Homework Help is Here – Start Your Trial Now!
Science
Advanced Physics
From Eq. (4.64), derive the heat diffusion equation: kV²T = pc₂² at

From Eq. (4.64), derive the heat diffusion equation: kV²T = pc₂² at

icon
Related questions
Question

Equation is attached

**Transcription for Educational Website** From Eq. (4.64), derive the heat diffusion equation: \[ k \nabla^2 T = \rho c_v \frac{\partial T}{\partial t} \] This equation represents the heat diffusion or conduction in a material, where: - \( k \) is the thermal conductivity of the material. - \( \nabla^2 \) (del squared) is the Laplace operator, which indicates the divergence of the gradient of the temperature field, \( T \). - \( \rho \) is the density of the material. - \( c_v \) is the specific heat capacity at constant volume. - \( \frac{\partial T}{\partial t} \) is the partial derivative of temperature with respect to time, representing the rate of change of temperature. This equation is central to thermal analysis and describes how heat moves through a medium over time.
Transcribed Image Text:**Transcription for Educational Website** From Eq. (4.64), derive the heat diffusion equation: \[ k \nabla^2 T = \rho c_v \frac{\partial T}{\partial t} \] This equation represents the heat diffusion or conduction in a material, where: - \( k \) is the thermal conductivity of the material. - \( \nabla^2 \) (del squared) is the Laplace operator, which indicates the divergence of the gradient of the temperature field, \( T \). - \( \rho \) is the density of the material. - \( c_v \) is the specific heat capacity at constant volume. - \( \frac{\partial T}{\partial t} \) is the partial derivative of temperature with respect to time, representing the rate of change of temperature. This equation is central to thermal analysis and describes how heat moves through a medium over time.
After it is simplified using the continuity equation, we have \[ \rho \frac{Du}{Dt} = -\nabla \cdot \mathbf{J}_E = \{ p_{ij} \} : \nabla \mathbf{v}_B \] (4.64) **Explanation:** This equation represents a form of the continuity equation used in fluid dynamics or continuum mechanics. The terms can be broken down as follows: - \(\rho\) is the density of the fluid. - \(\frac{Du}{Dt}\) is the material derivative, indicating the change of velocity \(u\) with respect to time. - \(-\nabla \cdot \mathbf{J}_E\) represents the divergence of the energy flux \(\mathbf{J}_E\). - \(\{ p_{ij} \} : \nabla \mathbf{v}_B\) is a tensor operation involving the gradient of velocity \(\nabla \mathbf{v}_B\) and a stress or pressure tensor \(\{ p_{ij} \}\). Equation (4.64) is used in contexts where the conservation of momentum or energy is being analyzed, particularly in scenarios involving complex fluids or materials.
Transcribed Image Text:After it is simplified using the continuity equation, we have \[ \rho \frac{Du}{Dt} = -\nabla \cdot \mathbf{J}_E = \{ p_{ij} \} : \nabla \mathbf{v}_B \] (4.64) **Explanation:** This equation represents a form of the continuity equation used in fluid dynamics or continuum mechanics. The terms can be broken down as follows: - \(\rho\) is the density of the fluid. - \(\frac{Du}{Dt}\) is the material derivative, indicating the change of velocity \(u\) with respect to time. - \(-\nabla \cdot \mathbf{J}_E\) represents the divergence of the energy flux \(\mathbf{J}_E\). - \(\{ p_{ij} \} : \nabla \mathbf{v}_B\) is a tensor operation involving the gradient of velocity \(\nabla \mathbf{v}_B\) and a stress or pressure tensor \(\{ p_{ij} \}\). Equation (4.64) is used in contexts where the conservation of momentum or energy is being analyzed, particularly in scenarios involving complex fluids or materials.
Expert Solution
steps

Step by step

Solved in 3 steps with 4 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer