Hydrodynamic equations of motion

Adding the equations for the y- and -components of the mean velocity, we obtain the hydrodynamic equation of motion ... 

For the case of creeping flow, that is flow at very low velocities relative to the sphere, the drag force F on the particle was obtained in 1851 by Stokes(1) who solved the hydrodynamic equations of motion, the Navier-Stokes equations, to give ... 

The hydrodynamic equation of motion (Navier-Stokes equation) for the stationary axial velocity, vfr), of an incompressible fluid in a cylindrical pore under the influence of a pressure gradient, dP /dz, and an axial electric field, E is... 

Accdg to Dunkle (Ref 28), Brode (Ref 14), in order to solve detonation problems without recourse to empirical values derived from explosion measurements, integrated the hydrodynamical equations of motion (which constitute a set of nonlinear partial... 

Sect. 6 the hydrodynamic equation of continuity for each species and a formal expression for the mass flux vector of each species Sect. 7 the hydrodynamic equation of motion for the liquid mixture and a formal expression for the stress tensor Sect. 8 the energy equation for the liquid and a formal expression for the heat flux vector... 

The Hydrodynamic Equation of Motion and the Stress Tensor (DPL, Sect. 17.2b)... 

If we form the cross product of the position vector r with the hydrodynamic equation of motion in Eq. (7.1), the following equation is obtained ... 

To calculate the particle velocities V (n = l,2,... N) we have to know the fluid velocity v(r) created by the external forces acting on the particles. In the usual condition of Brownian motion, the relevant hydrodynamic equation of motion is that of the low Reynolds number... 

For finite wavelengths, the collective dynamics of bulk nematics can be described within the hydrodynamic equations of motion introduced by Ericksen [4-8] and Leslie [9-11]. A number of alternate formulations of hydrodynamics [12-18] leads essentially to the equivalent results [19]. The spectrum of the eigenmodes is composed of one branch of propagating acoustic waves and of two pairs of overdamped, nonpropagating modes. These can be further separated into a low- and high-frequency branches. The branch of slow modes corresponds to slow collective orientational relaxations of elastically deformed nematic structure, whereas the fast modes correspond to overdamped shear waves, which are similar to the shear wave modes in ordinary liquids. 

However, an exact solution to the problem of convective diffusion to a solid surface requires first the solution of the hydrodynamic equations of motion of the fluid (the Navier-Stokes equations) for boundary conditions appropriate to the mainstream velocity of flow and the geometry of the system. This solution specifies the velocity of the flrrid at any point and at any time in both tube and yam assembly. It is then necessary to substitute the appropriate values for the local fluid velocities in the convective diffusion equation, which must be solved for boundary cortditiorts related to the shape of the package, the mainstream concentration of dye and the adsorptions at the solid surface. This is a very difficrrlt procedure even for steady flow through a package of simple shape. " ... 

The solution for Eqs 4.32 and 4.36 cannot be obtained until the values of w, v and p are known. This presupposes that the hydrodynamic equations of motion of the fiuid are already solved, or that a particular form of the solution is assumed. Therefore, a clear description of flow behaviour before the entrance of the yam assembly (i.e. liquor in the tube) is required. 

Assuming that the resistance to fluid flow between the core and pres-surizer is proportional to the square of the average fluid velocity, and that the fluid is incompressible, the one-dimensional hydrodynamic equation of motion is given by... 

Strongly non-linear rheology is characteristic of soft matter. In simple fluids, it is difficult to observe any deviations from Newtonian behavior, which is well described by the hydrodynamic equations of motion with linear transport coefficients that depend only on the thermodynamic state. Indeed, Molecular Dynamics simulations [9] have revealed that a hydrodynamic description is valid down to astonishingly small scales, of the order of a few collisions of an individual molecule. This means that one would have to probe the system with very short wave lengths and very high frequencies, which are typically not accessible to standard experiments (with the exception of neutron scattering [10]), and even less in everyday life. However, in soft-matter systems microstructural components (particles and polymers for example) induce responses that depend very much on frequency and length scale. These systems are often referred to as complex fluids. ... 

Alternatively the hydrodynamic equations of motion can determine directly the time evolution of the correlation functions instead of the mesoscopic variables, according to the Mori-Zwanzig model [42,52]. These equations are the starting point of the Mode-Coupling Theories (MCT) [60-62]. 

---