Steady-state drag force

Suppose there are two canonical types of inertial flow around a particle with each its own fluid-particle interaction force, such as a steady-state drag and a history related force. Each of these forces is due to a particular fluid flow field around the particle in question. Unless the principle of separation of scales applies and/or inertial effects can safely be ignored, the Hnear addition of two flow fields each described by its own Navier—Stokes equation does not yield the total flow field, just because of the convective or inertial terms in the Navier—Stokes equation, e.g.. 

At t = 0, V = 0 and the drag force is zero. As the particle accelerates, the drag force increases, which decreases the acceleration. This process continues until the acceleration drops to zero, at which time the particle falls at a constant velocity because of the balance of forces due to drag and gravity. This steady-state velocity is called the terminal velocity of the body and is given by the solution of Eq. (11-8) with the acceleration equal to zero ... 

Another approach is to consider the particle for which the drag force of the gas at the edge of the core where the velocity is maximum just balances the centrifugal force. This reduces Eq. (12-42) to a steady state, with no net acceleration or velocity of this particle. The maximum velocity is given by Eq. (12-41) applied at the edge of the core Vl r2 = Encore- When this is introduced into Eq. (12-42), the result is... 

To estimate the available interaction time in a free-fall droplet experiment the steady-state balance between gravity and Stokes drag can be analyzed. A free-falling droplet will be subject to both gravitational and drag forces. Assuming that... 

Nguyen et al. [205] used a technique in which a constant mass flow rate of water-saturated air was forced through a water-saturated sample. It was explained that the shear force of the gas flow dragged water out of the sample. In addition, the saturated air was needed in order to prevent water loss from the sample by evaporation. Once a steady state was achieved, the pressure difference between the inlet and outlet of the apparatus was recorded. After the tests were completed, the sample was weighed to obtain its water content. Thus, the relative permeability was calculated from the pressure drop, the water content in the sample, and the mass flow rate [205]. 

A molecule in a centrifuge is acted upon not only by the applied centrifugal force but also by an opposing buoyant force that depends upon the difference in density of the sedimenting particles and the solvent and by a frictional drag, which is proportional to a frictional coefficient/. Setting the sum of these forces to zero for the hydrodynamic steady state yields Eq. 3-13, which defines the sedimentation constant s. 

The Basset force can be substantial when the particle is accelerated at a high rate. The total force on a particle in acceleration can be many times that in a steady state [Hughes and Gilliland, 1952]. In a simple model with constant acceleration, the ratio of the Basset force to the Stokes drag, / gs> was derived [Wallis, 1969] and rearranged to [Rudinger, 1980]... 

At steady state, that is when the terminal velocity is attained, the accelerating force due to gravity must equal the drag force on the particle F, or (nd3/6)(px — p)g = 3npdun where no is the terminal velocity of the particle. 

We utilize the physics of rolling particles on a surface as developed by Bhattacharya and Mittal. Our treatment differs from that of Bhattacharya and Mittal in that we provide a more detailed description of the turbulent boundary layer which is formed in the steady state when a fluid flows over a flat plate.The drag force we use is consistent with the treatment of Gim et al. ... 

EXTENSIONAL FLOW. In steady extensional flows, such as uniaxial extension, the single-relaxation-time Hookean dumbbell model and the multiple-relaxation-time Rouse and Zimm models predict that the steady-state extensional viscosity becomes infinite at a finite strain rate, s. With the dumbbell model, this occurs when the frictional drag force that stretches the dumbbell exceeds the contraction-producing force of the spring—that is, when the extension rate equals the critical value Sc. ... 

The advancement of a flow through the duct is accompanied by its deceleration near the walls and by the EPR drag force, thus displacing the liquid to the center of the duct. This leads to the surprising maxima on the longitudinal velocity profiles, Fig. 3.12. The flow in the middle constantly accelerates, but its value tends to a certain limit reached in the main steady-state region. For this laminar case, v = const, the dimensionless axial velocity is known to reach the value 1.5 irrespective to Re [380], if the EPR is absent. In the case of interest, it depends upon all the parameters A, 6, and Re, can be... 

Fig. 8.9. Axial velocity-, gas voidage- and turbulent viscosity profiles as a function of column radius at the axial level z = 2.0 (m) after 80 (s) (steady-state) employing the steady drag and added mass forces. Crosses experimental data [61], continuous line standard k-s model, case (a), dotted hne standard k-s model plus Sato model, case (b), dashed line extended k-s model, case (c). Grid resolution 20x72, time resolution 2 10 " (s). Reprinted with permission from [66]. Copyright 2005 American Chemical Society.

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