Irrotation

The function (p is called the potential of an irrotational vector field and the function is called the potential of a solenoidal vector field. 

Braun and Hauck [3] discovered that the irrotational and solenoidal components of a 2-D vector field can be imaged separately using the transverse and longitudinal measurements, respectively. This result has a clear analogy in a 2-D tensor field. We can distinguish three types of measurements which determine potentials of the symmetric tensor field separately ... 

According to the Helmholtz theorem, the two-dimensional vector field can be represented as a sum of an irrotational field and of a solenoidal one... 

Here (0 is the magnitude of the vorticity vector, which is directed along the z axis. An irrotational flow is one with zero vorticity. Irro-tational flows have been widely studied because of their useful mathematical properties and applicability to flow regions where viscous effects m be neglected. Such flows without viscous effec ts are called in viscid flows. 

It is possible to use directly as a measure of the irrotational part of the deformation, but it is more convenient to use the strain tensor... 

The components of strain ej- relative to the unrotated spatial configuration are shifted to components of strain relative to the reference configuration by the stretch U, or to components of strain Cy relative to the current spatial configuration by the rotation R. The tensors E, e, and e all are measures of the same irrotational part of the deformation, but with components relative to different configurations. 

The seeond Helmholtz law states that the vortieity of a frietionless fluid does not ehange with time. Henee, if the flow at the inlet to an impeller is irrotational, the absolute flow must remain irrotational throughout the impeller. As the impeller has an angular veloeity lu, the fluid must have an angular veloeity—lu relative to the impeller. This fluid motion is ealled the relative eddy. If there were no flow through the impeller, the fluid in the... 

Flows may be subdivided into steady and unsteady, uniform and nonuniform, laminar and turbulent, and rotational and irrotational flows. 

Irrotational flow occurs when the fluid motion rotates about its axis (e.g., water flowing in a bend in a pipe). 

For an irrotational, incompressible, and frictionless fluid flow there exists a scalar velocity potential 4> such that the velocity vector V is... 

Concerning a liquid droplet deformation and drop breakup in a two-phase model flow, in particular the Newtonian drop development in Newtonian median, results of most investigations [16,21,22] may be generalized in a plot of the Weber number W,. against the vi.scos-ity ratio 8 (Fig. 9). For a simple shear flow (rotational shear flow), a U-shaped curve with a minimum corresponding to 6 = 1 is found, and for an uniaxial exten-tional flow (irrotational shear flow), a slightly decreased curve below the U-shaped curve appears. In the following text, the U-shaped curve will be called the Taylor-limit [16]. 

Figure 9 We - 8 plot comparison of effect of viscosity ratio 8 on critical shear We.cn in rotational and irrotational shear fields [18].

In contrast to rotational shear flow, deformation and breakage occurs over the whole range of viscosity ratio in an irrotational (extension) flow produced, for example, in a 4-roll apparatus (Fig. 23) from which the data shown in Fig. 21 were obtained [76]. Comparing the critical conditions for breakage by shear and by elongation. Fig. 23 shows that for equal deformation rates, irrotational flow tends to be more damaging than rotational flow. 

When the IRP is traced, successive points are obtained following the energy gradient. Because there is no external force or torque, the path is irrotational and leaves the center of mass fixed. Sets of points coming from separate geometry optimizations (as in the case of the DC model) introduce the additional problem of their relative orientation. In fact, the distance in MW coordinates between adjacent points is altered by the rotation or translation of their respeetive referenee axes. The problem of translation has the trivial solution of centering the referenee axes at the eenter of mass of the system. On the other hand for non planar systems, the problem of rotations does not have an analytical solution and must be solved by numeiieal minimization of the distanee between sueeessive points as a funetion of the Euler angles of the system [16,24]. 

Hence curl grad = 0 for all (j>. Again, conversely may be inferred that if 6 is a vector function with identically zero curl, then 6 must be a gradient of some scalar function. Vector functions with identically zero curl are said to be irrotational. 

Irrigation, in the United States, 26 4 Irritants, 5 823-824 21 836 Irrotational flow, 11 742-743 Ir-selective surfaces, cooling power of, 23 14... 

When all three components of the vorticity are zero the flow is said to be irrotational. In irrotational flow the effects of viscosity disappear as will be... 

This is a statement of Bernoulli s theorem the quantity v2l2+Plp+gh is constant throughout the fluid for steady, irrotational flow. Equation A.33 is the same as equation 1.11. It will be recalled that, for rotational flow with friction, the engineering form of Bernoulli s equation applies only along a streamline and allowance must be made for frictional losses. 

For potential flow, ie incompressible, irrotational flow, the velocity field can be found by solving Laplace s equation for the velocity potential then differentiating the potential to find the velocity components. Use of Bernoulli s equation then allows the pressure distribution to be determined. It should be noted that the no-slip boundary condition cannot be imposed for potential flow. 

The force due to the movement of the liquid surrounding the bubble is m (dt>ldt). For a sphere moving in an infinite medium of an inviscid fluid, the mass of the liquid m is equal to half the mass of the displaced liquid. The authors, however, assumed merely a direct proportionality between m and the mass of the displaced fluid, instead of the above relationship, because they considered their flow not to be irrotational. 

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