Analytical Solution for Laminar Viscous Flow inthe Gap Between Two Parallel Rotary Disks

Aim To get the analytical for laminar viscous flow in the gap of two parallel rotating disks. Methods By estimating the order of magnitude of each term in the Navier-Stokes equations to drop small terms and achieve the required simplified differential equations, and by integrating the equations to obtain the solution for theflow between two rotary disks. Results Parameters related to the laminar viscous flow in the gap between two parallel rotary disks, such as the velocity, the pressure, the flowrate, the force, the shearing stress, the torque and the power derived. Conclusion The result provides a theoretical basis and an effective method for the designs of the devices connected with the laminar viscous flow in the gap between two parallel rotary disks.

THEORETICAL ANALYSIS ON HYDRAULIC TRANSIENT RESULTED BY SUDDEN INCREASE OF INLET PRESSURE FOR LAMINAR PIPELINE FLOW

Hydraulic transient, which is resulted from sudden increase of inlet pressure for laminar pipeline flow, is studied. The partial differential equation, initial and boundary conditions for transient pressure were constructed, and the theoretical solution was obtained by variable-separation method. The partial differential equation, initial and boundary conditions for flow rate were obtained in accordance with the constraint correlation between flow rate and pressure while the transient flow rate distribution was also solved by variable-separation method. The theoretical solution conforms to numerical solution obtained by method of characteristics (MOC) very well.

Extraction Kinetics of Lanthanum in Chloride Medium by Bifunctional Ionic Liquid [A336][CA-12] Using a Constant Interfacial Cell with Laminar Flow

The extraction kinetics of La(III) from aqueous chloride solutions into n-heptane solutions of bifunctional ionic liquid extractant [A336][CA-12](tricaprylmethylammonium sec-octylphenoxy acetic acid) was investigated using a constant interfacial cell with laminar flow. The effects of stirring speed, temperature and specific interfacial area on the extraction rate were examined. The results indicate that mass transfer kinetics of La(III) is a mixed-controlled process influenced by interfacial reaction. On the basis of mass transfer kinetic results in the extraction of La(III) by [A336][CA-12], the extraction rate equation of La(III) is proposed in terms of pseudo-firstorder constants, which is supported by the measured thermodynamic equations. The mass-transfer kinetic model deduced from the rate controlling step is adequate to interpret the experimental data qualitatively.

Physics-informed deep learning for incompressible laminar flows

Physics-informed deep learning has drawn tremendous interest in recent years to solve computational physics problems,whose basic concept is to embed physical laws to constrain/inform neural networks,with the need of less data for training a reliable model.This can be achieved by incorporating the residual of physics equations into the loss function.Through minimizing the loss function,the network could approximate the solution.In this paper,we propose a mixed-variable scheme of physics-informed neural network(PINN)for fluid dynamics and apply it to simulate steady and transient laminar flows at low Reynolds numbers.A parametric study indicates that the mixed-variable scheme can improve the PINN trainability and the solution accuracy.The predicted velocity and pressure fields by the proposed PINN approach are also compared with the reference numerical solutions.Simulation results demonstrate great potential of the proposed PINN for fluid flow simulation with a high accuracy.

THE LOSS OF STABILITY OF LAMINAR FLOW IN OPEN CHANNEL AND THE MECHANISM OF SAND RIPPLE FORMATION

In the flow on a mobile bed in an open channel, sand ripple often appears after the sediment begins to move. Different scholars have different views on the formation of sand ripples. This paper holds that as the ripple in general is very small, its formation is due to the instability of the laminar flow or the evolution of the small-scale coherent structures in the sublayer adjacent to the wall of the open channel. When the shear stresses caused by the disturbing waves or the coherent structure near the bed surface boundary and the water flow itself are greater than the shields stresses, responses on the bed surface appear and the sand ripple forms. If the frequency of the shear stress caused by the disturbance is close to the natural frequency of the sand grains that produced resonance, such a phenomenon is called the 'detection property' of the sediment. It is at this point that the maximum resonance appears and the sand ripple develops rapidly.

Robust design of natural laminar flow supercritical airfoil by multi-objective evolution method

Abstract A transonic, high Reynolds number natural laminar flow airfoil is designed and studied. The γ-θ transition model is combined with the shear stress transport （SST） k-w turbulence model to predict the transition region for a laminar-turbulent boundary layer. The non-uniform free-form deformation （NFFD） method based on the non-uniform rational B-spline （NURBS） basis function is introduced to the airfoil parameterization. The non-dominated sorting genetic algorithm-II （NSGA-II） is used as the search algo- rithm, and the surrogate model based on the Kriging models is introduced to improve the efficiency of the optimization system. The optimization system is set up based on the above technologies, and the robust design about the uncertainty of the Mach number is carried out for NASA0412 airfoil. The optimized airfoil is analyzed and compared with the original airfoil. The results show that natural laminar flow can be achieved on a supercritical airfoil to improve the aerodynamic characteristic of airfoils.

