Is This Rectangular Plate BVP Correct?

In summary, the conversation discusses the steady state distribution of temperature in 2-dimensional plates or 3-dimensional regions. The problem presented is a rectangular plate with boundary conditions involving poor insulation and fixed temperature values on the other sides. The solution involves writing a BVP equation and using the law of Fourier to find the heat flux on the boundaries. The final step is to write the complete BVP.
  • #1
jc2009
14
0
THe following exercise deal with the steady state distribution of the temperature in either 2-dimensional plates or 3-dimensional regions.
Problem: A 10X20 rectangular plate with boundary conditions . at the lower side where there is poor insulation the normal derivative of the temperature is equal to 0.5 times the temperature

The rectangular 10 X 20 plate is as follows

bottom part : poor insulation
left side: Insulation
right side: Kept at 10 degrees
top side: Kept at 20 degrees

Write the BVP

Solution: What i found was the expression for poor insulation but i think this is more for one space variable, [tex]u_{x}(0,t)[/tex] + au(0,t) = 0
i wrote some BVP:
[tex]u_{x}(0,y) = 0[/tex]
u(x,0) = poor insulation ( i don't know how to do this part)
u(10,y) = 10
u(x,20) = 20

Is this even right?
any help would be appreciated
 
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  • #2
jc2009 said:
Write the BVP

Solution: What i found was the expression for poor insulation but i think this is more for one space variable, [tex]u_{x}(0,t)[/tex] + au(0,t) = 0
i wrote some BVP:
[tex]u_{x}(0,y) = 0[/tex]
u(x,0) = poor insulation ( i don't know how to do this part)
u(10,y) = 10
u(x,20) = 20

Hello jc2009,
The following is correct:
u(10,y) = 10
u(x,20) = 20
The left hand side is insulated, this means that no heat passes through this boundary and thus the heat flux is zero. This is given by the law of Fourier, or:
[tex]\frac{\partial u}{\partial x}=0[/tex]
The bottom is now easily found as:
[tex]\frac{\partial u}{\partial y}=\frac{1}{2}\cdot u[/tex]
Keep in mind the meaning of the normal for the two boundaries.
The PDE of the BVP problem is:
[tex]\frac{\partial u}{\partial t}=k\cdot \left[\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right][/tex]
Now it should be possible to write down the complete BVP.
Hope this helps
 

Related to Is This Rectangular Plate BVP Correct?

What is the "Rectangular Plate BVP problem"?

The Rectangular Plate BVP (Boundary Value Problem) problem is a mathematical problem that involves finding a solution to the differential equations that describe the behavior of a rectangular plate under certain constraints. This problem is commonly used in the field of engineering and physics to model the behavior of structures such as bridges, buildings, and aircraft wings.

What are the main challenges of solving the "Rectangular Plate BVP problem"?

One of the main challenges of solving the Rectangular Plate BVP problem is the complex nature of the differential equations involved. These equations often have no analytical solution and require numerical methods to approximate the solution. Additionally, the boundary conditions for the problem can be difficult to determine, making it challenging to find an accurate solution.

How is the "Rectangular Plate BVP problem" solved?

The Rectangular Plate BVP problem is typically solved using numerical methods such as finite difference or finite element methods. These methods involve dividing the plate into smaller elements and using iterative calculations to approximate the solution. Advanced computational techniques, such as the use of supercomputers, are often necessary to solve this problem accurately.

What are some real-world applications of the "Rectangular Plate BVP problem"?

The Rectangular Plate BVP problem has many real-world applications in engineering and physics. It is used to model the behavior of structures such as bridges, buildings, and aircraft wings under various load conditions. It is also used in the design and optimization of mechanical components, such as heat sinks and circuit boards.

What are the limitations of the "Rectangular Plate BVP problem"?

One limitation of the Rectangular Plate BVP problem is that it assumes the plate is made of a homogeneous material and has uniform thickness. In reality, most structures are made of different materials and have varying thicknesses, which can affect the accuracy of the solution. Additionally, the problem does not take into account any non-linear effects, such as large deformations or material failure, which may be present in some real-world scenarios.

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