Work - Energy Principle Application to Fluid Flow

Question

Work - energy principle states that work done by net force acting on the body equals the change in kinetic energy of the body. We are talking about continuum mechanics. This principle is usually introduced in the mechanics of solid bodies. For us to describe the motion of the body, it is enough to know how the center of mass of the body moves in time and space. For example, we can conclude that body accelerates, if its center of mass has a different velocity at two points in time.

I am not sure how to adequately apply this principle to fluids for reasons I'll explain.

Consider fluid flowing in a pipe like in a scheme.

When fluid starts entering a narrower section of the pipe, it accelerates. Newton's 2nd law states that in that case, resultant force must act on the fluid. We can see that this force originates from the difference in pressure of the surrounding fluid or pressure gradient. If we take some volume of fluid between two cross sections of a non-equal area in a narrowing region of the pipe (control volume), we can draw a free-body diagram to show all forces acting on that control volume.

Doing so comes with a problem because the fluid doesn't move like a solid body, it flows. The concept of drawing all forces acting on control volume seems to have no sense in fluids because control volume doesn't move in space and time as a solid body does. Its center of mass doesn't move in space and time like in solid bodies, but rather fluid has different velocities on different cross sections or points in the pipe.

If this is true, how should we apply Newton's 2nd law or work-energy principle to fluids? On what fluid element or control volume should we draw force diagrams and apply Newton's 2nd law? I am thinking we should probably take some differential volume element and if we want to know how much its velocity changes between two cross-sections (for inviscid fluid), we would need to calculate the line integral of pressure gradient along the streamline or fluid element path. Usually, the pressure gradient is constant, so the line integral is equal to the pressure gradient times the distance between two cross - sections.

What is strange to me is that Bernoulli's equation/principle is commonly derived by work - the energy principle where force diagrams are drawn for some fluid volume of finite size. This derivation seems wrong to me given what I said above and force balance in fluids can only be done for differential volume elements on a particular streamline. Do you agree?

What are your thoughts?

Answer 1

Bernoulli's equation assumes conservation of energy and and a friction-less fluid (which is never true). It is at best a first approximation. The forces (associated with pressure) are assumed to act on pistons (real or imagined) which move with the fluid. In a real fluid flowing in a pipe, there is a radial velocity gradient and a loss of pressure due to friction with the walls of the pipe.

Answer 2

The problem with your reasoning is that you assume that the fluid in the control volume is fixed. In reality, you have to consider the closed system formed by the fluid that coincides with the control volume at time $t$. Between $t$ and $t + dt$, this closed system evolves, its kinetic energy changes, and there is no problem to make an energy balance.

Answer 3

I'll list here the process that is usually done when you deal with balance equations for fluids (or continuum in general)

• integral balance, Lagrangian description: you usually start from the integral balances for a close volume, precisely a material volume of fluid that doesn't exchange mass through its boundaries, and this it is composed by the same fluid particles during its evolution;

• integral balance, Eulerian description you get equations for a steady control volume, applying Reynolds's transport theorem;

• differential balance equations: assuming that the fields are regular and no discontinuity occurs in the domain, you can get the differential form of the equations applying Gauss'theorem and divergence theorem.

I don't know what your level in Mathematics is, but you can get a tour in handwritten notes about it, starting from here https://basics.altervista.org/test/Physics/CM/BalanceEquations/main.html

If needed, I'll try to give you a sketch of the derivation of Bernoulli theorem, from the differential form of balance equations. Please note that Bernoulli's theorems are theorems, so they are valid under some assumpitons (broadly speaking conservation of energy for compressible flows, no viscosity and no vorticity for incompressible flows)