Existence of multiple solutions for the laminar flow in a porous channel with suction at both slowly expanding and contracting walls

The asymptotic behavior of solutions of a similarity equation for the laminar flow in a porous channel with suction at both expanding and contracting walls has been obtained by using a singular perturbation method.However,in the matching process,this solution neglects exponentially small terms.To take into account these exponentially small terms,a method involving the inclusion of exponentially small terms in a perturbation series was used to find two of the solutions analytically.The series involving the exponentially small terms and expansion ratio predicts dual solutions.Furthermore,the result indicates that the expansion ratio has much important influence on the solutions.

Numerical Simulation of Laminar Flow Field in a Stirred Tank

Stirred tanks are used extensively in process industry and one of the most commonly used impellers in stirred tanks is the R.ushton disk turbine. Surprisingly few data are available regarding flow and mixing in stirred-tank reactors with Rushton turbine in the laminar regime, in particular the laminar flow in baffled tanks.In this paper, the laminar flow field in a baffled tank stirred by a standard R.ushton turbine is simulated with the improved inner-outer iterative method. The non-inertial coordinate system is used for the impeller region, which is in turn used as the boundary conditions for iteration. It is found that the simulation results are in good agreement with previous experiments. In addition, the flow number and impeller power number calculated from the simulated flow field are in satisfactory agreement with experimental data. This numerical method allows prediction of flow structure requiring no experimental data as the boundary conditions and has the potential of being used to scale-up and design of related process equipment.

Homotopy analysis solutions for the asymmetric laminar flow in a porous channel with expanding or contracting walls

In this paper, the asymmetric laminar flow in a porous channel with expanding or contracting walls is investigated. The governing equations are reduced to ordinary ones by using suitable similar transformations. Homotopy analysis method （HAM） is employed to obtain the expres- sions for velocity fields. Graphs are sketched for values of parameters and associated dynamic characteristics, especially the expansion ratio, are analyzed in detail.

Surface wettability and flow properties of non-metallic pipes in laminar flow

In this paper,three liquids flowing in five pipes with the same inner diameter of 14 mm were studied to determine the relationship between the surface wettability and flow properties in laminar flow(Re b 2000).This was motivated by oilfield observations of increased pressure drops in non-metallic pipes compared to those in metal pipes,which was contrary to expectations.A new expression for the frictional coefficient that considers the Reynolds number and contact angleθin laminar flow for non-metallic pipes was proposed based on the experimental results of single-phase flow using dimension and regression analyses.The solutions of the anomalous phenomenon were proposed from the perspectives of the pipe diameter,contact-angle difference,and the compatibility between flexible composite pipe and JLHW105 oil according to the new formula.The surprising finding was that the surface wettability could control the frictional resistance by the critical contact angle(39.9°)obtained at the same Reynolds number.If 0°bθ≤39.9°,the frictional coefficient increased as the contact angle increased.In contrast,if 39.9°bθb 180°,the frictional coefficient decreased with increasing contact angle.The influences of the pipe diameter and contactangle difference on the pressure drop difference of JLHW105 oil showed an inversely proportional relation.A series of materials and liquids were tested.The selection of pipe material for transporting a given fluid can be based on the contact angle,surface tension,and critical limit of the contact angle obtained.The research results are expected to provide some guidelines for the selection of the appropriate pipe material for a given set of fluids.

STUDIES ON THE LAW OF LAMINAR HELICAL FLOW OF POWER-LAW FLUID IN ECCENTRIC ANNULI

The flow of mud under a constant axial pressure gradient in eccentric annuli was studied extensively in this paper. Assumed that the inner wall was rotating at a constant angular velocity, then the constitutive equation of power law fluid and the equation of motion were given. The expressions of apparent viscosity, velocity, flow rate and pressure loss were obtained. In Some numerical results were given by alternation-substitution method. The conclusions are that the viscosity difference between wide and narrow clearance is reduced with increasing the rotation of inner cylinder and/or decreasing the eccentric distance. The flow rate will be increased with increasing the rotation of inner cylinder and eccentric distance, etc.. The pressure loss in eccentric flow is quite large than in concentric axial flow. The results are useful for calculating hydraulic parameters in drilling engineering.

