Derive results (i), (ii), and (iii) in the last paragraph of Box 14.4. Box 14.4. BOX 14.4.

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)