# Why can a fluid that satisfies the continuity equation cross streamlines?

• tracker890 Source h
tracker890 Source h
Homework Statement
Confusing in the Conservation of Mass Flow Rate and mass cross streamlines
Relevant Equations
Continuity Equation

reference
reference_2
Q： Why can a fluid that satisfies the continuity equation for mass conservation cross streamlines?

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Mass can cross streamline, thats basically the definition of turbulent flow i.e. flow with heavy cross-stream mixing. It's not clear (to me) what you are taking issue with in this example?

berkeman
erobz said:
Mass can cross streamline, thats basically the definition of turbulent flow i.e. flow with heavy cross-stream mixing. It's not clear (to me) what you are taking issue with in this example?
But this case is a steady flow, not a turbulent flow.
So I don't understand why fluid elements in steady flow can cross streamlines.

tracker890 Source h said:
But this case is a steady flow.
So your issue is with what is happening to the flow as it passes around the sphere. Where the streamlines are being squeezed together?

erobz said:
So your issue is with what is happening to the flow as it passes around the sphere. Where the streamlines are being squeezed together?
Why can ##\dot{m}_{AD}## and ##\dot{m}_{BC}## cross streamlines instead of being equal to zero?

I don't think fluid elements are crossing streamlines here. Streamlines are being divided, that not necessarily mass exchange between layers.

Imagine what is happening between streamlines. In steady flow a group of molecules are all together doing the same thing as the others in the streamline. The streamline is a boundary that is saying things inside it are doing the same thing on average. Then, they are presented with an obstruction, something that forces convective acceleration. That group of fluid particles is forced to divide into new groups, or layers such that the mass flowrate of that original grouping is conserved. In reality there is always some small turbulence from the change, but in theory the layer that "was one homogeneous unit" splits into new layers each having their own homogeneous velocity. Past the obstruction, they may settle back into the original formation.

Thats what I think is happening.

erobz said:
I don't think fluid elements are crossing streamlines here. Streamlines are being divided, that not necessarily mass exchange between layers.

Imagine what is happening between streamlines. In steady flow a group of molecules are all together doing the same thing as the others in the streamline. The streamline is a boundary that is saying things inside it are doing the same thing on average. Then, they are presented with an obstruction, something that forces convective acceleration. That group of fluid particles is forced to divide into new groups, or layers such that the mass flowrate of that original grouping is conserved. In reality there is always some small turbulence from the change, but in theory the layer that "was one homogeneous unit" splits into new layers each having their own homogeneous velocity. Past the obstruction, they may settle back into the original formation.

Thats what I think is happening.
If fluid elements do not cross streamlines, then the diagram below would conflict with the principle of mass conservation. Is this an issue with the question or with my understanding?

tracker890 Source h said:
If fluid elements do not cross streamlines, then the diagram below would conflict with the principle of mass conservation. Is this an issue with the question or with my understanding?
View attachment 331872
There are other streamlines in between the two drawn on the diagram.

erobz said:
There are other streamlines in between the two drawn on the diagram.
Is it related to ##\dot{m}_{AD}## and ##\dot{m}_{BC}## not being equal to zero?

tracker890 Source h said:
Is it related to ##\dot{m}_{AD}## and ##\dot{m}_{BC}## not being equal to zero?
I think the diagram is just saying the total mass flowrate splits around the sphere equally. I think they are just trying to illustrate it with the angled arrows. half goes up around, half goes down around.

erobz said:
I think the diagram is just saying the total mass flowrate splits around the sphere equally. I think they are just trying to illustrate it with the angled arrows. half goes up around, half goes down around.
I hope you can draw a simple diagram to assist in explaining, for better communication and understanding.

tracker890 Source h said:
I hope you can draw a simple diagram to assist in explaining, for better communication and understanding.

tracker890 Source h
erobz said:
Thank you for the excellent and detailed explanation, along with my understanding diagram.

tracker890 Source h said:
Thank you
tracker890 Source h said:
for the excellent and detailed explanation, along with my understanding diagram.
I'm not saying our diagrams are entirely accurate, but I think you have understood the idea.

tracker890 Source h
tracker890 Source h said:
Why can ##\dot{m}_{AD}## and ##\dot{m}_{BC}## cross streamlines instead of being equal to zero?
The obstruction in the flow forces some of the mass to spill out of the perfectly cylinder shape that has been represented.
While it recovers back to the initial state, the flow downstream the ball slowdowns.

That slowing moving mass becomes an additional physical obstacle, around which the surrounding fluid tries to move.
Pushing the layers away from the obstacle is the easiest way, but those (also moving) layers have inertia and resist that push (perpendicular to the flow).

Then, most of our molecules (that could not spill out into the outer layers), caught in between obstacle and outer layers, must increase velocity to keep mass flow balance (upstream-downstream).

Flow mass AB cross section = Flow mass AD-BC cylinder wall + Flow mass CD cross section

Note the conical constant volume (cv) streamline shape represented in figure P3.44 in one of your reference links.

The closer to the obstruction the limits of the control volume are established, the more mass will cross it out in a transversal way (just to eventually cross back to the fill the void and return to be calmed still air).

erobz and tracker890 Source h

## 1. What is the continuity equation for mass conservation in fluid dynamics?

The continuity equation for mass conservation in fluid dynamics states that the mass of fluid entering a control volume must equal the mass of fluid leaving the control volume plus any change in mass within the control volume. Mathematically, it is often expressed as ∂ρ/∂t + ∇·(ρv) = 0, where ρ is the fluid density and v is the velocity vector.

## 2. What are streamlines in fluid flow?

Streamlines are imaginary lines in a fluid flow field that represent the trajectory that a fluid particle will follow. At any given point on a streamline, the tangent to the streamline is parallel to the local velocity vector of the fluid.

## 3. How can a fluid cross streamlines if it satisfies the continuity equation?

A fluid can cross streamlines if there are changes in the flow field, such as varying velocity or pressure gradients, that cause the fluid particles to change direction. This can occur in unsteady flows, where the flow properties change with time, or in the presence of external forces or boundary conditions that alter the fluid's path.

## 4. Does crossing streamlines violate the continuity equation?

No, crossing streamlines does not violate the continuity equation. The continuity equation ensures mass conservation in the flow, but it does not impose restrictions on the path that individual fluid particles must follow. Streamlines can change shape and direction due to various factors without violating mass conservation.

## 5. Can you provide an example where fluid crosses streamlines?

An example of fluid crossing streamlines is in a vortex flow, where fluid particles move in circular paths around a central point. In such a flow, the streamlines are concentric circles, but the fluid particles are constantly changing their position relative to the streamlines due to the rotational motion.

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