Homogeneous Differential Equation Solution of Laplace’s Homogeneous Differential Equation SOLO HERMELIN Updated: 17.03.07

Laplace’s Homogeneous Differential Equation SOLO Laplace’s Homogeneous Differential Equation TABLE OF CONTENT Laplace’s Homogeneous Differential Equation Green’s Identities Green’s Function Solution of the Laplace’s Homogeneous Differential Equation Boundary Conditions 1. Dirichlet Problem 2. Neumann Problem Uniqueness of a Laplace Solution that satisfies Dirichlet or Neumann Boundary Conditions Properties of Irrotational Fluids References

Laplace’s Homogeneous Differential Equation SOLO Laplace’s Homogeneous Differential Equation The Laplace’s Homogeneous Differential Equation for the Potential in a irrotational, homentropic (constant entropy) fluid is: Pierre-Simon Laplace 1749-1827 We want to find the Potential Φ at the point F (field) due to all the sources (S) in the volume V, including its boundaries . F inside V F on the boundary of V Therefore is the vector from S to F. Let define the operator that acts only on the source coordinate .

Laplace’s Homogeneous Differential Equation SOLO Laplace’s Homogeneous Differential Equation To find the solution we need to prove the following: GREEN’s IDENTITY GREEN’s FUNCTION This Green’s Function is a particlar solution of the following Poisson’s Non-homogeneous Differential Equation: Siméon Denis Poisson 1781-1840

Laplace’s Homogeneous Differential Equation SOLO Laplace’s Homogeneous Differential Equation GREEN’s IDENTITY Let start from the Gauss’ Identity where is any vector field (function of position and time) continuous and differentiable in the volume V. Let define . We have Karl Friederich Gauss 1777-1855 Then First Green’s Identity If we interchange G with Φ we obtain Subtracting the second equation from the first we obtain Second Green’s Identity

Laplace’s Homogeneous Differential Equation SOLO Laplace’s Homogeneous Differential Equation GREEN’s FUNCTION Define the Green’s Function is a particlar solution of the following Poisson’s Non-homogeneous Differential Equation: where δ (x) is the Dirac function Let use the Fourier Transformation to write where

Laplace’s Homogeneous Differential Equation SOLO Laplace’s Homogeneous Differential Equation Jean Baptiste Joseph Fourier 1768 - 1830 GREEN’s FUNCTION (continue – 1) Let use the Fourier Transformation to write Hence or

Laplace’s Homogeneous Differential Equation SOLO Laplace’s Homogeneous Differential Equation GREEN’s FUNCTION (continue – 2) Let compute: Therefore: Because this is true for all k, we obtain

Laplace’s Homogeneous Differential Equation SOLO Laplace’s Homogeneous Differential Equation GREEN’s FUNCTION (continue – 3) Let use spherical coordinates relative to vector:

Laplace’s Homogeneous Differential Equation SOLO Laplace’s Homogeneous Differential Equation GREEN’s FUNCTION (continue – 4) where we used (see next slide) Therefore

Poisson’s Non-homogeneous Differential Equation SOLO Poisson’s Non-homogeneous Differential Equation GREEN’s FUNCTION (continue – 5) Let compute: For this use the integral: since z = 0 is outside the region of integration M.R. Spiegel, “Complex Variables”, Schaum’s Outline Series, 1964, pg.184 Therefore:

Laplace’s Homogeneous Differential Equation SOLO Laplace’s Homogeneous Differential Equation GREEN’s FUNCTION (continue – 6) Hence a Green’s Function for the Poisson’s Non-homogeneous Differential Equation This solution is not unique since we can add any function that satisfies the Laplace’s Equation (1782) Therefore we have the following Green’s Function

Laplace’s Homogeneous Differential Equation SOLO Laplace’s Homogeneous Differential Equation Solution of the Laplace’s Homogeneous Differential Equation Let return to the Laplace’s Homogeneous Differential Equation for the Potential Φ: Pierre-Simon Laplace 1749-1827 We want to find the Potential Φ at the point F (field) due to all the sources (S) in the volume V, including its boundaries Solutions of the Laplace’s equation are known as harmonic functions. F inside V F on the boundary of V

Laplace’s Homogeneous Differential Equation SOLO Laplace’s Homogeneous Differential Equation Solution of the Laplace’s Homogeneous Differential Equation (continue -1) is the vector from S to F. Let define the operator that acts only on the source coordinate . Since is no defined at r = 0 we define the volume V’ as the volume V minus a small sphere of radius and surface around the point F, when F is inside V, or a semi-sphere of radius and surface around the point F, when F is on the boundary of V.

