Is irrotational flow field a conservative vector field?

In summary, the conversation discussed the relationship between a flowing fluid with a constant velocity and a conservative vector field. It was also mentioned that a vector field in a simply connected domain with zero rotor will have a potential function. The potential function was defined as a function whose gradient yields a constant velocity field. There was a discussion about whether the potential gradient of a conservative field has to be a force field or if it can be just a constant velocity field. It was clarified that potential is not always a short name for potential energy and can have broader applications. The famous example of the potential vortex was also mentioned, which is a potential in any domain with some half plane taken out. The conversation also touched upon the meaning of "some half plane along z
  • #1
Adel Makram
635
15
For a flowing fluid with a constant velocity, will this field be described as conservative vector field? If it is a conservative field, what will be the potential of that field?
 
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  • #2
I suspect that "irrotational flow field" means "##\mathrm{rot} =0"## but not necessarily
Adel Makram said:
with a constant velocity

Assume that a vector field v is defined in simply connected domain and its rotor equals zero. Then v has a potential function.

Adel Makram said:
what will be the potential of that field?
this you will see from definition of the potential function
 
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  • #3
wrobel said:
this you will see from definition of the potential function
What sort of potential function that its gradient yields a constant velocity field? If I integrate a constant velocity ##v(x)=c## with respect to ##x##, this gives ##cx##. So what physical potential has this form?
 
  • #4
Adel Makram said:
What sort of potential function that its gradient yields a constant velocity field? If I integrate a constant velocity ##v(x)=c## with respect to ##x##, this gives ##cx##. So what physical potential has this form?

Didn't you just answer your own question? And potential of the form ##\phi = Ax + By## will give you a constant velocity field.

Regarding your irrotationality question: saying a flow field is irrotational is equivalent to saying it is conservative. Consider that a conservative field can always be described as a gradient of a potential, and rotational is measured by the curl. Therefore,
[tex]\nabla \times \nabla\phi \equiv 0[/tex]
 
  • #5
boneh3ad said:
Didn't you just answer your own question? And potential of the form ##\phi = Ax + By## will give you a constant velocity field.

Regarding your irrotationality question: saying a flow field is irrotational is equivalent to saying it is conservative. Consider that a conservative field can always be described as a gradient of a potential, and rotational is measured by the curl. Therefore,
[tex]\nabla \times \nabla\phi \equiv 0[/tex]
So if fluid is pumped through a pipe and flows at a constant velocity, what is the name of physical potential which is measured at any point along the length of the pipe? In general, does the potential gradient of a conservative field have to be a force field or it may be just a constant velocity field?
 
  • #6
boneh3ad said:
saying a flow field is irrotational is equivalent to saying it is conservative.
this is wrong. The standard counterexample is as follows. Consider a domain
$$D=\{(x,y,z)\in\mathbb{R}^3\mid 1<x^2+y^2<2\}$$ and the following field ##v## in it
$$v=\Big(-\frac{y}{x^2+y^2},\frac{x}{x^2+y^2},0\Big).$$ It is easy to see that ##\mathrm{rot}\,v=0##. However there is no function ##f## in ##D## such that ##\mathrm{grad}\,f=v##
 
  • #7
wrobel said:
this is wrong. The standard counterexample is as follows. Consider a domain
$$D=\{(x,y,z)\in\mathbb{R}^3\mid 1<x^2+y^2<2\}$$ and the following field ##v## in it
$$v=\Big(-\frac{y}{x^2+y^2},\frac{x}{x^2+y^2},0\Big).$$ It is easy to see that ##\mathrm{rot}\,v=0##. However there is no function ##f## in ##D## such that ##\mathrm{grad}\,f=v##

That's fair. I took it a step too far with the "iff" relationship. I'll amend my original post to say that any flow field that can be described by a potential function is also, by definition, irrotational. Irrotationality is necessary but not sufficient for a field to be expressible as a potential.

Adel Makram said:
So if fluid is pumped through a pipe and flows at a constant velocity, what is the name of physical potential which is measured at any point along the length of the pipe? In general, does the potential gradient of a conservative field have to be a force field or it may be just a constant velocity field?

