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Exercises
1.
An open, cylindrical tank having a diameter d is supported around its bottom
circumference and is filled to a depth h with a liquid having specific weight  . The
vertical deflection  , of the center of the bottom is a function of d , h , t ,  and E ,
where t is the thickness of the bottom and E is the modulus of elasticity of the bottom
material. Establish the relation among the pi terms by dimensional analysis.
2.
A spherical particle of diameter d falls very slowly at a velocity V through a
viscous fluid of viscosity  . Determine the functional relation for the drag force acting
on the particle with the help of dimensional analysis.
3.
The drag force D of a plane during flight depends on the length of the aircraft l ,
velocity V , air density  and the bulk modulus of air K . Using the dimensional
analysis, determine the functional relationship between these variables and the drag force.
Also, explain the physical meaning of the dimensional group.
4.
The thrust developed by a propeller P depends on the angular velocity  ,
velocity V , diameter d , viscosity  , density  and elasticity of fluid medium, which
can be expressed by the speed of the sound in the medium C . Find the suitable
parameters to present the thrust developed by the propeller.
5.
The efficiency  of a fan depends on the density  , viscosity  of the fluid,
angular velocity  , diameter d of the rotor and the discharge Q . Express  in terms of
dimensionless parameters.
6. The pressure change across a diffuser of circular cross-section depends on the
discharge Q , inlet and outlet diameters D1 and D2 , diffuser length l , density 
and viscosity of the fluid  . Perform the dimensional analysis.
7. Form dimensionless parameters among the variables:
(a) F ,  , U , l ; (b) U ,  ,  , l ; (c)  , p, l ; (d)
du
,  ,  , y ; (e) F , U ,  , l ; (f) , , t ,  ;
dy
(g) f ,U , l ; (h)  , t , l ; (i) p,  , U , C ; (j) F ,  ,  ; (k)  , g ,  ,  ; (l)  ,U ,  ,
p
x
where the parameters and symbols can be denoted as follows;
C is the velocity of pressure wave, f is frequency, g is acceleration due to gravity, F
is the force, l is length, p is the pressure, t is the time, u is the velocity in y direction,
x, y is the distance, U is free stream velocity,  is the density,  is specific gravity, 
is dynamic viscosity,  is the kinematics viscosity,  is the angular velocity,  is
circulation,  is surface tension and  is the boundary layer thickness
8.
Check whether the following equations are dimensionally homogeneous or not.
Convert them into equations among dimensionless parameters and verify Bucking ham’s
Pi theorem.
(a) p 
U
32 Ul
;
D2
(b) h 
flU 2
where f is the dimensionless friction factor; (c)
2 gD
C 23 12
h S where d and h are length parameter, S is the slope, C is a constant;
d1 6
(d)  0  C
U 3 
x
9.
The capillary rise  h  of a liquid in a tube varies with the tube diameter  d  ,
gravity
 g,
fluid density    , surface tension   and contact angle   . Find the
dimensionless relation. If h = 9cm in a given experiment, then what will be the h in a
similar case for which the diameter and surface tension is halved, density being twice
with contact angle being the same.
(Ref. 5; e.g. 5.6; pp. 273)
10.
A large hydraulic turbine is to generate 300kW at 1000rpm under a head of 40m.
For initial testing, a 1:4 scale model of the turbine operates under a head of 10m. Find the
power generated by the model. (Ans. 2.34kW; GATE 2006)