The Inviscid Limit

As stated at the outset, our goal is to investigate the fate of an infinitesimal perturbation to the base state, which in this case is a stationary circular cylinder of constant radius in which the pressure inside differs from the air pressure by a/a (i.e., the dimensionless pressure difference is one). Hence we consider an initial perturbation in all of the independent variables (including the shape function/ which we have already introduced as ef)  

The evolution of this initial perturbation is governed by the equations and boundary conditions (12—14)—(12—18). However, because the magnitude of this perturbation is small, we can obtain approximate equations by substituting (12-19) into (12 14)—(12— 18), and retaining only the terms that are 0(e). As stated above, we also consider only the limiting case Re 1, and thus we also neglect all terms of ()(Re ). The final result for the governing equations is 

The limit in this case can be seen to reduce the equations to the linearized stability equations for an inviscid fluid. As a consequence, not all of the interface boundary conditions can be satisfied. Our experience from Chap. 10 shows that we should not expect the solution to satisfy the zero-shear-stress condition, which will come into play only if we were to 

The equations (12-20)-( 12-24) are the so-called linear stability equations for this problem in the inviscid fluid limit. We wish to use these equations to investigate whether an arbitrary, infinitesimal perturbation will grow or decay in time. Although the perturbation has an arbitrary form, we expect that it must satisfy the linear stability equations. Thus, once we specify an initial form for one of the variables like the pressure p, we assume that the other variables take a form that is consistent with p by means of Eqs. (12-20)-(12-24). Now the obvious question is this How do we represent a disturbance function of arbitrary form For this, we take advantage of the fact that the governing equations and boundary conditions are now linear, so that we can represent any smooth disturbance function by means of a Fourier series representation. Instead of literally studying a disturbance function of arbitrary form, we study the dynamics of all of the possible Fourier modes. If any mode is found to grow with time, the system is unstable because, with a disturbance of infinitesimal amplitude, every possible mode will always be present. 

Now the most convenient starting point is to assume that the cylinder suffers a infinitesimal perturbation in its shape of the form3 

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In the inviscid limit, the general linear stability problem takes the following simpler form. First, the governing equations, (12-65a) and (12-65b), are reduced to a pair of second-order DEs ... 

A key idea from the preceding analysis, as well as the analysis of capillary instability in section A, is that viscous effects often cannot change the conditions for instability of a rest state, but only moderate the rate of growth or decay of disturbances. In such cases, analysis of the stability (or instability) of the inviscid limit can be extremely useftd in identifying the conditions for instability. In view of the fact that the inviscid analysis is very much simpler, this is an important observation. 

Both correlations collapse to the Chen and Middleman result in the inviscid limit. 

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... 

Because the class of problems that we can currently study is restricted to flows in which the Reynolds number is very small, Re . 1, and because Pe = RePr, an obvious question is whether the combination Re PP 1 and Pe 1 is achievable in real systems. The key is to remember that the Prandtl number is an independent material parameter. For gases, for which Pr < 0(1), and for relatively inviscid liquids such as water that have Pr 0(1) or slightly larger, Pe will always be small when Re is small. On the other hand, viscous oils and greases can have Pr 103 — 106, and for these fluids Pe may be large, even though Re is small. This provides one clear motivation for studying heat transfer for the dual limit Re -PP 0(1) andPe > 0(1). 

Clearly, in the limit Re y> 1, the leading-order approximation for the solution to this problem is identical to the inviscid flow problem for a solid sphere. Although the no-slip boundary condition has been replaced in the present problem with the zero-shear-stress condition, (10-197), this has no influence on the leading-order inviscid flow approximation because the potential-flow solution can, in any case, only satisfy the kinematic condition u n = 0 at r = 1. Hence the first approximation in the outer part of the domain where the bubble radius is an appropriate characteristic length scale is precisely the same as for the noslip sphere, namely, (10-155) and (10-156). However, this solution does not satisfy the zero-shear-stress condition (10-197) at the bubble surface, and thus it is clear that the inviscid flow equations do not provide a uniformly valid approximation to the Navier-Stokes... 

