Inertial term fluid

In this chapter, we have described a novel design approach to correct flow nonuniformities caused by four types of production variations in a linearly tapered coat-hanger die. The theoretical approach is based on the onedimensional lubrication approximation and can be used to predict the taper function of an adjustable choker bar. Once a choker bar is constructed based on the prediction of the mathematical model, the flow nonuniformities can be properly eliminated by inserting this bar into the die. A choker bar can be tapered in different ways as indicated in Fig. 3. The shape displayed in Fig. 3a may be easier to machine and was selected for illustration. The four production variations we have considered include (i) enlarging the manifold, (ii) including the fluid inertial terms, (iii) varying the viscosity of the polymeric liquids, and (iv) narrowing the liquid film width to meet production requirements. All these four production variations can be properly handled, but if the fluid inertia becomes dominant, or the Reynolds number is not small, the present method may not be applicable. 

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... 

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 other method is the velocity head method. The term V2/2g has dimensions of length and is commonly called a velocity head. Application of the Bernoulli equation to the problem of frictionless discharge at velocity V through a nozzle at the bottom of a column of liquid of height H shows that H = V2/2g. Thus II is the liquid head corresponding to the velocity V. Use of the velocity head to scale pressure drops has wide application in fluid mechanics. Examination of the Navier-Stokes equations suggests that when the inertial terms dominate the viscous terms, pressure gradients are expected to be proportional to pV2 where V is a characteristic velocity of the flow. 

The second term in the right-hand side of Eq. (5.392) represents the fluid inertia, whereas the first term represents the viscous contribution to the pressure drop. At low Reynolds numbers, the Ergun equation can be simplified by neglecting the inertial term. Under this condition, Eq. (5.392) can be expressed as... 

For very viscous fluids the inertial term pv7 dv,/dz) is negligible, thus... 

The viscous effects dominate the fluid-particle interaction of small particles (below 100 pm), thus the inertial term of the Ergun equation can be neglected. Hence, the minimum fluidisation velocity can be obtained from ... 

Abu-Hijleh, B.A., and Al-Nimr, M.A. (2001) The effect of the local inertial term on the fluid flow in channels filled with porous materials Heat and Mass Transfer 44, 1565-1572. 

In the steady, unidirectional flow problems considered in this section, the acceleration of a fluid element is identically equal to zero. Both the time derivative du/dt and the nonlinear inertial terms are zero so that Du/Dt = 0. This means that the equation of motion reduces locally to a simple balance between forces associated with the pressure gradient and viscous forces due to the velocity gradient. Because this simple force balance holds at every point in the fluid, it must also hold for the fluid system as a whole. To illustrate this, we use the Poiseuille flow solution. Let us consider the forces acting on a body of fluid in an arbitrary section of the tube, between z = 0, say, and a downstream point z = L, as illustrated in Fig. 3-4. At the walls of the tube, the only nonzero shear-stress component is xrz. The normal-stress components at the walls are all just equal to the pressure and produce no net contribution to the overall forces that act on the body of fluid that we consider here. The viscous shear stress at the walls is evaluated by use of (3 44),... 

Summarizing the results for Couette flow, we have seen that the fluid moves in circular paths and thus the fluid particles are being accelerated. As a consequence, the inertial terms in the equations of motion are nonzero clearly the flow is not unidirectional. However, this centripetal acceleration does not produce a secondary flow because the nonlinear... 

The Reynolds number, which characterizes the importance of inertial forces compared to viscous forces is around unity. This implies that the non-linear terms in the Navier-Stokes equations are weak, easing the task of solving these equations. For some systems, the assumption of Stokes flow may be reasonable, i.e., the inertial terms are set equal to zero this affords a significant simplification of the fluid flow problem [160]. The Reynolds number is independent of pressure, when everything else is held constant. 

The solution of the Rubinow-Keller problem had previously been attempted by Garstang (Gla) on the basis of the Oseen equations. His result for the lift force is larger than (216) by a factor of 4/3. But as Garstang himself pointed out, his result was not unequivocal. Rather, different results were obtained according as the integration of the momentum flux was carried out at the surface of the sphere or at infinity. Garstang s paradox is clearly due to the fact that the term U-Vv does not represent a uniformly valid approximation of the inertial term v Vv throughout all portions of the fluid, at least not to the first order in R. 

The Reynolds number measures the relative importance of inertial terms to friction (viscosity) terms. It is defined by Re = (lvp)/rj, where is a typical length scale. When Re 1 one is describing a high viscosity flow (laminar flow). When Re I the inertial terms are dominant (potential flow). An ideal fluid has = 0 and only shows potential flow. Near walls, which are important for surface and interface problems, the velocity drops to zero so the important regime is that of low Re. 

Eor this purpose, an incremental time stepping is suggested. At each time increment, the fluid flow will be assumed to be steady. This assumption holds if the particle dimensions are sufficiently small when compared with the characteristic dimensions of the problem (e.g., channel wall dimensions) and the change of location of the particle within one time increment is not significant. In this case, the inertial terms in the fluid flow equation can be neglected, and a steady-state assumption can be made. 

When the flow of gas is induced by a driving force of total pressure gradient, the flow is called the viscous flow. For typical size of most capillaries, we can ignore the inertial terms in the equation of motion, and turbulence can be ignored. The flow is due to the viscosity of the fluid (laminar flow or creeping flow) and the assumption of no slip at the surface of the wall. 

If one further chooses the time unit as to = 3Txa r s/kT and parameters rjs, m, a typical of a realistic ER (MR) fluid, the inertial term in Eq. 53 turns out to be negligible [270-272] and the equation reduces to a first order differential equation... 

As an example of unorthodox behavior, consider the large velocity (large Reynolds number) flow over a sphere as shown in Fig. 6.5. We might assume that the characteristic length scale in this problem is the sphere diameter d), that is, the Reynolds number, which is the ratio of fluid inertial to viscous terms, is postulated to be... 

Problem By writing the Stokes equation (Navier-Stokes equation without the inertial term) in cylindrical coordinates, show that the velocity profile in the tube is parabolic.Designate by G the pressure gradient responsible for the transport of fluid (in the present problem, G is generated by the Laplace underpressure existing at the upstream interface), and set the velocity at the solid/liquid interface equal to zero. Under these conditions, deduce the average velocity in the tube (Poiseuille s law) and, from there, the viscous force F that opposes the progress of the fluid [equation (5.40)]. 

An inertial term, piVpdv/dt, is added to the buoyancy force, which is important when the fluid is accelerated. When particles approach the gas-liquid interface, which is identified as 0 < ot(x, t) < I, the surface tension force acts on the particles through the liquid film. Since the size of computational cell is larger than the thickness of the gas-liquid interface film, a bubble-induced force model (BIF) is applied to the particle ... 

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