A Course in Fluid Mechanics with Vector Field Theory

By Prieve, Dennis C.

Description
Physics Literature

Table of Contents
Table of Contents ALGEBRA OF VECTORS AND TENSORS ..............................................................................................................................1 VECTOR MULTIPLICATION.....................................................................................................................................................1 Definition of Dyadic Product.............................................................................................................................................2 DECOMPOSITION INTO SCALAR COMPONENTS....................................................................................................................3 SCALAR FIELDS..........................................................................................................................................................................3 GRADIENT OF A SCALAR..........................................................................................................................................................4 Geometric Meaning of the Gradient.................................................................................................................................6 Applications of Gradient....................................................................................................................................................7 CURVILINEAR COORDINATES..................................................................................................................................................7 Cylindrical Coordinates ....................................................................................................................................................7 Spherical Coordinates........................................................................................................................................................8 DIFFERENTIATION OF VECTORS W.R.T. SCALARS................................................................................................................9 VECTOR FIELDS........................................................................................................................................................................ 11 Fluid Velocity as a Vector Field.....................................................................................................................................11 PARTIAL & MATERIAL DERIVATIVES................................................................................................................................. 12 CALCULUS OF VECTOR FIELDS...........................................................................................................................................14 GRADIENT OF A SCALAR (EXPLICIT)................................................................................................................................... 14 DIVERGENCE, CURL, AND GRADIENT................................................................................................................................... 16 Physical Interpretation of Divergence...........................................................................................................................16 Calculation of ¥.v in R.C.C.S........................................................................................................................................16 Evaluation of ¥?v and ¥v in R.C.C.S............................................................................................................................18 Evaluation of ¥.v, ¥?v and ¥v in Curvilinear Coordinates ...................................................................................19 Physical Interpretation of Curl.......................................................................................................................................20 VECTOR FIELD THEORY..........................................................................................................................................................22 DIVERGENCE THEOREM.......................................................................................................................................................... 23 Corollaries of the Divergence Theorem.........................................................................................................................24 The Continuity Equation..................................................................................................................................................24 Reynolds Transport Theorem...........................................................................................................................................26 STOKES THEOREM................................................................................................................................................................... 27 Velocity Circulation: Physical Meaning .......................................................................................................................28 DERIVABLE FROM A SCALAR POTENTIAL.......................................................................................................................... 29 THEOREM III............................................................................................................................................................................. 31 TRANSPOSE OF A TENSOR, IDENTITY TENSOR.................................................................................................................... 31 DIVERGENCE OF A TENSOR .................................................................................................................................................... 32