Neumann solution

Recently Ruoff (Rll) has rederived the Stefan and Neumann solutions using the Boltzmann transformation. 

In this example, we consider the classic Stefan-Neumann solution. The solid is initially at a constant temperature Tq. At time t — 0 the surface temperature is raised to T, which is above the melting point, Tm. The physical properties of each phase are different, but they are temperature independent, and the change in phase involves a latent heat of fusion A. After a certain time t, the thickness of the molten layer is Xi(t) in each phase there is a temperature distribution and the interface is at the melting temperature Tm (Fig. E5.4). 

Neumann Solution. The instantaneous temperature rise for arbitrary dimensionless time Fo > 0 at arbitrary radius rla > 1 is given by the integral solution ... 

K. J. Negus and M. M. Yovanovich, Constriction Resistance of Circular Flux Tubes With Mixed Boundary Conditions by Linear Superposition of Neumann Solutions, ASME 84-HT-84,1984. 

In the context of this meeting which is devoted to the recollection of the beginnings of engineering education and to the progress of this education, the present introduction cannot be concluded without the remark that also the "changed view" on the Stefan problem can find its precipitate in elementary textbooks on transport phenomena where only the Neumann solution was presented so far, if heat transfer accompanied by a phase change was discussed at all. This will be demonstrated in Section A. 

As an example of the application of the method, Neumann and Tanner [54] followed the variation with time of the surface tension of aqueous sodium dode-cyl sulfate solutions. Their results are shown in Fig. 11-15, and it is seen that a slow but considerable change occurred. 

Altematively, in the case of incoherent (e.g. statistical) initial conditions, the density matrix operator P(t) I 1>(0) (v(01 at time t can be obtained as the solution of the Liouville-von Neumann equation ... 

We prove the existence of solutions for the three-dimensional elastoplastic problem with Hencky s law and Neumann boundary conditions by elliptic regularization and the penalty method, both for the case of a smooth boundary and of an interior two-dimensional crack (see Brokate, Khludnev, 1998). It is shown in particular that the variational solution satisfies all boundary conditions. 

In this section the existence of a solution to the three-dimensional elastoplastic problem with the Prandtl-Reuss constitutive law and the Neumann boundary conditions is obtained. The proof is based on a suitable combination of the parabolic regularization of equations and the penalty method for the elastoplastic yield condition. The method is applied in the case of the domain with a smooth boundary as well as in the case of an interior two-dimensional crack. It is shown that the weak solutions to the elastoplastic problem satisfying the variational inequality meet all boundary conditions. The results of this section can be found in (Khludnev, Sokolowski, 1998a). 

Because the Navier-Stokes equations are first-order in pressure and second-order in velocity, their solution requires one pressure bound-aiy condition and two velocity boundaiy conditions (for each velocity component) to completely specify the solution. The no sBp condition, whicn requires that the fluid velocity equal the velocity or any bounding solid surface, occurs in most problems. Specification of velocity is a type of boundary condition sometimes called a Dirichlet condition. Often boundary conditions involve stresses, and thus velocity gradients, rather than the velocities themselves. Specification of velocity derivatives is a Neumann boundary condition. For example, at the boundary between a viscous liquid and a gas, it is often assumed that the liquid shear stresses are zero. In numerical solution of the Navier-... 

The metal cluster will be modeled as an infinitely deep spherical potential well with the represented by an infinitely high spherical barrier. Let us place this barrier in the center of the spherical cluster to simplify the calculations. The simple Schrodinger equation, containing only the interaction of the electrons with the static potential and the kinetic energy term and neglecting any electron-electron interaction, can then be solved analytically, the solutions for the radial wave functions being linear combinations of spherical Bessel and Neumann functions. 

Von Neumann recognized this problem, of course. His solution was to essentially use the cooperative action of several automata to effectively copy a machine s blueprint. He first introduced a copier automaton M that copies whatever blueprint B it is given. Next, he defined an automaton A" that inserts a copy of B into the... 

The deviation of the heat capacity of a solid solution from the heat capacity calculated by the additivity hypothesis (Kopp-Neumann rule), a quantity of importance for the evaluation of the... 

Within esqjlicit schemes the computational effort to obtain the solution at the new time step is very small the main effort lies in a multiplication of the old solution vector with the coeflicient matrix. In contrast, implicit schemes require the solution of an algebraic system of equations to obtain the new solution vector. However, the major disadvantage of explicit schemes is their instability [84]. The term stability is defined via the behavior of the numerical solution for t —> . A numerical method is regarded as stable if the approximate solution remains bounded for t —> oo, given that the exact solution is also bounded. Explicit time-step schemes tend to become unstable when the time step size exceeds a certain value (an example of a stability limit for PDE solvers is the von-Neumann criterion [85]). In contrast, implicit methods are usually stable. 

P. Imas, B. Bar-Yosef, U. Kakafi, and R. Ganmore-Neumann, Carboxylic anions and proton secretion by tomato roots in response to ammonium/nitrate ratio and pH in nutrient. solution. Plant Soil 191 (1997). 

Applied Surface Thermodynamics, edited by A. 14/. Neumann and Jan K. Spelt Surfactants in Solution, edited by Arun K. Chattopadhyay and K. L. Mittal Detergents in the Environment, edited by Milan Johann Schwuger Industrial Applications of Microemulsions, edited by Conxita Solans and Hironobu Kunieda... 

Hummer, G. Soumpasis, D. M. Neumann, M., Computer simulations do not support Cl-Cl pairing in aqueous NaCl solution, Mol. Phys. 1993, 81, 1155-1163. 

Prior to an effective Hamiltonian analysis it is, in order to get this converging to the lowest orders, typical to remove the dominant rf irradiation from the description by transforming the internal Hamiltonian into the interaction frame of the rf irradiation. This procedure is well established and also used in the most simple description of NMR experiments by transforming the Hamiltonian into the rotating frame of the Zeeman interaction (the so-called Zeeman interaction frame). In the Zeeman interaction frame the time-modulations of the rf terms are removed and the internal Hamiltonian is truncated to form the secular high-field approximated Hamiltonian - all facilitating solution of the Liouville-von-Neumann equation in (1) and (2). The transformation into the rf interaction frame is given by... 

By analogy the propagation of a density matrix, which corresponds to the solution of the Liouville-von Neumann equation 231... 

We shall now make use of Neumann s formula to derive the form of the second solution of Legendre s cqnntion in the neighbourhood of the points ft = 1, From (18.1) wc have the result that, if fi > 1,... 

The function Y0 at) so obtained is called Neumann s Bessel function of llio second kind of zero order. Obviously if we add to Yn f) a function which is a constant multiple of >/0(.t) the resulting function is also a solution of the differential equation... 

Many electron systems such as molecules and quantum dots show the complex phenomena of electron correlation caused by Coulomb interactions. These phenomena can be described to some extent by the Hubbard model [76]. This is a simple model that captures the main physics of the problem and admits an exact solution in some special cases [77]. To calculate the entanglement for electrons described by this model, we will use Zanardi s measure, which is given in Fock space as the von Neumann entropy [78]. 

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