Steady irrotational flow

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. 

Another formulation of Bernoulli s theorem exists when the flow is irrotational (i.e. the curl of the velocity is zero at every point in the domain). This allows application to unsteady flows such as the propagation of waves on the sea. For a steady, irrotational flow, the head is the same at every point in the domain, and it is no longer necessary to check that the two points between which Bernoulli s theorem is being applied are coimected by a streamline. However, the assumption of an irrotational flow is a substantial restriction it precludes the recirculation depicted in Figure 2.2, for which the problem posed for the application of Bernoulli s theorem was already indicated. 

In equation [5.36] is the kinetic energy of unit mass of liquid while S2 is the potential energy of unit mass of liquid. The term /dp/p may be termed pressure energy, and arises because any liquid particle may do work on its surroundings. Under steady irrotational flow conditions the sum of the three energies has the same constant value at all points in this liquid. Equation [5.36] is equivalent to the theorem of the conservation of energy in mechanics (see Qassical Mechanics by B.P. Cowan in this series). 

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

For CO 0, Eq. (11-7) reduces to the stream function for steady creeping flow past a rigid sphere, i.e., Eq. (3-7) with k = co. The parameter 3 may be regarded as a characteristic length scale for diffusion of vorticity generated at the particle surface into the surrounding fluid. When co is very large, 3 is small, and the flow can be considered irrotational except in the immediate vicinity of the particle. In the limit co go, Eq. (11-7) reduces to Eq. (1-29), the result for potential flow past a stationary sphere. 

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 LHS is the sum of the classical Euler terms of inviscid flow. The RHS vanishes if the vorticity is zero, regardless of the value of viscosity. Thus, if tu = 0 which is the classic assumption of irrotational flow, the steady momentum equation reduces to the Bernoulli equation for steady incompressible flow ... 

It is emphasized that the Bernoulli equation (1.243), as derived from the Navier-Stokes equation, is still restricted to steady-, irrotational- and incompressible flow. 

Yih CH (1995) Kinetie-energy mass, momentum mass, and drift mass in steady irrotational subsonic flow. J Huid Mech 297 29-36... 

The stream function is very useful because its physical significance is that in steady flow lines defined by ip= constant are streamlines which are the actual curves traced out by the particles of fluid. A stream function exists for all two-dimensional, steady, incompressible flow whether viscous or inviscid and whether rotational or irrotational. 

An example of the use of the stream function is in obtaining the flow pattern for inviscid, irrotational flow past a cylinder of infinite length. The fluid approaching the cylinder has a steady and uniform velocity of in the x direction. Laplace s Equation... 

Figure 3.9-2. Streamlines (ip = constant) and constant velocity potential lines (4> = constant)/ /- the steady and irrotational flow of an inviscid and incompressible fluid about an infinite circular cylinder.

Equation (1.249) is seemingly exactly the same form as (1.244), but is not limited to application along a streamline. It is emphasized that the Bernoulli equation (1.249), as derived from the Navier-Stokes equation, is still restricted to steady-, irrotational- and incompressible flow. It may also be worthwhile noting that many more classical flow... 

If the liquid density is constant, that is, for steady, irrotational, incompressible flow, equation [5.36] becomes ... 

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. 

One of the most important textural and rheological properties of lipid solutions is viscosity, or consistency. The evaluation of viscosity can also be demonstrated by reference to the evalnation of creaminess, spreadability, and pour-ability characteristics. These characteristics primarily depend on the shear rate and are affected by viscosity and different flow conditions. If related to steady flow, this means that at any point, the velocity of snccessive flnid particles is the same at successive periods for the entire food system. Thns, the velocity is constant with respect to time, but may vary at different points with respect to distance. Flow is unsteady when conditions at any point in a fluid food system change with time. Lipid flow can be either nniform or nonnniform laminar or tnrbnlent one-, two-, or three-dimensional and rotational or irrotational. 

Tuck established a mathematical expression for squat with a slender body theory. The slender body theory assumes that the beam, draft, and water depth are very small relative to ship length. This theory uses potential flow where the continuity equation becomes Laplace s equation. The flow is taken to be inviscid and incompressible and is steady and irrotational. In restricted water, the problem is divided into the inner and the outer problems, following a technique of matched asymptotic expansions to construct an approximate solution. The inner problem deals with flow very close to the ship. The potential is only a function of y and 2 in the Cartesian coordinate system. In the cross-flow sections, the potential function... 

A second, very classical rotating flow is the vortex flow, of which a practical example in hydrocyclones will be seen in section 17.2.3. For now, let us seek a simple soluhon of the Navier-Stokes equations, in the form given by [17.5], which is steady and irrotational. The general solution of [17.9b] for the azimuthal component of the velocity can be written as ... 

Assume that the liquid is incompressible and that the effects of body forces may be neglected. Then, if the flow is steady and irrotational, the Bernoulli equation in the form of equation [5.37] gives ... 

Another instrument for determining liquid velocity is the Pitot tube. This consists of a tube with an opening facing the liquid, and connected to a manometer, as shown schematically in Fig. 5.4(a). The point S is a stagnation point in the liquid where the velocity is zero. Assume again that the liquid is incompressible, that the flow is steady and irrotational and that the effects of body forces can be neglected. Then, as for the Venturi meter, equation [5.37] gives ... 

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