Development of the laminar flow cooling control system of the No.3 hot strip rolling line at Baosteel

The coiling temperature of the hot strip rolling process plays an important role in strip quality control. The newly-built laminar flow cooling control system was independently developed by Baosteel. L2 conducts totally dynamic control （L1 only executes the result from L2, and does not conduct any dynamic control ）. The system has many advantages, such as quick real-time control, accurate model calculation, quick response after self-adaptation, etc. Compared with the 2050 mm hot strip rolling line,the new system can satisfy the strict cooling requirements of high strength steel, silicon steel and heavy plates because of its newly-developed technologies of cooling rate control, strip cooling by sections and proportion-integral-differential coefficient （PID） feedback fuzzy control.

EXTRACTION KINETICS OF RARE EARTH（Ⅰ） A Constant Interfacial Cell with Laminar Flow

A technique was developed to study mass trans fer process and inter facial reaction in two phases system. This constant inter facial cell with a laminar flow was made into a cubic structure. The two fluids were continuously recycled and mixed. The concentration of each liquid could be monitored by two different methods. This kind of structure made both flows near the inter face flow parallel to the inter face. The inter face was smooth and steady. The mass trans fer rate could be judged by the linear velocity of the flows. The technique can be used for the analyses of the control step in both phases near the inter face in a diffusion control process. A preliminary hydrodynamics and mass trans fer study on the cell was presented, which ensures the distinguishing between a diffusion and a chemical reaction control process. A simplified mass transfer equation,N =0.5303D 1 /2* (Ci- Cb)* (V / B) 1/2, was achieved.

A GPU-Accelerated Discontinuous Galerkin Method for Solving Two-Dimensional Laminar Flows

A graphics processing unit(GPU)-accelerated discontinuous Galerkin(DG)method is presented for solving two-dimensional laminar flows.The DG method is ported from central processing unit to GPU in a way of achieving GPU speedup through programming under the compute unified device architecture(CUDA)model.The CUDA kernel subroutines are designed to meet with the requirement of high order computing of DG method.The corresponding data structures are constructed in component-wised manners and the thread hierarchy is manipulated in cell-wised or edge-wised manners associated with related integrals involved in solving laminar Navier-Stokes equations,in which the inviscid and viscous flux terms are computed by the local lax-Friedrichs scheme and the second scheme of Bassi&Rebay,respectively.A strong stability preserving Runge-Kutta scheme is then used for time marching of numerical solutions.The resulting GPU-accelerated DG method is first validated by the traditional Couette flow problems with different mesh sizes associated with different orders of approximation,which shows that the orders of convergence,as expected,can be achieved.The numerical simulations of the typical flows over a circular cylinder or a NACA 0012 airfoil are then carried out,and the results are further compared with the analytical solutions or available experimental and numerical values reported in the literature,as well as with a performance analysis of the developed code in terms of GPU speedups.This shows that the costs of computing time of the presented test cases are significantly reduced without losing accuracy,while impressive speedups up to 69.7 times are achieved by the present method in comparison to its CPU counterpart.

Microchannel heat sink;Nanofluids;Entropy generation;Numerical analysis;Laminar flow

Based on the Second Law of Thermodynamics, the entropy generation is studied for laminar forced convection flow of different nanoparticles(Al_2 O_3, CuO and SiO_2) mixed with water through a hexagon microchannel heat sink(HMCHS). The effects of different heat fluxes and Reynolds numbers on the entropy generation for different nanofluids, volume fractions and nanoparticles diameter are investigated. The heat flux is in the range of 125 to 500 kW·m^(-2) and the Reynolds numbers vary between 200 and 1500. The thermal, frictional and total entropy generations are calculated by integrating the volumetric rate components over the entire HMCHS. The results clearly show that the rise in the heat flux leads to an increase in the thermal entropy generation for nanofluids and pure water but they don't have any influence on the frictional entropy generation. Moreover, when the Reynolds number increases, the frictional entropy generation increases while the thermal entropy generation decreases. The results revealed that at low heat fluxes and high Reynolds numbers, pure water gives the lowest entropy generation, while at high heat flux the nanofluid has to be used in order to lower the overall irreversibility.

