*Fluids in Motion*

*Fluids in Motion*

Fluid motions are quite complex and require rather advanced mathematics to describe them if all details are to be included. Simplifying assumptions can reduce the mathematics required, but even then the problems can get rather involved mathematically. To describe the motion of air around an airfoil, a tornado, or even water passing through a valve, the mathematics becomes quite sophisticated and is beyond the scope of an introductory course. We will, however, derive the equations needed to describe such motions, and make simplifying assumptions that will allow a number of problems of interest to be solved. These problems will include ﬂow through a pipe and a channel, around rotating cylinders, and in a thin boundary layer near a ﬂat wall. They will also include compressible ﬂows involving simple geometries.

The pipes and channels will be straight and the walls perfectly ﬂat. Fluids are all viscous, but often we can ignore the viscous effects. If viscous effects are to be included, we can demand that they behave in a linear fashion, a good assumption for water and air. Compressibility effects can also be ignored for low velocities such as those encountered in wind motions (including hurricanes) and ﬂows around airfoils at speeds below about 100 m/s (220 mph).

In this chapter we will describe ﬂuid motion in general, classify the different types of ﬂuid motion, and also introduce the famous Bernoulli’s equation along with its numerous assumptions.

__3.1 Fluid Motion__

__3.1 Fluid Motion__

*LAGRANGIAN AND EULERIAN DESCRIPTIONS*

Motion of a group of particles can be thought of in two basic ways: focus can be on an individual particle, such as following a particular car on a highway (a police patrol car may do this while moving with trafﬁc), or it can be at a particular location as the cars move by (a patrol car sitting along the highway does this). When analyzed correctly, the solution to a problem would be the same using either approach (if you’re speeding, you’ll get a ticket from either patrol car).

When solving a problem involving a single object, such as in a dynamics course, focus is always on the particular object: the *Lagrangian description of motion*. It is quite difﬁcult to use this description in a ﬂuid ﬂow where there are so many particles. Let’s consider a second way to describe a ﬂuid motion.

At a general point (*x*, *y*, *z*) in a ﬂow, the ﬂuid moves with a velocity **V**(*x*, *y*, *z*, *t*).

The rate of change of the velocity of the ﬂuid as it passes the point is *∂ ***V**/*∂ **t*, *∂ ***V**/*∂ **y*,

*∂ ***V**/*∂ **z*, and it may also change with time at the point: *∂ ***V**/*∂ **t. *We use partial derivatives here since the velocity is a function of all four variables. This is the *Eulerian description of motion*, the description used in our study of ﬂuids. We have used rectangular coordinates here but other coordinate systems, such as cylindrical coordinates, can also be used. The region of interest is referred to as a *ﬂow ﬁeld *and the velocity in that ﬂow ﬁeld is often referred to as the *velocity ﬁeld. *The ﬂow ﬁeld could be the inside of a pipe, the region around a turbine blade, or the water in a washing machine.

If the quantities of interest using the Eulerian description were not dependent on time *t*, we would have a *steady ﬂow*; the ﬂow variables would depend only on the space coordinates. For such a ﬂow,

to list a few. In the above partial derivatives, it is assumed that the space coordinates remain ﬁxed; we are observing the ﬂow at a ﬁxed point. If we followed a particular particle, as in a Lagrangian approach, the velocity of that particle would, in general, vary with time as it progressed through a ﬂow ﬁeld. But, using the Eulerian description, as in Eq. (3.1), time would not appear in the expressions for quantities in a steady ﬂow, regardless of the geometry.

__PATHLINES, STREAKLINES, AND STREAMLINES__

__PATHLINES, STREAKLINES, AND STREAMLINES__

Three different lines can be deﬁned in a description of a ﬂuid ﬂow. The locus of points traversed by a particular ﬂuid particle is a *pathline*; it provides the history of the particle. A time exposure of an illuminated particle would show a pathline. A *streakline *is the line formed by all particles passing a given point in the ﬂow; it would be a snapshot of illuminated particles passing a given point. A *streamline *is a line in a ﬂow to which all velocity vectors are tangent at a given instant; we can- not actually photograph a streamline. The fact that the velocity is tangent to a streamline allows us to write

since **V **and *d***r **are in the same direction, as shown in Fig. 3.1; recall that two vectors in the same direction have a cross product of zero.

In a steady ﬂow, all three lines are coincident. So, if the ﬂow is steady, we can photograph a pathline or a streakline and refer to such a line as a streamline. It is the streamline in which we have primary interest in our study of ﬂuids.

