Consider the flow field $u_r=$Vcos$(1-\frac{a^2}{r^2}),u_{}=-$Vsin$(1+\frac{a^2}{r^2})($flow over a circular cylinder$).$ Calculate the two linear strain rates in the $r-$ plane; i$.$e$.,$

calculate ${}_{rr}$ and ${}_{}$ and choose if the flow is compressible or incompressible. $($Hint: The strain rate tensor in cylindrical coordinates is

$\{$:${}_{ij}=([{}_{rr},{}_r,{}_{rx}],[{}_r,{}_{},{}_x],[{}_{xr},{}_x,{}_{}])=([\frac{del{u}_r}{\text{d}\text{e}\text{l}\text{r}},\frac{1}{2}(r\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(\frac{u_{}}{r})+\frac{1}{r}\frac{del{u}_r}{\text{d}\text{e}\text{l}}),\frac{1}{2}(\frac{del{u}_r}{\text{d}\text{e}\text{l}\text{x}}+\frac{del{u}_x}{\text{d}\text{e}\text{l}\text{r}})],[\frac{1}{2}(r\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(\frac{u_{}}{r})+\frac{1}{r}\frac{del{u}_r}{\text{d}\text{e}\text{l}}),\frac{1}{r}(\frac{del{u}_{}}{\text{d}\text{e}\text{l}}+\frac{u_r}{r}),\frac{1}{2}(\frac{del{u}_x}{\text{d}\text{e}\text{l}}+\frac{del{u}_{}}{\text{d}\text{e}\text{l}\text{x}})],[\frac{1}{2}(\frac{del{u}_r}{\text{d}\text{e}\text{l}\text{x}}+\frac{del{u}_x}{\text{d}\text{e}\text{l}\text{r}}),\frac{1}{2}(\frac{del{u}_x}{\text{d}\text{e}\text{l}}+\frac{del{u}_{}}{\text{d}\text{e}\text{l}\text{x}}),\frac{del{u}_x}{\text{d}\text{e}\text{l}\text{x}}]).)$