Hydrodynamics drag force

The behavior of a bead-spring chain immersed in a flowing solvent could be envisioned as the following under the influence of hydrodynamic drag forces (fH), each bead tends to move differently and to distort the equilibrium distance. It is pulled back, however, by the entropic need of the molecule to retain its coiled shape, represented by the restoring forces (fs) and materialized by the spring in the model. The random bombardment of the solvent molecules on the polymer beads is taken into account by time smoothed Brownian forces (fB). Finally inertial forces (f1) are introduced into the forces balance equation by the bead mass (m) times the acceleration ( ) of one bead relative to the others ... 

In a recent version of the Tobolsky and Eyring formulation, the rate of mechanochemical degradation was considered as a Thermally Activated Barrier to Scission (TABS) process. The elastic energy function f(v /) was explicitly considered in terms of the frictional hydrodynamic drag force acting over the entire macromolecule [100]. A more detailed account of this model will be presented in Sect. 5.1. 

Eq. (2.119). A diffusion equation of the form given in Section IV is recovered if and only if we identify (Fa )f as a hydrodynamic drag force, and (as for the rigid system) assume that it may be described by a generalized Stokes equation of the form given in Eq. (2.74), where U is defined for a stiff system by Eq. (2.106). 

Each particle, moving at a velocity it7, is acted upon by the hydrodynamic drag force, which has the form... 

The characterization and control of electrostatic forces are of particular interest. Electrostatic forces depend on the electric charge and potential at the particle surfaces. When subjected to a uniform, unidirectional electric field E. charged colloidal particles accelerate until the electric body force balances the hydrodynamic drag force, so that the particles move at a constant average velocity v. This motion is known as electrophoresis, and v is the electrophoretic velocity. 

A tribology study [28] focusing on the coefficient of friction by varying the instrumental parameters of scrubbers suggested that the hydrodynamic drag force is a very important factor for particle removal [29]. Busnaina et al. [30] found that full contact between a brush and a particle is necessary to lift or roll particles smaller than 0.1 mm off a substrate. 

The simplest form for the hydrodynamic drag force F, namely Stokes drag, is, like the point-dipole approximation, strictly valid only when the particles are widely spaced. In the Stokes -drag approximation, for a shearing flow at shear rate y, we have... 

In the presence of non-uniform AC electric field, colloidal particles suspended in an aqueous medium experience electrokinetic forces including electrophoresis (EP), dielectrophoresis (DEP), and hydrodynamic drag force due... 

When a nondeformable object is implanted in the flow field and the streamlines and equipotentials are distorted, the nature of the interface does not affect the potential flow velocity profiles. However, the results should not be used with confidence near high-shear no-slip solid-liquid interfaces because the theory neglects viscous shear stress and predicts no hydrodynamic drag force. In the absence of accurate momentum boundary layer solutions adjacent to gas-liquid interfaces, potential flow results provide a reasonable estimate for liquid-phase velocity profiles in Ihe laminar flow regime. Hence, potential flow around gas bubbles has some validity, even though an exact treatment of gas-Uquid interfaces reveals that normal viscous stress is important (i.e., see equation 8-190). Unfortunately, there are no naturally occurring zero-shear perfect-slip interfaces with cylindrical symmetry. 

Gravitational force Buoyant force Hydrodynamic drag force ... 

Now, the hydrodynamic drag force can be expressed explicitly in terms of physical properties of the fluid and solid ... 

Answer First, one must replace Rsphere by RbubWe, but this is a minor change. Second and most important, the hydrodynamic drag force... 

For flow around submerged objects, a = 1 in the creeping flow regime, and a = 0 for turbulent flow. Since the hydrodynamic drag force exerted by fluid B on solid particle A acts in the opposite direction of va when the fluid is stationary, and the gravitational force acts downward, these two forces are balanced ... 

An optional mode of performing a binary particle separation utilizes the mobility of the particle species to be separated. The motion of each particle is defined by its individual primary acoustic radiation force and the opposing hydrodynamic drag force. Equation 44.4, where v is the fluid velocity, p, is the fluid viscosity, and Dp is the particle diameter ... 

Cumulative hydrodynamic drag force on each bead... 

Maximum hydrodynamic drag force at the midpoint of a linear chain... 

Additional hydrodynamic drag force from side chain... 

At rest, the conformation of a flexible chain in dilute solution looks like a coil with spherical symmetry in the long-term. However, its instantaneous shape is asymmetric [125], which means the chain rotates along a streamline of a flow field with velocity gradient. The hydrodynamic drag force from the friction between the chain segments and the solvent molecules can deform the coil from its equilibrium shape. On the other hand, the conformation of the polymer chain is variable and changing all the time because of thermal fluctuation (Brownian motion of the solvent). So the shape of the chain in the flow field depends on how quickly the solvodynamic force deforms the chain and how slow the whole chain relaxes. This evolves two timescales. 

As shown in Fig. 10, there is a crossover point for QSSF (dashed line) and FTF (solid line). For below the crossover point, f in FTF is smaller than in QSSF. However, the calculated frictional force in FTF is orders of magnitude less than the breaking strength of the covalent bond. In QSSF, the hydrodynamic drag force corresponds well to the bond strength [129, 135]. 

Shown in Fig. 14 is a chain with A = 2m-i-1 beads in the completely stretched conformation. Each bead experiences a hydrodynamic drag force fi = Vi = i eb, where Vi is the velocity for given bead i relative to solvent, is the hydrodynamic drag coefficient, and b is the length of chain segment. 

Fig. 18 Activation of mechanophore in cyclic polymers, (a) A mechanophore is incorporated into a ring chain, (b) The mechanophores form a block of the ring chain. Red dots represent probable positions of maximum hydrodynamic drag force. Blue and green ovals denotes mechanophores...

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