Stress tensor bulk fluids

The two expansion coefficients ijs and rjv are called the shear and bulk viscosities, respectively. The shear viscosity is all that is required to describe our gedanken experiment. The bulk viscosity describes the viscous or dissipative part of the response to a compression. This linear constitutive relation is called the Newtonian stress tensor. A fluid correctly described by this form is called a Newtonian fluid.16... 

Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

The second approach used in first-principles tribological simulations focuses on the behavior of the sheared fluid. That is, the walls are not considered and the system is treated as bulk fluid, as discussed. These simulations are typically performed using ab initio molecular dynamics (AIMD) with DFT and plane-wave basis sets. A general tribological AIMD simulation would be run as follows. A system representing the fluid would be placed in a simulation cell repeated periodically in all three directions. Shear or load is applied to the system using schemes such as that of Parrinello and Rahman, which was discussed above. In this approach, one defines a (potentially time-dependent) reference stress tensor aref and alters the nuclear and cell dynamics, such that the internal stress tensor crsys is equal to aref. When crsys = aref, the internal and external forces on the cell vectors balance, and the system is subject to the desired shear or load. 

Here ma is the bulk solid-fluid interaction force, T.s the partial Cauchy stress in the solid, p/ the hydrostatic pressure in the perfect fluid, IIS the second-order stress in the solid, ha the density of partial body forces, ta the partial surface tractions, ts the traction corresponding to the second-order stress tensor in the solid and dvs/dn the directional derivative of v.s. along the outward unit normal n to the boundary cXl of C. 

In preparation for our subsequent treatment, we now consider a macroscopic two-phase fluid system with a planar interface of area s separating the bulk phases a and j. For the description of surface phenomena, the concentrations of the various components, p, and the components of the stress tensor d must be specified. In the interior of the bulk phases, the concentrations, be... 

In sec. 1.6b we have used a similar matrix notation for the stress tensor t imd in sec. I.app.yf we did so for the polarizability tensor a. We assume the system to be at mechanical equilibrium, i.e. the fluids are at rest. This means that the isotropic pressures in the bulk phases must the same everywhere (p = p ). Then... 

We will presume a Newtonian fluid and neglect the normally very small bulk viscosity. The statement (3.56) for the stress tensor is then transformed into Stokes formulation, the so-called Stokes hypothesis ... 

In the remainder of this section, we dehberately chose to deviate from our standard treatment of mechanical work in terms of stresses rather than pressure tensor elements. Even though we foster the former treatment throughout this book, the latter seems a bit more intuitive in the current context, especially with regard to existing literature on the bulk DSS fluid. However, we remind the reader that pressure tensor P and stress tensor r are trivially related through the relation P = —r. 

The dependence of the viscous stress on the velocity gradient in the fluid is a constitutive law, which is usually called the bulk rheological equation. The general linear relation between the viscous stress tensor, T, and the rate of strain tensor,... 

To close the system of equations for the fluid motion the tangential stress boundary condition and the force balance equation are used. The boundary condition for the balance of the surface excess linear momentum, see equations (8) and (9), takes into account the influence of the surface tension gradient, surface viscosity, and the electric part of the bulk pressure stress tensor. In the lubrication approximation the tangential stress boundary condition at the interface, using Eqs. (17) and (18), is simplified to... 

Momentum can be transported by convection and conduction. Convection of momentum is due to the bulk flow of the fluid across the surface associated with it is a momentum flux. Conduction of momentum is due to intermolecular forces on each side of the surface. The momentum flux associated with conductive momentum transport is the stress tensor. The general momentum balance equation is also referred to as Cauchy s equation. The Navier-Stokes equations are a special case of the general equation of motion for which the density and viscosity are constant. The well-known Euler equation is again a special case of the general equation of motion it applies to flow systems in which the viscous effects are negligible. 

In an Interface between pure fluids relcixation processes proceed so fast that, in the absence of temperature and pressure gradients. Interfaces may be considered as being homogeneous and likewise the interfacial tension. We exclude the extremely d5mamic situations considered in sec. 1.14a. Then the shear components of the interfacial tension tensor will also vanish and the normal or symmetric components are, except for the sigh, identical to the Interfacial tension, which is the same everywhere and, hence, no stresses can be built up in the interface. Any motion of, and in, such interfaces is entirely determined by the momentum transport of the adjacent bulk phases. For an illustration see sec. I.6.4d, example 3. 

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