Nonlinear flow

Fig. 11 presents the results of some 30 simulations for various conditions and two impeller types in terms of the mean agglomeration rate constant observed in the various simulations vs. the vessel-averaged shear rate found in the simulations. The simulations all started from the dotted curve for relating local agglomeration rate constant to local shear rate. A clear decrease in the maximum of /i0 as well as a shift toward higher average shear rates was found which are caused by the local nature of the nonlinear flow interactions only. These... 

When motion of the fluid consists of only small fluctuations about a state of near-rest, Lhe continuum equations are linearized by neglecting nonlinear terms and they become lhe equalions of acoustics. A large variety of fluid motions are described as sound waves when the small-motion or acoustic description can be used, the principle of superposition is valid. This powerful principle allows addition of simple simultaneous motions to represent a more complex motion, such as the sound reaching lhe audience from the instruments of a symphony orchestra. The superposition principle does not apply to large-scale (nonacoustical) motions, and the subject of fluid dynamics (in distinction from acoustics) treats nonlinear flows. i.e.. those that cannot be described as superpositions of other flows. 

These measures of solute segregation are closely related to the spatial and temporal patterns of the flow in the melt. Most of the theories that will be discussed are appropriate for laminar convection of varying strength and spatial structure. Intense laminar convection is rarely seen in the low-Prandtl-number melts typical of semiconductor materials. Instead, nonlinear flow transitions usually lead to time-periodic and chaotic fluctuations in the velocity and temperature fields and induce melting and accelerated crystal growth on the typically short time scale (order of 1 s) of the fluctuations. 

Large amplitude modulations induced a nonlinear flow response which has been studied by few authors [15, 32]. 

W.F. Florez. Multi-domain dual reciprocity method for the solution of nonlinear flow problems. PhD thesis, University of Wales, Wessex Institute of Technology, Southampton, 2000. 

W.F. Florez. Nonlinear flow using dual reciprocity. WIT press, Southampton, 2001. 

When polymer chains become sufficiently concentrated, entanglement interactions lead to cooperative relaxation phenomena that can dominate the dynamical response. The relaxation times dramatically slow down with concentration and nonlinear flow phenomena... 

Lange, J.L. and Antohe, B.V. 2000. Darcy s experiments and the deviation to nonlinear flow regime. ASME Journal of Fluids Engineering, 122 619-25. 

There are many ways to fabricate and model viscoelastic polymeric materials [22-32]. Fabrication often involves nonlinear flows that are spatially inhomogeneous, nonisothermal, and temporally complex. The flows also may involve material phase changes, and/or a wide range (1-5 decades) of strains and strain rates. Rheology is often the bridge between resin design and fabrication performance, and remains an active area of research [22]. 

Polymer blends and solutions are subjected to nonlinear flows when processed, and this can have important effects on the lengthscale of the ultimate morphology. Flow-concentration couplings in polymer solutions due to molecular deformation is an old problem, and much is known experimentally and... 

Rheometers are currently under development that will enable the anisotropic stress tensor of anisotropic complex fluids such as block copolymer melts and solutions to be probed, even during large amplitude shear. Here, a small amplitude probe waveform is applied orthogonal to the primary large amplitude shear flow. This could provide the linear dynamic modulus of an anisotropic system under nonlinear flow. 

The considered radial process in the bentonite annulus is a complicated one with coupled, highly nonlinear flows that involve many things. There are liquid flow and vapor flow as well as conductive and convective heat flow depending on gradients in pressure, water vapor density and temperature. The flow coefficients depend on water properties such as saturation water vapor pressure and dynamic viscosity of water. They also depend on the properties of bentonite water retention curve, hydraulic conductivity and water vapor diffusion coefficient, and thermal conductivity, all of which are functions of degree of water saturation. 

Some modifications of the melt flow behavior of thermoplastics that can be observed depending on filler concentration are a yield-like behavior (i.e., in these cases, there is no flow until a finite value of the stress is reached), a reduction in die swell, a decrease of the shear rate value where nonlinear flow takes place, and wall slip or nearwall slip flow behavior [14, 27, 46]. Other reported effects of flllers on the rheology of molten polymers are an increase of both the shear thinning behavior and the zero-shear-rate viscosity with the filler loading and a decrease in the dependence of the filler on viscosity near the glass transition temperature [18, 47-49]. 

Nonlinear costs could arise when the production cost function exhibits economies of scale and is concave. Another example of nonlinear flow costs is in the area of transportation and distribution planning. Models for analyzing the traffic congestion on a road network usually have arc travel costs that are convex functions of the arc flow in order to represent congestion effects on the network. These models also generally have multiple commodities to represent the various origin-destination characteristics of the network users. 

In Eq. (1), the permeability K is an intrinsic property of porous media. If the velocity becomes large enough, the Darcy s law is no longer sufficient. To describe the nonlinear flow in porous media, a quadratic term was introduced by Dupuit... 

A simple generalisation of the Bingham plastic model to embrace the nonlinear flow curve (for tyx > Tq ) is the three constant Herschel-Bulkley fluid... 

