On the Formulation of Mass,Momentum and Energy Conservation in the KdV Equation

The Korteweg-de Vries (KdV) equation is widely recognized as a simple model for unidirectional weakly nonlinear dispersive waves on the surface of a shallow body of fluid. While solutions of the KdV equation describe the shape of the free surface, information about the underlying fluid flow is encoded into the derivation of the equation, and the present article focuses on the formulation of mass, momentum and energy balance laws in the context of the KdV approximation. The densities and the associated fluxes appearing in these balance laws are given in terms of the principal unknown variable η representing the deflection of the free surface from rest position. The formulae are validated by comparison with previous work on the steady KdV equation. In particular, the mass flux, total head and momentum flux in the current context are compared to the quantities Q, R and S used in the work of Benjamin and Lighthill (Proc. R. Soc. Lond. A 224:448–460, 1954) on cnoidal waves and undular bores.     

Rigorous estimates on mechanical balance laws in the Boussinesq–Peregrine equations

It is shown that the Boussinesq–Peregrine system, which describes long waves of small amplitude at the surface of an inviscid fluid with variable depth, admits a number of approximate conservation equations. Notably, this paper provides accurate estimations for the approximate conservation of the mechanical balance laws associated with mass, momentum, and energy. These precise estimates offer valuable insights into the behavior and dynamics of the system, shedding light on the conservation principles governing the wave motion.     

Mathematical issues concerning the Boussinesq approximation for thermally coupled viscous flows

The flow of a viscous fluid driven by buoyancy forces is governed by balance equations for momentum, mass, and internal energy. Frequently, the Boussinesq approximation is employed to simplify the system, even in situations where dissipative heating cannot be neglected. The resulting equations violate the principle of conservation of total energy, which causes significant mathematical problems. We discuss these problems and possible remedies. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Conservation laws of turbulence models

The conservation of mass, momentum, energy, helicity, and enstrophy in fluid flow are important because these quantities organize a flow, and characterize change in the flow's structure over time. In turbulent flow, conservation laws remain important in the inertial range of wave numbers, where viscous effects are negligible. It is in the inertial range where energy, helicity (3d), and enstrophy (2d) must be accurately cascaded for a turbulence model to be qualitatively correct. A first and necessary step for an accurate cascade is conservation; however, many turbulent flow simulations are based on turbulence models whose conservation properties are little explored and might be very different from those of the Navier-Stokes equations.We explore conservation laws and approximate conservation laws satisfied by LES turbulence models. For the Leray, Leray deconvolution, Bardina, and Nth order deconvolution models, we give exact or approximate laws for a model mass, momentum, energy, enstrophy and helicity. The possibility of cascades for model quantities is also discussed.     

重建极性连续统理论的基本定律和原理(Ⅰ)——微极连续统

在对现有微极连续统理论已进行过再研究的基础上重新建立较为完整的微极连续统理论的基本均衡定律和方程体系。在此重建的新体系中不但考虑了由于动量引起的附加动量矩、面力引起的附加面矩和体力引起的附加体矩,而且还考虑了微角速度引起的附加速度,从而可以建立起耦合型的动量、动量矩和能量的均衡定律。从这些新的基本均衡定律可以很自然地推导出相应的局部和非局部均衡方程。通过对比可以清楚地看到这些新结果较之现有的结果都完整。     

On turbulence in fractal porous media

We examine three fundamental equations governing turbulence of an incompressible Newtonian fluid in a fractal porous medium: continuity, linear momentum balance and energy balance. We find that the Reynolds stress is modified when a local, rather than an integral, balance law is considered. The heat flux is modified from its classical form when either the integral or local form of the energy density balance law is studied, but the energy density is always unchanged. The modifications of Reynolds stress and heat flux are expressed directly in terms of the resolution length scale, the fractal dimension of mass distribution and the fractal dimension of a fractal’s surface. When both fractal dimensions become integer (respectively 3 and 2), classical equations are recovered.      

Effect of pressure-dependent viscosity on the exiting sheet thickness in the calendering of Newtonian fluids

An analysis of calendering for an incompressible Newtonian fluid flow, with pressure-dependent viscosity is studied theoretically under assumptions of isothermal conditions. We predict the influence of pressure-dependent viscosity on the exiting sheet thickness of the sheet of fluid from the gap. The dimensionless mass and momentum balance equations, which are based on lubrication theory, were solved for the velocity and pressure fields by using perturbation techniques, where the exiting sheet thickness represents an eigenvalue of the mathematical problem. When the above variables were obtained, the dimensionless exiting sheet thickness was determined by considering the influence of the pressure variations in the calendering process. Moreover, quantities of engineering interest are also calculated, which include the cylinder-separating force and power required to drive both cylinders in terms of the geometrical and kinematical parameters of the system. The results show that the inclusion of pressure-dependent viscosity effect increases the leave-off distance and consequently the dimensionless exiting sheet thickness in comparison with the case of pressure-independent viscosity.     

