Derive results (i), (ii), and (iii) in the last paragraph of Box 14.4. Box 14.4. BOX 14.4.
Question:
Derive results (i), (ii), and (iii) in the last paragraph of Box 14.4.
Box 14.4.
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BOX 14.4. STREAM FUNCTION FOR A GENERAL, TWO-DIMENSIONAL, INCOMPRESSIBLE FLOW ™Z Consider any orthogonal coordinate system in flat 3-dimensional space, for which the metric coefficients are independent of one of the coordinates, say, x3: ds² = 9₁₁(x₁, x₂) dx + 922(x₁, x₂) dx² +933(x₁, x₂) dx². The most common examples are Cartesian coordinates {x, y, z) with 911 922 = 933 = 1; cylindrical coordinates (, z, o) with 9₁1 = 922 = 1 and 933 = ²; and spherical coordinates {r, 0, 0) with 911 = 1, 922 = r², and 933 = r² sin² 0. Suppose the velocity field is also independent of x3, so it is effectively 2-dimensional (translation invariant for Cartesian coordinates; axisymmetric for cylindrical or spherical coordinates). Because the flow is incompressible, V .v = 0, we can write the velocity as the curl of a vector potential: v=V x A(t, x₁, x₂). By imposing the Lorenz gauge on the vector potential (i.e., making it divergence free, as is commonly done in electromagnetism), we can ensure that its only nonvanishing component is A3 = A. e3, where e3 is the unit vector pointing in the x3 direction. Now, a special role is played by the vector that generates local translations along the x3 direction (i.e., that generates the flow's symmetry). If we write a location P in space as a function of the coordinates P(x₁, x2, x3), then this generator is P/ax3 = √933 €3. We define the flow's stream function by (t, x₁, x₂) = A.ap/ax3, (1) (2)
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ISBN: 9780691159027
1st Edition
Authors: Kip S. Thorne, Roger D. Blandford
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