Navier Stokes Characteristic

Experimental and numerical study of the pressure drop and heat transfer in a single-phase micro-channel heat sink by Qu and Mudawar (2002a,b) demonstrated that the conventional Navier-Stokes and energy equations can adequately predict the fluid flow and heat transfer characteristics. 

Experimental and numerical analyses were performed on the heat transfer characteristics of water flowing through triangular silicon micro-channels with hydraulic diameter of 160 pm in the range of Reynolds number Re = 3.2—84 (Tiselj et al. 2004). It was shown that dissipation effects can be neglected and the heat transfer may be described by conventional Navier-Stokes and energy equations as a common basis. Experiments carried out by Hetsroni et al. (2004) in a pipe of inner diameter of 1.07 mm also did not show effect of the Brinkman number on the Nusselt number in the range Re = 10—100. 

Historically, gas lubrication theory was developed from the classical liquid lubrication equation—Re5molds equation [4]. The first gas lubrication equation was derived by Harrison [5] in 1913, taking the compressibility of gases into account. Because the classical gas lubrication equation is based on the Navier-Stokes equation, it does not incorporate some gas flow characteristics rooted in the rarefaction effects of dilute gases. As early as 1959, Brunner s experiment [6] showed that the classical gas lubrication equation was... 

In fluid dynamics the behavior in this system is described by the full set of hydrodynamic equations. This behavior can be characterized by the Reynolds number. Re, which is the ratio of characteristic flow scales to viscosity scales. We recall that the Reynolds number is a measure of the dominating terms in the Navier-Stokes equation and, if the Reynolds number is small, linear terms will dominate if it is large, nonlinear terms will dominate. In this system, the nonlinear term, (u V)u, serves to convert linear momentum into angular momentum. This phenomena is evidenced by the appearance of two counter-rotating vortices or eddies immediately behind the obstacle. Experiments and numerical integration of the Navier-Stokes equations predict the formation of these vortices at the length scale of the obstacle. Further, they predict that the distance between the vortex center and the obstacle is proportional to the Reynolds number. All these have been observed in our 2-dimensional flow system obstructed by a thermal plate at microscopic scales. ... 

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 Navier-Stokes equations equations involve the pressure gradient, but the pressure itself does not appear explicitly. As a result a further simplification is often available and useful. Assuming nominal atmospheric pressure (patm 105 N/m2), pressure variations associated with the characteristic velocity scales are very often quite small. For air at standard atmospheric conditions, the sound speed is a0 350 m/s. The pressure variations for a low-speed atmospheric flow, say u0 = 10 m/s, are around p p0u20 100, which is three orders of magnitude lower than p0. Thus the pressure field can be usefully separated into two components [255,303] as... 

Our intent here is not to suggest a solution method but rather to use the stream-function-vorticity formulation to comment further on the mathematical characteristics of the Navier-Stokes equations. In this form the hyperbolic behavior of the pressure has been lost from the system. For low-speed flow the pressure gradients are so small that they do not measurably affect the net pressure from a thermodynamic point of view. Therefore the pressure of the system can simply be provided as a fixed parameter that enters the equation of state. Thus pressure influences density, still accommodating variations in temperature and composition. Since the pressure or the pressure gradients simply do not appear anywhere else in the system, pressure-wave behavior has been effectively filtered out of the system. Consequently acoustic behavior or high-speed flow cannot be modeled using this approach. 

The steady-state heat equation (Eq. 3.284) is often used as the model equation for an elliptic partial-differential equation. An important property of elliptic equations is that the solution at any point within the domain is influenced by every point on the boundary. Thus boundary conditions must be supplied everywhere on the boundaries of the solution domain. The viscous terms in the Navier-Stokes equations clearly have elliptic characteristics. 

