Inertial terms

Solution of the algebraic equations. For creeping flows, the algebraic equations are hnear and a linear matrix equation is to be solved. Both direct and iterative solvers have been used. For most flows, the nonlinear inertial terms in the momentum equation are important and the algebraic discretized equations are therefore nonlinear. Solution yields the nodal values of the unknowns. 

In the equation shown above, the first term—including p for density and the square of the linear velocity of u—is the inertial term that will dominate at high flows. The second term, including p. for viscosity and the linear velocity, is the viscous term that is important at low velocities or at high viscosities, such as in liquids. Both terms include an expression that depends on void fraction of the bed, and both change rapidly with small changes in e. Both terms are linearly dependent on a dimensionless bed depth of L/dp. 

The Reynolds number is very small (i.e. creeping flow) inertial terms in the equation of motion are neglected. 

Henry [ 157] solved the steady-flow continuity and Navier-Stokes equations in spherical geometry, neglecting inertial terms but including pressure and electrical force terms, coupled with Poisson s equation. The electrical force term in Henry s analysis consisted of the sum of the externally applied electric field and the field due to the double layers. His major assumptions are low surface potential (i.e., potentials less than approximately 25 mV) and undistorted double layers. The additional parameter ku appearing in the Henry... 

For axial capillary flow in the z direction the Reynolds number, Re = vzmaxI/v = inertial force/viscous force , characterizes the flow in terms of the kinematic viscosity v the average axial velocity, vzmax, and capillary cross sectional length scale l by indicating the magnitude of the inertial terms on the left-hand side of Eq. (5.1.5). In capillary systems for Re < 2000, flow is laminar, only the axial component of the velocity vector is present and the velocity is rectilinear, i.e., depends only on the cross sectional coordinates not the axial position, v= [0,0, vz(x,y). In turbulent flow with Re > 2000 or flows which exhibit hydrodynamic instabilities, the non-linear inertial term generates complexity in the flow such that in a steady state v= [vx(x,y,z), vy(x,y,z), vz(x,y,z). ... 

In the first part to follow, the equations of motion of a soft solid are written in the harmonic approximation. The matrices that describe the potential, and hence the structure, of the material are then considered in a general way, and their properties under a normal mode transformation are discussed. The same treatment is given to the dissipation terms. The long wavelength end of the spectral density is of interest, and here it seems that detailed matrix calculations can be replaced by simple scaling arguments. This shows how the inertial term, usually absent in molecular problems, is magnified to become important in the continuum limit. 

Many engineering operations involve the separation of solid particles from fluids, in which the motion of the particles is a result of a gravitational (or other potential) force. To illustrate this, consider a spherical solid particle with diameter d and density ps, surrounded by a fluid of density p and viscosity /z, which is released and begins to fall (in the x = — z direction) under the influence of gravity. A momentum balance on the particle is simply T,FX = max, where the forces include gravity acting on the solid (T g), the buoyant force due to the fluid (Fb), and the drag exerted by the fluid (FD). The inertial term involves the product of the acceleration (ax = dVx/dt) and the mass (m). The mass that is accelerated includes that of the solid (ms) as well as the virtual mass (m() of the fluid that is displaced by the body as it accelerates. It can be shown that the latter is equal to one-half of the total mass of the displaced fluid, i.e., mf = jms(p/ps). Thus the momentum balance becomes... 

Consider the limit of high particle Reynolds numbers where the inertial term in the Ergun equations dominates. 

There is no general solution of the Navier-Stokes equations, which is due in part to the non-linear inertial terms. Analytical solutions are possible in cases when several of the terms vanish or are negligible. The skill in obtaining analytical solutions of the Navier-Stokes equations lies in recognizing simplifications that can be made for the particular flow being analysed. Use of the continuity equation is usually essential. 

The 2-component Navier-Stokes equation is equation A.25. Each of the inertial terms is zero, the reasons being respectively that the flow is steady, vr = 0, the flow is axisymmetric and the flow is fully developed. The second and third viscous terms vanish because the flow is axisymmetric and fully-developed. The flow being horizontal, gz = 0. 

It is to be noted that on expressing the motion of the center in terms of the motion of the base of the bubble, the various terms of Eq (47) have been split into two parts, viz. (i) that associated with the movement of the base due to the free motion of the bubble (this would be the only term if the bubble does not change its size during its movement), and (ii) that associated with the movement of the bubble center due to expansion. The inertial term has split into left-hand-side terms and the second term on the right-hand side, whereas the viscous drag term has split into the third and fourth terms on the right-hand side of Eq. (48). 

If the inertial terms on the right-hand side of equation 3.108 are neglected, then ... 

Most drop situations in extraction are far above the upper limit of application of the preceding equations. A drop moving through a liquid at a velocity such that the viscous forces could be termed negligible can not exist. It will break up into two or more smaller droplets (HIO, K5). Most real situations involve both viscous and inertial terms, and the Navier-Stokes equations can not then be solved. 

The above equations are limited to the creeping-flow range. For large drops moving under conditions such that inertial terms are not negligible, an empirical equation equation based on experimental data (S12) is... 

Equation (11-11) depends on neglect of inertial terms in the Navier-Stokes equation. Neglect of inertia terms is often less serious for unsteady motion than for steady flow since the convective acceleration term is small both for Re 0 (Chapters 3 and 4), and for small amplitude motion or initial motion from rest. The second case explains why the error in Eq. (11-11) can remain small up to high Re, and why an empirical extension to Eq. (11-11) (see below) describes some kinds of high Re motion. Note also that the limited diffusion of vorticity from the particle at high cd or small t implies that the effects of a containing wall are less critical for accelerated motion than for steady flow at low Re. 

It is difficult to solve the system of Eqs. (39)—(41) for these boundary conditions. However, certain simplifying assumptions can be made, if the Prandtl number approaches large values. In this case, the thermal boundary layer becomes very thin and, therefore, only the fluid layer near the plate contributes significantly to the heat transfer resistance. The velocity components in Eq. (41) can then be approximated by the first term of their Taylor series expansions in terms of y. In addition, because the nonlinear inertial terms are negligible near the wall, one can further assume that the combined forced and free convection velocity is approximately equal to the sum of the velocities that would exist when these effects act independently. Therefore, for assisting flows at large Prandtl numbers (theoretically for Pr -> oo), Eq. (41) can be rewritten in the form ... 

The first term on the left is called the inertial term, and the second arises from the temporal variation in the velocity at any given position. For low velocities, the former may be neglected. 

Under conditions for which the inertial term can be neglected compared to the other terms in the equation, Equation (28a) becomes... 

In addition, the simple phenomenological relation (6.1.4), with a constant electro-osmotic coefficient lc, was replaced by a more elaborate one, accounting for the w dependence on the flow rate and the concentrations Ci, C2 via a stationary electro-osmotic calculation. This approach was further adopted by Meares and Page [7] [9] who undertook an accurate experimental study of the electro-osmotic oscillations at a Nuclepore filter with a well-defined pore structure. They compared their experimental findings with the numerically found predictions of a theoretical model essentially identical to that of [5], [6]. It was observed that the actual numerical magnitude of the inertial terms practically did not affect the observable features of the system concerned. 

It should be noted that if the inertial terms are omitted, the above relations will return to the original form of the EMMS model (Li and Kwauk, 1994). For specified conditions Us, Gs, and g, this set of 10... 

If we neglect the effect inertia has on the stretching fiber and drop the inertial term in the above equation, it can be solved as... 

Neglecting the inertial terms and using eqn. (8.96), the momentum equations are... 

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