Dispersed phase normal stresses

Attempts to introduce dispersed phase normal stresses on a purely phenomenological basis [32,43] or by means of simple mechanical models [51,52,56] are heuristic by their very nature and in no way help to solve the problem. Therefore, the results of the corresponding stability studies, and especially the studies treating... 

Beginning with the paper by Jackson [20], disturbance stabilization in a fluidized bed is usually associated with the action of specific normal stresses inherent to the dispersed phase. These stresses impede volume deformations of the dispersed phase. Despite this fact having been understood for a long time, comprehensive development of a stability theory is hindered by the almost total absence of reliable information concerning the dependence of dispersed phase stresses (or of the corresponding bulk moduli of dispersed phase elasticity) on the suspension concentration and on the physical parameters. This lack of information partly invalidates all theoretical inferences bearing upon hydrodynamic stability in suspension flow. 

Genovese, D. B. and Rao, M. A. 2003. Apparent viscosity and first normal stress of starch dispersions role of continuous and dispersed phases, and prediction with the Goddard-Miller model. Appl. Rheol. 13(4) 183-190. 

Han and King (22) had shown that concentrated emulsions can exhibit viscoelastic behavior even though the dispersed and the continuous phases are both Newtonian. For a given shear stress, the primary normal stress... 

Furtheron, the dispersed droplets are the smaller the closer to unity the viscosity ratio of the components is (62-64). Their sizes decrease also if the first normal stress difference of the dispersed phase becomes smaller than that of the matrix (61). The droplet size, moreover, is influenced by the tendency to further break down of elongated particles due to capillary instabilities (61) as well as by coalescence via an interfacial energy driven viscous flow mechanism. All these procedures and dependences affect the structure formation within their typical time scales (61,62). 

It is important to stress here that in Eq. (7.145) there are explicit dependences on spatial coordinates in the normalized NDF F and fluid velocity V, but also implicit dependences on the rates of change of internal coordinates the rate of formation of the disperse phase J, and the kernels jS and b. In fact, as described in Chapter 5, these rates and the kernels depend on the flow properties, which change from point to point in the system. 

Vrc y,y r 11 o - volume fraction of dispersed and matrix phase, respectively - volume fraction of the crosslinked monomer units - volume fraction of phase i at phase inversion - maximum packing volume fraction - percolation threshold - shear strain and rate of shearing, respectively - viscosity - zero-shear viscosity - hrst and second normal stress difference coefficient, respectively... 

Thus, in order to render the stability theory completely determinate, we need to specify in an unequivocal form both the conservation equations governing macroscopic suspension flow and all the rheological equations of state. This is easily seen to be possible for coarse dispersions of small particles. For such dispersions, normal stresses in the dispersed phase may be approximately described in terms of the particulate pressure as explained in Section 4, and this pressure can be evaluated for uniform dispersion states with the help of Sections 7 and 8. As a result, particulate pressure appears to be a single-valued function of mean variables characterizing the uniform dispersion state under study and of the physical properties of its phases. This single-valued function involves neither unknown quantities nor arbitrary parameters. On the other hand, if the particle Reynolds number is small, all interphase interaction force constituents also can be expressed in an explicit consummate form with help from the theory in reference [24]. This expression for the fluid-particle interaction force recently has been employed as well in stability studies for flows of collisionless finely dispersed suspensions [15,60]. 

Another factor that may affect the rheology of emulsions is the viscosity of the disperse droplets. This is particularly the case when the viscosity of the droplets is comparable to or lower than that of the dispersions medium. This problem was considered by Taylor (17), who extended the Einstein hydrodynamic treatment for suspensions for the case of droplets in a liquid medium. Taylor (17) assumed that the emulsifier film around the droplets would not prevent the transmission of tangential and normal stresses form the continuous phase to the disperse phase and that there was no slippage at the o/w interface. These stresses produce fluid circulation within the droplets, which reduces the flow patterns around them. Taylor derived the following expression for 11 ... 

During the flow the interfacial tension coefficient is not constant. Its value depends on the total interfacial area as well as on the difference of the first normal stresses of the dispersed and continuous phase. The latter dependence was formulated by Van Oene for the deformation under simple shear stress [10] ... 

In addition to the dispersion processes, these of coalescence must be taken into account. Both processes dispersion and coalescence are simultaneous. The coalescence depends on the concentration of the dispersed phase, the mean drop size and the molecular mobility of the interface between the matrix and dispersed phase. The viscosity ratio, 8, is essential. Thus an increase of the matrix viscosity results in better dispersion since the coalescence is hindered. In the opposite case, the coalescence increases, and the effect is intensified by the normal stress effects. The drops moving in a capillary are also subjected to radially variable stresses, that create a concentration gradient over the capillary cross-section, what leads to enhanced coalescence in the middle of the strand. The number of collisions per unit volume and time can be expressed as [15] ... 

In systems where with a > 0 there will always be some characteristic size below which the flow forces cannot reduce the dispersed phase the inherent dispersion limit. Regardless of whether the fluids are Newtonian or viscoelastic, the capillary number represents the competition of flow forces which tend to deform and break domains relative to forces which oppose deformation and breakup. This limit typically depends on the rheological characteristics of the components, such as the viscosity ratio and first normal stress differences. 

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