**9. References**

32 Hydrodynamics – Advanced Topics

6 12 3 45

In the system of equations (AII-8), generated from equations (AII-4) through (AII-6), *x* = *z*

The system of equations (AII-1) through (AII-6) was solved using a spreadsheet for Microsoft Excel®, available at www.stoa.usp.br/hidraulica/files/. Two initial values were fixed and one was calculated. Note that in the present study it was intended to verify if the method furnishes a viable profile, so that boundary or initial values obtained from the experimental data were assumed as adequate. The first was *n*(0)=1. The second was *n'*(0)=-3, corresponding to the experiments of Janzen (2006). The third information did not constitute an initial value, and was *n*(1)=0 or 0<*n*(1)<0.01 (threshold value corresponding to the definition of the boundary layer). As the Runge-Kutta methods need initial values, this information was used to obtain *n''*(0), the remaining initial value needed to perform the calculations. With the aid of the Newton (or quasi-Newton) method, it was possible to

The derivative of *n* at *z*=0 is generally unknown in such mass transfer problems. In this case, solutions must be found considering, for example, *n*(0)=1, 0<*n*(1)<0.01 and *n'*(1)=0 (three reasonable boundary conditions), for which another scheme must be developed to calculate the first and second derivatives at the origin. As mentioned, the aim of this study was to verify the applicability of the method. The details of solutions for different purposes must be

i. determine the initial values: *n*(0) = 1, *n'*(0) = -3 (or other appropriate value) *n''*(0) =

ii. Compute ���� and ����, the function values *f*1, *f*2 e *f*3 with the initial values, and then ����. In the variable ����, *i* = 1,2,...,6 and *j* = 1,2,3, the first index corresponds to the six stages of the method and the second to the order of the ODE that generated the original

iii. With the values calculated in (ii), calculate now *n*k+(1/4)����� Δ*z*, *j*k+(1/4) ���� Δ*z* and

The spreadsheet available at www.stoa.usp.br/hidraulica/files/ presents some suggestions that simplify some items of the above described steps (some manual work is simplified). The

 

 

3 2 12 12 8 , 77 7 7 7

 

(AII-8)

*f x xy x x x x x*

 

in which

<sup>1</sup>

 

initial guess;

system to be solved;

*fx y*

,

*k k*

2 1

*k k*

*k k*

*k k*

*k k*

*k k*

and *y* = *n* , following the representation used in this chapter.

3 1 2

 

1 11 , 4 88

 

*f x xy x x*

*f x xy x x*

3 39 , 4 16 16

*f x xy x x*

obtain values for *n''*(0) that satisfied the third condition imposed at *z* = 1.

The construction of the spreadsheet is described in the following steps:

*w*k+(1/4) ���� Δ*z*. The following steps are similar until *j* = 6; iv. Equation AII-7 (a system) is then used to advance in space *z*.

estimate of n"(0), for example, is obtained following simplified procedures.

considered by the researchers interested in that solution.

1 1 , 4 4

*f x xy x*

4 2 3

1 1 , 2 2

 

5 1 4


**2** 

*Russia* 

German A. Maximov

*N. N. Andreyev Acoustical Institute* 

**Generalized Variational Principle for Dissipative** 

A system of hydrodynamic equations for a viscous, heat conducting fluid is usually derived on the basis of the mass, the momentum and the energy conservation laws (Landau & Lifshitz, 1986). Certain assumptions about the form of the viscous stress tensor and the energy density flow vector are made to derive such a system of equations for the dissipative viscous, heat conductive fluid. The system of equations based on the mass, the momentum and the energy conservation laws describes adequately a large set of hydrodynamical phenomena. However, there are some aspects which suggest that this system is only an

For example, if we consider propagation of small perturbations described by this system, then it is possible to separate formally the longitudinal, shear and heat or entropy waves. The coupling of the longitudinal and heat waves results in their splitting into independent acoustic-thermal and thermo-acoustic modes. For these modes the limits of phase velocities tends to infinity at high frequencies so that the system is in formal contradiction with the requirements for a finite propagation velocity of any perturbation which the medium can undergo. Thus it is possible to suggest that such a hydrodynamic equation system is a mere low frequency approximation. Introducing the effects of viscosity relaxation (Landau & Lifshitz, 1972), guarantees a limit for the propagation velocity of the shear mode, and the introduction of the heat relaxation term (Deresiewicz, 1957; Nettleton, 1960; Lykov, 1967) in turn ensures finite propagation velocities of the acoustic-thermal and thermo-acoustic modes. However, the introduction of such relaxation processes requires serious effort with

Classical mechanics provides us with the Lagrange's variational principle which allows us to derive rigorously the equations of motion for a mechanical system knowing the forms of kinetic and potential energies. The difference between these energies determines the form of the Lagrange function. This approach translates directly into continuum mechanics by introduction of the Lagrangian density for non-dissipative media. In this approach the dissipation forces can be accounted for by the introduction of the dissipation function derivatives into the corresponding equations of motion in accordance with Onsager's

**1. Introduction** 

approximation.

motivation.

**Hydrodynamics: Shear Viscosity from Angular** 

**Momentum Relaxation in the Hydrodynamical** 

**Description of Continuum Mechanics** 

and PDE, June 27 to July 1, Institute of Mathematics, Statistics and Scientific Computation, Campinas, Brazil.

