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at x = XA at time t = 0 (Fig. P4 the flow to some new location x is, the fluid particle remains on on of the fluid particle at some a. other words, develon an expressi

at x = XA at time t = 0 (Fig. P4 the flow to some new location x is, the fluid particle remains on on of the fluid particle at some a. other words, develon an expressi

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4-51 Converging duct flow is modeled by the steady, two-dimensional velocity field of Prob.4–16. A fluid particle (A) is located on the x-axis at x = X4 at time t = 0 (Fig. P4-51). At some later time t, the fluid particle has moved downstream with the flow to some new location x = x4', as shown in the figure. Since the flow is symmetric about the x-axis, the fluid particle remains on the x-axis at all times. Generate an analytical expression for the x-location of the fluid particle at some arbitrary time t in terms of its initial location x, and constants U and b. In other words, develop an expression for x4. (Hint: We know that u = dxparticte/dt following a fluid particle. Plug in u, separate variables, and integrate.) Fluid particle at some later time t Fluid particle at time i-0 FIGURE P4-51 4-63 A general equation for a steady, two-dimensional velocity field that is linear in both spatial direction (x and y) is: V = (u, v) = (U + a,x+ b,y)i+ (V + azx + bąy)j - azx- Where U and V and the coefficients are constant. Their dimensions are assumed to be appropriately defined. Calculate the shear strain rate in the xy-plane.
Transcribed Image Text:4-51 Converging duct flow is modeled by the steady, two-dimensional velocity field of Prob.4–16. A fluid particle (A) is located on the x-axis at x = X4 at time t = 0 (Fig. P4-51). At some later time t, the fluid particle has moved downstream with the flow to some new location x = x4', as shown in the figure. Since the flow is symmetric about the x-axis, the fluid particle remains on the x-axis at all times. Generate an analytical expression for the x-location of the fluid particle at some arbitrary time t in terms of its initial location x, and constants U and b. In other words, develop an expression for x4. (Hint: We know that u = dxparticte/dt following a fluid particle. Plug in u, separate variables, and integrate.) Fluid particle at some later time t Fluid particle at time i-0 FIGURE P4-51 4-63 A general equation for a steady, two-dimensional velocity field that is linear in both spatial direction (x and y) is: V = (u, v) = (U + a,x+ b,y)i+ (V + azx + bąy)j - azx- Where U and V and the coefficients are constant. Their dimensions are assumed to be appropriately defined. Calculate the shear strain rate in the xy-plane.
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