Drag force with deformation

Under a pressure gradient, if the particle packing arrangement in the cake can sustain the drag force without deformation, the cake is regarded as incompressible. However, stresses developed in the particulate structure usually lead to deformation and compression with substantial changes in the porosity and permeability as a load (fluid drag) is applied. This kind of filter cake is known as a compressible cake. 

The term (v — f ) is the difference between the statistically averaged velocity of the bead (fj) and the local velocity (v) of the continuum background at the position of the bead. The drag force is zero when (v — fj) = 0, in which case the beads deform affinely with the surrounding medium. 

For very thin films V/t Newtonian melt being film cast nonisothermally has been treated by Pearson [J. R. A. Pearson, Mechanics of Polymer Processing, Elsevier, New York, 1985], (a) For thicker films the deformation of the melt can be considered as one where only h = h(z) and w = w0. Use the continuity and z-momentum equations, neglecting inertial terms, gravity, and air-drag forces, to obtain the following expressions for h(z) and v- (z) ... 

Here, a and b are the semiaxes of the elUpse for a circular disk (or lihn), we have a = b = R. By combining Equations 5.297 and 5.298 with Equation 5.302, we can derive an expression for the net drag force experienced by the deformed particle (the inset in Figure 5.49) when it moves along the y axis with a linear velocity U ... 

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. 

Abstract In this chapter the basic physics and methods of calculation of the effective drag forces acting on drops in isolated-drop and multidrop configurations relevant to sprays are provided. The effect of various physical phenomena such as drop deformation, nonuniformity of the incoming flow, drop-drop interactions, drop-gas interactions, and evaporation on the drag coefficient on the drop, with special focus on the underlying physics, is highlighted. 

The mobilizing drag, lift, and inertia forces contributing to sliding. The correction factors of the drag force Fd and lift force Fl (KScd and KScl) are obtained as the ratios of the areas As and At with and without deformation effect. The correction factor KScm is assumed to be 1.0 since the container volume V remains constant. 

The mobilizing moment induced by the drag, lift, and inertia forces. The correction factor for the moments induced by the drag and lift forces (KOcd and KOcl ) area is obtained as the ratio of describing the changes of both surface areas (As and At) and lever arms (r and Vs) of the drag and lift forces Fd and Fl. The correction factor KOcm is obtained as the ratio of lever arms Vsn of the inertia force with and without deformation effect. 

KScd, KScl, KScm and KSr = Deformation factors for the drag force, lift force, inertia force and resisting forces associated with sliding stability... 

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