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Introduction to Partial Differential Equations

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Presentation on theme: "Introduction to Partial Differential Equations"— Presentation transcript:

1 Introduction to Partial Differential Equations
P M V Subbarao Professor Mechanical Engineering Department I I T Delhi PDEs are the language of nature !!!

2 Introductory Remarks PDEs stand alone as one of the greatest intellectual achievements of the human race in its attempt to understand the physical world. PDEs are the transcriptions of the behaviour of solid and fluid mechanics, quantum mechanics and general relativity. What is the mathematical origin of PDEs?

3 What is a Partial Differential Equation?
A partial differential equation for a function y of the independent variables x1, x2, ,xn (n > 1) is a relation of the form y = f(x1,x , xn) is said to be a solution of above, when this function, satisfies the PDE identically within a given domain of the independent variables x1,x2,…..xn.

4 Universality of Vector Calculus
Theorem: Gradient, divergence, curl and Laplacian are coordinate-free.

5 Divergence of a vector field
Divergence is the outflow of flux from a small closed surface area (per unit volume) as volume shrinks to zero. Divergence of a vector field: In an orbitrary coordinate systems: In Cartesian coordinate system:

6 Definition of Divergence in fundamental CoS
In Cartesian coordinate system: In Cylindrical coordinate system: In Spherical coordinate system:

7 What is divergence : Application to Rivers / Natural Reservoirs
Think of a vector field as a velocity field for a moving fluid. The divergence measures sources and drains of flow: The divergence measures the expansion or contraction of the fluid. A vector field with constant positive or negative value of divergence. Presence of A source Presence of A Drain

8 Vector Calculus for Study/Quantification of a River

9 Common Knowledge about Flow Rate of a River
The flow is higher close to the center and slower at the edges. Let us insert small paddle wheels in a flowing river. V Wheels close to the edges will rotate due to difference in velocities. A wheel close to the center (of a river) will not rotate since velocity of water is the same on both sides of the wheel. The curl operation determines the direction and the magnitude of rotation.

10 Flow Measurement in a River

11 Curl of a vector field The curl of vector field at a point in a medium is a measure of the net rotation of the vector as the area shrinks to zero. Curl of a vector field: If (in a Cartesian system) the vector is defined as

12 Thermofluid Significance of Curl
Circulation is the amount of force that pushes along a closed boundary or path. It's the total "push" you get when going along a path, such as a circle. Curl is simply circulation per unit area, circulation density, or rate of rotation (amount of twisting at a single point) Curl is a vector field with magnitude equal to the maximum "circulation" at each point and oriented perpendicularly to this plane of circulation for each point. More precisely, the magnitude of curl is the limiting value of circulation per unit area.

13 The Curl of Velocity Field
Define the vorticity vector as being the curl of the velocity vorticity vector in cylindrical co-ordinates: vorticity vector in spherical co-ordinates:

14 Irrotational Flow Field
Flows with non zero vorticity are said to be rotational flows. Flows with zero vorticity are said to be irrotational flows. If the velocity is exactly equal to gradient of a scalar, the flow filed is obviously irrotational. If an application calls for an irrotational flow, the problem is completely solved by finding a scalar field, .

15 Irrotational Solenoidal Field
Irrotational Field Solenoidal Field Irrotational, Solenoidal Field Laplacian Field

16 The Laplacian Operator
Laplace’s equation is named for Pierre-Simon Laplace, a French mathematician. It’s in electricity. It’s in magnetism. It’s in fluid mechanics. It’s in gravity. It’s in heat. It’s in soap films. It’s everywhere. In 1799, Laplace proved that the solar system was stable over astronomical timescales. It is in contrary to what Newton had thought a century earlier. In the course of proving Newton wrong, Laplace investigated the equation that bears his name.

17 The Original Laplacian Operator
It has just five symbols. With just these five symbols, Laplace read/mimicked the universe. There’s an upside-down triangle called a nabla that’s being squared, the squiggly Greek letter phi, an equals sign, and a zero. Phi is the thing you’re interested in. It’s a potential. Potential is something thermo fluid experts confidently pretend to understand.

18 Laplacian of A scalar field
The Laplacian is a scalar operator. If it is applied to a scalar field, it generates a scalar field. The Laplacian operator is defined as: The equation 2f = 0 is called Laplace’s equation. This is an important equation in thermofluids.


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