Fluid Kinematics: Worked Examples

Water flows steadily through a full pipe that contracts from $D_1=200\text{ mm}$ to $D_2=100\text{ mm}$. The upstream mean velocity is $V_1=2.00\text{ m/s}$. Determine the discharge and downstream mean velocity.

A main pipe delivers $0.100\text{ m}^3/\text{s}$ to a junction. Branch 1 has $D_1=150\text{ mm}$ and $V_1=2.00\text{ m/s}$. Branch 2 has $D_2=200\text{ mm}$. Determine the Branch 2 discharge and mean velocity.

Air enters a duct with $\rho_1=1.20\text{ kg/m}^3$, $A_1=0.500\text{ m}^2$, and $V_1=10.0\text{ m/s}$. It exits where $\rho_2=0.900\text{ kg/m}^3$ and $A_2=0.300\text{ m}^2$. For steady flow with no leakage, determine mass flow rate and exit velocity.

A steady velocity field is $u=2xy$ and $v=x^2-y^2$, with coordinates in meters and velocities in $\text{m/s}$. Determine the acceleration vector at $(x,y)=(1.00,2.00)$.

A one-dimensional unsteady velocity field is $u=x(1+2t)$, with $x$ in meters and $t$ in seconds. Determine local, convective, and total acceleration at $x=2.00\text{ m}$ and $t=1.00\text{ s}$.

A two-dimensional incompressible flow has $\psi=3xy$ in $\text{m}^2/\text{s}$. Using $u=\partial\psi/\partial y$ and $v=-\partial\psi/\partial x$, determine the velocity at $(2.00,1.00)$ and discharge per unit depth between $\psi=3$ and $\psi=12\text{ m}^2/\text{s}$.

For the two-dimensional field $u=4x$ and $v=-4y$, determine whether the flow is compatible with constant-density incompressibility and whether it is irrotational.

A vertical tank has constant plan area $A_t=10.0\text{ m}^2$. Water enters at $0.0800\text{ m}^3/\text{s}$ and leaves at $0.0300\text{ m}^3/\text{s}$. Determine the water-level rise rate and the rise after $5.00\text{ min}$ if both flows remain constant.

A steady two-dimensional field is $u=2x$ and $v=-y$. Determine the streamline passing through $(x,y)=(1.00,2.00)$ and find $y$ where that streamline reaches $x=4.00$.

A proposed velocity potential is $\phi=2x^2-2y^2$ in $\text{m}^2/\text{s}$. Determine the velocity at $(1.00,2.00)$ and verify incompressibility.

A two-dimensional isotropic seepage flow net has hydraulic conductivity $k=2.00\times10^{-5}\text{ m/s}$, total head loss $H=6.00\text{ m}$, $N_f=4$ flow channels, and $N_d=12$ equipotential drops. Determine seepage discharge per unit thickness.