Basic Equations of Fluid Mechanics

In terms of the formalism presented in the previous section, the boundary conditions derived in Chapter 5 (Section 2) are rewritten for the particular situation of interest here, which is the case of two immiscible fluid media, labeled A and B, separated by an interface. The shape of the interface is given vectorially as a(r, t) and the rate at which it moves is given by a = da/di. For simplicity, the fluids are chosen to be incompressible and Newtonian, thus the equations of motion and continuity become 

Expressions involving the surface gradient operator in simple coordinate 

Equations 7.12 and 7.13 are subject to appropriate boundary conditions away from the intraface, while at the interface we have, first of all. 

This equation is obtained on neglecting terms in T in the interfacial mass balance equation (Equation 5.32). The two terms are equal to the flux (mass transfer) through the interface, in the absence of which Equation 7.14 reduces to 

The tangential velocities are continuous at the interface, the generalization of Equation 5.40, 

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Aris, R., 1989. Vectors, Tensors and the Basic Equations of Fluid Mechanics, Dover Publications, New York. 

The constitutive relations along with the conservation equations give the basic equations of fluid mechanics, which are a set of five nonlinear partial differential equations involving the seven variables, p, g,e, P, and T. Because five equations [Eqs. (1), (2), (3), (5), and (6)] cannot determine seven quantities, the equations are closed by expressing any two variables of the set (p,e,P,T) in terms of the other two remaining variables. This is done by using the assumption of local equilibrium and thermodynamic equations of state. 

Aris R. Vectors, tensors and the basic equations of fluid mechanics. New York Dover Publications 1990. 

A quantitative treatment of acoustic and thermal wave generation is usually possible only for simple excitation geometries, as can be realized by laser excitation, using the basic equations of fluid mechanics and thermodynamics. 

In this appendix, we will derive the essential equations for swirling flow from the basic equations of fluid mechanics the Navier-Stokes equations. The Navier-Stokes equations are derived in most textbooks on fluid mechanics, for instance Bird et al. (2002). 

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