Laminar Flows. Navier-Stokes Equations

laminar flows of fluids are considered. For brevity, in what follows we often refer to laminar flows simply as flows.  

Navier-Stokes equations. The closed system of equations of motion for a viscous incompressible Newtonian fluid consists of the continuity equation 

Equations (1.1.1) and (1.1.2) are written in an orthogonal Cartesian system X, Y, and Z in physical space t is time gx, gy, and gz are the mass force (e.g., the gravity force) density components v = p/p is the kinematic viscosity of the fluid. The three components of the fluid velocity Vx,Vy,Vz, and the pressure P are the unknowns. 

By introducing the fluid velocity vector V = i Vx + iy Vy + izVz, where ix, iy, and iz are the unit vectors of the Cartesian coordinate system, and by using the symbolic differential operators 

The continuity and Navier-Stokes equations in cylindrical and spherical coordinate systems are given in Supplement 5. 

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See also -> convection, -> Grashof number, - Hagen-Poiseuille, -> hydrodynamic electrodes, -> laminar flow, - turbulent flow, -> Navier-Stokes equation, -> Nusselt number, -> Peclet number, -> Prandtl boundary layer, - Reynolds number, -> Stokes-Einstein equation, -> wall jet electrode. 

Exact Solutions to the Navier-Stokes Equations. As was tme for the inviscid flow equations, exact solutions to the Navier-Stokes equations are limited to fairly simple configurations that aHow for considerable simplification both in the equation and in the boundary conditions. For the important situation of steady, fully developed, laminar, Newtonian flow in a circular tube, for example, the Navier-Stokes equations reduce to... 

Velocity Profiles In laminar flow, the solution of the Navier-Stokes equation, corresponding to the Hagen-PoiseuiUe equation, gives the velocity i as a Innction of radial position / in a circular pipe of radius R in terms of the average velocity V = Q/A. The parabolic profile, with centerline velocity t ce the average velocity, is shown in Fig. 6-10. 

The hydrauhc diameter method does not work well for laminar flow because the shape affects the flow resistance in a way that cannot be expressed as a function only of the ratio of cross-sectional area to wetted perimeter. For some shapes, the Navier-Stokes equations have been integrated to yield relations between flow rate and pressure drop. These relations may be expressed in terms of equivalent diameters Dg defined to make the relations reduce to the second form of the Hagen-Poiseulle equation, Eq. (6-36) that is, Dg (l2SQ[LL/ KAPy. Equivalent diameters are not the same as hydraulie diameters. Equivalent diameters yield the correct relation between flow rate and pressure drop when substituted into Eq. (6-36), but not Eq. (6-35) because V Q/(tiDe/4). Equivalent diameter Dg is not to be used in the friction factor and Reynolds number ... 

Assuming laminar flow for a linear momentum equation in the a direction (an approximation from the Navier-Stokes equations) gives... 

The steady, laminar, incompressible fluid flow in cyclone collectors is governed by the 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 Spite of the existence of numerous experimental and theoretical investigations, a number of principal problems related to micro-fluid hydrodynamics are not well-studied. There are contradictory data on the drag in micro-channels, transition from laminar to turbulent flow, etc. That leads to difficulties in understanding the essence of this phenomenon and is a basis for questionable discoveries of special microeffects (Duncan and Peterson 1994 Ho and Tai 1998 Plam 2000 Herwig 2000 Herwig and Hausner 2003 Gad-el-Hak 2003). The latter were revealed by comparison of experimental data with predictions of a conventional theory based on the Navier-Stokes equations. The discrepancy between these data was interpreted as a display of new effects of flow in micro-channels. It should be noted that actual conditions of several experiments were often not identical to conditions that were used in the theoretical models. For this reason, the analysis of sources of disparity between the theory and experiment is of significance. 

Wu and Cheng (2003) measured the friction factor of laminar flow of de-ionized water in smooth silicon micro-channels of trapezoidal cross-section with hydraulic diameters in the range of 25.9 to 291.0 pm. The experimental data were found to be in agreement within 11% with an existing theoretical solution for an incompressible, fully developed, laminar flow in trapezoidal channels under the no-slip boundary condition. It is confirmed that Navier-Stokes equations are still valid for the laminar flow of de-ionized water in smooth micro-channels having hydraulic diameter as small as 25.9 pm. For smooth channels with larger hydraulic diameters of 103.4-103.4-291.0pm, transition from laminar to turbulent flow occurred at Re = 1,500-2,000. 

There is an analytical solution of the Navier-Stokes equations for the flow between two rotating cylinders with laminar flow (see e.g. [37]). The following equation applies for the velocity gradient in the annular gap in the general case of rotation of the outer cylinder (index 2) and the inner cylinder (index 1) ... 

