmetric condition, in a cylindrical annulus between the small intestine and the endoscope. We obtain novel mathematical expressions for the pressure drop, forces exerted by the endoscope on the flow of chyme, and the force exerted by the chyme on the intestine for one wave length of the peristaltic rush wave. We also investigate and calculate the flow velocity and pressure for different flow rates and the wave lengths. The results are presented, and discussed for the cases and conditions under which pressure, pressure drop can be positive or negative and the forces can be acted either by the intestine or endoscope on the flow or vice-versa.

Hall Current Effects on Unsteady MHD Flow in a Rotating Parallel Plate Channel Bound-ed by Porous Bed on the Lower Half—Darcy Lapwood Model

We discussed the unsteady flow of an incompressible viscous fluid in a rotating parallel plate channel bounded on one side by a porous bed under the influence of a uniform transverse magnetic field taking hall current into account. The perturbations are created by a constant pressure gradient along the plates in addition to the non-torsional oscillations of the upper plate. The flow in the clean fluid region is governed by Navier-Stoke’s equations while in the porous bed the equations are based on Darcy-Lapwood model. The exact solutions of velocity in the clean fluid and the porous medium consist of steady state and transient state. The time required for the transient state to decay is evaluated in detail and ultimate quasi-steady state solution has been derived analytically and also its behaviour is computationally discussed with reference to different flow parameters. The shear stresses on the boundaries and the mass flux are also obtained analytically and their behaviour is computationally discussed.