How Does the Bernoulli Equation Determine Axial Pressure in Jet Flows?

[Lucus]

Okay, here's the question:

A jet of incompressible fluid emerges from a horizontal channel into an environment of the same fluid. The momentum of the jet at the exit of the channel is measured to be M. The exit momentum is the source of a downstream flow which spreads gradually with distance.

The fluid outside the jet flow is at rest. Use the Bernoulli equation to determine the axial (x) pressure beyond the boundary of the jet flow. How can that result, and the properties of the boundary layer equations be used to determine the axial pressure gradient inside the jet?

So there's the question. I'm not really sure where to start. Any help would be much appreciated. Thanks!

 

[Clausius2]

Okay, here's the question:
A jet of incompressible fluid emerges from a horizontal channel into an environment of the same fluid. The momentum of the jet at the exit of the channel is measured to be M. The exit momentum is the source of a downstream flow which spreads gradually with distance.
The fluid outside the jet flow is at rest. Use the Bernoulli equation to determine the axial (x) pressure beyond the boundary of the jet flow. How can that result, and the properties of the boundary layer equations be used to determine the axial pressure gradient inside the jet?
So there's the question. I'm not really sure where to start. Any help would be much appreciated. Thanks!

Nice problem, although the statement makes no sense. You can't use Bernoulli equation to determine the axial pressure assuming viscous flow as you are doing. The jet core developed in a non dimensional distance of order $$Re_j^{-1}$$ is dominated by viscous forces. You only can apply Bernouilli to calculate external boundary layer pressure, but as it is an stagnant atmosphere, this pressure is trivially constant $$P_a$$.
Usual calculations of incompressible jets assume negligible pressure gradients, in part because the boundary layer is so thin at large $$Re$$ that transversal pressure gradients are very small, and also because by this argument the external pressure impose an uniform pressure across the symmetry axis.

 

[piercas]

I would first start by defining some key terms and concepts to better understand the question. In this case, we are dealing with fluid dynamics, specifically the flow of an incompressible fluid. The Bernoulli equation is a fundamental equation in fluid dynamics that relates the pressure, velocity, and elevation of a fluid. It states that in a steady flow, the sum of pressure energy, kinetic energy, and potential energy is constant.

In this scenario, we have a jet of incompressible fluid emerging from a horizontal channel into an environment of the same fluid. This means that the fluid properties, such as density and viscosity, are constant throughout the flow. The momentum of the jet at the exit of the channel is measured to be M, and this momentum is the source of a downstream flow.

Now, to use the Bernoulli equation to determine the axial pressure beyond the boundary of the jet flow, we need to consider the energy changes at different points along the flow. At the exit of the channel, the fluid has a certain velocity and pressure. As the fluid spreads out and moves downstream, its velocity decreases due to the conservation of mass. This decrease in velocity leads to an increase in pressure according to the Bernoulli equation. Therefore, the pressure at the boundary of the jet flow will be higher than at the exit of the channel.

Next, we need to consider the properties of the boundary layer equations. The boundary layer is the thin layer of fluid that forms along a solid surface in a flow. This layer has different properties compared to the bulk flow, such as a slower velocity and higher pressure. By analyzing the boundary layer equations, we can determine the axial pressure gradient inside the jet. This gradient is the change in pressure along the length of the jet and is affected by factors such as fluid viscosity and velocity.

In summary, to determine the axial pressure beyond the boundary of the jet flow, we can use the Bernoulli equation to consider the energy changes along the flow. And to determine the axial pressure gradient inside the jet, we can use the properties of the boundary layer equations. I hope this helps to provide some guidance on how to approach this question.