In the steady one$-$dimensional incompressible flow situation, shown in Fig. Q$4,$ the velocity is evaluated at locations $A,B,$ and C $,$ whereas the pressure is evaluated at locations $1,2$ and $3.$ The velocity$-$correction is given as:

$u=u^{**}+d(p_i^{\text{'}}-p_{i+1}^{\text{'}})$

where locations i and $i+1$ lie on either side of the location for $u.$ The value of $d$ is $2\mathrm{.}0\frac{m}{s.Pa}.$ The boundary conditions are $u_A=10\mathrm{.}0\frac{m}{s}$ and $p_3^{\text{'}}=0\mathrm{.}0Pa.$

$($a$)$ If at a given stage of the iteration process, the momentum equations provide $u_B^{**}=8\mathrm{.}0\frac{m}{s}$ and $u_C^{**}=11\mathrm{.}0\frac{m}{s},$ calculate the values of $p_1^{\text{'}}$ and $p_2^{\text{'}}.$

$(10$ marks$)$

$($b$)$ Please evaluate the values of $p_1^{\text{'}}$ and $p_2^{\text{'}}$ if the right$-$hand side boundary condition is $u_C=10\mathrm{.}0\frac{m}{s}$ instead of $p_3^{\text{'}}=0\mathrm{.}0$Pa$.$ Please provide your answer with explanations.

$(5$ marks$)$