A Numerical Study of the Exact Evolution Equations for Surface Waves in Water of Finite Depth

We describe a pseudo-spectral numerical method to solve the systems of one-dimensional evolution equations for free surface waves in a homogeneous layer of an ideal fluid. We use the method to solve a system of one-dimensional integro-differential equations, first proposed by Ovsjannikov and later derived by Dyachenko, Zakharov, and Kuznetsov, to simulate the exact evolution of nonlinear free surface waves governed by the two-dimensional Euler equations. These equations are written in the transformed plane where the free surface is mapped onto a flat surface and do not require the common assumption that the waves have small amplitude used in deriving the weakly nonlinear Korteweg–de Vries and Boussinesq long-wave equations. We compare the solution of the exact reduced equations with these weakly nonlinear long-wave models and with the nonlinear long-wave equations of Su and Gardner that do not assume the waves have small amplitude. The Su and Gardner solutions are in remarkably close agreement with the exact Euler solutions for large amplitude solitary wave interactions while the interactions of low-amplitude solitary waves of all four models agree. The simulations demonstrate that our method is an efficient and accurate approach to integrate all of these equations and conserves the mass, momentum, and energy of the Euler equations over very long simulations.     

Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition

In this paper the boundary layer flow over a flat plat with slip flow and constant heat flux surface condition is studied. Because the plate surface temperature varies along the x direction, the momentum and energy equations are coupled due to the presence of the temperature gradient along the plate surface. This coupling, which is due to the presence of the thermal jump term in Maxwell slip condition, renders the momentum and energy equations non-similar. As a preliminary study, this paper ignores this coupling due to thermal jump condition so that the self-similar nature of the equations is preserved. Even this fundamental problem for the case of a constant heat flux boundary condition has remained unexplored in the literature. It was therefore chosen for study in this paper. For the hydrodynamic boundary layer, velocity and shear stress distributions are presented for a range of values of the parameter characterizing the slip flow. This slip parameter is a function of the local Reynolds number, the local Knudsen number, and the tangential momentum accommodation coefficient representing the fraction of the molecules reflected diffusively at the surface. As the slip parameter increases, the slip velocity increases and the wall shear stress decreases. These results confirm the conclusions reached in other recent studies. The energy equation is solved to determine the temperature distribution in the thermal boundary layer for a range of values for both the slip parameter as well as the fluid Prandtl number. The increase in Prandtl number and/or the slip parameter reduces the dimensionless surface temperature. The actual surface temperature at any location of x is a function of the local Knudsen number, the local Reynolds number, the momentum accommodation coefficient, Prandtl number, other flow properties, and the applied heat flux.     

On turbulence in fractal porous media

We examine three fundamental equations governing turbulence of an incompressible Newtonian fluid in a fractal porous medium: continuity, linear momentum balance and energy balance. We find that the Reynolds stress is modified when a local, rather than an integral, balance law is considered. The heat flux is modified from its classical form when either the integral or local form of the energy density balance law is studied, but the energy density is always unchanged. The modifications of Reynolds stress and heat flux are expressed directly in terms of the resolution length scale, the fractal dimension of mass distribution and the fractal dimension of a fractal’s surface. When both fractal dimensions become integer (respectively 3 and 2), classical equations are recovered.     

Numerical scheme for multilayer shallow-water model in the low-Froude number regime

The aim of this note is to present a multi-dimensional numerical scheme approximating the solutions to the multilayer shallow-water model in the low-Froude-number regime. The proposed strategy is based on a regularized model where the advection velocity is modified with a pressure gradient in both mass and momentum equations. The numerical solution satisfies the dissipation of energy, which acts for mathematical entropy, and the main physical properties required for simulations within oceanic flows.     

重建极性连续统理论的基本定律和原理(Ⅶ)——增率型

目的是建立微极连续统增率型的较为完整的运动方程,边界条件和能率方程.为此,先给出较为完整的变形梯度及其逆的定义.接着推导出各种应力率和偶应力率间的新关系式.最后,作为一种特殊情形得到连续统力学的耦合的增率型运动方程、边界条件和能率方程.     

重建极性连续统理论的基本定律和原理(Ⅵ)——质量和惯性守恒定律

重建极性连续统理论的耦合型质量和惯性的守恒定律和局部守恒方程以及跳变条件.为此推导出新的变形梯度、线元、面元和体元的物质导数,并给出广义Reynolds输运定理.把这些结果和作者以前推导出的耦合型动量、动量矩和能量的基本定律和有关原理结合在一起就大体上构成极性连续统理论相当完整的耦合型基本定律、局部守恒和均衡方程及原理体系.从此体系可以根据常用的局部化方法给出耦合型的非局部质量和惯性守恒方程以及动量、动量矩和能量均衡方程.     

Weak solution of incompressible Euler equations with decreasing energy

Weak solution of incompressible Euler equations are L²-vector fields, satisfying integral relations, which express the mass and momentum balance. They are believed to describe the turbulent fluid motion at high Reynolds numbers. We justify this conjecture by constructing a weak solution with decreasing kinetic energy. The construction is based on Generalized Flows, introduced by Y. Brenier.     

