Module 2
Mathematical Classification
of Partial differential
equations
∂ρ ∂ρu ∂ρv ∂ρw
+ + + =0
∂t ∂x ∂y ∂z "
∂ρu ∂ρu2 ∂ρuv ∂ρuw
#
∂p ∂τxx ∂τyx ∂τzx
+ + + =− + + +
∂t ∂x ∂y ∂z ∂x " ∂x ∂y ∂z #
∂ρv ∂ρuv ∂ρv 2 ∂ρvw ∂p ∂τxy ∂τyy ∂τzy
+ + + =− + + +
∂t ∂x ∂y ∂z ∂x " ∂x ∂y ∂z #
2
∂ρw ∂ρuw ∂ρvw ∂ρw ∂p ∂τxz ∂τyz ∂τzz
+ + + =− + + +
∂t ∂x ∂y ∂z ∂x ∂x ∂y ∂z
Navier stokes equations which govern the flow of fluids are par-
tial differential equations. A differential equation involving partial
derivatives of a dependent variable(one or more) with more than
one independent variable is called a partial differential equation
Order of a partial differential equation
The order of the highest derivative term in the equation is called
the order of the PDE
1
Mathematical Classification
of Partial differential
equations
∂ρ ∂ρu ∂ρv ∂ρw
+ + + =0
∂t ∂x ∂y ∂z "
∂ρu ∂ρu2 ∂ρuv ∂ρuw
#
∂p ∂τxx ∂τyx ∂τzx
+ + + =− + + +
∂t ∂x ∂y ∂z ∂x " ∂x ∂y ∂z #
∂ρv ∂ρuv ∂ρv 2 ∂ρvw ∂p ∂τxy ∂τyy ∂τzy
+ + + =− + + +
∂t ∂x ∂y ∂z ∂x " ∂x ∂y ∂z #
2
∂ρw ∂ρuw ∂ρvw ∂ρw ∂p ∂τxz ∂τyz ∂τzz
+ + + =− + + +
∂t ∂x ∂y ∂z ∂x ∂x ∂y ∂z
Navier stokes equations which govern the flow of fluids are par-
tial differential equations. A differential equation involving partial
derivatives of a dependent variable(one or more) with more than
one independent variable is called a partial differential equation
Order of a partial differential equation
The order of the highest derivative term in the equation is called
the order of the PDE
1
