Coordinate system moving

The starting point for obtaining quantitative descriptions of flow phenomena is Newton s second law, which states that the vector sum of forces acting on a body equals the rate of change of momentum of the body. This force balance can be made in many different ways. It may be appHed over a body of finite size or over each infinitesimal portion of the body. It may be utilized in a coordinate system moving with the body (the so-called Lagrangian viewpoint) or in a fixed coordinate system (the Eulerian viewpoint). Described herein is derivation of the equations of motion from the Eulerian viewpoint using the Cartesian coordinate system. The equations in other coordinate systems are described in standard references (1,2). 

In the coordinate system moving upward with velocity the particle is falling as if the air around it were at rest. In other words, the free-falling veloc-itv is the sum of the two velocities... 

For one-dimensional dispersion in soils, llie describing equation for a conservative species and/or pollutant, c, is a cartesian (rectatigular) coordinate system moving with velocity v is... 

Momentum equations for the liquid film and the gas above it (relative to a coordinate system moving with a velocity u,) ... 

The primary difference between the two equations is the unsteady term in equation (E2.2.2) and the convective term in equation (E5.3.2). Now, let s convert our coordinate system of Example 2.2 to a moving coordinate system, moving at the bulk velocity, U, which suddenly experiences a pulse in concentration as it moves downstream. This is likened to assuming that we are sitting in a boat, moving at a velocity U, with a concentration measuring device in the water. The measurements would be changing with time, as we moved downstream with the flow, and the pulse in concentration would occur at x = 0. We can therefore convert our variables and boundary conditions as follows ... 

We will convert our fixed coordinate system to a coordinate system moving at velocity U through the change of variables, x = x - Ut. Then, equation (6.39) is given as... 

Let us compare the apparent distortion of flat surfaces moving past an observer at increasing relativistic velocities. Figure 17b represents an orthogonal coordinate system moving from right to left at the speed of 0.6c past the stationary observer at B. As the velocity is increased and approaches that of... 

The condition at x = 0 states that the net flux of solute at the boundary is zero. To solve for the unknown Ce, we use the approach of van Genuchten and Parker (1984) by assuming that the solute concentration is continuous across the boundary at x=L. Furthermore, if we let y = x-ut, the problem is the same as diffusion in a stagnant fluid when viewed in a coordinate system moving at a speed u (Fisher et al., 1979). Consequently, the solution to equations (4) and (5) can be obtained by substituting y = x-ut to the solution of equations (6) and (7). 

Fig. 13.15 Schematic representation of the flow pattern in the central portion of the advancing front between two parallel plates. The coordinate system moves in the x direction with the front velocity. Black rectangles denote the stretching deformation the fluid particles experience. [Reprinted by permission from Z. Tadmor, Molecular Orientation in Injection Molding, J. Appl. Polym. Sci., 18, 1753 (1974).]...

The translation and diffusion processes can be separated and the mathematics simplified by a change in coordinates. We define a coordinate system moving with the center of gravity of the zone, located at x = X, and thus in the case under consideration, moving forward at a velocity of W with respect to the old coordinate system. The distance along the axis of translation in the new coordinate system y is related to x by... 

The symmetric random walk used here is one in which forward and backward steps are equally probable. All positive and negative steps are considered as displacements with respect to the zone center. Therefore in chromatography where the zone as a whole is in a state of motion, we must consider all displacements with respect to a coordinate system moving with the zone center at velocity Y = Rv. 

The mass flow of component i, p,v is a vector showing the flow of a component relative to a motionless coordinate system. On the other hand, diffusion flow shows the transport of a component relative to a coordinate system moving at the reference velocity vr. The diffusion flow relative to the center-of-mass velocity v (or mass average velocity) is... 

Other regional transport models, such as the Regional Lagrangian Model of Air Pollution (RELMAP Eder et al., 1986), use a different computational scheme than Eulerian models. In a Lagrangian model, the coordinate system moves with a parcel of air and mass balance of pollutant concentrations is computed on a parcel as it moves through space. 

The fictitious forces are conventionally derived with the help of the framework of classical mechanics of a point particle. Newtonian mechanics recognizes a special class of coordinate systems called inertial frames. The Newton s laws of motion are defined in such a frame. A Newtonian frame (sometimes also referred to as a fixed, absolute or absolute frame) is undergoing no accelerations and conventionally constitute a coordinate system at rest with respect to the fixed stars or any coordinate system moving with constant velocity and without rotation relative to the inertial frame. The latter concept is known as the principle of Galilean relativity. Speaking about a rotating frame of reference we refer to a coordinate system that is rotating relative to an inertial frame. 

