Laplacian Operator : A Mathematical Key to Thermofluids P M V Subbarao Professor Mechanical Engineering Department I I T Delhi A Beautiful Cipher to tie pure mathematics to any branch of physics your heart might desire ….

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.

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.

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.

Irrotational Field Irrotational Field Physicists named it as Lamellar Field Theorem: A vector Field is defined and continuously differentiable throughout a simply connected domain D is conservative if and only if it is irrotational in D.

Irrotational Solenoidal Field Consider a Solenoidal Field  An irrotational- solenoidal field is defined as: Laplacian Scalar Field

Important Vector Calculus Identities

Laplacian of product of Scalar fields If a field may be written as a product of two scalar functions, then:

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

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

The Laplacian of Vector Fields If the scalar Laplacian operator is applied to a vector field, it acts on each component in turn and generates a vector field. In electromagnetic theory, in particular, one often finds 2 operator acting on a vector field A class of functions, known as “Harmonic Functions” satisfy what is known as the Laplacian equation Solenoidal-rotational Vector Fields….

Laplacian Scalar Fields in Thermofluids The mathematical theory of heat conduction, is due principally to Jean Baptiste Joseph Fourier. Fourier was the first mathematician, who brought order out of the confusion in which the experimental physicists had left the subject. Fourier, given these theories to the world through the French Academy in his “Theorie analytique de la Chaleur.” with solutions of problems naturally arising from it.

Three Fold Novelty of Fourier Work Great mathematicians generally admire what to praised of the three great achievements of Fourier thru this work. whether the uniquely original quality, or Their transcendently intense mathematical interest, or their perennially important instructiveness for physical science.

The Fourier Conduction Equation This equation expresses the conditions which govern the flow of heat in a body. The solution of any particular problem in heat conduction must first of all satisfy this equation, either as it stands, or in a modified form. For the steady state, Fourier's becomes;

Heat Conduction in A Plane Take the plate as the x-y plane with the base on the x-axis and one side as the y-axis. A Thin Circular plate in the r- plane. An axi-symmetric conduction problem as Thin Circular plate in the r- plane.

Types of boundary condition The value of T(x,y) is specified at each point on the boundary: “Dirichlet conditions” The derivative of T normal to the boundary is specified at each point of the boundary: “Neumann conditions”

Elementary potential flows-1 One dimensional Laplacian flow in Spherical Coordinate system: