Question

Which of these equations is not parabolic?

(a) \(\rho\dot{q}\frac{\partial}{\partial x}(k\frac{\partial T}{\partial x})+\frac{\partial}{\partial y}(k\frac{\partial T}{\partial y})+\frac{\partial}{\partial z}(k\frac{\partial T}{\partial z})=\rho\frac{\partial e}{\partial t}\)

(b) \(\rho\dot{q}+\frac{\partial}{\partial x}(k\frac{\partial T}{\partial x})=\rho\frac{\partial e}{\partial t}\)

(c) \(\frac{1}{\rho C_v}[\frac{\partial}{\partial x}(k\frac{\partial T}{\partial x})+\frac{\partial}{\partial y}(k \frac{\partial T}{\partial y})+\frac{\partial}{\partial z}(k\frac{\partial T}{\partial z})]=\frac{\partial T}{\partial t} \)

(d) \(\frac{\partial}{\partial x}(k\frac{\partial T}{\partial x})+\frac{\partial}{\partial y}(k \frac{\partial T}{\partial y})+\frac{\partial}{\partial z}(k\frac{\partial T}{\partial z})=0\)

This question was addressed to me in homework.

My question is based upon Mathematical Behaviour of PDE in section Mathematical Behaviour of Partial Differential Equations of Computational Fluid Dynamics

Answer 1

The correct answer is:

(d) ∂/∂x(k ∂T/∂x) + ∂/∂y(k ∂T/∂y) + ∂/∂z(k ∂T/∂z) = 0

Explanation:

In the context of partial differential equations (PDEs) in computational fluid dynamics, equations are classified into three categories based on their characteristics: elliptic, parabolic, and hyperbolic.

• A parabolic equation typically represents a time-dependent process with a diffusive nature, such as heat conduction, and it has a characteristic form where the highest derivative is in time (first derivative) and the spatial derivatives (second derivatives) indicate diffusion. This results in a situation where the behavior of the solution evolves over time based on spatial diffusion.

Looking at each equation:

1. (a) ρq˙∂∂x(k∂T∂x)+∂∂y(k∂T∂y)+∂∂z(k∂T∂z)=ρ∂e∂tρq˙​∂x∂​(k∂x∂T​)+∂y∂​(k∂y∂T​)+∂z∂​(k∂z∂T​)=ρ∂t∂e​
   This is a heat conduction equation with a time-dependent term, and it behaves like a parabolic equation.

2. (b) ρq˙+∂∂x(k∂T∂x)=ρ∂e∂tρq˙​+∂x∂​(k∂x∂T​)=ρ∂t∂e​
   This is another form of the heat conduction equation. It contains a time derivative and spatial derivatives that indicate a diffusive process, making it a parabolic equation.

3. (c) 1ρCv[∂∂x(k∂T∂x)+∂∂y(k∂T∂y)+∂∂z(k∂T∂z)]=∂T∂tρCv​1​[∂x∂​(k∂x∂T​)+∂y∂​(k∂y∂T​)+∂z∂​(k∂z∂T​)]=∂t∂T​
   This equation is another form of a parabolic equation governing heat transfer, where the temperature changes over time based on the spatial gradients of the heat flux.

4. (d) ∂∂x(k∂T∂x)+∂∂y(k∂T∂y)+∂∂z(k∂T∂z)=0∂x∂​(k∂x∂T​)+∂y∂​(k∂y∂T​)+∂z∂​(k∂z∂T​)=0
   This is a steady-state equation, which has no time derivative. It represents a condition for the temperature field in the absence of time dependence and is not parabolic. In fact, it is more of an elliptic equation because it describes equilibrium heat conduction where the solution is time-independent.

Thus, equation (d) is not parabolic; it is steady-state and elliptic.