Continuity equation cylindrical coordinates

Solution In a pipe flow we have, in principle, three velocity components vz, vg, and vr. The equation of continuity in cylindrical coordinates is given in Table 2.1. For an incompressible fluid, this equation reduces to... 

The Equation of Continuity by Differential Mass Balance Derive the equation of continuity in cylindrical coordinates by making a mass balance over the differential volume Ar(rA6)Az. 

Also, at steady state dfi/df = 0. The equation of continuity in cylindrical coordinates is ... 

Solution On physical grounds the fluid moves in a circular motion and the velocity in the radial direction is zero and in the axial direction is zero. Also, dp/dt = 0 at steady state. There is no pressure gradient in the 9 direction. The equation of continuity in cylindrical coordinates as derived before is... 

Derivation of Equation of Continuity in Cylindrical Coordinates. By means of a... 

In an axisymmetric flow regime all of the field variables remain constant in the circumferential direction around an axis of symmetry. Therefore the governing flow equations in axisymmetric systems can be analytically integrated with respect to this direction to reduce the model to a two-dimensional form. In order to illustrate this procedure we consider the three-dimensional continuity equation for an incompressible fluid written in a cylindrical (r, 9, 2) coordinate system as... 

Continuity and rate equations can also be written in a cylindrical coordinate system for the two-dimensional annular chromatography. Assuming steady state and neglecting velocity and concentration variations in the radial direction, the above-mentioned equations may then be written as... 

Written out in cylindrical coordinates, using the definition of the substantial derivative, the continuity equation is given as... 

For steady-state (no time variation) two-dimensional flows, the notion of a streamfunction has great utility. The stream function is derived so as to satisfy the continuity equation exactly. In cylindrical coordinates, there are two two-dimensional situations that are worthwhile to investigate the r-z plane, called axisymmetric coordinates, and the r-0 plane, called polar coordinates. 

Discuss the behavior of the continuity equation (and by analogy the other governing equations) as the toroid radius R becomes large compared to the channel radius r,. Under these circumstances, show how the curvilinear equation becomes closer and closer to the regular cylindrical-coordinate equations. 

In all the above derivations in this section, the influence of viscosity is neglected so that analytical solutions for velocity and pressure profiles can be obtained. When the viscosity of fluid is taken into account, it is difficult to obtain any analytical solution. Kuts and Dolgushev [35] solved numerically the flow field in the impingement of two axial round jets of a viscous impressible liquid ejected at the same velocity from conduits with the same diameter and located very close to each other. The mathematical formulation incorporated the complete Navier-Stokes equations transformed into stream and velocity functions in cylindrical coordinates r and z, with the assumption that the velocity profiles at the entrance and the exit of the conduit were parabolic. The continuity equation is given by Eq. (1.22) and the equations for motion in dimensionless form are ... 

Table 5.1 presents the continuity equation in the Cartesian, cylindrical and spherical coordinate systems. 

Another commonly used model is based on the general differential balance of mass and momentum [Burgers, 1948]. Consider a steady, incompressible, and axially symmetric flow in which the body forces are negligible. In cylindrical coordinates, the equation of continuity of the fluid can be given as... 

The equation of continuity with the preceding assumptions gives dvz/dz — 0. The equation of motion in the cylindrical coordinates reduces to... 

The equations of continuity, momentum and energy are summarized in Tables 6.1, 6.2 and 6.3, respectively. The vectorial forms in Tables 6.1 and 6.3 are provided for scholars of heat transfer that would like to go to three dimensional applications. The equations in cylindrical coordinates may be obtained from the rectangular coordinate equations by the use of the appropriate transformations. 

Consider a cylindrical elemental control volume of dimensions Ar, rA0, and Az in the r, 0 and z directions, respectively. Derive the continuity equation in cylindrical coordinates. 

Write out the continuity, Navier-Stokes, and energy equations in cylindrical coordinates for steady, laminar flow with constant fluid properties. The dissipation term in the energy equation can be ignored. Using this set of equations, investigate the parameters that determine the conditions under which similar" velocity and temperature fields will exist when the flow over a series of axisymmetrie bodies of the same geometrical shape but with different physical sizes is considered. 

Liquid Phase Continuity Equation in Cylindrical Coordinates... 

Depth average the continuity equation in cylindrical coordinates to show that... 

To analyze the linear stability of a Couette flow, we begin with the Navier Stokes and continuity equations in a cylindrical coordinate system. The frill equations in dimensional form can be found in Appendix A. We wish to consider the fate of an arbitrary infinitesimal disturbance to the base flow and pressure distributions (12 114) and (12 116). Hence we consider a linear perturbation of the form... 

In axisymmetric problems, all variables are independent of the axial coordinate Z in the cylindrical coordinates TZ, 6, Z. The continuity equation has the form (both sides are multiplied by TV)... 

Let us use the cylindrical coordinate system TZ, ip, Z, where the coordinate Z is measured from the disk surface along the rotation axis. Taking account of the problem symmetry (the unknown variables are independent of the angular coordinate Navier-Stokes equations in... 

In this section we present the equations of motion and heat transfer for incompressible non-Newtonian fluids governed by the rheological equation of state (7.1.1) when the apparent viscosity p = p(h, T) arbitrarily depends on the second invariant I2 of the shear rate tensor and on the temperature T. This section contains some material from the books [47, 320, 443], For the continuity equation in cylindrical and spherical coordinates, see Supplement 5.3. 

The solvent is assumed to be in solid body rotation at an angular speed (o, and the solute is assumed to move circumferentially with the solvent. A single solute is considered, that is, a binary mixture, and a cylindrical coordinate system rotating with the angular speed (o is adopted. The solute concentration is then a function only of the time t and radial distance r from the rotation axis. The continuity (diffusion) equation (Eq. 3.3.15) can therefore be written... 

The second step is the linearization of the governing flow equations and boundary conditions assuming the flow to be only slightly disturbed. The momentum equation so linearized is given by Eq. (10.4.3), with the excess or disturbance pressure and u the disturbance velocity. The continuity equation for the disturbance velocity is as before, V (/> = 0. The natural coordinates for the problem are the cylindrical coordinates (r, 6, z) in which Laplace s equation takes the form... 

Answer Invoke incompressibility because extensional flow occurs at constant volnme when Poisson s ratio is 5. In cylindrical coordinates with no flow in the 9 direction, the steady-state equation of continuity reduces to... 

Answer Begin with the equation of continuity and the mass transfer equation in cylindrical coordinates with two-dimensional flow (i.e., Vr and vq) in the mass transfer boundary layer and no dependence of Ca on z because the length of the cylinder exceeds its radius by a factor of 100. Heat transfer results will be generated by analogy with the mass transfer solution. The equations of interest for an incompressible fluid with constant physical properties are... 

Among the first to consider root hairs in their model were Bhat et al. (1976) and Itoh and Barber (1983), while attempting to explain experimentally obtained P uptake values exceeding those calculated by one of the previously available models. Three principal approaches to integrate root hairs into a rhizosphere model are found in the literature (1) The boundary where exudation and uptake occurs is extended by the length of the root hairs (e.g. Kirk, 1999) (2) The continuity equation for root uptake is extended with a separate sink term (e.g. Geelhoed et al., 1997) and (3) The transport equation is solved in a three-dimensional model with cylindrical coordinates (Geelhoed et al., 1997). 

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