MECH $342$ Assignment $4$: Differential Relations

Q$1$: Which of the following velocity fields are physically possible for incompressible flow? $($Show your

working$)$

a$)$ vec$(V)=sin(x)hat()-ycos(x)hat()+5$hat$(k)$

b$)$ vec$(V)=(2x+1)hat()-(y+1)hat()+(z+1)hat(k)$

c$)$ vec$(V)=x^2$hat$()-y^2$hat$()-2z(x-y)hat(k)$

Q$2$: What are the two main assumptions used to derive the Navier$-$Stokes Equations $($excluding

incompressibility$)?$ Can you think of a practical example where the assumptions do not hold?

Q$3$: Below is the x $-$momentum equation simplified to $1$ D :

$[\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{t}}+u\frac{\text{d}\text{e}\text{l}\text{u}}{dx}]=-\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{x}}+(\frac{de{l}^{2}u}{del{x}^2})$

Consider the $1$D velocity vector: vec$(V)=$uhat$()=(Kx)hat()$

Use the x$-$momentum equation above to obtain an expression for the pressure $P.$

$[$Hint: Firstly, try to simplify the respective terms. Then integrate the pressure gradient.$]$