1 Equations of Motion September 15 Part 1

2

3 Continuum Hypothesis  Assume that macroscopic behavior of fluid is same as if it were perfectly continuous  Newton’s 2 nd Law: Acceleration of a particle is proportional to the sum of the forces acting on that particle F=ma

4 R-H Cartesian f-plane or  -plane y, v x, u z, w N x: eastward direction u: eastward velocity y: northward direction v: northward velocity z: local vertical w: vertical velocity R: vector distance from center of earth  : Earth’s rotation vector  R

5 x-component of acceleration: or, for fluids: Other components:

6  Important kinds of forces acting on a fluid particle: Wind stress Wind stress Gravitational Gravitational Pressure gradient Pressure gradient Frictional Frictional Coriolis Coriolis

7 Acceleration  Two kinds: - Particle acceleration – acceleration measured following a particle (Langrangian) - Local acceleration – acceleration seen at a fixed point in space (Eulerian)

8 A B vel. u dist. x Particle undergoes acceleration, but velocity measured at point A or B would not change with time

9 particle accerleration local acceleration + field acceleration

10 A v u y x At point A, if there were no other forces, we would see a local acceleration due to movement of the velocity field

11 Figure 7.2 in Stewart Consider the flow of a quantity q in into and q out out of the small box sketched in Figure 7.2. If q can change continuously in time and space, the relationship between qin and qout is

12 Total Derivative

13 Momentum Equation

14 Coriolis  Coriolis arises because we measure a relative to coordinates fixed to the surface of a rotating earth i.e. accelerating – easier to measure

15 The acceleration of a parcel of fluid in a rotating system, can be written: R = vector distance from the center of the Earth Ω = angular velocity vector of Earth u = velocity of the fluid parcel in coordinates fixed to Earth (2Ω × u) = the Coriolis force Ω × (Ω × R) = centrifugal acceleration Coriolis and centrifugal accelerations are “Fictitious” – arise only because of choice of coordinate frame Coriolis exists only if there is a velocity – no velocity, no “force”

16 Gravity Term in Momentum Equation The gravitational attraction of two masses M 1 and m is R = distance between the masses G = gravitational constant F g = vector force along the line connecting the two masses force per unit mass due to gravity is M E = mass of the Earth

17 Adding the centrifugal acceleration to previous equation gives gravity g Figure 7.4 in Stewart

18 Momentum Equation in Cartesian Coordinates incompressible – no sound waves allowed

19 Figure 7.3 in Stewart

20 Derivation of Pressure Term Consider the forces acting on the sides of a small cube of fluid (Figure 7.3). The net force δF x in the x direction is But Therefore

21 Divide by the mass of the fluid (δm) in the box, the acceleration of the fluid in the x direction is

22  Can solve for u, v, w, p as a function of x, y, z, t  Need boundary conditions: - u, v, w, p must behave at boundaries i.e. no flow through boundaries no “slip” along boundaries no “slip” along boundaries