# 6: Fluid Dynamics

Fluid dynamics (as opposed to kinematics) inquires into the causes of fluid motion. Our discussion will be based on the two great theories of classical physics: Newton’s laws of motion and the laws of thermodynamics. Newton’s laws were designed to apply to rigid objects like apples and planets. How do we apply them to an object that not only accelerates but also changes its shape in response to an applied force? That challenge will occupy us for the first half of this chapter. Newton’s laws lead naturally to a consideration of kinetic and potential energy. To these concepts we will add internal energy in accordance with the laws of thermodynamics. Finally, the addition of an equation of state (through which we describe the thermodynamic properties of the specific fluid we are interested in) results in a complete set of equations that will, we hope, describe the motion of real fluids.

Our approach to applying classical laws of physics to fluid motion involves the use of conservation laws. Specifically, we assume that

• The mass of a fluid parcel does not change.

• The momentum of a fluid parcel changes due to the applied forces in accordance with Newton’s second law.

• The total energy of a parcel changes due work done and heat added in accordance with the first law of thermodynamics.

Mathematically, conservation laws for a fluid parcel take the Lagrangian form:

$\frac{D}{D t} \int_{V_{m}(t)}(\text { something }) d V=\cdots,\label{eqn:1}$

where $$V_m(t)$$ is the fluid parcel: a volume of space that changes in time but always contains the same fluid. (The subscript m indicates a material volume; another term for a fluid parcel.) The equation says that the net amount of some property contained by the fluid parcel (e.g., mass, momentum, energy) changes in time according to whatever is on the right hand side. While the physical meaning of Equation $$\ref{eqn:1}$$ is easy to understand, the result is an integro-differential equation whose solution would be quite daunting. We’ll start by establishing Leibniz’ rule, which allows us to convert Lagrangian conservation laws of the form Equation $$\ref{eqn:1}$$ into Eulerian partial differential equations, which we at least have some hope of solving.