Water enters a 5-ft-wide, 1-ft-deep channel as shown in the figure. Across the inlet the water velocity is $6 \mathrm{ft} / \mathrm{s}$ in the center portion of the channel and $1 \mathrm{ft} / \mathrm{s}$ in the remainder of it. Farther downstream the water flows at a uniform $2 \mathrm{ft} / \mathrm{s}$ velocity across the entire channel. The fixed control volume $A B C D$ coincides with the system at time $t=0$. Make a sketch to indicate (a) the system at time $t=0.5 \mathrm{~s}$ and (b) the fluid that has entered and exited the control volume in that time period.

The given figure shows a fixed control volume. It has a volume $V_0$=1.0 ft$^3$, a flow area A=1.0 ft$^2$, and a length $\ell_0$=1.0 ft. Position x represents the center of the control volume where the fluid velocity $V_0$=1.0 ft/s and the density $\rho_0$=1.8 slug /ft$^3$. Also, at position, x the fluid density does not change locally with time but decreases in the axial direction at the linear rate of 0.25 slug/ ft$^4$. Employ the system or Lagrangian approach to evaluate $DV/Dt$.

A layer of oil flows down a vertical plate as shown in the figure with a velocity of $\mathrm{V}=\left(V_0 / h^2\right)\left(2 h x-x^2\right) \mathrm{j}$ where $V_0$ and $h$ are constants. (a) Show that the fluid sticks to the plate and that the shear stress at the edge of the layer $(x=h)$ is zero. (b) Determine the flowrate across surface $A B$. Assume the width of the plate is $b$. (Note: The velocity profile for laminar flow in a pipe has a similar shape.

The lower yield point for an iron that has an average grain diameter of $5 \times 10^{-2} \mathrm{~mm}$ is 135 MPa $(19,500 \mathrm{psi})$. At a grain diameter of $8 \times$ $10^{-3} \mathrm{~mm}$, the yield point increases to $260 \mathrm{MPa}$ $(37,500 \mathrm{psi})$. At what grain diameter will the lower yield point be $205 \mathrm{MPa}(30,000 \mathrm{psi})$ ?

A certain flow field has the velocity vector

${V}=\frac{-2 x y z}{\left(x^2+y^2\right)^2} {i}+\frac{\left(x^2-y^2\right) z}{\left(x^2+y^2\right)^2} {j}+\frac{y}{x^2+y^2} {k}$.

Determine the acceleration vector for this flow.

When determining the thrust developed by a jet engine, a wise choice of control volume is(a) Fixed control volume(b) Moving control volume(c) Deforming control volume(d) Moving or deforming control volume(e) None of these

The fluid velocity along the $x$ axis shown in the given figure changes from $6 \mathrm{~m} / \mathrm{s}$ at point $A$ to $18 \mathrm{~m} / \mathrm{s}$ at point $B$. It is also known that the velocity is a linear function of distance along the streamline. Determine the acceleration of points $A, B$, and $C$. Assume steady flow.

In the region just downstream of a siuice gate, the water may develop a reverse flow region as is indicated in the figure. The velocity profile is assumed to consist of two uniform regions, one with velocity $V_{\mathrm{at}}=10 \mathrm{fps}$ and the other with $V_b=3 \mathrm{fps}$. Determine the net flowrate of water across the portion of the control surface at section (2) if the channel is $20 \mathrm{ft}$ wide.

The velocity of air in the diverging pipe is given by $V_{1}=4 t \ \mathrm{ft} / \mathrm{s}$ and $V_{2}=2 t \ \mathrm{ft} / \mathrm{s},$ where t is in seconds. (a) Determine the local acceleration at points (1) and (2). (b) Is the average convective acceleration between these two points negative, zero, or positive? Explain.

A tornado has the following velocity components in polar coordinates:

$$V_r=-\frac{C_1}{r} \quad \text { and } \quad V_\theta=-\frac{C_2}{r}$$

Note that the air is spiraling inward. Determine an equation for the streamlines. r and $\theta$ are polar coordinates.

A velocity field is given by

$$\mathbf{V}=x \hat{\mathbf{i}}+x(x-1)(y+1) \hat{\mathbf{j}}$$

where u and v are in ft/s and x and y are in feet. Plot the streamline that passes through x = 0 and y = 0. Compare this streamline with the streakline through the origin.

A two-dimensional, unsteady velocity field is given by

$$u=5 x(1+t) \text { and } v=5 y(-1+t),$$

where u is the x-velocity component and v the y-velocity component. Find x(t) and y(t) if x=$x_0$ and y=$y_0$ at t=0. Do the velocity components describe an Eulerian description or a Lagrangian description?

Nitrogen $(N_2)$ undergoes an internally reversible process from 6 bar, $247^{\circ} \mathrm{C}$ during which $p v^{1.20}=\text { constant }.$The initial volume is $0.1 m^3$ and the work for the process is 121.14 kJ. Assuming ideal gas behavior, and neglecting kinetic and potential energy effects, determine heat transfer, in kJ, and the entropy change, in kJ/K. Show the process on a T-s diagram.

The components of a velocity field are given by $u=x+y, v=x y^{3}+16,$ and w = 0. Determine the location of any stagnation points $(\mathbf{V}=0)$ in the flow field.