Problem 1: Continuity through a Converging Pipe

Water flows through a pipe that contracts from $200\text{ mm}$ diameter to $100\text{ mm}$. The upstream velocity is $2.0\text{ m/s}$. Determine the discharge and downstream velocity.

Problem 2: Flow Division at a Junction

A pipe carries $0.100\text{ m}^3/\text{s}$ and divides into two branches. Branch 1 has diameter $150\text{ mm}$ and velocity $2.0\text{ m/s}$. Branch 2 has diameter $200\text{ mm}$. Determine the flow and velocity in Branch 2.

Problem 3: Compressible Mass Continuity

Air enters a duct with density $1.20\text{ kg/m}^3$, area $0.50\text{ m}^2$, and velocity $10\text{ m/s}$. It exits where density is $0.90\text{ kg/m}^3$ and area is $0.30\text{ m}^2$. Determine mass flow rate and exit velocity.

Problem 4: Acceleration from a Two-Dimensional Velocity Field

A steady velocity field is

Determine acceleration at $(x,y)=(1,2)$.

Problem 5: Local and Convective Acceleration

A one-dimensional velocity field is $u=x(1+2t)$, where $x$ is in meters and $t$ in seconds. Determine local, convective, and total acceleration at $x=2\text{ m}$ and $t=1\text{ s}$.

Problem 6: Velocity and Flow from a Streamfunction

A two-dimensional incompressible flow has streamfunction $\psi=3xy$ in $\text{m}^2/\text{s}$. Find the velocity at $(2,1)$ and the discharge per unit width between streamlines $\psi=3$ and $\psi=12$.

Problem 7: Incompressibility and Irrotationality Check

For the velocity field $u=4x$ and $v=-4y$, determine whether the two-dimensional flow is incompressible and irrotational.

Problem 8: Unsteady Storage in a Tank

A vertical tank has constant plan area $10\text{ m}^2$. Water enters at $0.080\text{ m}^3/\text{s}$ and leaves at $0.030\text{ m}^3/\text{s}$. Determine the water-level rise rate and the rise after $5\text{ min}$.