The both considered limit situations can be encountered in numerous problems of convective heat transfer they are schematically shown in Figure 3.1. One can see that in the case Pr — 0, which approximately takes place for liquid metals (e.g., mercury), one can neglect the dynamic boundary layer in the calculation of the temperature boundary layer and replace the velocity profile v(x, y) by the velocity v<, (x) of the inviscid outer flow. As Pr-)- oo, which corresponds to the case of strongly viscous fluids (e.g., glycerin), the temperature boundary layer is very thin and lies inside the dynamic boundary layer, where the velocity increases linearly with the distance from the plate surface. 

It is emphasized that (1.52) is equivalent to the solution of (1.39), and, in the inviscid case, to the long-wave limit of the Rayleigh result. 

Based on the comparison of three predictive equations as shown in Fig. 6, the modified Mendelson equation, which is valid only under the inviscid condition, has limited predictive capability at the low temperature... 

The Darcy flow limit. In reservoir engineering, Equations 1-1 and 1-2, known as Darcy s equations, apply (Muskat, 1937). Historically, they were determined empirically by the French engineer Henri Darcy, who observed that the inviscid, high Reynolds number models then in vogue did not describe hydraulics problems. Darcy s laws do not follow immediately from Equations 1-21 and 1-22, but they can be derived through an averaging process taken over many pore spaces and, then, only in the low Reynolds number limit (Batchelor, 1970). If Equations 1-1 and 1-2 are substituted in Equation 1-20 and if constant viscosities and constant isotropic permeabilities are further assumed, Laplace s equation 5 p/9x -l- 5 p/9y = 0 for reservoir pressure p(x,y) follows. Now let us derive the Laplace s equation used in aerodynamics. 

The aerodynamic limit. Inviscid aerodynamics, the study of nonviscous flow, is obtained by contrast in the limit of infinite Reynolds number. In this limit. Equations 1-21 and 1-22 become... 

Validity of Laplace s equation. Sinee Laplaee s equation, that is. Equation 1-13 for the Darcy pressure and Equation 1-30 for the inviscid aerodynamic potential, arise in both problems as a result of different physical limits, it is of interest to ask when the approximate models apply and why. This understanding is crucial to the translation process alluded to earlier, so that fixes used in aerodynamics, which may be inappropriate to Darey flows, can be removed if and when they are present. It is especially important because the analogies presupposed by nonspecialists are sometimes not analogous at all. 

A common approximation in many flow field computations at high fluid velocities is to consider that inertial forces dominate the flow and to neglect viscous forces (inviscid approximation). Since solvent viscosity is a variable in some of the experiments discussed here, the above approximation may be not be valid throughout and viscous forces are explicitly considered in the flow equations. Results of computations showed, nevertheless, that even with viscous solvents such as bis-(2-ethyl-hexyl)-phtalate with qi = 65 mPa s, viscous forces do not affect the flow field unless tbe fluid velocity drops below a few m s"1 at the orifice. This limit is generally more than one order of magnitude lower than the actual range used in the present investigations. 

As the Reynolds number increases, dV/dz is approximately a constant over a greater region. In the limit of infinite Reynolds number, the entire gap flow is inviscid. These solutions are represented as dashed lines in Fig. 6.7. When the viscous term vanishes, the... 

This behavior stems from the fact that there is an essentially inviscid region between the inlet manifold and viscous boundary layer near the surface. As the Reynolds number increases, the viscous layer becomes thinner. As the Reynolds number decreases below around 10, the viscous layer fills the entire gap. For sufficiently low Reynolds number, the fluid flow becomes negligible and the heat transfer is characterized by thermal conduction. In that limit, Nu = 1. 

Discuss the limit > ->- 0. Show that there is a contribution to F for o> 1 that is independent of the viscosity. This term is known as the added-mass contribution and is identical to the force on an accelerating sphere in an inviscid fluid (that is, a fluid with /x = 0). It predicts that there is a contribution to the force that has the effect of adding an additional mass to the sphere that is equal to 1 /2 of the sphere volume times the fluid density. 

We shall see that the fact that the fluids are viscous does not play a critical role in determining whether a pair of fluids with different density is unstable or not. We begin in this section by solving the linear stability problem in the limit where both fluids are assumed to be inviscid. Then we will return in Subsection 2 to consider how the problem is changed when the fluid viscosity is not neglected. 

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