Navier Stokes Equation - Flow of waves

In summary, for an incompressible flow, the velocity potential can be written as the gradient of some potential, and the Navier-Stokes equation can be used to show that the potential satisfies the equation ##\nabla^2 \phi = 0##. The boundary conditions for velocity are zero at ##z=0##, and the given form of ##\phi## can be derived by substituting it into the equation. However, further progress may be needed to fully verify the form of ##\phi##.
  • #1
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Homework Statement


[/B]
(a) Show that for an incompressible flow the velocity potential satisfies ##\nabla^2 \phi = 0##. Show further the relation for the potential to be ## \frac{\partial \phi}{\partial t} + \frac{\nabla \phi \dot \nabla \phi}{2} + \frac{p}{\rho} + gz = const.##
(b)Write out the boundary conditions on velocity and show ##\phi$## is of the form: ## \phi = -\frac{\omega n_0}{k} cos(kx-\omega t) \frac{cosh kz}{sinh kH}## subject to ## \omega^2 = gk \space tanh \space (kH)##
(c) How does the frequency of the waves vary after the boat is gone?

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Homework Equations

The Attempt at a Solution



Part (a)[/B]

For an irrotational flow, ## \nabla \cdot \vec u = 0## so this means that velocity can be written as gradient of some potential ## u = \nabla \phi##.

Therefore ## \nabla \cdot \vec u = \nabla^2 \phi = 0##.

Navier stokes equation is given by:
[tex] \frac{\partial \vec u}{\partial t} + (\vec u \cdot \nabla) \vec u + \frac{1}{\rho} \nabla p + g \vec k = \nu \nabla^2 \vec u [/tex]

Using the identity and ## \nabla^2 \vec u = 0 ##, ## \nabla \times \vec u \times \vec u = 0## I get the answer.

Part (b)

The boundary condition is velocity at ##z=0## is zero (non-slip).

I don't think this question wants me to simply substitute the given form of ##\phi## to verify it is right. I am expected to derive it.

Substituting, I get
[tex] Z \frac{\partial X}{\partial t} + \frac{ (Z\nabla X + X \nabla Z) \cdot (Z\nabla X + X \nabla Z) }{2} + \frac{p}{\rho} + gz = const. [/tex]

This leads to nowhere. Any idea how to proceed?
 
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  • #3
Any help for part (b)??
 
  • #5


For part (b), you can use the boundary condition at ##z=0## to solve for the constant in the potential equation. This will give you a relation between the potential and the boundary conditions. Then, you can use this relation to verify that the given form of the potential satisfies the boundary conditions.

For part (c), the frequency of the waves will decrease after the boat is gone because the boat's motion will no longer be a source of energy for the waves. This means that the energy of the waves will decrease, causing a decrease in frequency. Additionally, the boat's motion may have caused some dispersion of the waves, which can also contribute to a decrease in frequency.
 

1. What is the Navier-Stokes Equation?

The Navier-Stokes Equation is a mathematical equation that describes the motion of fluid particles in a given space. It takes into account factors such as viscosity, pressure, and velocity to determine the flow of the fluid.

2. What are the applications of the Navier-Stokes Equation?

The Navier-Stokes Equation has various applications, including predicting weather patterns, designing aircraft and vehicles, and understanding ocean currents. It is also used in computer simulations to model fluid behavior in engineering and scientific research.

3. What is meant by "flow of waves" in the context of the Navier-Stokes Equation?

The "flow of waves" refers to the movement of fluid particles, which can be seen as waves in the fluid. The Navier-Stokes Equation can be used to predict and analyze the behavior of these waves, which is essential in understanding fluid dynamics.

4. What are the limitations of the Navier-Stokes Equation?

While the Navier-Stokes Equation is a powerful tool for studying fluid dynamics, it has some limitations. It assumes that fluids are continuous, incompressible, and have constant viscosity, which may not always be the case in real-world scenarios. Additionally, it cannot accurately predict turbulent flows, which are common in many fluid systems.

5. How does the Navier-Stokes Equation relate to other equations in physics?

The Navier-Stokes Equation is a set of partial differential equations that describe the motion of fluids. It is often used in combination with other equations, such as the continuity equation and the energy equation, to fully understand the behavior of fluids in different scenarios. It is also related to other fundamental equations in physics, such as the conservation of mass and momentum principles.

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