A *streamtube *is a tube whose walls are streamlines. A pipe is a streamtube, as is a channel. We often sketch a streamtube in the interior of a ﬂow for derivation purposes.

__ACCELERATION__

__ACCELERATION__

To make calculations for a ﬂuid ﬂow, such as forces, it is necessary to describe the motion in detail; the expression for the acceleration is usually needed. Consider a ﬂuid particle having a velocity **V**(*t*) at an instant *t*, as shown in Fig. 3.2. At the next instant, *t *+ Δ *t*, the particle will have velocity **V**(*t *+ Δ *t*), as shown. The acceleration of the particle is

**F****i****g****u****re 3.1 **A streamline.

where (*u*, *v*, *w*) are the velocity components of the particle in the *x*-, *y*-, and *z*-directions, respectively, and **i**, **j**, and **k **are the unit vectors. For the particle at the point of inter- est, we have

In Eq. (3.8), the time derivative of velocity represents the *local acceleration *and the other three terms represent the *convective acceleration*. Local acceleration results if the velocity changes with time (e.g., startup), whereas convective acceleration results if velocity changes with position (as occurs at a bend or in a valve).

It is important to note that the expressions for the acceleration have assumed an inertial reference frame, i.e., the reference frame is not accelerating. It is assumed that a reference frame attached to the earth has negligible acceleration for problems of interest in this book. If a reference frame is attached to, say, a dishwasher spray arm, additional acceleration components, such as the Coriolis acceleration, enter the expression for the acceleration vector.

The vector equation (3.8) can be written as the three scalar equations:

where *D*/*D**t *is called the *material*, or *substantial derivative *since we have followed a material particle, or the substance. In rectangular coordinates, the material derivative is

It can be used with other quantities of interest, such as the pressure: *Dp*/*Dt *would represent the rate of change of pressure of a ﬂuid particle at some point (*x, y, **z*).

The material derivative and acceleration components are presented for cylindrical and spherical coordinates in Table 3.1 at the end of this section.

__ANGULAR VELOCITY AND VORTICITY__

__ANGULAR VELOCITY AND VORTICITY__

Visualize a ﬂuid ﬂow as the motion of a collection of ﬂuid elements that deform and rotate as they travel along. At some instant in time, we could think of all the elements that make up the ﬂow as being little cubes. If the cubes simply deform and don’t rotate, we refer to the ﬂow, or a region of the ﬂow, as an *irrotational ﬂow*.

**F****i****g****u****re 3.3 **The rectangular face of a ﬂuid element.

We are interested in such ﬂows in our study of ﬂuids; they exist in tornados away from the “eye” and in the ﬂow away from the surfaces of airfoils and automobiles. If the cubes do rotate, they possess *vorticity*. Let’s derive the equations that allow us to determine if a ﬂow is irrotational or if it possesses vorticity.

Consider the rectangular face of an inﬁnitesimal volume shown in Fig. 3.3. The *angular velocity *Ω about the *z*-axis is the average of the angular velocity of segments *AB *and *AC*, counterclockwise taken as positive:

These three components of the angular velocity represent the rate at which a ﬂuid particle rotates about each of the coordinate axes. The expression for Ω_{2} would predict the rate at which a cork would rotate in the *xy-*surface of the ﬂow of water in a channel.

The *vorticity vector **w *is deﬁned as twice the angular velocity vector: *w *= 2Ω. The vorticity components are

__EXAMPLE 3.1__

A velocity ﬁeld in a plane ﬂow is given by **V **= 2*yt***i **+ *x***j**. Find the equation of the streamline passing through (4, 2) at *t *= 2.

__Solution__

Equation (3.2) can be written in the form

The constant is evaluated at the point (4, 2) to be *C *= 0. So, the equation of the streamline is

Distance is usually measured in meters and time in seconds so that velocity would have units of m/s.

For the velocity ﬁeld **V **= 2*xy***i **+ 4*t**z*^{2} **j **− *y**z***k**, ﬁnd the acceleration, the angular velocity about the *z*-axis, and the vorticity vector at the point (2, −1, 1) at *t *= 2.

__Solution__

The acceleration is found, using *u *= 2*x**y*, *v *= 4*t**z*2 , and *w *= − *y**z*, as follows:

he angular velocity component Ω_{z} is

At the point (2, −1, 1) and *t *= 2, it is

ω = (−1 − 16)**i **− 4**k **= −17**i **− 4**k**

Distance is usually measured in meters and time in seconds. Thus, angular velocity and vorticity would have units of m/s/m, i.e., rad/s.