Dimensionless form Navier-Stokes equations

Before discussing the on.set, and nature, of fluid turbulence, it is convenient to first recast the Navier-Stokes equations into a dimensionless form, a trick first used by Reynolds in his pioneering experimental work in the 1880 s. In this form, the Navier-Stokes equations depend on a single dimensionless number called Reynolds number, and fluid behavior from smooth, or laminar, flow to chaos, or turbulence,... 

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 ... 

It is possible to solve a flow problem in either dimensional or dimensionless form. The variables can be assigned values using a consistent set of dimensions, which must be the SI system for turbulent flow. The dimensional formula is convenient since the problem is usually specified in that way, but in some cases the iterations may not converge. Alternatively, the equations can be made dimensionless. The dimensionless formulations are good when you are having trouble getting the iterations to converge, since you have a better sense of the problem when you specify the Reynolds number. This section takes the dimensional Navier-Stokes equation, Eq. (10.40), and derives two different dimensionless versions ... 

When the dimensionless form of the thermal energy equation is compared with the dimensionless Navier-Stokes equation, it is clear that the Peclet number plays a role for heat transfer that is analogous to the Reynolds number for fluid motion. Thus it is natural to seek approximate solutions for asymptotically small values of the Peclet number, analogous to the low-Reynolds-number approximation of Chaps. 7 and 8. 

These also result even if the motion is unsteady, providing that ajxV and the other dimensionless terms remain finite in the limit R = 0 Equations (7) and (8) are then referred to as the quasi-static or quasi-steady Stokes equations. In this case the time variable enters the equations of motion only in an implicit form. The precise relationship between the solutions of Eqs. (7) and (8) and the asymptotic solutions of the Navier-Stokes equations at small Reynolds numbers is discussed in Section III. 

We have now derived a dimensionless variable equivalent to each dimensionalized variable in the Navier—Stokes equation. Thus we can convert the Navier—Stokes equation to a dimensionless form. Substituting the dimensionless variables for the dimensional variables in the Navier—Stokes equation for our model yields... 

Equation 1.28 is the dimensionless form of the Navier—Stokes equation for our model. The dimensionless form of the Navier—Stokes equation for our prototype is... 

Let the velocity field be u = uer + ve + Oe = (u, v, 0) and denote the pressure by P. If we nondimesionalize lengths with the bubble radius a, velocities with f/oo> bulk concentration of surfactant with Coo, and pressure with then the dimensionless equations in the bulk, written in vector notation (for the component form of the equations see the [2], [18]), are the Navier-Stokes equations and a convection diffusion equation for the concentration ... 

The full mathematical formulation of the problem in dimensionless form has been justified in paper Sh. Navier-Stokes equations in dimensionless form are written as... 

Because axisymmetric bubble motion will be considered in this paper, the streamfunction-vorticity equations will be used instead of the Navier-Stokes equation. In dimensionless form, the equations for the vorticity, (0, 0, —to), and the streamfunction, ij/, are as follows ... 

Dimensional analysis of the equations that govern a system enables the identification of the relevant dimensionless numbers for the problem. It also ensures that all parameters have been taken into account, and that they are independent. We illustrate this principle below with the example of Navier-Stokes equations, written in vector form ... 

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 ... 

Because of the difficulties inherent in solving the Navier-Stokes and continuity equations, as discussed above, the mass-transfer properties of any given geometry are commonly expressed in the form of empirical correlations between dimensionless groups. These correlations commonly (but not always) take the form... 

With dimensionless variables defined as in Eq. (4), the steady-state Navier-Stokes and continuity equations may be written in the nondimensional forms... 

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