Homework 8 1. A gas-­‐filled pneumatic strut in an automobile suspension system behaves like a piston-­‐cylinder apparatus. At one instant when the piston is L = 0.15 m away from the closed end of the cylinder, the gas density is uniform at ρ = 18 kg/m3 and the piston begins to move away from the closed end at V = 12 m/s. The gas velocity is one-­‐dimensional and proportional to distance from the closed end; it varies linearly from zero at the end to u = V at the piston. a) Obtain an expression for the average density as a function of time. (20 pts) b) Evaluate the rate of change of gas density at t = 0 (10 pts) c) Graph density as a function of time (10 pts) ∂ρu ∂ρv ∂ρw ∂ρ
+
+
+
=0
dx
dy
dz
dt
∂ρu ∂ρ
+
=0
dx
dt
∂ρu
∂ρ
∂u
∂ρ
=−
= −ρ
−u
dx
dt
dx
dx
∂ρ
dρ
∂u
=0
= −ρ
∂x
dt
∂x
x
∂u V dρ
V
u =V ,
=
= −ρ
L
∂x L dt
L
L = Lo + Vt
ρ
∫
ρ
o
dρ
ρ
t
= −∫
0
V dt
Lo + Vt
ln
Lo
ρ
= ln
ρo
Lo + Vt
⎡
⎤
1
ρ (t ) = ρ o ⎢
⎥
⎣1 + Vt / Lo ⎦
At t = 0 ∂ρ
V
= − ρ o =-­‐1440 kg/(m3-­‐s) ∂t
L
0.25 0.2 0.15 0.1 0.05 0 0 5 10 15 20 25 2. Fluid flows down an inclined plane surface in a steady, fully-­‐developed laminar film of thickness h. Simplify the continuity and Navier-­‐Stokes equations to model this flow field. Obtain expressions for: a) the liquid velocity profile (20 pts) b) the shear stress distribution (5 pts) c) the volume flow rate, and (5 pts) d) the volume flow rate in a film of water 1 mm thick on a surface 1 m wide, inclined at 15o to the horizontal.(5 pts) a) Asssumptions: a) steady flow – all t terms gone b) incompressible – ρ = constant c) w = 0, d/dz = 0 d) fully developed – d/dx = 0 Using continuity: dv/dy = 0 According to c) dv/dz = 0 According to d) dv/dx = 0 therefore v = constant. v = 0 at surface, therefore v = 0 everywhere. Final simplification: ∂p
∂y
0 = ρg y −
0 = ρg x + µ
∂ 2u
∂y 2
d 2u
sin θ
= − ρg
2
µ
dy
du
sin θ
= − ρg
y + c1
dy
µ
u = − ρg
sin θ y 2
+ c1 y + c2
µ 2
Boundary conditions: at y = 0, u = 0, therefore c2 = 0 at y = h, du/dy = 0, therefore c1 = ρg
sin θ
µ
h Therefore: u = − ρg
b) τ = µ
sin θ y 2
sin θ
sin θ ⎛
y 2 ⎞
⎜⎜ hy − ⎟⎟ + ρg
hy = ρg
µ 2
µ
µ ⎝
2 ⎠
du
= ρg sin θ (h − y) dy
h
h
c) Q = u dA = u b dy = ρg
∫
A
d) Q = ρg
∫
0
∫
0
sin θ ⎛
y 2 ⎞
sin θ b ⎛ h 3 ⎞
⎜⎜ hy − ⎟⎟dy = ρg
⎜ ⎟ µ ⎝
2 ⎠
µ ⎜⎝ 3 ⎟⎠
sin θ b ⎛ h 3 ⎞
1
(.001) 3
⎜⎜ ⎟⎟ = 999 * 9.81* sin(15 o ) *1*
X
*1000 = 0.846L / s µ ⎝ 3 ⎠
3
1.00 *10 −3
3. Verify that the stream function in cylindrical coordinates satisfies the continuity equation. (5 pts) 4. Find the velocity distribution for two-­‐dimensional flow of a viscous fluid between wide parallel plates. Determine the corresponding stream function and velocity potential if possible. (15 pts) 5. Determine whether the following stream functions for an incompressible, two-­‐dimensional flow field are irrotational. Assume a and b are nonzero constants. (5 pts) ψ = ay + by 3
ψ = ay 2 − bx