$(60$pt$)$

$($a$)(20$pt$)$ Consider a linear diffusion equation

$($delu$)/($delt$)=\backslash$alpha $($del$^(2)$u$)/($delx$^(2))$

Discretize it using a numerical scheme of your choice. Perform the consistency analysis, obtaining an

expression for the truncation error.

$($b$)(20$pt$)$ Describe the amplification factor $($G$)$ in von Newman stability analysis. Consider a linear convection

equation as$,$

$($delu$)/($delt$)+$a$($delu$)/($delx$)=0.$

Apply a numerical scheme where you discretize the both time and space terms using the second order

central difference formula. Perform the von Neumann stability analysis to discuss its stability.

$($c$)(20$pt$)$ Consider a $1$ D Couette$-$Poiseuille flow as shown in the Fig. $1.$ The flow model can be given by

Eq$.1$ and Eq$.2$ as$,$

Momentum equation:

$($d$)/($dy$)[($

u $+$

u $\_(()$t$))($du$)/($dy$)]=(1)/(\backslash$rho $)($dp$)/($dx$)$

$($d$)/($dy$)[($

u $+$

u $\_(()$t$))$u$($du$)/($dy$)+($c$\_($p$))/($Pr$)($

u $+()/(0\mathrm{.}9)$

u $\_(()$t$))($delT$)/($dely$)]=0$

Couette flow $($dp$)/($dx$)=0,$v$\_($w$)!=0$

Poisenille flow $($dP$)/($dx$)!=0$

Conette$-$poisenille flow $($dp$)/($dx$)!=0,$V$\_($u$)!=0$

Figure $1$: Couette$-$Poiseuille Flow

where

u and

u $\_(()$t$)$ are the kinematic flow and turbulent viscosities, respectively. c$\_($p$)$ and Pr are the constant$-$

pressure heat capacity and the Prandtl number, respectively. u$($y$)$ and T$($y$)$ are the streamwise velocity

and temperature, respectively, varying in the wall normal direction $($y$)$ only. H is the distance between

the two walls, while V$\_($w$)$ is the wall velocity. $($dp$)/($dx$)$ is a constant streamwise pressure gradient. The other

velocity components are v$=0$y directionw$=0$z direction