396
Solutions Manual x Fluid Mechanics, Fifth Edition
Solution: (a) Since all terms in the equation contain C, we establish the dimensions of k and by comparing {k} and {w2/w x2} to {uw/w x}:
­ w2 ½
­1½
­w ½
{k} {} ® 2 ¾ {} ® 2 ¾ {u} ® ¾
¯L ¿
¯w x ¿
¯ w x °¿
hence { k}
­1½
® ¾ and { }
¯T ¿
­L ½­ 1 ½
® ¾ ® ¾,
¯T ¿ ¯ L ¿
­ L2 ½
® ¾ Ans. (a)
¯ T °¿
(b) To non-dimensionalize the equation, define u* u/V , t* Vt /L , and x*
into the basic partial differential equation. The dimensionless result is
u*
x/L and sub-stitute
w C § · w 2 C § kL ·
wC
VL
C
, where
mass-transfer Peclet number Ans. (b)
¨©
¸¹
¸
2 ¨
w x* VL w x* © V ¹
w t*
5.46 The differential equation for compressible inviscid flow of a gas in the xy plane is
2
2
w 2I w 2 2
w 2I
2
2 w I
2
2 w I
(u X ) (u a ) 2 (X a ) 2 2uX
w xw y
w t2 w t
wx
wy
0
where I is the velocity potential and a is the (variable) speed of sound of the gas.
Nondimensionalize this relation, using a reference length L and the inlet speed of sound ao as
parameters for defining dimensionless variables.
Solution: The appropriate dimensionless variables are u* u/a o , t* a o t/L, x* x/L,
a* a/a o , and I * I /(a o L) . Substitution into the PDE for I as above yields
2
2
w 2I * w
w 2I *
2
2
2
2 w I*
2
2 w I*
u* v* u* a* v* a* 2u*v*
w x*w y*
w t*2 w t*
w x*2
w y*2
0 Ans.
The PDE comes clean and there are no dimensionless parameters. Ans.
5.47 The differential equation for small-amplitude vibrations y(x, t) of a simple beam is given by
UA
where U
A
I
E
beam material density
cross-sectional area
area moment of inertia
Young’s modulus
w 2y
w 4y
EI
w t2
w x4
0