MECH $342$ Assignment $1$: Viscosity and Streamlines

Q$1$: A two$-$dimensional velocity field is described by vec$(V)=(x^2+1)hat()-2$xyhat$().$

a$)$ Determine the streamline equation for the velocity field. $[3]$

b$)$ Plot the streamline which passes through $x=0,y=1.[2]$

Q$2$: The velocity profile for laminar flow in a circular pipe is given by $u=u_c[1-(\frac{r}{R}{)}^2],$ where $u_c$ is the

centerline velocity, $R$ is the radius, and $r$ is the radial position. Considering a pipe with diameter $30$ cm and

centerline velocity $6\frac{m}{s}$ :

a$)$ Calculate the shear stress exerted on the walls of the pipe by the fluid. Assume the pipe contains water at

$20C.[4]$

b$)$ Calculate the force due to friction in a section of pipe with length $10$m$.[1]$