Irrotational flows

The condition of irrotational flow is very important and needs further consideration. First, the equation for general irrotational flow will be derived and then the simplification introduced by treating the liquid as incompressible will be considered. 

Making use of equation [5.34], the general equation of motion of a liquid particle (equation [5.16]) may be written in the form  

Substitution of these two expressions in equation [5.42], together with the condition for irrotational flow, gives  

Under conditions of irrotational flow the liquid velocity u may be written in the form  

Using equation [5.45] the velocity field for the irrotational flow of an incompressible liquid can be determined without using the Euler equations, though these are needed to determine the pressure distribution. 

Equations 10.7, which define the velocity potential, have an interesting consequence A flow which obeys them must be irrotational. If = 4 (x, y) has continuous derivatives, then the order of differentiation is immaterial, and 

How Eq. 10.29 is related to rotation may be seen by viewing a body of fluid rotating in fwo-dimensional, rigid-body rotation about the origin see Fig. 10.9. If, as shown, the fluid is rotating in rigid-body rotation with angular 

These calculations were carried out for any rigid-body rotation, so we see that dVytdx - dVJdy is exactly twice the angular velocity. This quantity is.given the name vorticity in theoretical fluid mechanics  

We see above that for simple, rigid-body rotation BVylBx - BVjBy is not zero. Thus it is impossible to find any potential function f which, when substituted in Eqs. 10.7, will describe such a flow. This does not mean that there can be no potential flows which have circular motion. Only those circular motions which have zero vorticity are irrotational and hence can be potential flows. For a flow to be irrotational, the two derivatives BVylBx and BVjBy must 

Solid-body rotation (sometimes called forced vortex). 

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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. 

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). 

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. 

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. 

Treatment of liquid drops is considerably more complex than bubbles, since the internal motion must be considered and internal boundary layers are difficult to handle. Early attempts to deal with boundary layers on liquid drops were made by Conkie and Savic (C8), McDonald (M9), and Chao (C4, W7). More useful results have been obtained by Harper and Moore (HIO) and Parlange (PI). The unperturbed internal flow field is given by Hill s spherical vortex (HI3) which, coupled with irrotational flow of the external fluid, leads to a first estimate of drag for a spherical droplet for Re 1 and Rep 1. The internal flow field is then modified to account for convection of vorticity by the internal fluid to the front of the drop from the rear. The drag coefficient. 

Generally speaking viscous fluid flow is not irrotational. Nevertheless, in regions of irrotational flow there is a great simplification of the acceleration vector. Referring back to Eqs. 2.55 and 2.56, note that for irrotational flow... 

The third term clearly vanishes for an irrotational flow. Explain why it also vanishes if the displacement vector dx is in the flow direction, namely along a streamline. 

The governing equation is therefore identical with that for the irrotational flow of an ideal fluid through a circular aperture in a plane wall. The stream lines and equipotential surfaces in this rotationally symmetric flow turn out to be given by oblate spheroidal coordinates. Since, from Eq. (157), the rate of deposition of filter cake depends upon the pressure gradient at the surface, the governing equation and boundary conditions are of precisely the same form as in the quasi-steady-state approximation... 

Example of a Rotational Constant The ground-state rotational band of 152Gd is shown in Figure 6.11. Use the energy separation between the 2+ and 0+ levels to estimate the rotational constant in keV, the moment of inertia in amu-fm2, and then compare your result to that obtained to the rigid-body result with a deformation parameter of 3 = 0.2. Finally, evaluate the irrotational flow moment of inertia for this nucleus. 

R is the radius of the conduits n the velocity at the outlet of the conduit, i.e., the impinging velocity. The velocity components in terms of stream function were given by Eq. (1.26) while the conditions of irrotational flow were determined by Eq. (1.21). 

As is well known, the regularity of the particle motion is a result of the combined action of various forces. In the stream(s) three forces act on the particle (1) The field force, which is the gravitational force in irrotational flows (2) The buoyancy force... 

M 54] [P 48] CFD simulations for the flow in the separation-layer micro mixer predict a stable, almost irrotational flow pattern in the inlet region, which is in line with the experimental findings of a transparent region mentioned above [39], This pattern is maintained until the droplet end cap. Changes only occur when the droplet breaks up and falls, inducing rotational flow. 

These two equations resemble the pair of coupled differential field equations of hydrodynamics, which describe the irrotational flow of a compressible fluid by [40] ... 

Tlie hypothetical line of u 0.99V divides the flow over a plate into two regions the boundary layer region, in which the viscous effects and the velocity changes are significant, and tlie irrotational flow region, in which the frictional effects are negligible and the velocity remains essentially constant. 

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