1. Simple ODEs to Study Thermofluids
P M V Subbarao
Professor
Mechanical Engineering Department
I I T Delhi
Understanding of A Single DoF System ….

2. The Laplacian Operator
In Cartesian Coordinates
In Cylindrical Coordinates
In Spherical Coordinates

3. Elementary potential flows-1
One dimensional Laplacian flow in Spherical Coordinate system:
where m and C are integration constants.

4. Application in Hydrodynamics
This potential is considered as velocity potential, it produces the three-dimensional radial velocity.
m is called as fluid strength.
If m > 0 this is called as a point source or point sink, if m < 0.
By convention the constant m = Q/4 where Q is the flow rate or volume flux.

5. Application in Conduction
This potential is considered as Temperature, it describes radial heat conduction in a spherical solid.
Boundary conditions:
Dirichlet conditions : Temperatures specified at r0, r1, r2 and r3.
Neumann conditions t conditions : T/r specified at r0, r1, r2 and r3.

6. Elementary Vector Calculus Functions for Thermofluids
Solenoidal Flow Fields :
Irrotational (Lamellar) Flow Fields :
Potential Flow Fields, Steady state conduction & Steady state mass diffusion :
Beltrami Flow Fields :
Complex Lamellar Field

7. Assignment 1 Submit a research report on:
Two applications of Beltrami Flows.
Two applications of Complex Lamellar Flows.
Date of Submission: 20th August 2018.
Original Reports will be given highest marks.
All copied will be given zero marks.

8. Vectorially Described Thermofluid Fields

9. It is Essential to know Solutions of ODEs to Solve True Thermofluid Mathematical Models

10. First Order Differential Equations for Thermofluids
The general first-order differential equation for the function y = y(x) is written as
where f (x, y) can be any function of the independent variable x and the dependent variable y.
It is not always possible to find an analytical solution.

11. Separable First Order Differential Equations
A first-order ode is separable if it can be written in the form
where the function v(y) is independent of x and u(x) is independent of y.
Homogeneous first order ODE:
Inhomogeneous first order ODE:

12. Solution of Separable Homogeneous First ODE
Divide through by v(y) to obtain
Proceed to integrate both sides of this equation with respect to x, to get
Variables are separated, because
the left-hand side contains only the variable y and
the right-hand side contains only the variable x.
It can be tried to integrate each side separately.
If required integration is actually performed, it is possible to obtain a relationship between y and x.

13. Design of Thermometers in Thermofluids
Zeroth Law of Thermodynamics
Every system in this universe spontaneously move towards equilibrium with surroundings.
If A and C are in thermal equilibrium with B, then A is in thermal equilibrium with C. Maxwell [1872]
Design Rule:
Every thermal instrument must be designed to reach thermal equilibrium with the system as fast as possible.
Independent variable : time t
Dependent variable : Temperature T
Engineering Mathematics: Obtain a suitable ODE for this application.

14. Construction of A Thermometer

15. Options for Thermometers
Liquid in Glass Thermometer
Thermocouple Thermometer

16. Design Criteria : How Long it takes to achieve Zeroth Law?
Conservation of Energy during a time dt
Heat in = Change in energy of bulb
= Instantaneous Temperature of the bulb

17. Governing ODE for Thermometer Bulb
Define Time constant

18. Design for Constant System Temperature
Define relative temperature of the bulb as
Initial condition
Equilibrium condition

19. Instantaneous Relative Temperature of Bulb
Initial condition

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

21. Need for Inhomogeneous First Order ODE
Ts(t)=Ct

22. 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 .

23. 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: