Advancements in Numerical Solutions:Fractal Runge-Kutta Approach to Model Time-Dependent MHD Newtonian Fluid with Rescaled Viscosity on Riga Plate

Fractal time-dependent issues in fluid dynamics provide a distinct difficulty in numerical analysis due to their complex characteristics,necessitating specialized computing techniques for precise and economical solutions.This study presents an innovative computational approach to tackle these difficulties.The main focus is applying the Fractal Runge-Kutta Method to model the time-dependent magnetohydrodynamic(MHD)Newtonian fluid with rescaled viscosity flow on Riga plates.An efficient computational scheme is proposed for handling fractal time-dependent problems in flow phenomena.The scheme is comprised of three stages and constructed using three different time levels.The stability of the scheme is shown by employing the Fourier series analysis to solve scalar problems.The scheme’s convergence is guaranteed for a time fractal partial differential equations system.The scheme is applied to the dimensionless fractal heat and mass transfer model of incompressible,unsteady,laminar,Newtonian fluid with rescaled viscosity flow over the flat and oscillatory Riga plates under the effects of space-and temperature-dependent heat sources.The first-order back differences discretize the continuity equation.The results show that skin friction local Nusselt number declines by raising the coefficient of the temperature-dependent term of heat source and Eckert number.The numerical simulations provide valuable insights into fluid dynamics,explicitly highlighting the influence of the temperature-dependent coefficient of the heat source and the Eckert number on skin friction and local Nusselt number.

The Magnetic Field as Trigger for Disturbances of a Flow of a Newtonian Fluid through a Cylindrical Pipe with a Horizontal Axis

A Fourrier Petrov Galerkin spectral method is described for high accuracy computation of linearized dynamics for flow in a circular pipe. The code used here is based on solenoidal velocity variables. It is written in FORTRAN. Systematic studies are presented on the dependence of eigenvalues and other quantities on the axial and azimuthal wave number;the reynolds’ number Re and a new none-dimensional number Ne. The flow will be considered stable if all the real parts of the eigenvalues are negative and unstable if only one of them is positive.

MHD three-dimensional flow of Jeffrey fluid with Newtonian heating

The magnetohydrodynamic(MHD) three-dimensional flow of Jeffrey fluid in the presence of Newtonian heating is investigated. Flow is caused by a bidirectional stretching surface. Series solutions are constructed for the velocity and temperature fields. Convergence of series solutions is ensured graphically and numerically. The variations of key parameters on the physical quantities are shown and discussed in detail. Constructed series solutions are compared with the existing solutions in the limiting case and an excellent agreement is noticed. Nusselt numbers are computed with and without magnetic fields. It is observed that the Nusselt number decreases in the presence of magnetic field.

Modeling the Unsteady Flow of a Newtonian Fluid Originating from the Hole of an Open Cylindrical Reservoir

This work deals with the modeling of the unsteady Newtonian fluid flow associated with an open cylindrical reservoir.This reservoir presents a hole on the right bottom wall.Fluid volume variation,heat and mass transfers are neglected.The unsteady governing equations are based on the conservation of mass and momentum.A finite volume technique is used to solve the non-dimensional equations and related boundary conditions.The algebraic system of equations resulting from the discretization process are solved by means of the THOMAS algorithm.For pressure-velocity coupling,the SIMPLE algorithm(Semi Implicit Method for Pressure Linked Equations)is used.Results for laminar flow(Re<1000),including the pressure and velocities profiles as well as the streamlines in the reservoir are presented.Moreover,the effects of the D/d and H0/D ratios and Reynolds number Re on the fluid flow are discussed.It is shown that the velocities and pressure depend essentially on the reservoir size.To validate the model,the present results have been compared with Zhou et al.’s results,Poiseuille’s and Bernoulli’s exact solution.

BASIC EQUATIONS OF TURBULENT FLOW FOR VARIABLE DENSITY AND VARIABLE VISCOSITY NEWTONIAN FLUID IN OPEN CHANNEL

In this paper, using Navier-Stokes equations and Reynolds time-averaged rules, the turbulent motional differential equations of variable density and variable viscosity Newtonian fluid have been presented, and the turbulent motional differential equations of variable density and variable viscosity Newtonian fluid in open channel have been further proposed. The concepts of the density turbulence stress and the viscosity turbulence stress have been firstly presented in the paper.

