Laplace’s formula

FinaUy, under the pressure P, at the temperature t, sound is propagated in the gas considered with a velocity V, which, by Laplace s formula, has the value... 

Substituting in (9.18) gives Laplace s formula for the surface pressure ... 

Panicle huental Pressure Laplace s Formula 257 Limit of Applicability of Kelvin Relation 258 Hygroscopic Particle-Vapor Equilibrium 259 Charged Particle-Vapor Equilibria 263 Solid-Particle-Vapor Equilibrium 265... 

The pressure in the liquid bridge (Figure. 62.1a) is negative if r2 < ri (liquid is in tension), or positive if r2 > r, (liquid is in compression). These effects follow from the Laplace s formula ... 

If ai and (Fig- IV.6.a) are the radii of curvature of the water interlayer, the capillary pressure is determined by Laplace s formula ... 

The methods now being used to determine disjoining pressure have made it possible to obtain isotherms relating the disjoining pressure to the thickness of the liquid layer separating the boundary phases [172]. The magnitude of the disjoining pressure for a liquid interlayer located between an air bubble and a solid substrate can be calculated from Laplace s formula... 

Laplace s formula gives the jump in pressure across a general curved interface ... 

Fig. 1.4. Laplace s formula definition of the curvature C. (a) A spherical droplet of radius R C = 2/R= const.), (b) An interface with varying curvature (a pear). C M) = 1/Ri + I/R2...

Fig. 1.5. Illustration of Laplace s formula (a) small bubbles empty into larger ones (b) a soap film drawn between two circular frames defines a surface of zero curvature (c) the wavelike shape of drop of liquid on a fibre...

Jurin s law describes the spontaneous ascent of a liquid up a fine capillary tube (radius R < n ) (see Fig. 1.10). Dip the end of an extremely clean glass capillary tube (cleaned with sulfochromic acid) into water. The water will rise up to a height h and, as the water totally wets the glass, the meniscus will be hemispherical. The pressure at point A, just inside the liquid, can be calculated using Laplace s formula. If Pq is the outside pressure, then... 

Minimal surfaces can be calculated with the help of Laplace s formula [equation (1.6)]. Alternatively, one can minimize the surface, keeping the volume constant, by means of Euler-Lagrange s equations. 

We have just rediscovered Laplace s formula 7(- + ) = A, which proves that A = Ap. 

This relation is called Laplace s formula, among other Laplace law or relations. ... 

Recall that the determinant of the N x N matrix can be represented as the sum of the Nl products of the matrix elements. According to Laplace s formula, a determinant can be expanded along a row or a column. Thus, calculation of the ( Fo di T > integrals, when the occupied orbital a in To has been exchanged with the virtual orbital p in Tf, can be performed by expanding the determinant To along the o-th row and the determinant T along the p-th row ... 

In this chapter, analytical solutions were obtained for parabolic and elliptic partial differential equations in semi-infinite domains. In section 4.2, the given linear parabolic partial differential equations were converted to an ordinary differential equation boundary value problem in the Laplace domain. The dependent variable was then solved in the Laplace domain using Maple s dsolve command. The solution obtained in the Laplace domain was then converted to the time domain using Maple s inverse Laplace transform technique. Maple is not capable of inverting complicated functions. Two such examples were illustrated in section 4.3. As shown in section 4.3, even when Maple fails, one can arrive at the transient solution by simplifying the integrals using standard Laplace transform formulae. 

We shall now illustrate the use of dimensional analysis with the appHcation of Laplace s law to mammalian hearts. The beat-to-beat pumping ability of the mammalian heart is determined by its force-generating capability and the lengths of its constituent muscle fibers, as governed by the Starling s experimental observations on the heart. The formula for calculating force or tension, however, has been based on the law of Laplace... 

In establishing practical procedures for tidal reductions of levellings and ellipsoidal height differences observed by GPS the choice of the quasi-stationary state ("zero" or "mean" state) is decisive. Another feature of the reduction procedure concerns the type of representation of the tidal potential, i. e. either Laplace s tidal formula (6) or the harmonic development (8). In the folowing the resulting four alternatives are presented. 

Using the Fourier-Laplace transforms (x, t) (k, s), we derive from the balance equations (8.40) and (8.42) expressions for sp ik, s) - pf k). Using the Fourier-Laplace inversion formula, we obtain a system of integro-differential equations. 

Discontinuous properties. The preceding formulas assume continuously varying formation properties. Problems for discontinuous, but piece-wise constant properties, are also easily solved. For such problems, Laplace s equation (see Equation 6-1) apphes in each separate flow annulus it is convenient to append a subscript i or j corresponding to each pressure function. Each solution then takes the form Pq = + log r. The integration constants... 

It is worth noting here that the same estimate for no( ) was established before for ATM with optimal set of Chebyshev s parameters, but other formulas were used to specify r] in terms of and A - If R = —A, where A is the difference Laplace operator, and the Dirichlet problem is posed on a square grid in a unit square, then... 

---