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AA210AFundamentals of Compressible Flow
Chapter 3 - Control volumes, vector calculus
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3.1 Control volume definition
The control volume is a closed, simply connected region in space.
A(t)
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3.2 Vector calculus
Gradient operator
Gradient of a scalar
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Gradient of a vector.
Divergence of a vector.
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The Kronecker unit tensor.
The dot product of a vector and a tensor.
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Curl of a vector.
Curl in index notation
The alternating unit tensor (Levi-Civita tensor)
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Useful identities involving the Kronecker and alternating tensors
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Some useful vector identities.
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Some more vector identities - this time involving second derivatives.
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The Laplacian.
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3.3 Gauss’ theoremThis famous theorem in vector calculus can be used to convert a volume integral involving the gradient to a surface integral.
The variable F can be a scalar, vector or tensor.
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A volume integral involving the curl can be converted to a surface integral.
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Recall the development of the continuity equation
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3.4 Stokes’ theorem
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3.5 Problems
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