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Unformatted text preview: Chapter 6 Miscellaneous Topics 6.1 Boundary Conditions After determining the governing equations one must determine appropriate boundary and initial conditions which make the problem physically meaningful and mathematically possible to solve. Mathematically we say the problem (the governing equations and associated initial and boundary conditions) must be well-posed , that is, be such that the solution exists (it’s nice to know that if we are going to try and solve a problem that a solution exists), the solution is unique (there is only one solution) and that the solution doesn’t change too much if one of the coefficients or boundary or initial conditions (the data ) is slightly changed. It is not always possible for a physical problem to be well-posed: take weather predictions. It turns out that the equations used to model the pressure/temperature/wind velocity/humidity are notoriously ill-posed: change the initial conditions by just a bit (say due to limited accuracy of the equipment measuring the temperature, or due to the limited amount of data available to get a good initial condition everywhere) and the prediction for 24 hours later can go from predicting 3 inches of snow to 10. We will not address well-posedness here (for this see a text on Partial Differential Equations). What we will address is what are physically meaningful boundary condi- tions. If the boundary conditions are posed correctly then the resulting mathematical problem (governing equations with associated boundary and initial conditions) is in general well-posed (at lease existence and uniqueness). 6.1.1 Mathematical Development To begin we go back to the global form of the balance equations, (3.1). Now suppose that instead of a continuous body, V ( X , t ), the body has a discontinuity. Let S ( X , t ) be the two-dimensional manifold boundary of V ( X , t ), and σ ( t ) be the two-dimensional manifold surface (see Figure 6.1). Examples of the discontinuity include shock waves (CHECK), where the pressure is not continuous across σ , a solid-liquid interface such as sand and water, and an ice-water interface where, due to melting or freezing, the interface is moving. At this point we do not consider surfaces for which the surface 115 116 CHAPTER 6. MISCELLANEOUS TOPICS v t=0 (reference) t>0 S X ( V ) σ σ (t) σ n w +--- + s s v Figure 6.1: Body of Material with Discontinuity itself has its own thermodynamical properties such as a gas-liquid interface which is a miniscus and has its own tension and energy. The interface at the reference configuration ( t = 0) is denoted as σ . If the disconti- nuity is not a material discontinuity (an interface which is determined by the material such as a wood-metal interface), the interface may move relative to the body, V , and the discontinuity in the reference configuration is a function of time, σ = σ ( X , t ). In the Eulerian configuration the interface is denoted as σ = σ ( x , t ). We can use σ to partition the body into two pieces, the left side denoted as...
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This note was uploaded on 11/11/2009 for the course MATH 6735 taught by Professor Bennethum during the Fall '06 term at University of Colombo.
- Fall '06