Engineering Mathematics 1: Bernoulli's Equation and Fluid Flow

This assignment presents a solution to a fluid dynamics problem using Bernoulli's equation and the principle of continuity. It assumes incompressible and inviscid flow to analyze the drainage of a conical funnel. The solution derives an equation for the time taken to drain the funnel based on its dimensions and the initial height of the fluid. The detailed steps involve equating mass flow rates at different points, integrating a differential equation, and applying relevant geometric relationships. References to several fluid mechanics textbooks are also included to support the methodology and theoretical background used in solving the problem.

Engineering Mathematics 1
ENGINEERING MATHEMATICS
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Engineering Mathematics 2
Problem 1
The assumptions made are:
 Incompressible flow
 In viscid flow
Using Bernoulli’s equation
p1
r + v1
2
2 g + z1 = p2
r + v2
2
2 g + z2
Here, P1=P2=0
As v2=−dh
dt ≪ <v1 , theequation becomes
z1= v2
2
2 g + z2; z2= v1
2
2 g + z1
v2
2=2 g (z1 −z2 ) v1
2=2 g (z2 −z1 )
v2=2 g v1 = [ 2 g .( z2−z1 ) ]1
2

Engineering Mathematics 3
By the equation of continuity, the mass flow rate in the downward direction at point 2 is the
same as the mass flow at point 1
m=ρ V 1 A1=ρ V 2 A2
V 1 A1=V 2 A2
As R=h tan Ɵ, [ 2 gh (t) ] 1
2 π 92=−dh
dt . π R2, where 9 is the diameter of the smaller circular
part of the cone as shown in the diagram at point 1.
[ 2 gh (t) ] 1
2 π 92=−dh
dt [ π h2 ( t ) . tanθ ]
− ( 2 g )
1
2 . 92
tan θ ∫
0
t
dt= 2
5 ( h1
5
2 −ht
5
2 )
t=
2
5 ( h1
5
2 −ht
5
2 ) tan2 θ
92 √ 2 g
Sinceθ=45 °,
t=
2
5 ( h1
5
2 −ht
5
2 ) tan2 45 °
92 √ 2 × 9.8
t=0.09 √ hi ( hi
9 )
2
seconds
Since R=h tan 45
R=h= hi

Engineering Mathematics 4
Hence the time taken to drain the conical funnel is
t=0.09 × √R . ( R
9 )2
t=0.09 × √R . ( R
9 )2
seconds

Engineering Mathematics 5
References
Bird, J., 2013. Understanding Engineering Mathematics. 9th ed. London: Routledge.
Couling, S., 2015. Measurement of Airborne Pollutants. 3rd ed. New York: Elsevier.
Katz, J., 2010. Introductory Fluid Mechanics. 2nd ed. Cambridge: Cambridge University Press.
Kaven Baessler, B.S.K.L.B.K.M.S.L.S., 2010. Pelvic Floor Re-education: Principles and
Practice. 2nd ed. Oxford: Springer Science & Business Media.
Keisler, H.J., 2012. Elementary Calculus: An Infinitesimal Approach. 5th ed. Kansas: Courier
Corporation.
Mitchell, J.W., 2013. Fundamentals of Fluid Mechanics. 4th ed. London: John Wiley & Sons.
Pritchard, P.J., 2015. Fox and McDonald's Introduction to Fluid Mechanics, 9th Edition. 4th ed.
London: Wiley.
Robert W. Fox, A.T.M.P.J.P., 2009. Introduction to Fluid Mechanics. 7th ed. New York: Wiley.