I am not sure that I fully understand what you are asking here. What do you mean by the "name of the physical potential"? Further, you can't really measure potential in a fluid flow. A potential gradient is, by definition (at least in this case) a velocity field (##\vec{v} = \nabla \phi##). The velocity in field does not have to be constant.
 
  • #8
boneh3ad said:
I am not sure that I fully understand what you are asking here. What do you mean by the "name of the physical potential"? Further, you can't really measure potential in a fluid flow. A potential gradient is, by definition (at least in this case) a velocity field (##\vec{v} = \nabla \phi##). The velocity in field does not have to be constant.
The convention in physics is that the potential is just a short name of potential energy. But the unite of the potential in this example is velocity times distance which is not a unite of energy? So my question, what does this potential represent?
 
  • #9
Potential is not, in general, a short name for potential energy. It often works out that way (an started out that way), but that is not a general rule, as scalar potentials have much broader application than just gravity and electrostatics. A scalar potential is a scalar-valued function that can be used to completely describe a conservative vector field.
 
  • #10
wrobel said:
this is wrong. The standard counterexample is as follows. Consider a domain
$$D=\{(x,y,z)\in\mathbb{R}^3\mid 1<x^2+y^2<2\}$$ and the following field ##v## in it
$$v=\Big(-\frac{y}{x^2+y^2},\frac{x}{x^2+y^2},0\Big).$$ It is easy to see that ##\mathrm{rot}\,v=0##. However there is no function ##f## in ##D## such that ##\mathrm{grad}\,f=v##
This is the famous example of the potential vortex. It has a potential in any domain with some half plane along the z-axis taken out. Such a potential is given in cylinder coordinates by
$$V=-\varphi$$
since then
$$-\vec{\nabla}V=1/r \vec{e}_{\varphi}=(-y,x,0)/(x^2+y^2).$$
Depending on which open interval of length ##2\pi## you have taken out a corresponding half-plane, which restricts the domain to a single connected part.
 
  • #11
vanhees71 said:
This is the famous example of the potential vortex. It has a potential in any domain with some half plane along the z-axis taken out. Such a potential is given in cylinder coordinates by
$$V=-\varphi$$
since then
$$-\vec{\nabla}V=1/r \vec{e}_{\varphi}=(-y,x,0)/(x^2+y^2).$$
Depending on which open interval of length ##2\pi## you have taken out a corresponding half-plane, which restricts the domain to a single connected part.
I don`t understand this famous example because I am confusing about definitions. First what is the meaning of "some half plane along z taken out"? If the vector field can be represented as a gradient of a potential as required by the definition of conservative field, why isn`t it conservative?
Also, wrobel said that the rot of that field =0 but in wikipedia is equal to ##2\pi##.
Finally, by Stokes theorem the microcirculation in the form of curl should equal to macrocirculation in the form of rot, but here it is not the case?
 

1. What is an irrotational flow field?

An irrotational flow field is a type of vector field in which the curl, or rotational component, of the vector field is equal to zero at every point. This means that the flow of the field is smooth and does not have any swirling or rotating motion.

2. What is a conservative vector field?

A conservative vector field is a type of vector field in which the line integral of the vector field along any closed path is equal to zero. This means that the work done by the field on any closed path is zero, and the path taken does not affect the final result.

3. How are irrotational flow fields and conservative vector fields related?

An irrotational flow field is always a conservative vector field. This is because the curl of an irrotational flow field is equal to zero, and the line integral of a conservative vector field is also equal to zero. Therefore, all irrotational flow fields are conservative vector fields, but not all conservative vector fields are irrotational flow fields.

4. What are some examples of irrotational flow fields?

Some examples of irrotational flow fields include the velocity field of a non-spinning object, the flow of an ideal fluid, and the flow of heat in a stationary medium.

5. Why is the concept of irrotational flow fields important?

The concept of irrotational flow fields is important in many areas of science and engineering, such as fluid mechanics, electromagnetism, and thermodynamics. It helps us understand and analyze the behavior of vector fields in these fields, and allows us to make accurate predictions and calculations.

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