# Fluid mechanics: Separation surface of a source in a parallel flow (3D)

• steem84
In summary, the conversation discusses a problem in fluid mechanics involving a source with strength Q and a parallel flow U. The velocity potential is described in cartesian and cylindrical coordinates, and the surface rho(x) is used to separate the fluid. The law of volume conservation is used to calculate the surface, but there is uncertainty about how to solve the integral. Figure 3 shows an integration with respect to rho and theta, but the second expression should not have a theta term. The conversation concludes with a clarification and gratitude for the explanation.
steem84
Hello,

I have the following problem with respect to fluid mechanics:

A source of strength Q is in a 3D space and subjected to a parallel flow U along the x-axis. The position of the source is at (xq,yq,zq). This will lead to the following velocity potential in cartesian and cylindrical coordinates

figure 1

With the velocity in the x-direction determined to be

figure 2

To calculate the surface rho(x) which separates the fluid (coming from the source) from the fluid, (coming from the parallel flow), one can use the law of volume conservation (because of constant density). In that body which is described by that surface, the flow through any plane x=constant should equal to Q

figure 3

I know this is the correct way to calculate the separation surface, but from this point I can not go further: how do I solve the integral? Or can I use some trick?

Btw: sorry for the clumsy format, but I can’t LateX

Thanks!

Steven

If this is the wrong sub-forum, please let me know

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Figure 3 shows an integration with respect to rho and theta then shows an integral with respect to rho after integrating with with respect to theta. This second expression should not have a theta term, namely cos(theta - thetaq). This is a single variable integral with two independent variables. Evaluate the first integral for theta taking into account the cosine term.

yes, that does make sense. Thank you

## 1. What is the separation surface in fluid mechanics?

The separation surface in fluid mechanics is a theoretical boundary that separates the flow of a fluid into two regions: the upstream region and the downstream region. It is defined as the surface on which the velocity of the fluid is zero, meaning that there is no flow across this boundary.

## 2. What is a source in a parallel flow?

A source in a parallel flow is a type of fluid flow in which the fluid particles move in the same direction and at the same speed. In this type of flow, the fluid particles do not converge or diverge, but maintain a constant distance from each other.

## 3. How is the separation surface of a source in a parallel flow determined?

The separation surface of a source in a parallel flow is determined by solving the governing equations of fluid mechanics, such as the Navier-Stokes equations. The location of the separation surface can also be influenced by factors such as the shape of the object or the viscosity of the fluid.

## 4. What is the significance of the separation surface in fluid mechanics?

The separation surface has significant implications for the study of fluid mechanics, as it can affect the behavior and characteristics of fluid flow. For example, the presence of a separation surface can cause changes in pressure and velocity distribution, and can also lead to the formation of vortices and other flow patterns.

## 5. Is the separation surface of a source in a parallel flow always fixed?

No, the separation surface of a source in a parallel flow can change depending on various factors, such as the speed of the flow, the fluid properties, and the geometry of the object. In some cases, the separation surface may even disappear entirely, leading to a different type of flow behavior.

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