It is clear that sound, meaning pressure waves, travels at finite speed. Thus some of the hyperbolic—wavelike-characteristics associated with pressure are in accord with everyday experience. As a fluid becomes more incompressible (e.g., water relative to air), the sound speed increases. In a truly incompressible fluid, pressure travels at infinite speed. When the wave speed is infinite, the pressure effects become parabolic or elliptic, rather than hyperbolic. The pressure terms in the Navier-Stokes equations do not change in the transition from hyperbolic to elliptic. Instead, the equation of state changes. That is, the relationship between pressure and density change and the time derivative is lost from the continuity equation. Therefore the situation does not permit a simple characterization by inspection of first and second derivatives. 

Scaling arguments are used to establish the circumstances where the boundary-layer behavior is valid. These arguments, which are usually made for external flows over surfaces, may be found in many texts on fluid mechanics (e.g., [350]). The essential feature of the boundary-layer approximation is that there is a principal flow direction in which the convective effects significantly dominate the diffusive behavior. As a result the flow-wise diffusion may be neglected, while the cross-flow diffusion and convection are retained. Mathematically this reduction causes the boundary-layer equations to have essentially parabolic characteristics, whereas the Navier-Stokes equations have essentially elliptic characteristics. As a result the computational simulation of the boundary-layer equations is much simpler and more efficient. 

The other method is the velocity head method. The term V2/2g has dimensions of length and is commonly called a velocity head. Application of the Bernoulli equation to the problem of frictionless discharge at velocity V through a nozzle at the bottom of a column of liquid of height H shows that H = V2/2g. Thus II is the liquid head corresponding to the velocity V. Use of the velocity head to scale pressure drops has wide application in fluid mechanics. Examination of the Navier-Stokes equations suggests that when the inertial terms dominate the viscous terms, pressure gradients are expected to be proportional to pV2 where V is a characteristic velocity of the flow. 

Because the Reynolds number is much smaller than 1 and a Newtonian flow behavior is being observed in the first place, the Navier-Stokes equations convert to Stokes equations, and we obtain a system of linear equations for the flow calculations. It therefore follows that there must be a linear relationship both between the flow rate and the pressure and between the flow rate and the power. This is demonstrated in Figs. 8.10 and 8.11 in which the dimensionless conveying and power characteristic are illustrated, respectively. The red lines reflect the Newtonian cases. As expected, a linear relationship is revealed. 

In problems of heat convection, the most complex equations to solve are the fluid flow equations. Often times, the governing equations for the fluid flow are the Navier-Stokes equations. It is useful, therefore, to study a model equation that has similar characteristics to the Navier-Stokes equations. This model equation has to be time-dependent and include both convection and diffusion terms. The viscous Burgers equation is an appropriate model equation. In the first few sections of this chapter, several important numerical schemes for the Burgers equation will be discussed. A simple physical heat convection problem is solved as a demonstration. 

It is also possible to derive the Reynolds number by dimensional analysis. This represents a more analytical, but less intuitive, approach to defining the condition of similar fluid flow and is essentially independent of particular shape. In this approach, variables in the Navier-Stokes equation (relative particle-fluid velocity, a characteristic dimension of the particle, fluid density, and fluid viscosity) are combined to yield a dimensionless expression. Thus... 

The Navier-Stokes equations are valid when A is much smaller than the characteristic flow dimension L. When this condition is violated, the flow is no longer near equilibrium and the linear relations between stress and rate of strain and the no-slip velocity condition are no longer valid. Similarly, the linear relation between heat flux and temperature gradient and the no-jump temperature condition at a solid-fluid interface are no longer accurate when A is not much smaller than L. The different Knudsen number regimes are delineated in Fig. 2. 

It was shown that, down to the characteristic dimension of 7 (im and for the flnids nsed, the hydrodynamics obey the conventional theories deduced from the Navier - Stokes equations. The effect of roughness on the flow behaviour needs complementary work. 

As far as convective heat transfer is concerned, liquid and gaseous flows musf be considered separately. Liquid flow has been investigated experimentally, whereas analytical, numerical and molecular simulation techniques have been applied to understand the characteristics of gaseous flow and heat transfer. While the Navier-Stokes equations can still be applied, due to the small size of microchannels, some deviations from the conventionally sized applications have been observed. Flow regime boundaries are significantly different, as well as flow and heat transfer characteristics. 

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