Differential Equation with forcing/fluids question

In summary, the conversation discusses a research issue involving a differential equation with a forcing function of the form f(t) = γe^(-t/ε), where γ is a constant and L is a linear second order operator. The goal is to determine the behavior of the particular solution as ε approaches 0 while keeping the integral of the forcing function constant. It is suggested that the forcing function may turn into a boundary condition, but the speaker is unsure how to make this rigorous. The issue arises from a specific example involving the linearized Navier Stokes equation for a viscous rotating fluid with forcing. The speaker is seeking guidance and advice on how to approach this problem.
  • #1
nickthequick
53
0
Hi,
An issue has come up in my research: I think the problem can be phrased as such:

Given a differential equation of the form
[tex]Lu(t) = f(t)[/tex]
Where the forcing function is of the form [tex]f(t)=\gamma e^{-t/\epsilon},\gamma[/tex] is a constant, and L is some linear second order operator.

We want to see what happens to the particular solution in the limit as [tex]\epsilon \to 0[/tex] while keeping
[tex]\int_{0}^{\infty}f(t')dt' = const[/tex]

Physically, it seems as if the forcing function will turn into a boundary condition, although I am not sure how to make this rigorous. Is this always the case? When I go through and evaluate the limits for the particular solution I am finding results that do not make sense.

In case a particular example helps, this is where this problem comes from.

Consider the time independent linearized Navier Stokes equation for a viscous rotating fluid with forcing. The equation governing the dynamics is:
[tex] u_{zz}-\frac{if}{\nu} u =-\frac{1}{\nu} A(z) [/tex]
where [tex]A(z)=\gamma e^{-z/\epsilon}[/tex]
I am getting results that do make sense physically for particular types of forcing functions A, so I would like to see what happens when A turns into a surface condition as opposed to a forcing function, since I know what to expect in that case.
 
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  • #2
I am not sure how to approach this problem and would really appreciate any guidance or advice. Thank you in advance.
 

1. What is a differential equation with forcing in the context of fluids?

A differential equation with forcing in the context of fluids is a mathematical model used to describe the behavior of a fluid under the influence of an external force or pressure. It takes into account factors such as the fluid's velocity, pressure, and density, as well as the forces acting on it, to determine how the fluid will flow and change over time.

2. How is a differential equation with forcing used to study fluids?

A differential equation with forcing is used to study fluids by providing a mathematical framework for understanding and predicting fluid behavior. By solving the equation, scientists can determine how a fluid will respond to various external forces and conditions, such as changes in pressure or flow rate. This information is crucial for applications in fields such as engineering, meteorology, and oceanography.

3. What are some common types of forcing in differential equations for fluids?

Some common types of forcing in differential equations for fluids include external pressure, gravity, and surface tension. Other types of forcing can include magnetic or electric fields, rotation, and chemical reactions. These forces can have a significant impact on the behavior of fluids and are essential to consider in fluid dynamics studies.

4. How are differential equations with forcing solved?

Differential equations with forcing can be solved through various methods, depending on the specific equation and problem being studied. Some common approaches include analytical solutions, where the equation is solved using mathematical techniques, and numerical solutions, where the equation is approximated using computational methods. The choice of method will depend on the complexity of the equation and the desired level of accuracy.

5. What are some real-world applications of differential equations with forcing in fluids?

Differential equations with forcing in fluids have numerous real-world applications, including predicting weather patterns, designing aircraft and other vehicles, and understanding ocean currents. They are also used in the study of groundwater flow, blood flow in the human body, and the behavior of fluids in industrial processes. These equations play a crucial role in many fields and have a wide range of practical applications.

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