Question 1

Which of the following conditions is never a boundary condition in fluid mechanics?
(A) The velocity is zero at a boundary
(B) The normal component of velocity is zero in an inviscid flow at a boundary
(C) The tangential component of velocity is zero in an inviscid flow at a boundary
(D) The pressure is zero at a boundary

Accepted Answer

The tangential component of velocity is zero in an inviscid flow at a boundary
The tangential component of velocity must be tangential to the boundary at a solid
boundary, but usually, if not always, in an inviscid flow it is nonzero. A porous
boundary would provide a normal component of velocity to the fluid at the
boundary.
The pressure is zero (gage) at a boundary if the boundary is a free surface, the
atmosphere.
In a viscous flow, the fluid sticks to the boundary so it takes on the velocity of
the boundary, which is usually zero.

Question 2

$\text {If the $x$-component of velocity $u$ depends only on $y$ in a plane incompressible flow, the }$$y \text {-component of velocity } v \text { is: }$ &#10;A) f(y)&#10;B) const&#10;C) f(x)&#10;D) 0

Accepted Answer

$f ( x )$
$\text { The differential continuity equation } \partial u / \partial x+\partial v / \partial y=0 \text { is used: }$ 
$$\frac { \partial u } { \partial x } = 0 = - \frac { \partial v } { \partial y } . \quad \therefore v \neq f ( y ) \text { and } \underline { v = f ( x ) }$$
$\text { It could be a constant, or } 0 \text {, but not necessarily so. It is not a function of } y \text {. }$

Question 3

$\text {What four equations provide for the four unknowns $u , v , w , p$ that are most often of interest in}$ $\text {fluid mechanics?}$&#10;A) Continuity, momentum&#10;B) Continuity, momentum, energy&#10;C) Continuity, momentum, energy, equation of state&#10;D) Continuity, momentum, energy, enthalpy relation

Accepted Answer

$\text {Continuity, momentum}$
$\text {The momentum equation is a vector equation containing the three velocity components}$ $\text {$u , v , w$, and $p$. Continuity provides the fourth equation. The energy differential equation}$ $\text {enters only when temperature is an unknown.}$

Question 4

Three measurements are made of the x-component velocity in the diffuser of an incompressible plane flow to be 32 m/s, 28 m/s and 20 m/s. The measurement points are
Along the centerline of the symmetrical diffuser and are 4 cm apart. The y-component of the
Velocity 2 cm above the centerline is approximated to be:

A) 2 m/s
B) 3 m/s
C) 4 m/s
D) 5 m/s

Accepted Answer

$3 \mathrm {~m} / \mathrm { s }$
$\text {We chose to use the more accurate central difference to approximate the derivative }$$\text {since the information given allows such an approximation:}$
$$\frac { \partial u } { \partial x } \cong \frac { \Delta u } { \Delta x } = \frac { 20 - 32 } { 0.08 } = - 150 \frac { \mathrm { m } / \mathrm { s } } { \mathrm { m } }$$
$\text {The continuity equation then provides}$
$$\frac { \partial v } { \partial y } = - \frac { \partial u } { \partial x } = 150 \quad \text { or } \quad \frac { \partial v } { \partial y } \cong \frac { \Delta v } { \Delta y } = \frac { v _ { 2 } - 0 } { 0.02 } = 150 . \quad \therefore v _ { 2 } = \underline { 3 \mathrm {~m} / \mathrm { s } }$$

Deck 12: Turbomachinery