Stability Condition and Wave Motion for Superposed Fluids

3 Stability Condition and Wave Motion for Superposed Fluids 

Equation 5.34, the hst of unknown constants in these solutions given at the beginning of Seetion 2b has beoi reduced to A, Aj, A3, A4, and p. MoreovCT, we derived four applicable boundary eonditions in Section 2.2 (Equations 5.34, 5.37,5.38, and 5.40). When the velocity and pressure distributions are substituted into these equations, we find the following relationships among the unknown constants  

Note that these are four linear, homogeneous equations in the four unknowns D[ through D4. An obvious solution, but an uninteresting one because it implies no interfacial deformation and no flow, is the trivial solution Z)i = A = A = A = 0. According to the theory of linear equations, the condition for a nontrivial solution to exist is that the determinant of coefficients vanishes for Equations 5.43 through 5.46. This relationship, which of course does not contain any of the D S, may be solved to obtain the time factor Pas a function of the wavenumber a and the physical properties of the fluids. If the real part of pis positive, the 

It is instructive to consider some special cases. First let fluid A be a gas of negligible density and viscosity. Then Equations 5.43 and 5.46, which require continuity of normal and tangential velocity at the interface, are not needed. If the derivation leading to Equation 5.42 is repeated for fluid B, the quantity Z)i -r in that equation is replaced by + D. Thus, if A is a gas. Equation 5.44 becomes 

Noting that terms in D and 2 also disappear from Equation 5.45, we find, after some manipulation, that the determinant of coefficients of Equations 5.45 and 5.47 simplifies to 

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