Forces viscous

A polymer chain can be approximated by a set of balls connected by springs. The springs account for the elastic behaviour of the chain and the beads are subject to viscous forces. In the Rouse model [35], the elastic force due to a spring connecting two beads is f= bAr, where Ar is the extension of the spring and the spring constant is ii = rtRis the root-mean-square distance of two successive beads. The viscous force that acts on a bead is... 

In connection with Eq. (2.6), we used the fact that the product of a viscous force and a velocity gives a rate of energy dissipation, so F j v j + Fy j Vy j equals the rate of energy dissipation by segment i. Thus the energy loss per second for the ith segment (AW/At)j is... 

Comparing this result with Eq. (3.74) shows that elastic and viscous forces-occurring separately-are 90° out of phase. In the more general case of the two occurring together, these two components are out of phase by some angle 5. 

The displacement of beads representing subchains is resisted by viscous forces which follow Eq. (2.47). 

The phenomena under discussion are viscoelastic we have only considered the elastic forces. Next we must incorporate viscous forces. As indicated above, we use Eq. (2.47) to express the proportionality between the viscous resistance to displacement and the velocity of the bead, dZj/dt ... 

From Eq. (9.1) we see that the viscous force associated with this motion equals [i7(dv/dr)] (area), where the pertinent area is proportional to the surface of the sphere and varies as. This qualitative argument suggests that the viscous force opposing the relative motion of the liquid and the sphere is propor tional to [t7(v /R)] (R ). The complete solution to this problem reveals that both pressure and shear forces arising from the motion are proportional tc 77Rvj., and the total force of viscous resistance is given by... 

Figure 2.1 served as the basis for our initial analysis of viscosity, and we return to this representation now with the stipulation that the volume of fluid sandwiched between the two plates is a unit of volume. This unit is defined by a unit of contact area with the walls and a unit of separation between the two walls. Next we consider a shearing force acting on this cube of fluid to induce a unit velocity gradient. According to Eq. (2.6), the rate of energy dissipation per unit volume from viscous forces dW/dt is proportional to the square of the velocity gradient, with t]q (pure liquid, subscript 0) the factor of proportionality ... 

Figure 9.5a shows a portion of a cylindrical capillary of radius R and length 1. We measure the general distance from the center axis of the liquid in the capillary in terms of the variable r and consider specifically the cylindrical shell of thickness dr designated by the broken line in Fig. 9.5a. In general, gravitational, pressure, and viscous forces act on such a volume element, with the viscous forces depending on the velocity gradient in the liquid. Our first task, then, is to examine how the velocity of flow in a cylindrical shell such as this varies with the radius of the shell.

The net viscous force acting on this volume element is given by the difference between the frictional forces acting on the outer and inner surfaces of the shell ... 

Reynolds dumber. One important fluid consideration in meter selection is whether the flow is laminar or turbulent in nature. This can be deterrnined by calculating the pipe Reynolds number, Ke, a dimensionless number which represents the ratio of inertial to viscous forces within the flow. Because... 

Falling ball viscometers are based on Stokes law, which relates the viscosity of a Newtonian fluid to the velocity of the falling sphere. If a sphere is allowed to fall freely through a fluid, it accelerates until the viscous force is exactly the same as the gravitational force. The Stokes equation relating viscosity to the fall of a soHd body through a Hquid may be written as equation 34, where ris the radius of the sphere and d are the density of the sphere and the hquid, respectively g is the gravitational force and p is the velocity of the sphere. 

Electrokinetics. The first mathematical description of electrophoresis balanced the electrical body force on the charge in the diffuse layer with the viscous forces in the diffuse layer that work against motion (6). Using this force balance, an equation for the velocity, U, of a particle in an electric field... 

Figure 6-40 shows power number vs. impeller Reynolds number for a typical configuration. The similarity to the friction factor vs. Reynolds number behavior for pipe flow is significant. In laminar flow, the power number is inversely proportional to Reynolds number, reflecting the dominance of viscous forces over inertial forces. In turbulent flow, where inertial forces dominate, the power number is nearly constant. 

Porous Media Packed beds of granular solids are one type of the general class referred to as porous media, which include geological formations such as petroleum reservoirs and aquifers, manufactured materials such as sintered metals and porous catalysts, burning coal or char particles, and textile fabrics, to name a few. Pressure drop for incompressible flow across a porous medium has the same quahtative behavior as that given by Leva s correlation in the preceding. At low Reynolds numbers, viscous forces dominate and pressure drop is proportional to fluid viscosity and superficial velocity, and at high Reynolds numbers, pressure drop is proportional to fluid density and to the square of superficial velocity. 

Reynolds number is the ratio of the inertia forces to the viscous forces... 

Inertial forces are developed when the velocity of a fluid changes direction or magnitude. In turbulent flow, inertia forces are larger than viscous forces. Fluid in motion tends to continue in motion until it meets a sohd surface or other fluid moving in a different direction. Forces are developed during the momentum transfer that takes place. The forces ac ting on the impeller blades fluctuate in a random manner related to the scale and intensity of turbulence at the impeller. 

The deformation of soft surfaces can be minimized with SFM by selecting cantilevers having a low force constant or by operating in an aqueous environment. The latter eliminates the viscous force that arises from the thin film of water that coats most surfaces in ambient environments. This viscous force is a large contributor to the total force on the tip. Its elimination means that the operating force in liquid can be reduced to the order of 10 N. 

Darcy s law is considered valid for creeping flow where the Reynolds number is less than one. The Reynolds number in open conduit flow is the ratio of inertial to viscous forces and is defined in terms of a characteristic length perpendicular to flow for the system. Using four times the hydraulic radius to replace the length perpendicular to flow and conecting the velocity with porosity yields a Reynolds number in the form ... 

Surface forces pressure forces, viscous forces. 

Jnenial F orcei" Viscous Forces" Pressure gradient ri. 0 = f e 0... 

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