Laplace s law

A real foam has further degrees of freedom available for estabHshing local mechanical equiHbrium the films and Plateau borders may curve. In fact, curvature can be readily seen in the borders of Figure 1. In order to maintain such curvature, there must be a pressure difference between adjacent bubbles given by Laplace s law according to the surface free energy of the film and the principle radii of curvature of the film AP = ) Note that the... 

If the pressure inside the bubble is 2 7Jr (Laplace s law) and the laws of ideal gases are applicable, then with the volume V = (4/3) rr r3 the relation... 

According to Laplace s law, a reduction in ventricular pressure and heart size results in a decrease in the myocardial wall tension that is required to develop a given intraventricular pressure and therefore decreases oxygen requirement. Since blood flow to the subendocardium occurs primarily in diastole, the reduction in left ventricular end diastolic pressure induced by nitroglycerin reduces extravascular compression around the subendocardial vessels and favors redistribution of... 

LaPlace s Law Let s imagine a very thin film of a liquid, as shown in Figure 7.23. The surface tension in the liquid causes it to curl up. Since we have a two-dimensional surface, we can curl the liquid in only two dimensions. Let s call the radii of curling Rt and Rr The black arrows represent the x, y, and z axes of a Cartesian coordinate system. 

When the liquid is in contact with another surface, this curling creates a pressure difference, which is described by LaPlace s Law. The Greek letter gamma (7) is the surface tension, P is pressure, and R is the radius of curvature. 

We can use LaPlace s law to calculate the distance a fluid will rise in a capillary. In this system, we have two opposing forces the upward pressure due to LaPlace s law and the downward pressure due to gravity. The fluid will rise until these pressures are equal. We will begin with a model in which the meniscus is a sphere of air in contact with water in a capillary tube, as shown in Figure 7.25. 

In a medical setting, LaPlace s law explains why the surface tension on a blood vessel wall depends on the radius of the vessel. Figure 7.26 shows a (gray) blood vessel of radius R. The curved white arrows represent the tension across the interface between the blood and the blood vessel wall. 

Solubility 168 Surface Tension 169 LaPlace s Law 171 Surfactants 175 Monolayers 176 Bilayers 176 Micelles 177 Viscosity 178 Vapor Pressure 178 Boiling Point 182 Melting Point 184... 

Solvent can be associated with the solid in different chemical and physical states. It can be part of the crystalline structure, chemically adsorbed in solid interior or physically penetrated/adsorbed inside the interstices of the solid. The interstices of the solid are subjected to a large capillary tension upon drying. The force exerted to solid can be estimated by Laplace s law/ ... 

First, it is important to note that cracking is always related to the establishment of capillary forces, which appear when the solid part of the gel comes in contact with the vapor. The pressure change versus radius of the meniscus is given by Laplace s law, which stipulates that the pressure variation is related to the mean curvature x. 

Laplace s law indicates that any treatment that induces an increase in the pore size is beneficial to monolithicity. Several methods are available to enlarge the pore size. 

When identical dried gels were placed into liquids having different surface energies, they spontaneously crack. According to Laplace s law, micropores, which induce the highest stresses, should crack first. Thermoporometry measurements lead to the conclusion that cracking does not induce a change in the microporous or mesoporous volumes. This result does not depend on the na-... 

However, nucleation may be induced and the microcavities are efficient only if they have a radius R greater than a critical radius Rc defined by Laplace s law. Indeed, below this value, the excess pressure in the bubble is such that carbon dioxide passes from the gas phase to the liquid phase and so the bubble disappears. 

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... 

The larger the size of the mammalian heart, the greater the tension exerted on the myocardium. To sustain this greater amount of tension, the wall of the larger mammal must thicken proportionally with increasing radius of curvature. This results in a larger heart weight. The Lame relation that accounts for wall thickness, h, therefore substitutes Laplace s law ... 

They provide a description of the geometric and mechanical relations of the mammalian hearts and Laplace s Law is implicit in the ratio of the two. 

In general, both 7ti and 7t2 and their ratio, I, are not only dimensionless, they are also independent of mammalian body mass. That is, jt2 indicates that ratio of ventricular wall thickness to its radius, h/r, is invariant among mammals. This also establishes a scaling factor. They are thus considered invariant numbers, that is, of the form [M]°[L]°[T]° = a dimensionless constant. This invariance implies that Laplace s law applies to all mammalian hearts [Martin and Haines, 1970 Li, 1986a],... 

Li, J.K.-J. Comparative cardiac mechanics Laplace s law,/. Theor. Biol. 118 339-343,1986a. 

Martin, R.R. and H. Haines. Application of Laplace s law to mammalian hearts. Comp. Biochem. Physiol. 34 959,1970. 

If two profiles are taken perpendicular to each other, intersecting at a point at the liquid-gas interface, it is possible to fit them to polynomials of an appropriate order and calculate the 2-D curvature of the interface at that point. Using this knowledge, and Laplace s law, the pressure difference across the interface can be determined. If this proves successful it could give information, for example, on the pressure of the trapped air bubbles and whether this had a role in preventing complete wetting in some cases. 

Laplace law. The big law in these domains, which span from capillarity to wetting, through diphasic systems, is Laplace s law. This law is demonstrated here in the most general case, with a curvature of the interface separating the two phases that can be of any shape. Physically speaking, this law assumes a simple coupling relationship between efforts. 

Why do the holes expand at constant speed Figure 1.26 shows the profile of the film when a hole nucleated at time t = 0 opens out. The dry region is surrounded by a slightly swollen rim, extending between points A and B, which gathers the liquid from the dry zone. Viscous dissipation is concentrated at the contact lines A and B. The rest of the thick rim soon attains pressure equilibrium and Laplace s law tells us that if the pressure remains constant, then so will the curvature. The cross-section of the rim is then a circular arc and the angles a, are equal. The dynamical equations of motion of lines at points A and B can be written ... 

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