For homogeneous reacting systems, the reaction continues in the system, thereby connecting the reactor and the NMR probe. In this case, the time of the NMR acquisition is directly the time for which the solutions have reacted, and, hence the time used in the evaluation of the reaction kinetics (as long as the NMR data acquisition is started upon initialization of the reaction at t = 0). We determined the time point after the initialization of a reaction when reliable data acquisition could be started. It was shown experimentally that this time is of the order 2-4 min for the experiments carried out in this work, which is short compared to the reaction times of the experiments (about 10 min to 12 h). However, reliable NMR data can only be obtained after the delay time when the solution in the NMR probe is completely replaced by sample from the reactor after the perturbation. Effects of nonlinear flow and back mixing with educt solution in the transfer lines during initiation of the reaction can be neglected with respect to the small volume fraction of the NMR sample lines. 

However, planar geometrical configurations will be the main focus here, as they are commonly found in microfluidic devices due to their ease of fabrication. Rodd et al. [4] have summarized the nonlinear flow phenomena in microfluidic channels in the De-Re space (see Fig. 1). Care should be taken when relating various phenomena from axisynunetric to planar geometries at similar De-Re regimes. 

Physically, the hysteresis roots in that fact that the effect of the electric force on the stability of 1D conduction is different in different parts of the diffusion layer. Indeed, this force stabilizes ID conduction in the electroneutral bulk and in the quasi-equUibrium portion of EDL and destabilizes it in the ESC region. The nonlinear flow resulting from this instability reduces concentration polarization and, thus, weakens the hampering effect of the electric force in the bulk in the down way portion of the hysteresis loop. In order to verify this mechanism, a model electroosmotic formulation without electric force term in the Stokes equation was analyzed. As illustrated in Fig. 8, this modification results in shrinking of the hysteresis loop. The bifurcation still remains subcritical and the hysteresis loop still exists owing to the hampering effect of the electric force in the quasi-equilibrium portion of the EDL, implicit in the first term in the electroosmotic slip conditions (21). 

The possibility of nonlinear electroosmotic flow, varying as m oc E, seems to have been first described by Murtsovkin [1, 2], who showed that an alternating electric field can drive steady quadrupolar flow around a polarizable particle (Fig. la). This effect has recently been unified with other nonlinear electrokinetic phenomena in microfluidics [3], such as AC electro-osmotic flow (ACEO) at microelectrodes [4, 7, 8] (Fig. lb), DC electrokinetic jets at dielectric corners [5] (Fig. Ic), and nonlinear flows around metal posts [9] (Fig. Id-e). These are all cases of induced-charge electroosmosis (ICEO) - the nonlinear electroosmotic flow resulting firom the action of an electric field on its own induced diffuse charge near a polarizable surface. 

Induced-charge and second-kind electrokinetic phenomena arise due to electrohydrodynamic effects in the electric double layer, but the term nonlinear electrokinetic phenomena is also sometimes used more broadly to include any fluid or particle motion, which depends nonlinearly on an applied electric field, fit the classical effect of dielectrophoresis mentioned above, electrostatic stresses on a polarized dielectric particle in a dielectric liquid cause dielectro-phoretic motion of particles and cells along the gradient of the field intensity (oc VE ). In electrothermal effects, an electric field induces bulk temperature gradients by Joule heating, which in turn cause gradients in the permittivity and conductivity that couple to the field to drive nonlinear flows, e.g., via Maxwell stresses oc E Ve. In cases of flexible solids and emulsions, there can also be nonlinear electromechanical effects coupling the... 

Mmtsovkm VA (1996) Nonlinear flows near polarized disperse particles. Colloid J 58 341-349... 

The viscosity of a liquid is a parameter that measures the resistance of that liquid to flow. For example, water has a very low viscosity, while honey has a much larger or thicker viscosity. Newtonian fluids have constant values of viscosity, which means that the stress in a flowing liquid is proportional to the rate of strain of the flow. Non-Newtonian liquids do not have constant viscosity, but rather have viscosities that can be functions of the rate of strain, the total amount of strain, and other flow characteristics. Huids are usually non-Newtonian as a result of microscopic additives such as polymers or particles. These additives alter the viscosity of a liquid and impart nonlinear flow behavior, such as viscoelasticity. The non-Newtonian behavior of many complex liquids is described thoroughly in several texts, for example [1]. In this entry we focus on behavior and applications of polymer solutions in microfluidic devices. For example, DNA is a biopolymer that is common in microfluidics applications such as gene sequencing and amplification. 

Furthermore, let gi () denote the piecewise linear approximation of the nonlinear flow versus effort relation of the t-tti diode according to Shockley s equation implemented by the switch model Sw Z),. 

Though the elastic energy equation (10.140) is important in many nonlinear flow properties of low-molecular-weight nematics, its effect is less important in polymeric nematics since the stress is usually dominated by the viscosity in polymeric nematics. 

Questions concerning further experimental details including many on linear and nonlinear flow measurements can often be answered in References 62-68. 

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