A fluid of constant density and viscos- ity is contained in a very long pipe of...
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A fluid of constant density and viscos- ity is contained in a very long pipe of length L and radius R. Initially the fluid is at rest. At time t = 0, a pressure gradient (Po-P₁)/L is imposed on the system. Determine how the ve- locity profiles change with time. V₂ Vmax 1.0 0.8 0.6 0.4 0.2 Tube center 0 1.0 0.8 0.6 0.4 0.2 vt R² 0.2 0.15 0.1 0.05 81 Tube wall 0.5 0.4. 0.3 0 0.2 0.4 0.6 0.8 1.0 r/R-> Fig. 4D.2. Velocity distribution for the unsteady flow re- sulting from a suddenly impressed pressure gradient in a circular tube [P. Szymanski, J. Math. Pures Appl., Series 9, 11, 67-107 (1932)]. (a) Show that the relevant equation of motion can be put into dimensionless form as follows: 26 26 = 4 + ƏT 1 a } aξ (5. φ(ξ, τ) = (1 – ξ?) – 8 Σ #-1 ok in which & = r/R, T = μt/pR², and = [(P₁-P₁)R²/4µL]¯¹v₂. (b) Show that the asymptotic solution for large time is = 1 - ². Then define o, by (§, 7) = ..(E) - (E, T), and solve the partial differential equation for o, by the method of separation of variables. (c) Show that the final solution is Jo(α,§) 1a²³ J₁ (α) (4D.2-1) (4D.2-2) -exp(-ar) in which J() is the nth order Bessel function of , and the a,, are the roots of the equation Jo(a) = 0. The result is plotted in Fig. 4D.2. A fluid of constant density and viscos- ity is contained in a very long pipe of length L and radius R. Initially the fluid is at rest. At time t = 0, a pressure gradient (Po-P₁)/L is imposed on the system. Determine how the ve- locity profiles change with time. V₂ Vmax 1.0 0.8 0.6 0.4 0.2 Tube center 0 1.0 0.8 0.6 0.4 0.2 vt R² 0.2 0.15 0.1 0.05 81 Tube wall 0.5 0.4. 0.3 0 0.2 0.4 0.6 0.8 1.0 r/R-> Fig. 4D.2. Velocity distribution for the unsteady flow re- sulting from a suddenly impressed pressure gradient in a circular tube [P. Szymanski, J. Math. Pures Appl., Series 9, 11, 67-107 (1932)]. (a) Show that the relevant equation of motion can be put into dimensionless form as follows: 26 26 = 4 + ƏT 1 a } aξ (5. φ(ξ, τ) = (1 – ξ?) – 8 Σ #-1 ok in which & = r/R, T = μt/pR², and = [(P₁-P₁)R²/4µL]¯¹v₂. (b) Show that the asymptotic solution for large time is = 1 - ². Then define o, by (§, 7) = ..(E) - (E, T), and solve the partial differential equation for o, by the method of separation of variables. (c) Show that the final solution is Jo(α,§) 1a²³ J₁ (α) (4D.2-1) (4D.2-2) -exp(-ar) in which J() is the nth order Bessel function of , and the a,, are the roots of the equation Jo(a) = 0. The result is plotted in Fig. 4D.2.
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Fundamentals of Thermal-Fluid Sciences
ISBN: 978-0078027680
5th edition
Authors: Yunus A. Cengel, Robert H. Turner, John M. Cimbala
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