THE NUMERICAL SIMULATION OF TURBULENT FLOWS OF NEWTONIAN AND A SORT OF NON-NEWTONIAN FLUID IN 180-DEGREE SQUARE SECTIONED BEND

Using k- model of turbulence and measured wall functions, turbulent flows of Newtonian (pure water) andasort of non-Newtonian fluid (dilute, drag-reduction solution of polymer) in a 180-degree curved bend were simulated numerically. The calculated results agreed well with the measured velocity profiles. On the basis of calculation and measurement, appropriateness of turbulence model to complicated flow in which the large-scale vortex exists was analyzed and discussed.

TIME PERIODIC SOLUTIONS TO THE EVOLUTIONARY OSEEN MODEL FOR A GENERALIZED NEWTONIAN INCOMPRESSIBLE FLUID

In this paper we study a nonstationary Oseen model for a generalized Newtonian incompressible fluid with a time periodic condition and a multivalued,nonmonotone friction law.First,a variational formulation of the model is obtained;that is a nonlinear boundary hemivariational inequality of parabolic type for the velocity field.Then,an abstract first-order evolutionary hemivariational inequality in the framework of an evolution triple of spaces is investigated.Under mild assumptions,the nonemptiness and weak compactness of the set of periodic solutions to the abstract inequality are proven.Furthermore,a uniqueness theorem for the abstract inequality is established by using a monotonicity argument.Finally,we employ the theoretical results to examine the nonstationary Oseen model.

Impact of Radiation and Slip on Newtonian Liquid Flow Past a Porous Stretching/Shrinking Sheet in the Presence of Carbon Nanotubes

The impacts of radiation,mass transpiration,and volume fraction of carbon nanotubes on the flow of a Newtonian fluid past a porous stretching/shrinking sheet are investigated.For this purpose,three types of base liquids are considered,namely,water,ethylene glycol and engine oil.Moreover,single and multi-wall carbon nanotubes are examined in the analysis.The overall physical problem is modeled using a system of highly nonlinear partial differential equations,which are then converted into highly nonlinear third order ordinary differential equations via a suitable similarity transformation.These equations are solved analytically along with the corresponding boundary conditions.It is found that the carbon nanotubes can significantly improve the heat transfer process.Their potential application in cutting-edge areas is also discussed to a certain extent.

Dynamic Analysis of Single Pile in Liquefied Soils Considered as Newtonian Fluid

Case histories have shown that the liquefaction-induced soil lateral spreading is one of the main causes of damage to pile foundations subjected to seismic loading. Post-liquefaction soil behaves similarly to a viscous fluid. This study investigated the effect of soil lateral spreading on a single pile based on fluid mechanics in which the liquefied soils were treated as Newtonian fluids. A numerical simulation on a single pile embedded in a fully saturated sandy foundation was conducted and compared with shake table tests. The lateral flow effect and the effect of shear strain rate were discussed. After liquefaction, the acceleration of the foundation shows that there are no obvious spikes and finally reaches a stable state. The presented method can predict the pile response better than p-y curve method. A parametric study was performed to explore the effect of several influence factors on pile behaviors. The results show that the pile head displacement decreases and the maximum bending moment at pile bottom increases with the increase of bending stiffness. With the same pile bending stiffness, the displacement and bending moment of pile increase with the increase of soil viscosity and acceleration amplitude.

A hybrid finite volume/finite element method for incompressible generalized Newtonian fluid flows on unstructured triangular meshes

This paper presents a hybrid finite volume/finite element method for the incompressible generalized Newtonian fluid flow （Power-Law model）. The collocated （i.e. non-staggered） arrangement of variables is used on the unstructured triangular grids, and a fractional step projection method is applied for the velocity-pressure coupling. The cell-centered finite volume method is employed to discretize the momentum equation and the vertex-based finite element for the pressure Poisson equation. The momentum interpolation method is used to suppress unphysical pressure wiggles. Numerical experiments demonstrate that the current hybrid scheme has second order accuracy in both space and time. Results on flows in the lid-driven cavity and between parallel walls for Newtonian and Power-Law models are also in good agreement with the published solutions.

