Navier-Stokes equations in Cartesian coordinates

These are called the incompressible Navier-Stokes equations in Cartesian coordinates. 

Notice that in these equations the terms on the right-hand side are written with a certain resemblance to the rows of the stress tensor, Eq. 2.180. The pressure gradients have been written as a separate terms. In the r and 9 equations, the final term collects some of the left-overs in going from Eqs. 3.57 and 3.58, yet maintaining the other terms in a form analogous to the stress tensor. The z equation has no left-over terms, which is also the case for the Navier-Stokes equations in cartesian coordinates. 

The Navier-Stokes equations in Cartesian coordinates have the form of Eqs. (1.1.2). These are considered in conjunction with the continuity equation (1.1.1). 

These are the general Navier-Stokes equations in Cartesian coordinates. There are only three equations for the four variables p, u, v, and w, but a fourth relation is supplied by the continuity equation ... 

By replacing u with U + u etc. in the Navier-Stokes equation and taking time average, it can be shown that for the turbulent case the two-dimensional Navier-Stokes equation in cartesian coordinates becomes... 

Table 6.2.3. Navier-Stokes equation in Cartesian coordinates...

Let us consider a laminar steady-state fluid flow in a rectilinear tube of constant cross-section. The fluid streamlines in such systems are strictly parallel (we neglect the influence of the tube endpoints on the flow). We shall use the Cartesian coordinates X, Y, Z with Z-axis directed along the flow. Let us take into account the fact that the transverse velocity components of the fluid are zero and the longitudinal component depends only on the transverse coordinates. In this case, the continuity equation (1.1.1) and the first two Navier-Stokes equations in (1.1.2) are satisfied automatically, and it follows from the third equation in (1.1.2) that... 

The key of the simulation is to solve the continuity and navier-stokes equations in an Eulerian Cartesian coordinate system. Driving forces from the fluid flow are applied to the particles as body forces. These forces are also added to the fluid equations and cause change in momentum, as reflected by the change in the pressure gradient in the flow direction. 

To analyze the flow around the bubble, the Navier-Stokes equation must be solved. The Navier-Stokes equation in the Cartesian coordinates may be written as follows ... 

For a Newtonian fluid, the Navier-Stokes equations are derived by introducing the constitutive equation [1.7] into the fundamental law of mechanics. The Navier-Stokes equations in a Cartesian coordinate system are also shown in Table 1.1. 

For steady, incompressible fluid flow in a cyclone separator, the governing Navier-Stokes equations of motion are given, in a Cartesian coordinate system, by ... 

Next, we substitute these dimensionless variables into the incompressible Navier-Stokes equations (equation 9.16). In Cartesian coordinates, the T component of the first equation reads... 

Equation A.22 is the Navier-Stokes equation for the x-component of motion in rectangular Cartesian coordinates. The corresponding equations for they and z components are obvious. 

Equations (if.4) and (ff.S) are solved, along with the continuity equation (which does not change upon nondimensionalization), in a Cartesian coordinate system using the Fourier-Galerkin (spectral) technique under periodic boundary conditions in all three space dimensions. The scheme is similar to that used by Orszag [8] for direct solution of the incompressible Navier-Stokes equations. More details can be found in [9] and [7], and the scheme may be considered to be pseudospectral. ... 

Since velocity is a vector quantity, it is usually necessary to identify the component of the velocity, as was done for the rectangular Cartesian coordinate system in Eq. (1). The value of the integral as it differs from zero may be employed as a measure of the accuracy with which average characteristics (Kl) of the stream may be used to describe the macroscopic aspects of turbulence. Such methods do not yield results of practical significance when applied to the solution of the Navier-Stokes equations. 

The term (ui V) V, which is called vortex stretching, originates from the acceleration terms (2.3.5) in the Navier-Stokes equations, and not the viscous terms. In two-dimensional flow, the vorticity vector is orthogonal to the velocity vector. Thus, in cartesian coordinates (planar flow), the vortex-stretching term must vanish. In noncartesian or three-dimensional flows, vortex stretching can substantially alter the vorticity field. 

In this theory, equilibrium flow is obtained using thin shear layer (TSL) approximation of the governing Navier- Stokes equation. However, to investigate the stability of the fluid dynamical system the disturbance equations are obtained from the full time dependent Navier- Stokes equations, with the equilibrium condition defined by the steady laminar flow. We obtain these in Cartesian coordinate system given by. 

A.3 Navier-Stokes equations for an incompressible fluid of constant viscosity in cartesian coordinates... 

Equation (3-10), which we have derived from the Navier-Stokes equations, governs the unknown scalar velocity function for all unidirectional flows, i.e., for any flow of the form (3-1). However, instead of Cartesian coordinates (x, y, z), it is evident that we could have derived (3-10) by using any cylindrical coordinate system (q, 1/2, z) with the direction of motion coincident with the axial coordinate z. In this case,... 

Equations 6.30 through 6.32 represent the complete momentum or the Navier-Stokes equations written in the conservation form of the Cartesian coordinate system for time-dependent, compressible, and viscous flow. 

If we consider the flow with constant density, we can cancel items from the continuity equation and effect of terms due to constant property and neglect body forces. Thus, we can derive the momentum or Navier-Stokes equations written in the conservation form of the Cartesian coordinate system for time-dependent, incompressible, and viscous flow in terms of velocities. The x component of the momentum equations is... 

Expanded in fuU, the Navier-Stokes equations are three simultaneous, nonlinear scalar equations, one for each component of the velocity field. In Cartesian coordinates. Equation 7.3 takes the form... 

Navier-Stokes equations A set of mathematical expressions used to study the motion of fluids. They ate expressed in terms of velocity gradients for a Newtonian fluid with constant density and gradient. Using Cartesian or rectangular coordinates, the equations represent inertia of the left-hand side and body force, pressure, and viscous terms of the right-hand side ... 

In order to formulate the flow equations for a fluid, for instance, for the gas in the cyclone or swirl tube, we must balance both mass and momentum. The mass balance leads to the equation of continuity the momentum balance to the Navier-Stokes equations for an incompressible Newtonian fluid. When balancing momentum, we have to balance the x-, y- and -momentum separately. The fluid viscosity plays the role of the diffusivity. Books on transport phenomena (e.g. Bird et ah, 2002 Slattery, 1999) will give the full flow equations both in Cartesian, cylindrical and spherical coordinates. 

Navier-Stokes equation and the continuity equation for the fluid flow via isotropic porous medium are expressed in tensor notation and in Cartesian coordinates as... 

The Navier-Stokes equation is written here for a Cartesian two-dimensional coordinate system where i and j represent the two axes. Accordingly, vi and vj are the velocity components in the directions i and j. P is the hydrostatic pressure, and v and vt are the moleciflar and the turbulent kinematic viscosity, respectively. For systems involving forced convection, the fluid flow equations are typically decoupled from the electrochemical process, and can be solved separately. 

In this case, we must therefore begin with the full Navier Stokes and continuity equations for a 2D flow, (2 91) and (2 20). In terms of the Cartesian coordinate system described in Fig. 4-8, these are... 

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