Some Examples of the Poisson Equation  . (ku) = f

[Taken from J. N. Reddy's Book "Introduction to the Finite Element Method", J.N. Reddy, McGraw Hill Publishers, 2nd Edition]

Field of Application

Primary Variable u

Material Constant k

Source Variable f

Secondary Variables d, du/dx, du/dy 

Heat Transfer

Temperature T

Conductivity k

Heat Source Q

Heat Flow q [comes from conduction k T/n and convection h(T-T)

 

Irrotational Flow of an Ideal Fluid

 

Stream Function y

 

Velocity Potential f

 

Density r

 

Density r

 

Mass Production s (normally zero)

Mass Production s (normally zero)

 

Velocities:

y /x = -v;

y /y = u

f /x = -v;

f /y = u

 

Groundwater Flow

 

Piezometric Head f

 

Permeability K

 

Recharge Q

 

Seepage: q = k f/dn

Velocities:

u = -k f/dx ,

v = -k f/dy

 

Torsion of Members with Constant Cross-Section

 

Stress Function Y

 

k = 1

G = Shear Modulus

 

f = 2

q = angle of twist per unit length

 

Gqdf/dx = -syz

Gqdf/dx = -sxz

 

Electrostatics

 

Scalar Potential f

 

Dielectric Constant e

 

Charge Density r

 

Displacement Flux density Dn

 

Magnetostatics

 

Magnetic Potential f

 

Permeability m

 

Charge density r

 

Magnetic Flux density Bn

 

Transverse Deflection of Elastic Membranes

 

Transverse deflection u

 

Tension T in membrane

 

Transversely distributed Load

 

Normal force q

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