Steady-state incompressible flow

The RTD concept is now well established and widely used for designing, scaling-up and optimizing chemical reactors. This subject was recently reviewed by Nauman (1). Most classical textbooks (55) mainly deal with the simple case of steady state incompressible flow, single inlet and outlet, and purely convective motion... 

Steady-state incompressible flow 39 Since geometry and the conservation of mass imply that... 

The ribbon extrusion process is studied using the following assumptions steady state, incompressible flow, the velocity component in the stretching direction is the only function of this direction, each cross section remains... 

The governing equations for viscous fluid flow are conservation of mass (or continuity) and the Navier-Stokes equations (conservation of momentum or Newton s Second Law applied to a fluid). For steady-state, incompressible flow, these equations can be written as... 

The Entrance Velocity Field For an isothermal, steady state, incompressible flow of a Newtonian fluid being symmetrical in the azimuthal direction, the governing equations are the Navier-Stokes equations and the steady state continuity equation. In dimensionless form the equations are ... 

In the absence of body force, the dimensionless form of the governing model equations for two-dimensional steady-state incompressible creeping flow of a viscoelastic fluid are written as... 

The estimation of the diffusional flux to a clean surface of a single spherical bubble moving with a constant velocity relative to a liquid medium requires the solution of the equation for convective diffusion for the component that dissolves in the continuous phase. For steady-state incompressible axisym-metric flow, the equation for convective diffusion in spherical coordinates is approximated by... 

There is a class of flow situations, first identified by Jeffery [201] and Hamel [163], for which the flow has self-similar behavior. To realize the similar behavior leading to ordinary-differential-equation boundary-value problems, the analysis is restricted to steady-state, incompressible, constant property flows. After first discussing the classic analysis,... 

For steady-state, incompressible, and isothermal flows, the transport equation of the turbulent kinetic energy can be derived as (Problem 5.4)... 

Topakoglu, H.C. Steady state laminar flow in incompressible viscous fluid in curved pipes. Math Mech. 1967, 16, 1321-1337. 

Kaganov, S. A., On steady-state laminar flow of incompressible fluid in a plane channel and in a circular cylindrical tube with regard to heat friction and dependence of viscosity on temperature, J. Appl. Mech. Techn. Phys., No. 3, 1962. 

The sixth reactor design criterion requires that the pressure drop at the minimum residence time be less than 100 psi. For a small diameter channel, the flow through that channel wiU be laminar for all flow rates of interest for this particular applicahon. Neglecting end effects, the solutions to the equations of continuity and of motion for steady-state laminar flow of an incompressible Newtonian fluid are well-known, yielding a parabolic velocity distribution and the Hagen-Poiseuille equahon for pressure drop, as given in Eqs. (9) and (10) ... 

Silva and Nerba [39] also used the two-fluid approach and presented a mathematical model of drying in cyclone. Slip condition of particles on the wall, particles-wall heat transfer, and particles shrinkage were considered. The mathematical model considered a steady state, incompressible, two-dimensional, axisymmetric, turbulent gas-solids flow. The gravity force effect on the particles was neglected. The particles assumed to be spherical and distributed in a layer of uniform concentration... 

For steady-state incompressible Newtonian flows, the governing equations can be simplified as follows ... 

Topakoglu, H. C., Steady State Laminar Flows of an Incompressible Viscous Fluid in Curved Pipes, J. Math. Mech., 16, 1,321-1,337 (1967). 

Engineers often deal with the flow of fluids inside a circular conduit or pipe. In Fig. 2.9-1 we have a horizontal section of pipe in which an incompressible Newtonian fluid is flowing in one-dimensional, steady-state, laminar flow. The flow is fully developed i.e., it is not influenced by entrance effects and the velocity profile does not vary along the axis of flow in the x direction. 

EXAMPLE 3.8 3. Laminar Flow in a Circular Tube Derive the equation for steady-state viscous flow in a horizontal tube of radius, where the fluid is far from the tube inlet. The fluid is incompressible and n is axonstant. The flow is driven in one direction by a constant-pressure gradient. 

Equation 6.17 is the simplest form of the continuity equation in the 3D space of the Cartesian coordinate system, written for the steady-state, incompressible, and viscous flow. 

The vane system is analyzed assuming steady state, incompressible, isothermal liquid flow along wall-mounted vanes with no center post. The governing equations for the system are continuity and momentum ... 

For a steady-state ID flow of an incompressible fluid over a solid plane surface, the picture is as presented in Figures 5.2 and 5.3. Now, by plotting the velocity versus distance (in y direction), we observe the following ... 

For axisymmetric, inviscid, steady state potential flow of an incompressible fluid, the velocity distribution in the neighborhood of the stagnation point... 

Using different types of time-stepping techniques Zienkiewicz and Wu (1991) showed that equation set (3.5) generates naturally stable schemes for incompressible flows. This resolves the problem of mixed interpolation in the U-V-P formulations and schemes that utilise equal order shape functions for pressure and velocity components can be developed. Steady-state solutions are also obtainable from this scheme using iteration cycles. This may, however, increase computational cost of the solutions in comparison to direct simulation of steady-state problems. 

In deriving the equation for this case, Johnson et al. (J4) assume steady-state conditions with viscous, incompressible axisymmetric flow around single... 

For adiabatic, steady-state, and developed gas-liquid two-phase flow in a smooth pipe, assuming immiscible and incompressible phases, the essential variables are pu, pG, Pl, Pg, cr, dh, g, 9, Uls, and Uas, where subscripts L and G represent liquid and gas (or vapor), respectively, p is the density, p is the viscosity, cr is the surface tension, dh is the channel hydraulic diameter, 9 is the channel angle of inclination with respect to the gravity force, or the contact angle, g is the acceleration due to gravity, and Uls and Ugs are the liquid and gas superficial velocities, respectively. The independent dimensionless parameters can be chosen as Ap/pu (where Ap = Pl-Pg), and... 

To describe the velocity profile in laminar flow, let us consider a hemisphere of radius a, which is mounted on a cylindrical support as shown in Fig. 2 and is rotating in an otherwise undisturbed fluid about its symmetric axis. The fluid domain around the hemisphere may be specified by a set of spherical polar coordinates, r, 8, , where r is the radial distance from the center of the hemisphere, 0 is the meridional angle measured from the axis of rotation, and (j> is the azimuthal angle. The velocity components along the r, 8, and (j> directions, are designated by Vr, V9, and V. It is assumed that the fluid is incompressible with constant properties and the Reynolds number is sufficiently high to permit the application of boundary layer approximation [54], Under these conditions, the laminar boundary layer equations describing the steady-state axisymmetric fluid motion near the spherical surface may be written as ... 

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