Macroscopic velocity gradient

A phenomenogical expression for the hydrodynamic force F may be constructed by assuming that this force is linear in the flux velocities and in the strength of any applied flow field. We consider a system that is subjected to a macroscopic flow field v(r) characterized by a spatially homogeneous macroscopic velocity gradient Vv. We assume that Fa vanishes for all a = 1in the equilibrium state, where the flux velocities and the macroscopic... 

In this section, we use the Cartesian force of Section VI to derive several equivalent expressions for the stress tensor of a constrained system of pointlike particles in a flow field with a macroscopic velocity gradient Vv. The excess stress of any system of interacting beads (i.e., point centers of hydrodynamic resistance) in a Newtonian solvent, beyond the Newtonian contribution that would be present at the applied deformation rate in the absence of the beads, is given by the Kramers-Kirkwood expression [1,4,18]... 

Necessary definitions now exist at this stage to permit continuing the discussion preceding and following Eq. (2.5) for the case of infinitely extended suspensions, When N is infinite rather than finite, average macroscopic quantities must be prescribed in place of the prior asymptotic boundary condition (2.5). These must be defined as volume averages. For example, the macroscopic velocity gradient is defined as... 

This equation describes the change in the distribution function of a system under macroscopic velocity gradient. 

Equations (3.121) and (3.133) together may be regarded as a constitutive equation for a given macroscopic velocity gradient K p(t), the distribution function is obtained from eqn (3.121) and the stress is calculated using eqn (3.133). 

Fig. Geometrical meaning of eqn (8.11). If the rod follows the macroscopic velocity gradient, its direction dianges as = k - mxK)u. Hence the angular velocity aib is given by aiio= x = X (k ).

Substituting Eq. (6.44) into Eq. (6.27) gives, for the macroscopic velocity gradient problem... 

The last term is the rate of viscous energy dissipation to internal energy, Ev = jv <5 dV, also called the rate of viscous losses. These losses are the origin of frictional pressure drop in fluid flow. Whitaker and Bird, Stewart, and Lightfoot provide expressions for the dissipation function <5 for Newtonian fluids in terms of the local velocity gradients. However, when using macroscopic balance equations the local velocity field within the control volume is usually unknown. For such... 

The title of the book, Optical Rheometry of Complex Fluids, refers to the strong connection of the experimental methods that are presented to the field of rheology. Rheology refers to the study of deformation and orientation as a result of fluid flow, and one principal aim of this discipline is the development of constitutive equations that relate the macroscopic stress and velocity gradient tensors. A successful constitutive equation, however, will recognize the particular microstructure of a complex fluid, and it is here that optical methods have proven to be very important. The emphasis in this book is on the use of in situ measurements where the dynamics and structure are measured in the presence of an external field. In this manner, the connection between the microstructural response and macroscopic observables, such as stress and fluid motion can be effectively established. Although many of the examples used in the book involve the application of flow, the use of these techniques is appropriate whenever an external field is applied. For that reason, examples are also included for the case of electric and magnetic fields. 

Once the continuum hypothesis has been adopted, the usual macroscopic laws of classical continuum physics are invoked to provide a mathematical description of fluid motion and/or heat transfer in nonisothermal systems - namely, conservation of mass, conservation of linear and angular momentum (the basic principles of Newtonian mechanics), and conservation of energy (the first law of thermodynamics). Although the second law of thermodynamics does not contribute directly to the derivation of the governing equations, we shall see that it does provide constraints on the allowable forms for the so-called constitutive models that relate the velocity gradients in the fluid to the short-range forces that act across surfaces within the fluid. 

Dispersions. When particles are added to a liquid, the viscosity is increased. Near a particle the flow is disturbed, which causes the velocity gradient XF to be locally increased. Because the energy dissipation rate due to flow equals t] F2, more energy is dissipated, which becomes manifest as an increased macroscopic viscosity. The microscopic viscosity, as sensed by the particles, remains that of the solvent (pure liquid) rjs. For very dilute dispersions of solid spherical particles, Einstein derived... 

Although Ey and are analogous to fj. and v, respectively, in that all these quantities are coefficients relating shear stress and velocity gradient, there is a basic difference between the two kinds of quantities. The viscosities n and v are true properties of the fluid and are the macroscopic result of averaging motions and momenta of myriads of molecules. The eddy viscosity and the eddy diffusivity are not just properties of the fluid but depend on the fluid velocity and the geometry of the system. They are functions of all factors that influence the detailed patterns of turbulence and the deviating velocities, and they are especially sensitive to location in the turbulent field and the local values of the scale and intensity of the turbulence. Viscosities can be measured on isolated samples of fluid and presented in tables or charts of physical properties, as in Appendixes 8 and 9. Eddy viscosities and diffusivities are determined (with difficulty, and only by means of special instruments) by experiments on the flow itself. 

Where P is the pressure, L is the length along the macroscopic pressure gradient in porous media, is superficial velocity, p is dynamic viscosity of fluid, is porosity of porous media, and is average diameter of particles, p is density of fluid. The constants 150 and 1.75 in Eq. (3) were obtained from fitting 640 experimental data on spheres of different diameters. Compared Eq. (3) with Eq. (2), we get... 

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