Isothermal Incompressible Body

We can repeat the same consideration as shown in Sect. 4.4.1 with the free enthalpy G(r, p). The partial derivative of second order with respect to pressure is 

Therefore, the free energy is a function of the temperature alone. We emphasize that for the isothermal incompressible body the heat capacity at constant volume is not accessible. When we change the temperature, which is needed to measure the C then we would also change the volume, due to Eq. (4.46). An exception is the special case dy(r)/dr = 0 that occurs in liquid water at the density maximum. Therefore, C is not defined in general however, Cp is readily accessible. 

We restrict now Cp = Cp(T), the heat capacity should only be a function of the temperature. From 

Therefore, together with Eq. (4.46), the volume must be of the form V T) = AT - -B. Usually we write 

a is the thermal expansion coefficient. We do not need the special equation, Eq. (4.49), but we use the more general form from Eq. (4.46)  

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Considering an example of isothermal, incompressible body in elastic contact with the presumption that there are adequate molecular layers on the minimum film thickness spot, we will get the governing equations as follows... 

We have to distinguish between an isothermal incompressible body and an isentropic incompressible body. We characterize... 

We discuss finally the case of a thermal expansion coefficient of zero. In this case, a pressure change does not effect a temperature change of the material and also no change in volume. So no input of compression energy is achieved. In contrary to the adiabatic application of pressure in the isothermal case in fact no change in volume will occur. Therefore, the incompressible body should be more accurately addressed as the isothermal incompressible body. 

More precisely we must differentiate between an adiabatically incompressible boy and an isothermally incompressible body. Since the generalized susceptibilities must be positive for the sake of stability of matter, we have... 

Consider the isothermal incompressible smectic A liquid crystal in the absence of significant body forces due to electromagnetic or gravitational fields. The elastic strain for the liquid crystal can be descril d in terms of a single variable w (x, y, z ) that specifies the local displacement of the smectic layers. The theoretical pr iction [2,3 or 7] is that w (x, y, z ) satisfies the differential equation... 

Equation (2-91) would seem to imply that the gravity force pg has no direct effect on the velocity distribution in a moving fluid provided the fluid is incompressible and isothermal so that the density is constant. This is generally true. An exception occurs when one of the boundaries of the fluid is a gas-liquid or liquid-liquid interface. In this case, the actual pressure p appears in the boundary conditions (as we shall see), and the transformation of the body force out of the equations of motion by means of (2-90) simply transfers it into the boundary conditions. We shall frequently use the equations of motion in the form (2-91), but we should always keep in mind that it is the dynamic pressure that appears there. 

The isothermal flow of incompressible liquid is described by equations (5.13) and (5.21), and the viscosity coefficient n = const. Hence, there are four equations for four unknowns - the pressure p and three velocity components u, v, and w. Thus, the system of equations is a closed one. For its solution it is necessary to formulate the initial and boundary conditions. Let us discuss now possible boundary conditions. Consider conditions at an interface between two mediums denoted as 1 and 2. The form and number of boundary conditions depends on whether the boundary surface is given or it should be found in the course of solution, and also from the accepted model of the continuum. Consider first the boundary between a non-viscous liquid and a solid body. Since the equations of motion of non-viscous liquid contain only first derivatives of the velocity, it is necessary to give one condition of the impermeability u i = u 2 at the boundary S, where u is the normal component of the velocity. The equations of motion of viscous liquid include the second-order derivatives, therefore at the boundary with a solid body it is necessary to assign two conditions following from the condition of sticking u i = u 2, Wii = u i where u is the tangential to S component of the velocity. If the boundary S is an interface between two different liquids or a liquid and a gas, then it is necessary to add the kinematic condition Ui = U2 =... 

In capillary flows, the velocity profile and the resulting volumetric flow rate versus pressure drop behavior can be determined from Equations 15 and 16. There exist two regions of flow. Neglecting end effects and body forces, and assuming incompressibility, the isothermal solution can be obtained by using the boundary condition, V. (R) = Ug. In the... 

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