Laplace Equation

Overview

Engineering reference data for Laplace Equation in fluid mechanics.

Key Formulas

Reynolds Number

$Re = \frac{\rho v D}{\mu}$

Ratio of inertial to viscous forces — determines flow regime.

Bernoulli's Equation

$P + \frac{1}{2}\rho v^2 + \rho g h = \text{const}$

Conservation of energy for steady, inviscid, incompressible flow.

Continuity Equation

$A_1 v_1 = A_2 v_2$

Conservation of mass for incompressible flow.

Darcy-Weisbach

$\Delta P = f \frac{L}{D} \frac{\rho v^2}{2}$

Pressure drop due to friction in a pipe.

Variables

Symbol	Description	Unit
$Re$	Reynolds number	—
$\rho$	Fluid density	kg/m³
$v$	Flow velocity	m/s
$D$	Characteristic dimension	m
$\mu$	Dynamic viscosity	Pa·s
$P$	Pressure	Pa
$f$	Darcy friction factor	—

Applications

The Laplace equation is fundamental in multiple engineering and scientific disciplines:

• Electromagnetism: Governs electrostatic potentials in charge-free regions and magnetostatics in current-free regions
• Astronomy: Models gravitational potentials outside mass distributions
• Fluid Dynamics: Describes incompressible, irrotational flow (potential flow)

Solution Techniques

Laplace's equation can be solved using:

• Separation of variables
• Conformal mapping
• Green's functions
• Numerical methods (finite differences, finite elements)

Numerical Solution Example

For 2D problems with Dirichlet boundary conditions, a finite difference approximation yields:

$\frac{u_{i+1,j} - 2u_{i,j} + u_{i-1,j}}{(\Delta x)^2} + \frac{u_{i,j+1} - 2u_{i,j} + u_{i,j-1}}{(\Delta y)^2} = 0$

This creates a linear system $A\mathbf{u} = \mathbf{b}$ that can be solved iteratively.

References

• Original Source