Euler flow/viscosity

Equation (54) is derived from the so-called power law of viscosity which, strictly speaking, is not exact (see footnote 3 on p. 376). According to Mooney (1931) a general solution for a non-Newtonian flow in a rotational viscometer is not possible. Recently Krieger and Maron (1952) and Krieger and Elrod (1953) have proposed an approximate solution using Euler and Maclaurin s method. 

The most immediate way of calculating viscosities and studying flow properties by molecular dynamics is to simulate a shear flow. This can be done by applying the SLLOD equations of motion [8]. In angular space they are the same as the ordinary equilibrium Euler equations. In linear space one adds the streaming velocity to the thermal motion,... 

The LHS is the sum of the classical Euler terms of inviscid flow. The RHS vanishes if the vorticity is zero, regardless of the value of viscosity. Thus, if tu = 0 which is the classic assumption of irrotational flow, the steady momentum equation reduces to the Bernoulli equation for steady incompressible flow ... 

An important parameter is the Reynolds number. At Re 1 the viscous term in (5.107) is small in comparison with the inertial one. Neglecting it, one obtains the equations of motion of an ideal liquid (Euler s equations). These equations describe flow of liquid in a volume, with the exception of small regions, adjoining the surface of an immersed body. Near such surfaces, the viscosity force can be comparable with inertial force, which results in formation of a viscous boundary layer with thickness S I/(Re), where L is the characteristic size of the body. Approximation Re 1 leads to an inertialess flow described by Stokes equations. These equations follow from (5.107), in which the inertial terms are omitted. Such equations describe the problems of micro-hydrodynamics, for example, problems of the small particles motion in a liquid. 

Two hundred years after the early contributions of Newton and Hooke, various laws of real fluids emerged as well as a quantitative description of flow and the measurement of viscosity, including the work of Euler, Cauchy, Coulomb, Poiseuille, Hagen, Couette, Reynolds, and Bingham. In 1890, the first rotational rheometer was invented by Couette. In 1929, Reiner and Bingham founded the first rheological society. 

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. 

CFD simulation considered momentum equation of all phases, which can provide the velocity of every phase and the flow field. However, big reactor requires a lot of grid, leading to significant increase of the computation time. Therefore, it is necessary to establish a mesoscale model which combines multiscale drag force with viscosity model. The CFD model consists of Euler—Euler, Euler—Lagrange, and Lagrange—Lagrange methods. 

As long as the above equation is satisfied, even if the viscosity is finite, the flow is still potential. The Navier Stokes equation reduees to the irrotational Euler equation. 

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