A two$-$dimensional liquid fluid stream with height d $,$ uniform velocity $U_{},$ density $$ and dynamic viscosity $$ is incident on the top side of a flat plate as shown in the Figure. The gravitational acceleration is g and the atmospheric pressure is assumed equal to $p_a.$ A laminar boundary layer develops over the flat plate and its displacement thickness grows and finally it reaches the free surface at a distance L from the leading edge of the plate.

Denote by $(x)$ the displacement thickness of the boundary layer and by $U(x)$ the horizontal fluid velocity at the edge of the displacement thickness. Assume that the horizontal velocity profile $u(x,y)$ inside the boundary layer at a distance $x$ downstream of the leading edge is given by the expression

$a(x)b(x)U(L)U,d(L)\frac{u(x,y)}{U(x)}=a(x)(\frac{y}{(x)})+b(x)(\frac{y}{(x)}{)}^3,0$

Assume that $at$ the edge $of$ the displacement thickness $of$ the boundary layer the horizontal fluid velocity $is$ equal $to$ that $of$ the inviscid flow above $it.$

$(1)$ Determine $a(x)$ and $b(x)by$ providing appropriate physical justifications.

$(2)$ Deriv

A two$-$dimensional liquid stream with height d$,$ uniform velocity $U_{},$ density $$ and dynamic viscosity $$ is incident on the top side of a flat plate. The gravitational acceleration is $g$ and the atmospheric pressure is $p_a.$ A laminar boundary layer develops over the flat plate and its boundary layer thickness grows and finally it reaches the free surface at a distance from the leading edge of the plate.

The boundary layer thickness is denoted by $(x)$ and $U(x)$ is the horizontal fluid velocity at the edge of the boundary layer. Assume that the horizontal velocity profile $u(x,y)$ inside the boundary layer at a distance $x$ downstream of the leading edge is given by

$a(x)b(x)U(L)U_{},d(L)x=Lx=L$:$p(L,y),0.\frac{u(x,y)}{U(x)}=a(x)(\frac{y}{(x)})+b(x)(\frac{y}{(x)}{)}^3,0$

Assume that $at$ the edge $of$ the displacement thickness $of$ the boundary layer, the horizontal fluid velocity $is$ equal $to$ that $of$ the inviscid flow above $it.$

$a.$ Determine $a(x)$ and $b(x)by$ applying suitable boundary conditions.

$b.$ Derive $an$ expression relating $U(L)$ with $U_{},d$ and $(L).$

$c.$ Apply the mass conservation principle and derive $an$ expression for the displacement thickness $atx=L.$

$d.$ Determine the pressure variation $atx=L$:$p(L,y),0.ean$ expression relating $U(L)$ with $U,d$ and $(L).$