Solution of Laplace ’ s Homogeneous Differential Equation

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

2. 2. Neumann Problem 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 Uniqueness of a Laplace Solution that satisfies Dirichlet or Neumann Boundary Conditions Properties of Irrotational Fluids References

3. 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 . Therefore is the vector from S to F. Let define the operator F inside V F on the boundary of V that acts only on the source coordinate . 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

4. 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

5. where is any vector field (function of position and time) continuous and differentiable in the volume V. Let define . SOLO Laplace’s Homogeneous Differential Equation • GREEN’s IDENTITY Let start from the Gauss’ Identity 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

6. 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

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

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

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

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

11. Let compute: For this use the integral: Therefore: SOLO Poisson’s Non-homogeneous Differential Equation • GREEN’s FUNCTION (continue – 5) since z = 0 is outside the region of integration

12. 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

13. 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. Pierre-Simon Laplace 1749-1827 F inside V F on the boundary of V 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:

14. Since is no defined at r = 0 we define the volume V’ as the volume V minus a small sphere of radiusand 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. Let define the operator that acts only on the source coordinate . is the vector from S to F. SOLO Laplace’s Homogeneous Differential Equation Solution of the Laplace’s Homogeneous Differential Equation (continue -1)

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

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

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

18. 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

19. Let choose such that where 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. In this case Johann Peter Gustav Lejeune Dirichlet 1805-1859

20. Let choose such that where 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. Carl Neumann 1832-1925 In this case

21. in the volume V, including its boundaries . Suppose that thee exist two solutions and , and define 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: Suppose also that Dirichlet or Neumann conditionsor a combination of those, are specified. In this case the solution is unique (up to an additive constant). Proof We have

22. 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

23. 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. 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 2. The spatial derivative of Φ are also harmonic functions, that is they satisfy Laplace’s equation

24. we may choose instead of Φ the components of to obtain Since the velocity magnitude cannot be maximum inside the fluid. 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

25. 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

26. 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