Numerical Simulation of Blood Flow in Aorta with Dilation:A Comparison between Laminar and LES Modeling Methods

Computational modeling methods have been increasingly employed to quantify aortic hemodynamic parameters that are challenging to in vivo measurements but important for the diagnosis/treatment of aortic disease.Although the presence of turbulence-like behaviors of blood flow in normal or diseased aorta has long been confirmed,the majority of existing computational model studies adopted the laminar flow assumption(LFA)in the treatment of sub-grid flow variables.So far,it remains unclear whether LFA would significantly compromise the reliability of hemodynamic simulation.In the present study,we addressed the issue in the context of a specific aortopathy,namely aortic dilation,which is usually accompanied by disturbed flow patterns.Three patient-specific aortas with treated/untreated dilation of the ascending segment were investigated,and their geometrical models were reconstructed from computed tomography angiographic images,with the boundary conditions being prescribed based on flow velocity information measured in vivo with the phase contrast magnetic resonance imaging technique.For the modeling of blood flow,apart from the traditional LFA-based method in which sub-grid flow dynamics is ignored,the large eddy simulation(LES)method capable of incorporating the dissipative energy loss induced by turbulent eddies at the sub-grid level,was adopted and taken as a reference for examining the performance of the LFA-based method.Obtained results showed that the simulated large-scale flow patterns with the two methods had high similarity,both agreeing well with in vivo measurements,although locally large between-method discrepancies in computed hemodynamic quantities existed in regions with high intensity of flow turbulence.Quantitatively,a switch from the LES to the LFAbased modeling method led to mild(<6%)changes in computed space-averaged wall shear stress metrics(i.e.,SA-TAWSS,SA-OSI)in the ascending aortic segment where intensive vortex evolution accompanied by high statistical Reynolds stress was observed.In addition,comparisons among the three aortas revealed that the treatment status of aortic dilation or the concomitant presence of aortic valve disease,despite its remarkable influence on flow patterns in the ascending aortic segment,did not significantly affect the degrees of discrepancies between the two modeling methods in predicting SA-TAWSS and SA-OSI.These findings suggest that aortic dilation per se does not induce strong flow turbulence that substantially negates the validity of LFA-based modeling,especially in simulating macro-scale hemodynamic features.

A Laminar Flow Model for Mucous Gel Transport in a Cough Machine Simulating Trachea: Effect of Surfactant as a Sol Phase Layer

In this paper, a planar three layer quasisteady laminar flow model is proposed in a cough machine which simulates mucous gel transport in model trachea due to mild forced expiration. The flow is governed by the time dependent pressure gradient generated in trachea due to mild forced expiration. Mucous gel is represented by a viscoelastic Voigt element whereas sol phase fluid and air are considered as Newtonian fluids. For fixed airflow rate, it is shown that when the viscosity of mucous gel is small, mucous gel transport decreases as the elastic modulus increases. However, elastic modulus has negligible effect on large gel viscosity. It is also shown that for fixed airflow rate and fixed airway dimension, mucous gel transport increases with the thickness of sol phase fluid and this increase is further enhanced as the viscosity of sol phase fluid decreases. The effect of surfactant is studied by considering sol phase as surfactant layer which causes slip at the wall and interface of sol phase and mucous gel. It is found that in the presence of surfactant mucous gel transport is enhanced.

A Review of the Attachment Line Instability for Hybrid Laminar Flow Control

This paper focuses on the investigation of the attachment line instability for Hybrid Laminar Flow Control(HLFC),one of the most promising drag reduction technologies for modern transport aircraft respect to high Reynolds numbers and large sweep angles.The attachment line instability also plays an important role during laminar-turbulent transition control and HLFC design on a swept wing.The overview of historical research is presented and knowledge gaps are pointed out as the conclusion.

ANALYTIC SOLUTIONS OF INCOMPRESSIBLE LAMINAR FLOW IN THE INLET REGION OF A PIPE

The laminar analytic solutions of velocities and pressure in the central zone of the inlet region of pipe flow are given under the condition of uniform inflow, based on the Navier-Stokes equations of incompressible viscous flow.

THEORETICAL ANALYSIS ON HYDRAULIC TRANSIENT RESULTED BY SUDDEN INCREASE OF INLET PRESSURE FOR LAMINAR PIPELINE FLOW

Hydraulic transient, which is resulted from sudden increase of inlet pressure for laminar pipeline flow, is studied. The partial differential equation, initial and boundary conditions for transient pressure were constructed, and the theoretical solution was obtained by variable-separation method. The partial differential equation, initial and boundary conditions for flow rate were obtained in accordance with the constraint correlation between flow rate and pressure while the transient flow rate distribution was also solved by variable-separation method. The theoretical solution conforms to numerical solution obtained by method of characteristics (MOC) very well.