Steady incompressible flow

In which directions do the forces arising from the change of fluid momentum act for steady incompressible flow in the pipe-work shown in Figure 1.3 ... 

EXAMPLE 4.3 Plagan Poiseuille flow - laminar, steady incompressible flow in a long pipe with a linear pressure gradient (first-order, nonlinear solution to an ordinary differential equation)... 

Equation (7.5) is the continuity equation for a two-dimensional, steady, incompressible flow in rectangular coordinates. 

The mean flow equations are obtained by invoking boundary layer approximation to the above conservation equations. For the two-dimensional steady incompressible flow with constant properties and Boussinesq approximation, the non-dimensional equations are written in a Cartesian coordinate system, fixed at the leading edge of the semi-infinite horizontal flat plate as,... 

Using the boundary layer approximations and a similarity variable, these equations can be solved for parallel steady incompressible flow over a flat plate, witli the following results ... 

The maximum velocity is determined from the conservation of mass requirement for steady incompressible flow. For in-line arrangement, the maximum velocity occurs at the minimum flow area between the tubes, and the conservation of mass can be expressed as (see Fig. 7—26a) pVA, pV ArOt VSj- V np,(57- — D). Then the maximum velocity becomes... 

Within the flowing liquid, mass and momentum must be conserved for steady, incompressible flow these requirements in... 

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 ... 

For steady incompressible flow with one inlet and one outlet, both assumed ID, the integral form of (1.246) reduces to a simple relation between the flow at two cross section planes (1) and (2) as used in engineering calculations ... 

Incompressible steady flows are commonly solved by pressure-based methods and methods based on the concept of artificial compressibility [183, 45]. The extension of pressure correction methods to steady flows, generally elliptic equations, has been performed by Patankar and Spalding [140] and Patankar [141]. The artificial compressibility method for calculating steady incompressible flows was proposed by Chorin [29]. In this method, an artificial compressibility term is introduced in the continuity equation, and the unsteady terms in the momentum equations are retained. Hence, the system of equations becomes hj perbolic and many of the methods developed for h3rperbolic systems can be applied. 

The energy balance for steady, incompressible flow, which is called Bernoulli s equation, is probably the most useful single equation in fluid mechanics. 

The stream function ip x, y) is a convenient parameter by which we can represent two-dimensional, steady, incompressible flow. This stream function, tp in m /s, is related to the velocity components and Vy by... 

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. 

For steady boundary conditions and steady-flow sinks and sources, equation (19) indicates steady incompressible flow that is in equilibrium with such conditions. These conditions correspond to a stress event in MODFLOW. MODFLOW already can or does calculate steady flow across each face of every cell for any stress event. The programmer needs merely re-label and re-interpret these steady flow solutions as and q. Together they... 

For steady, incompressible flow into a filled vessel, the flow rate in is the same as that out. Denote this flow rate by q. A second assumption at this stage is that the vigorous fluid mixing provides for a uniform number density of the particles in the region consisting of the following components ... 

Before moving on to the chemical engineering contributions to fluid mechanics, it is of some interest to apply Eq. 2-23 to a steady, incompressible flow in terms of the stream tube illustrated in Fig. 2-2. For this particular situation, Eq. 2-21 takes the form... 

In this section a brief summary of the most general form of Bernoulli s equation for steady incompressible flows is introduced. Bernoulli s equation is composed of kinetic energy, potential energy, and internal energy. The energy equation is written between the inlet at point 1 and the exit of the pipeline at point 2 (Figure 2.1). The process flow diagram (PFD) represents sudden contraction (the exit of the first tank to the inlet of pipe 1), two 90° elbows, and sudden expansion (the exit of pipe 4 and the inlet of the second tank). 

For cylindrical geometry (axisymmetric) and steady, incompressible flow, Eq. 3.77 may be further simplified to yield ... 

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