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Engineering Tripos 1B
Paper 4
Fluid Mechanics
Lecture 1
- Picturing fluids
- Properties of a fluid
- Why we use partial derivatives
- The del operator
- The gradient of a scalar field
- The law of conservation of mass
- The divergence of a vector field
- Constant density flows
- The curl of a vector field
1B Thermofluids website
The 1B Thermofluids website contains extra explanations of the key concepts in the course and some worked examples:
- The address is: http://camtools.caret.cam.ac.uk/
- The course name is: Thermofluids : EngIB
- A Raven login is required.
Matthew Juniper [email protected]
1.1 Picturing fluids
Fluids are made up of a great number of molecules. It is useful to picture these molecules as little balls, each with a position and a velocity and each obeying New- ton’s laws of motion^1. In liquids, the molecules are in close contact with their neighbours. In gases, the molecules are normally well-separated, which means that they move in straight lines between one collision and the next with a mean free path that is much larger than the molecular diameter. We will consider gases because our picture works better when the molecules are well-separated. We assume that similar principles apply to liquids. Firstly, how many molecules are there in an average-sized piston?
In any real-life situation it would be impossible to follow every molecule. Instead, we zoom out and look at the average properties of the fluid. Normally we do this at a point in space. For instance, we average^2 all the molecular velocities, vi, around a point in space (x, y, z) and say that the fluid there has a velocity v(x, y, z).
So now we can think of the fluid as a continuous lump of stuff - with no gaps! - and say that it has a certain velocity field.
(^1) At the molecular scale, electrical forces and quantum mechanics become important. However,
classical mechanics gives us useful insight and this is (nearly) the last we will say on the subject! (^2) To be really careful we should define what we mean by ‘average’. It is v(x, y, z) = 1 N
∑N i=1 vi, where N is the number of molecules around that point in space and vi is the velocity vector of each molecule.
1.3 Why we use partial derivatives
In our molecular picture of a fluid, every molecule has a velocity vector, vi, and obeys Newton’s laws of motion. The velocity is held by the molecule so we use ordinary derivatives such as d/dt. If we knew exactly how all the molecules started we could march forwards in time solving ordinary differential equations for each molecule. However, this is impractical for more than a few million molecules.
In our continuum picture of a fluid, the velocity field is defined at points in space rather than on individual molecules. Newton’s laws of motion still work but we have to apply them to regions of space rather than to individual molecules. In a three- dimensional world we need to isolate changes in each of the three directions: (x, y, z) so we need to use partial derivatives such as (∂/∂x, ∂/∂y, ∂/∂z). By averaging over very many molecules and very many collisions we exchange an enormous number of ordinary differential equations for a few partial differential equations that describe the averaged fields.
This is a crucial conceptual leap for physicists and engineers.
“The application of Newton’s mechanics to continuously distributed masses led inevitably to the discovery and application of partial differential equa- tions, which in their turn first provided the language for the laws of the field-theory.”
Albert Einstein
1.4 The del operator
When dealing with point masses, we use ordinary differential operators such as d/dt. When dealing with fields we need to use partial differential operators such as ∂/∂t, ∂/∂x, ∂/∂y and ∂/∂z. Partial differential operators are not very useful when they act independently; they only produce the change in one direction and, even worse, that direction depends on the choice of coordinate system.
The real power of these partial differential operators arises when they are com- bined to form the del operator, which is given the symbol ∇ and is also called nabla. In Cartesian coordinates, ∇ is defined as:
∇ ≡ ˆex
∂x
∂y
∂z
The operation that is represented by ∇ is independent of the coordinate system. This means that ∇ is expressed differently in different coordinate systems. For in- stance, in cylindrical polars it is:
∇ ≡ ˆer
∂r
r
∂θ
∂z
In a Cartesian coordinate system, the unit vectors are the same everywhere. This means that, when ∇ acts on another vector, we do not need to worry about the effect of ∇ on the unit vectors because ∂ˆex/∂x, ∂ˆey/∂x etc. are all zero. (This is why the Cartesian shorthand works for ∇).
In other coordinate systems, the unit vectors are not the same everywhere. This means that, when ∇ acts on a vector, its effect on the unit vectors must also be taken into account. We encounter this in lecture 2 and there is also an example on the website.
1.6 The law of conservation of mass
You met the law of conservation of mass in 1A fluids. It was applied to a pipe flow:
We can also introduce the mass flux, defined as the flow of mass per unit area per unit time. For example, the mass flux at entry to the control volume above is:
In 1B we apply the same law to a volume of space. Here we apply it over depth δz into the page and assume that vz = 0.
In a time δt, the change in mass, δM , is given by
{
ρvxδyδz −
ρ +
∂ρ ∂x
δx
vx +
∂vx ∂x
δx
δyδz + ρvyδxδz −
ρ +
∂ρ ∂y
δy
vx +
∂vx ∂y
δy
δxδz
δt
δM δt
∂(ρvx) ∂x
∂(ρvy) ∂y
δxδyδz
But M = ρδxδyδz so the δxδyδz cancels and, as δt tends to zero, we obtain:
∂ρ ∂t
∂(ρvx) ∂x
∂(ρvy) ∂y
[
∂/∂x ∂/∂y
]
[
ρvx ρvy
]
This is the law of conservation of mass: the rate of change of mass per unit volume (density) is the net rate at which mass flows out of the volume. this method of deriving the formula is easy to understand physically but requires some messy maths. In section 8.2.1 of your Vector Calculus notes you will find an equivalent derivation, which is harder to visualize but which only takes three steps. It uses Gauss’ theorem.
1.7 The divergence of a vector field
When ∇ is dotted with a vector field a, it produces a scalar field equal to the net flux of a out of each point in space. This is known as the divergence of the field a.
For example, we saw in section 1.6 that the quantity ρv is a vector field that represents the mass flux. We know from the law of conservation of mass that:
∂ρ ∂t
= −∇ · (ρv)
If we were to heat up a fluid at a point in space, perhaps with a laser, its density would drop at that point (∂ρ/∂t negative), and the fluid would diverge away from that point - i.e. ∇ · (ρv) would be positive.
1.8 Constant density flows
Calculations in Fluid Mechanics become much easier if the density is constant. For example: ∂ρ ∂t
= −∇ · (ρv)
In the rest of this course we assume that flows have constant density. In other words, the equation of state (such as ρ = p/RT ) is replaced by ρ = constant. We call these flows ‘incompressible’. In lecture 10 we shall determine when this is a valid assumption.