A Navier–Stokes–Fourier system for incompressible fluids with temperature dependent material coefficients

We consider a complete thermodynamic model for unsteady flows of incompressible homogeneous Newtonian fluids in a fixed bounded three-dimensional domain. The model comprises evolutionary equations for the velocity, pressure and temperature fields that satisfy the balance of linear momentum and the balance of energy on any (measurable) subset of the domain, and is completed by the incompressibility constraint. Finding a solution in such a framework is tantamount to looking for a weak solution to the relevant equations of continuum physics. If in addition the entropy inequality is required to hold on any subset of the domain, the solution that fulfills all these requirements is called the suitable weak solution. In our setting, both the viscosity and the coefficient of the thermal conductivity are functions of the temperature. We deal with Navier’s slip boundary conditions for the velocity that yield a globally integrable pressure, and we consider zero heat flux across the boundary. For such a problem, we establish the large-data and long-time existence of weak as well as suitable weak solutions, extending thus Leray [J. Leray, Sur le mouvement d’un liquide visquex emplissant l’espace, Acta Math. 63 (1934) 193–248] and Caffarelli, Kohn and Nirenberg [L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations, Comm. Pure Appl. Math. 35 (6) (1982) 771–831] results, that deal with the problem in a purely mechanical context, to the problem formulated in a fully thermodynamic setting.     

The non-homogeneous flow of a thixotropic fluid around a sphere

The non-homogeneous flow of a thixotropic fluid around a settling sphere is simulated. A four-parameter Moore model is used for a generic thixotropic fluid and discontinuous Galerkin method is employed to solve the structure-kinetics equation coupled with the conservation equations of mass and momentum. Depending on the normalized falling velocity U*, which compares the time scale of structure formation and destruction, flow solutions are divided into three different regimes, which are attributed to an interplay of three competing factors: Brownian structure recovery, shear-induced structure breakdown, and the convection of microstructures. At small U*( ≪ 1), where the Brownian structure recovery is predominant, the thixotropic effect is negligible and flow solutions are not too dissimilar to that of a Newtonian fluid. As U* increases, a remarkable structural gradient is observed and the structure profile around the settling sphere is determined by the balance of all three competing factors. For large enough U*( ≫ 1), where the Brownian structure recovery becomes negligible, the balance between shear-induced structure breakdown and the convection plays a decisive role in determining flow profile. To quantify the interplay of three factors, the drag coefficient Cs of the sphere is investigated for ranges of U*. With this framework, the effect of the destruction parameter, the confinement ratio, and a possible nonlinearity in the model-form on the non-homogeneous flow of a thixotropy fluid have been addressed.     

Shallow water equations for power law and Bingham fluids

In this note,we provide a consistant thin layer theory for power law and Bingham incompressible fluids flowing down an inclined plane under the effect of gravity.The derivation of such equations is based on formal asymptotic expansions of solutions of Cauchy momentum equations in the shallow water scaling and in the neighbourhood of steady solutions so that we can close the average equations on the fluid height h and the total discharge rate q.     

Flow,heat, and species transfer over a stretching plate considering coupled Stefan blowing effects from species transfer

In this paper, we investigate the flow, heat and mass transfer of a viscous fluid flow over a stretching sheet by including the blowing effects of mass transfer under high flux conditions. Mass transfer in this work means species transfer and is different from mass transpiration for permeable walls. The new contribution from this work is, for the first time, to consider the coupled blowing effects from massive species transfer on flow, heat, and species transfer for a stretching plate. Based on the exact solutions of the momentum equations, which are valid for the whole Navier–Stokes equations, the energy and mass transfer equations are solved exactly and the effects of the blowing parameter, the Schmidt number, and the Prandtl number on the flow, heat and mass transfer are presented and discussed. The solution is given in terms of an incomplete Gamma function. It is found the coupled blowing effects due to mass transfer can have significant influences on velocity profiles, drag, heat flux, as well as temperature and concentration profiles. These solutions provide rare results with closed form analytical expressions and can be used as benchmark problem for numerical code validation.     

Numerical study of transient free convective mass transfer in a Walters-B viscoelastic flow with wall suction

A transient model for the free convective, nonlinear, steady, laminar flow and mass transfer in a viscoelastic fluid from a vertical porous plate is presented. The Walters-B liquid model is employed which introduces supplementary terms into the momentum conservation equation. The transformed conservation equations are solved using the finite difference method (FDM). The influence of viscoelasticity parameter (Γ), species Grashof number (Gc), Schmidt number (Sc), distance (Y) and time (t) on the velocity (U) and also concentration distribution (C) is studied graphically. Velocity is found to increase with a rise in viscoelasticity parameter (Γ) with both time and distances close to the plate surface. An increase in Schmidt number is observed to significantly decrease both velocity and concentration in time and also with separation from the plate. Increasing species Grashof number boosts the flow velocity through all time and causes a significant rise primarily near the plate surface. The study has applications in polymer materials processing.