Problem $2$

A flow of air is described by the velocity field,

vec$(V)(x,y,t)=$xhat$()+t^2$hat$()$

for $t0,$ where $x,y$ are in $[$m$],t$ is in $[$s$],$ and vec$(V)$ is in $[$m$/$s$].$ The fluid is stagnant before $t=0.$

a$)$ What is the velocity vector at the location $x=1,y=1$ at time $t=1?$

b$)$ On three separate graphs, sketch the velocity field in the $xy$ plane within the region $0x2,0y2$ at times $t=0,1,$ and $2.$ Each sketch should include $9$ vectors.

c$)$ Find an expression for the time dependent acceleration field.

d$)$ What is the acceleration vector at the location $x=1,y=1$ at time $t=1?$

e$)$ What are the convective and local components of the acceleration at $x=1,y=1$ at time $t=1?$

f$)$ On two separate graphs, sketch how the $x$ and $y$ components of acceleration vary with time at the location $x=1,y=1$ for $0t2.$

g$)$ On three separate graphs, sketch the acceleration field in the $xy$ plane within the region $0x2,0y2$ at times $t=0,1,$ and $2.$ Each sketch should include $9$ vectors.