Plane irrotational flow

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... 

The important idea of an irrotational flow is that at any point in the fluid the angular velocity about any axis is zero. This is shown for zero angular velocity about any axis perpendicular to the xy plane in Figs. 10.10 and 10.11. And it can be shown that for any three-dimensional flow which obeys Laplace s equation the angular velocity is zero about any axis. 

Figure 9.28 shows the stress train eurves of a whole eomplement of deformation modes with all flow stresses normalized with tq, giving the dependenees of e/ro, the normalized global equivalent deviatorie shear resistances, on Se, the global equivalent plastic strain. The predieted stress strain eurve for plane-strain eompression agrees well with the data points of the Gal ski et al. experiments. We note that the predicted response for uniaxial tension is also elose to the predietion for plane-strain compression and that these two, as examples of irrotational flow, differ markedly from the simple shear results and also from the experimental results and the predictions for uniaxial compression, in comparison with the experimental results of Bartczak et al. (1992b). 

In fact, when the contact radius r y is small compared to the particle radius R, the distribution of temperature inside the two particles is approximately the same as that of the velocity potential in irrotational flow of incompressible fluid through a circrflar hole in a plane wall. Based on this argument. 

Fluid flow may be steady or unsteady, uniform or nonuniform, and it can also be laminar or turbulent, as well as one-, two-, or three-dimensional, and rotational or irrotational. One-dimensional flow of incompressible fluid in food systems occurs when the direction and magnitude of the velocity at all points are identical. In this case, flow analysis is based on the single dimension taken along the central streamline of the flow, and velocities and accelerations normal to the streamline are negligible. In such cases, average values of velocity, pressure, and elevation are considered to represent the flow as a whole. Two-dimensional flow occurs when the fluid particles of food systems move in planes or parallel planes and the streamline patterns are identical in each plane. For an ideal fluid there is no shear stress and no torque additionally, no rotational motion of fluid particles about their own mass centers exists. 

In the extensional, irrotational field, under steady state conditions, the particles remain oriented in the direction of stress. In uniaxial flow, they align with the main axis in the flow direction, while in biaxial they lie on the stretch plane [Batchelor, 1970, 1971]. For dilute spherical suspensions in Newtonian liquid the extensional viscosity follows the Trouton rule, i.e., = 3q. 

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