1. First Order Inhomogeneous ODEs to Study Thermofluids
P M V Subbarao
Professor
Mechanical Engineering Department
I I T Delhi
Understanding of A Single DoF Systems Driven by External Drivers….

2. Response of A Thermometer bulb in A Constant Temperature System

3. Linear Inhomogeneous First Order ODEs
The first-order linear differential equation (linear in y and its derivative) can be written in the form
with the initial condition y(x0) = y0.
p(x) and g(x) are functions of x only.
It is defined as a linear equation, as each term involves y either as the derivative dy/dx OR
through a single factor of y .

4. Theory of Solution
A linear first order ODE can be solved using the integrating factor method.
The equation may be multiplied by the “Integrating Factor”, (x).
Multiply the original ODE with an integrating factor (x).
The IF is defined so that the equation becomes equivalent to:

5. The Solution in terms of IF
Where by integrating both sides with respect to x, gives:
Above equation is easily integrated using  (x0) = 0 and y(x0) = y0:

6. The Finale
It remains to determine (x).

7. The Integrating Factor
Above equation is separable:
It can be integrated to
it is customary to assign (x0)) = 1.

8. The Final Solution to Linear Inhomogeneous ODE
The Integrating Factor

10. Stokes Flow
In a remarkable 1851 scientific paper, Sir G. G. Stokes first derived the basic formula for the drag of a sphere.
Stokes solved a fourth order ODE to get the velocity field. CD Re
Derived a simple formula for drag using above velocity field.
The formula is strictly valid only for Re << 1 but agrees with experiment up to about Re = 1.

11. Important Applications of Stokes Solution in Thermofluids
What electric field is required to move a charged particle in electrophoresis?
What g force is required to centrifuge cells in a given amount of time?
What is the effect of gravity on the movement of a monocyte in blood?
How does sedimentation vary with the size of the sediment particles?
How rapidly do enzyme-coated beads move in a bioreactor?

12. Momentum Balance for The Falling Ball
Conservation Momentum:
Stokes drag

13. Inhomogeneous Liner ODE of Falling Ball Problem
Compare with general form of Linear Inhomogeneous first order ODE

14. Integrating Factor for Falling Ball

15. The Instantaneous Velocity of A falling Ball
Initial condition: ball stats from rest:

16. Inhomogeneous par of Falling Ball Problem

17. The Instantaneous Velocity of A falling Ball

18. Measurement of Temperature of an Harmonic System
If there is a system with sine-wave temperature in time, the output response of a thermometer is quite different.
The Integrating Factor

19. Measurement of Temperature of an Harmonic System

20. Response of A thermometer
Ts,max- Tb,max 

21. Measurement of Temperature of A Turbulent Flow
Fourier transform helps in transforming a random signal into a series of harmonic functions.
The Integrating Factor

22. Generation of Cheapest Initial Solutions….
A Travel into Space…..
Generation of Cheapest Initial Solutions….

23. Basic Forces Acting on A Rocket
T = Rocket thrust
D = Rocket Dynamic Drag
Vr = Velocity of rocket
mejects = Mass flow rate of ejects
mr= Mass of the rocket

24. MOMENTUM BALANCE FOR A ROCKET
Rocket mass X Acceleration = Thrust – Drag -gravity effect
Drag

25. First Order ODE for Travel of A Rocket
Conservation of mass:

26. Finite Duration of Flying
A rocket is to be designed for a finite duration of flying, known as time of burnout, tb.

27. Requirements to REACH An ORBIT
For a typical launch vehicle headed to an orbit, aerodynamic drag losses are in the order of 100 to 500 m/sec.
Gravitational losses are larger, generally ranging from 700 to 1200 m/sec depending on the shape of the trajectory to orbit.
By far the largest term is the equation for the space velocity increment.
The lowest altitude where a stable orbit can be maintained, is at an altitude of 185 km.

28. Geostationary orbit
A circular geosynchronous orbit in the plane of the Earth's equator has a radius of approximately 42,164 km from the center of the Earth.
A satellite in such an orbit is at an altitude of approximately 35,786 km above mean sea level.
It maintains the same position relative to the Earth's surface.
If one could see a satellite in geostationary orbit, it would appear to hover at the same point in the sky.
Orbital velocity is 11,066 km/hr= 3.07 km/sec.

29. Series Stage Rocket
3rd Stage
Thrusting

30. Clustered Rocket in First Stage

31. Travel Cycle of Modern Spacecrafts