FIGURE 5.3 Motion of particles in simple shear flow. The arrows indicate the direction of flow relative to the particles. In the central plane the flow velocity is zero or, in other words, the coordinate system moves with the geometric center of the particle. See text. 

Figure 15. Spherical bubble. The streamlines and the velocity vectors at steady-state in a coordinate system moving with the bubble centroid for (a) a clean bubble and (b) a contaminated bubble. Every third grid points are used in the velocity vector plots (Eo = 1 and Mo = 0.1).

Let us now derive the simplified form of (25.27) corresponding to a moving coordinate system. Let x , /, and z7 be the coordinates in the new system and u x, u y, and be the wind velocities with respect to this new moving coordinate system. The coordinate system moves horizontally with velocity equal to the wind speed and therefore ux = u y = 0, while u = u-. Therefore the second and third terms on the left-hand side [LHS] of (25.27) are zero in this case. Physically, the air parcel is moving with velocity equal to the windspeed, so there is no exchange of material with its surroundings by advection. The atmospheric diffusion equation then simplifies to... 

The discontinuities diagrammed in Fig. 5.4.1 are termed kinematic shocks in that they represent discontinuities in density. Let us calculate the speed at which the top discontinuity moves down and the bottom one up. For specificity consider a downward-moving shock. With respect to a coordinate system moving down with the speed of the discontinuity u (Fig. 5.4.2A), the flow is steady and conservation of mass for the one-dimensional picture considered gives... 

The flow is considered to be inertia free and to obey the Stokes equation, with the solid spherical particle taken to move within the cell with superficial velocity U. This is equivalent to using a coordinate system moving with velocity U. The appropriate boundary conditions are thus... 

With respect to a coordinate system moving down with the front speed the solution moves down with speed and the resin moves up with... 

We can Either simplify Eq. [80] by assuming a constant propagation velocity, a reasonable assumption because fronts observed in experimental systems typically exhibit constant wave speeds following a transient initiation period. Thus we introduce a coordinate system moving at the velocity of the propagating front. The new spatial coordinate, z = — cr, where c is the dimensionless wave velocity d / dr, allows the front to be described in terms of an ordinary differential equation... 

It is convenient to look for the solution in a coordinate system moving with velocity U relative to the pipe wall. In this system of coordinates, the equation of convective diffusion (6.118) subject to the inequalities L/R 1 and Pe 1 takes the form... 

The electrophoresis retardation is motion of ions in the double layer in direction opposite to particles motion. Due to forces of viscous friction, the ions cause the electroosmotic motion of liquid, which retards the particle s motion. Following the approach presented in [48], consider the electrophoresis motion of a particle, assuming that the double layer remains spherical during the motion, and the potential of the particle s surface is small enough, so Debye-Huckel approximation is valid. The motion is supposed to be inertialess. Introduce a coordinate system moving with the particle s velocity U so that in the chosen system of coordinates the particle is motionless, and the flow velocity at infinity is equal to - 17 (Fig. 9.2). 

Proceed now to the formulation of the hydrodynamical problem. The motion of the drop is supposed to be inertialess. Introduce a coordinates system moving with the drop. Then, by virtue of spherical symmetry, the problem would be similar to the problem of Stokesian flow around a liquid drop. In the spherical system of coordinates, the equations describing the flow inside and outside the drop... 

The method used to research stability of small perturbations of a cylindrical jet of non-viscous incompressible fluid is similar to the previously discussed method of handling the small perturbation problem. The main difference from the case of a plane surface is the axial symmetry of the problem, which consequently features the characteristic linear size a (radius of the jet). In a coordinate system moving with the jet s velocity, the jet itself is motionless. Let us neglect gravity and take into account only the force of surface tension. Then the pressure along the jet is equal to p = pa+ 2/a (since l i = oo, i 2 = a). Now proceed to linearize the equations and the boundary conditions. Assume the perturbations of the flow to be small and consider the equation of motion. After linearization, i.e. after rejecting the second order terms, one obtains Eq. (17.36) for velocity perturbations and full pressure. Since the flow is a potential one, any perturbation of the velocity potential satisfies the Laplace equation = 0, which in a cylindrical coordinate system (r, 0, z) is written as... 

To find the total fluxes, we have to decide on an appropriate reference or basis velocity. Because the reference velocity is defined as a velocity for which there is no convection, in the reference coordinate system the net flux of A plus B must be zero otherwise there would be convection. Thus, in the reference coordinate system moving at reference velocity v gf, by definition. 

---