Numerical Simulations of Polymer Melt Conveying in Co-Rotating Twin Screw Extruder

The 3 D non isothermal flow of non Newtonian viscous polymer melt in a co rotating twin screw extruder is modeled. The distributions of the velocity, temperature, pressure and the viscous dissipation in the flow domain are presented by using a fluid dynamics analysis package (Polyflow). The numerical results show that the temperatures are high in the intermeshing region and on the screw surface, the maximum pressure and the minimum pressure occur in the intermeshing region, and the flow rate is almost proportional to the screw speed.

Viscous falling film instability around a vertical moving cylinder

The present work discusses both the linear and nonlinear stability conditions of a viscous falling film down the outer surface of a solid vertical cylinder which moves in the direction of its axis with a constant velocity. After studying the linear conditions, a generalized nonlinear kine- matic model is then derived to present the physical system. Applying the boundary conditions, analytical solutions are obtained using the long-wave perturbation method. In the first step, the normal mode method is used to characterize the linear behaviors. In the second step, the nonlinear film flow model is solved by using the method of multiple scales, to obtain Ginzburg-Landau equation. The influence of some physical parameters is discussed in both linear and nonlinear steps of the problem, and the results are displayed in many plots showing the stability criteria in various param- eter planes.

Existence of Periodic Solution for a Dirichlet Problem

Aim To discuss the existence of periodic solutions for the first boundary problem of incompressible non Newtonian fluids, a problem arising from polymer processing and concerned with the first initial boundary value problem of nonstationary flow of the non Newtonian viscous incompressible fluids through slit dice. Methods The monotone operator theory and Schauders fixed point theorem were used. Results and Conclusion The existence theorem of periodic solutions of a Dirichlet problem is proved under reasonable conditions.

Differential transformation method for studying flow and heat transfer due to stretching sheet embedded in porous medium with variable thickness, variable thermal conductivity,and thermal radiation

This article presents a numerical solution for the flow of a Newtonian fluid over an impermeable stretching sheet embedded in a porous medium with the power law surface velocity and variable thickness in the presence of thermal radiation. The flow is caused by non-linear stretching of a sheet. Thermal conductivity of the fluid is assumed to vary linearly with temperature. The governing partial differential equations （PDEs） are transformed into a system of coupled non-linear ordinary differential equations （ODEs） with appropriate boundary conditions for various physical parameters. The remaining system of ODEs is solved numerically using a differential transformation method （DTM）. The effects of the porous parameter, the wall thickness parameter, the radiation parameter, the thermal conductivity parameter, and the Prandtl number on the flow and temperature profiles are presented. Moreover, the local skin-friction and the Nusselt numbers are presented. Comparison of the obtained numerical results is made with previously published results in some special cases, with good agreement. The results obtained in this paper confirm the idea that DTM is a powerful mathematical tool and can be applied to a large class of linear and non-linear problems in different fields of science and engineering.

Melting phenomenon in magneto hydro-dynamics steady flow and heat transfer over a moving surface in the presence of thermal radiation

The Lie group method is applied to present an analysis of the magneto hydro-dynamics（MHD） steady laminar flow and the heat transfer from a warm laminar liquid flow to a melting moving surface in the presence of thermal radiation.By using the Lie group method,we have presented the transformation groups for the problem apart from the scaling group.The application of this method reduces the partial differential equations（PDEs） with their boundary conditions governing the flow and heat transfer to a system of nonlinear ordinary differential equations（ODEs） with appropriate boundary conditions.The resulting nonlinear system of ODEs is solved numerically using the implicit finite difference method（FDM）.The local skin-friction coefficients and the local Nusselt numbers for different physical parameters are presented in a table.

