CIVL 2310 Fluid Mechanics Lecture 1: Flow Concepts and Kinematics

Chapter 3 of “Mechanics of Fluids,” Merle C. Potter and David C. Wiggert, Brooks/Cole, 2001. Topics

• Reference frames • Flowlines • Fluid processes and simplifications

1. Reference frames a) Rigid-body versus fluid mechanics (system vs control volume) Application of Newton’s second law (Force=mass·acceleration) to a rigid-body (system) in free fall gives: maF =∑

2

2

dtsdmmg =

using s=s0 at t=0

0==Vdtds at t=0

we get

20 2

1 gtss =−

Newton’s law is a powerful tool that can provide us with a description of the position of the body (and also the velocity) at each time. Applying the above concepts to fluids is difficult since the definition of the “body” is not always clear due to continuous deformation (see Fig. 1). Instead, we will concentrate on whole regions of fluid rather than on particular fluid elements. The definition of these regions is arbitrary and is called the control volume.

Figure 1: (a) solid and (b) fluid parcel paths (from “Fluid Mechanics,” Victor L. Streeter, E. Benjamin Wylie, Keith W. Bedford, McGraw-Hill, 1997). b) Lagrangian versus Eulerian reference frames A reference frame following a system is called Lagrangian reference frame. A reference frame using the control volume concept is called Eulerian. Key concept: in order to be able to apply the universal laws of mechanics to fluids, we need to adapt them to a suitable system of reference. 2. Flowlines: streamlines, pathlines, streaklines In an Eulerian approach, the velocity is a vector field (bold for vectors), i.e., v(x,y,z,t)=ui+vj+wk Streamline: continuous line drawn through the fluid which has the direction of the velocity vectors at every point.

Figure 2: streamline Since the displacement has the direction of the velocity, we can write:

wdz

vdy

udx

==

Pathline: the successive positions of a particle over time (history) is called a pathline. In unsteady flow a particle can follow different streamlines at different times.

Figure 3: pathlines in unsteady flow Streakline: position of several particles released from the same source point.

Figure 4: streaklines In steady flow all three (streamlines, pathlines and streaklines) coincide.

ds

3. Fluid processes and simplifications Shear stress: internal forces per unit area between neighboring fluid “particles”. General formulation for both laminar and turbulent flow is given by

yu∂∂

= µτ

where µ is the viscosity (laminar flow) or eddy viscosity (turbulent flow). Laminar flow: fluid moves in layers along smooth paths Turbulent flow: fluid moves in irregular swirling paths Ideal flow: frictionless and incompressible, with no viscosity Boundary-layer flow: flow close to boundaries in which viscosity can not be neglected. Steady flow: conditions at any point do not change in time, for example:

0=∂∂

tu

Uniform flow: conditions at a given time are the same regardless of position, for example:

0=dsdv