Laplace’s Homogeneous Differential Equation SOLO Laplace’s Homogeneous Differential Equation Using the Green’s Identity let compute

Laplace’s Homogeneous Differential Equation SOLO Laplace’s Homogeneous Differential Equation We obtain where Note If F is outside V from the Green’s Second Identity we obtain End Note

Laplace’s Homogeneous Differential Equation SOLO Laplace’s Homogeneous Differential Equation For we have where

Laplace’s Homogeneous Differential Equation SOLO Laplace’s Homogeneous Differential Equation Physical interpretation of Green’s Functions 1. Point Source The radial velocity is given by 2. Doublet The derivative of a source in any direction is called a doublet

Johann Peter Gustav Lejeune Dirichlet SOLO Laplace’s Homogeneous Differential Equation BOUNDARY CONDITIONS The General Green Function that is a class of bi-position function, and contains an arbitrary harmonic function (solution of the Laplace’s Equation) Let consider th following two simple cases (Dirichlet and Neumann Problems): Dirichlet Problem The potential is defined at the boundary S of the volume V. Let choose such that In this case Johann Peter Gustav Lejeune Dirichlet 1805-1859 where

Laplace’s Homogeneous Differential Equation SOLO Laplace’s Homogeneous Differential Equation BOUNDARY CONDITIONS (continues – 1) 2. Neumann Problem The potential derivative is defined at the boundary S of the volume V. Let choose such that In this case Carl Neumann 1832-1925 where

Laplace’s Homogeneous Differential Equation SOLO Laplace’s Homogeneous Differential Equation Uniqueness of a Laplace Solution that satisfies Dirichlet or Neumann Boundary Conditions Suppose that we have a solution Φ that satisfies the Laplace Homogeneous Differential Equation: in the volume V, including its boundaries . Suppose also that Dirichlet or Neumann conditions or a combination of those, are specified. In this case the solution is unique (up to an additive constant). Proof Suppose that thee exist two solutions and , and define We have

Laplace’s Homogeneous Differential Equation SOLO Laplace’s Homogeneous Differential Equation Uniqueness of a Laplace Solution that satisfies Dirichlet or Neumann Boundary Conditions (continue – 1) Proof (continue) If Dirichlet conditions are satisfied: If Neumann conditions are satisfied: Let use Green’s First Identity (with G = Φ) We have End of Proof

Laplace’s Homogeneous Differential Equation SOLO Laplace’s Homogeneous Differential Equation Properties of Irrotational Fluids (Karamcheti ,“Principles of Ideal Aerodynamics”, pp.269-273) 1. The potential Φ can neither be a maximum nor a minimum in interior of the fluid For any point inside the fluid we can choose an infinitesimal sphere δS centered at this point for which Since can not be either positive or negative inside the fluid, the maximum or minimum of the potential Φ can occur only at boundary of motion. 2. The spatial derivative of Φ are also harmonic functions, that is they satisfy Laplace’s equation

Laplace’s Homogeneous Differential Equation SOLO Laplace’s Homogeneous Differential Equation Properties of Irrotational Fluids (Karamcheti “Principles of Ideal Aerodynamics”, pp.269-272 (continue – 1) 3. The spatial derivative of Φ can neither be a maximum nor a minimum in interior of the fluid Follows from (1) and (2). 4. The velocity components can neither be a maximum nor a minimum in interior of the fluid Follows from (3). 5. The magnitude of the velocity cannot be a maximum in interior of the fluid Let use Green’s First Identity (with G = Φ) Since the velocity vector also satisfies the Laplace’s equation we may choose instead of Φ the components of to obtain Since the velocity magnitude cannot be maximum inside the fluid.

Laplace’s Homogeneous Differential Equation SOLO Laplace’s Homogeneous Differential Equation References H. Lass, “Vector and Tensor Analysis”, McGraw-Hill, 1950, pp.155 Karamcheti,“Principles of Ideal Aerodynamics”, pp.269-272

Israeli Institute of Technology Israeli Armament Development Authority SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA April 28, 2017