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Problem 5.85 - Problem 5.85[Difficulty 3 Given Find...

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(a) Expression for circulation about a closed contour of height h and length L (b) Evaluate part (a) for h = b/2 and h = b (c) Show that the same result is obtained from area integral of Stokes Theorem (Eq. 5.14) Problem 5.85 [Difficulty: 3] Given: Velocity field for pressure-driven flow between stationary parallel plates Find: Solution: We will apply the definition of circulation to the given velocity field. s d V Governing Equations: (Definition of circulation) s d V dA V A (Stokes Theorem) Assumptions: (1) Steady flow 4 3 2 1 s d V s d V s d V s d V From the definition of circulation we break up the integral: dx b y b y U j dy i dx i b y b y U s d V 1 ˆ ˆ ˆ 1 The integrand is equal to: Therefore, the circulation is equal to: Γ 0 L x U 0 b 1 0 b d L 0 x U h b 1 h b d U L
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