Calculation and Analysis of Velocity and Viscous Drag in an Artery with a Periodic Pressure Gradient

Blood as a fluid that human and other living creatures are dependent on has been always considered by scientists and researchers.Any changes in blood pressure and its normal velocity can be a sign of a disease.Whatever significant in blood fluid＇s mechanics is Constitutive equations and finding some relations for analysis and description of drag,velocity and periodic blood pressure in vessels.In this paper,by considering available experimental quantities,for blood pressure and velocity in periodic time of a thigh artery of a living dog,at first it is written into Fourier series,then by solving Navier-Stokes equations,a relation for curve drawing of vessel blood pressure with rigid wall is obtained.Likewise in another part of this paper,vessel wall is taken in to consideration that vessel wall is elastic and its pressure and velocity are written into complex Fourier series.In this case,by solving Navier-Stokes equations,some relations for blood velocity,viscous drag on vessel wall and blood pressure are obtained.In this study by noting that vessel diameter is almost is large（3.7 mm）,and blood is considered as a Newtonian fluid.Finally,available experimental quantities of pressure with obtained curve of solving Navier-Stokes equations are compared.In blood analysis in rigid vessel,existence of 48% variance in pressure curve systole peak caused vessel blood flow analysis with elastic wall,results in new relations for blood flow description.The Resultant curve is obtained from new relations holding 10% variance in systole peak.

A Novel Lattice Boltzmann Model For Reactive Flows with Fast Chemistry

A novel lattice Boltzmann model, in which we take the ratio of temperature difference in the temperature field to the environment one to be more than one order of magnitude than before, is developed to simulate two-dimensional reactive flows with fast chemistry. Different from the hybrid scheme for reactive flows [Comput. Phys. Commun. 129 （2000）267], this scheme is strictly in a pure lattice Doltzmann style （i.e., we solve the flow, temperature, and concentration fields using the lattice Boltzmann method only）. Different from the recent non-coupled lattice Boltzmann scheme lint. J. Mod. Phys. B 17（2003）197], the fluid density in our model is coupled directly with the temperature. Excellent agreement between the present results and other numerical data shows that this scheme is an efficient numerical method for practical reactive flows with fast chemistry.

Investigation on Liquid Holdup in Vertical Zero Net-Liquid Flow

Zero net-liquid flow (ZNLF) is a special case of upward gas-liquid two-phase flow. It is a phenomenon observed as a gas-liquid mixture flows in a conduit but the net liquid flow rate is zero. Investigation on the liquid holdup of ZNLF is conducted in a vertical ten-meter tube with diameter of 76 mm, both for Newtonian and nonNewtonian fluids. The gas phase is air. The Newtonian fluid is water and the non-Newtonian fluids are water-based guar gel solutions. The correlations developed for predicting liquid holdup on the basis of Lockhart-Martinelli parameter are not suitable to ZNLF. A constitutive correlation for the liquid holdup of vertical ZNLF was put forward by using the mass balance. It is found that the liquid holdup in ZNLF is dependent on both the gas flow rate and the flow distribution coefficient.

On the energy conversion in electrokinetic transports

Energy conversion in micro/nano-systems is a subject of current research,among which the electrokinetic energy conversion has attracted extensive attention.However,there exist two different definitions on the electrokinetic energy conversion efficiency in literature.A few researchers defined the efficiency using the pure pressure-driven flow rate,while other groups defined the efficiency based on the flow rate with the inclusion of the effect of the streaming potential field.In this work,both definitions are investigated for different fluid types under the periodic electrokinetic flow condition.For Newtonian fluids,the two definitions give similar results.However,for viscoelastic fluids,these two definitions lead to significant difference.The efficiency defined by the pure pressure-driven flow rate even exceeds 100%in a certain range of the parameters.The result shows that in the case of viscoelastic flow,it is incorrect to define the energy conversion efficiency by pure pressure-driven flow rate.At the same time,the reason for this problem is clarified through comprehensive analysis.

OBTAINING PRECISE CRITICAL DRAW RATIO OF DRAW RESONANCE IN MELT SPINING BY NUMERICAL SIMULATION OF DIFFERENCE EQUATIONS

Direct difference methods have been used to solve the simultaneous non-linear partial differential equations formelt spinning without recourse to linearisation or perturbation approximation.The stability of each difference schemes wasstudied by error analysis using the Taylor series,and by comparison of the results obtained from numerical simulation withthe logical value in melt spinning.It is found that computation with 19 digit long double precision has significantlysimplified the stability problem of difference equations.Using this method,the precise critical draw ratio of draw resonancein an isothermal and uniform tension spinning of Newtonian fluids can be obtained in between 20.218 and 21.219,a figureconsistent with 20.218 which was obtained by a linear perturbation approximation method by Kase and Denn.It thus haspaved the way to computation of full information for unsteady melt spinning processes using the difference method.