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Unformatted text preview: 2.5. CURVILINEAR COORDINATES 39 Since the equation (2.85) is a second order differential equation, we would need to specify temperature derivatives in addition to specifying temperature val- ues at the boundary. These conditions can be obtained from the consideration of energy conservation. In particular, heat flux across the boundary should be conserved. Thus from (2.85) we obtain (2.86) kT i n i q where n i is a boundary-normal unit vector, and q is the heat flux across the bound- ary. 2.5 Curvilinear Coordinates Physical laws should not depend on the choice of a coordinate system. This is expressed in the terminology of tensor calculus as coordinate invariance . Tensors are designed to be invariant under coordinate transformations (Remark A.3.3). Therefore, tensor relations provide a consistent way of writing physical laws. There are two aspects of expressing physical laws in tensor forms: identify- ing , physical components , and forming invariant expressions . 2.5.1 Invariant forms The scalar product (Definition A.3.4) was constructed to be invariant. By virtue of its invariance it represents a physical entity. Using the invariant forms of the scalar product (Corollary A.3.5), we can rewrite the expression for the substantial derivative (1.7) in invariant form: (2.87) d dt u i i Correspondingly, the mass conservation equation (2.2) will be expressed as (2.88) u i i 40 CHAPTER 2. FUNDAMENTAL LAWS and the momentum equation (2.22) becomes: (2.89) t u i u k u i k P i k i k where the covariant and contravariant velocities and stress tensors are linked by the conjugate tensor relation (A.38): u i g...
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