The most general equations for the laminar flow of a viscous incompressible fluid of constant physical properties are the Navier-Stokes equations. In terms of the rectangular coordinates x, y, z, these may be written ... 

It is assumed that the instantaneous Navier-Stokes equations for turbulent flows have the exact form of those for laminar flows. From the Reynolds decomposition, any instantaneous variable, (j>, can be divided into a time-averaged quantity and a fluctuating part as... 

We should note that the Navier-Stokes equation holds only for Newtonian fluids and incompressible flows. Yet this equation, together with the equation of continuity and with proper initial and boundary conditions, provides all the equations needed to solve (analytically or numerically) any laminar, isothermal flow problem. Solution of these equations yields the pressure and velocity fields that, in turn, give the stress and rate of strain fields and the flow rate. If the flow is nonisothermal, then simultaneously with the foregoing equations, we must solve the thermal energy equation, which is discussed later in this chapter. In this case, if the temperature differences are significant, we must also account for the temperature dependence of the viscosity, density, and thermal conductivity. 

Let us deal with the equation of motion for turbulent flow. In the case of laminar flow under the condition of constant density and constant viscosity, the equation of motion is expressed by the Navier-Stokes equation as... 

The difference between this equation for turbulent flow and the Navier-Stokes equation for laminar flow is the Reynolds stress/turbulent stress term —pujuj appears in the equation of motion for turbulent flow. This equation of motion for turbulent flow involves non-linear terms, and it is impossible to be solved analytically. In order to solve the equation in the same way as the Navier-Stokes equation, the Reynolds stress or fluctuating velocity must be known or calculated. Two methods have been adopted to avoid this problem—phenomenological method and statistical method. In the phenomenological method, the Reynolds stress is considered to be proportional to the average velocity gradient and the proportional coefficient is considered to be turbulent viscosity or mixing length ... 

Magnetic resonance imaging permitted direct observation of the liquid hold-up in monolith channels in a noninvasive manner. As shown in Fig. 8.14, the film thickness - and therefore the wetting of the channel wall and the liquid hold-up -increase nonlinearly with the flow rate. This is in agreement with a hydrodynamic model, based on the Navier-Stokes equations for laminar flow and full-slip assumption at the gas-liquid interface. Even at superficial velocities of 4 cm s-1, the liquid occupies not more than 15 % of the free channel cross-sectional area. This relates to about 10 % of the total reactor volume. Van Baten, Ellenberger and Krishna [21] measured the liquid hold-up of katapak-S . Due to the capillary forces, the liquid almost completely fills the volume between the catalyst particles in the tea bags (about 20 % of the total reactor volume) even at liquid flow rates of 0.2 cm s-1 (Fig. 8.15). The formation of films and rivulets in the open channels of the structure cause the further slight increase of the hold-up. 

Comparing these equations with the x- and y- Navier-Stokes equations for two-dimensional laminar flow shows that in turbulent flow extra terms arise due to the presence of the fluctuating velocity components. These extra terms, which arise because the Navier-Stokes equations contain nonlinear terms, are the result of the momentum transfer caused by the velocity fluctuating components and are often termed the turbulent or Reynolds stress terms because of their similarity to the viscous stress terms which arise due to momentum transfer on a molecular scale. This similarity can be clearly seen by noting that the x-wise momentum equation, for example, for laminar flow can be written as ... 

Compared to the x-wise Navier-Stokes equation for two-dimensional laminar flow, the equivalent equation for turbulent flow has, when the time averaged values of the... 

After Navier- Stokes equation has been written down in the first half of nineteenth century, few exact solutions were obtained for few fluid flows. In one such case, Stokes compared theoretical prediction with available experimental data for pipe flow and found no agreement whatsoever. Now we know that the theoretical solution of Stokes corresponded to undisturbed laminar flow, while the experimental data given to him corresponded to a turbulent flow. This problem was seized upon by Osborne Reynolds, who explained the reason for such mismatches by his famous pipe flow experiments (Reynolds, 1883). It was shown that the basic flow obtained as a... 

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. 

Classically, electrokinetic phenomena are described by the equation of Navier-Stokes along with the continuity equation [ 1-3J. The Navier-Stokes equation accounts for the balance of forces in the electrokinetic problem. For steady laminar fluid flow the Navier-Stokes equation takes the following form in electrokinetics ... 

A drawback to the rectangular cell design is that the hydrodynamic description of fluid flow is much more complicated than in the cylindrical cell. Though it is beyond the intent of this discussion to go into full detail, analytical solutions to the Navier-Stokes equation for steady laminar fluid flow have been derived that can be used for calculation of electro-osmosis at flat plates in the rectangular cell configuration where electro-osmosis may differ at the upper, lower, and side chamber walls [ 15). 

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