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 \( x = x_A \) 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 \( x = x_{A'} \), 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 \( x_A \), and constants \( U \) and \( b \). In other words, develop an expression for \( x_{A'} \). (Hint: We know that \( u = \frac{dx_{particle}}{dt} \) following a fluid particle. Plug in \( u \), separate variables, and integrate.)
**Diagram Explanation:**
The diagram labeled "FIGURE P4-5I" shows a schematic of a converging duct flow with the velocity field depicted by arrows pointing rightward, representing fluid motion. Initially, a fluid particle is placed at point \( A \) on the x-axis. As time progresses, the particle moves to point \( A' \) further right, staying on the x-axis due to symmetry in the flow.
**4-63**
A general equation for a steady, two-dimensional velocity field that is linear in both spatial direction (\( x \) and \( y \)) is:
\[
\vec{V} = (u, v) = (U + a_1x + b_1y)\hat{i} + (V + a_2x + b_2y)\hat{j}
\]
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F07c1f221-7c51-40ce-b048-691f836c8187%2F9c4dd8e0-f762-4357-a775-234fcb6bf3e7%2F18rhluj_processed.jpeg&w=3840&q=75)
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 \( x = x_A \) 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 \( x = x_{A'} \), 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 \( x_A \), and constants \( U \) and \( b \). In other words, develop an expression for \( x_{A'} \). (Hint: We know that \( u = \frac{dx_{particle}}{dt} \) following a fluid particle. Plug in \( u \), separate variables, and integrate.)
**Diagram Explanation:**
The diagram labeled "FIGURE P4-5I" shows a schematic of a converging duct flow with the velocity field depicted by arrows pointing rightward, representing fluid motion. Initially, a fluid particle is placed at point \( A \) on the x-axis. As time progresses, the particle moves to point \( A' \) further right, staying on the x-axis due to symmetry in the flow.
**4-63**
A general equation for a steady, two-dimensional velocity field that is linear in both spatial direction (\( x \) and \( y \)) is:
\[
\vec{V} = (u, v) = (U + a_1x + b_1y)\hat{i} + (V + a_2x + b_2y)\hat{j}
\]
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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