229

Motivation. In analyzing fluid motion, we might take one of two paths: (1) seekingan estimate of gross effects (mass flow, induced force, energy change) over a finiteregion or control volume or (2) seeking the point-by-point details of a flow patternby analyzing an infinitesimal region of the flow. The former or gross-average view-point was the subject of Chap. 3.

This chapter treats the second in our trio of techniques for analyzing fluid motion:small-scale, or differential, analysis. That is, we apply our four basic conservationlaws to an infinitesimally small control volume or, alternately, to an infinitesimal fluidsystem. In either case the results yield the basic differential equations of fluid motion.Appropriate boundary conditions are also developed.

In their most basic form, these differential equations of motion are quite difficultto solve, and very little is known about their general mathematical properties. How-ever, certain things can be done that have great educational value. First, as shown inChap. 5, the equations (even if unsolved) reveal the basic dimensionless parametersthat govern fluid motion. Second, as shown in Chap. 6, a great number of useful solu-tions can be found if one makes two simplifying assumptions: (1) steady flow and (2)incompressible flow. A third and rather drastic simplification, frictionless flow, makesour old friend the Bernoulli equation valid and yields a wide variety of idealized, orperfect-fluid, possible solutions. These idealized flows are treated in Chap. 8, and wemust be careful to ascertain whether such solutions are in fact realistic when com-pared with actual fluid motion. Finally, even the difficult general differential equa-tions now yield to the approximating technique known as computational fluid dynam-ics (CFD) whereby the derivatives are simulated by algebraic relations between afinite number of grid points in the flow field, which are then solved on a computer.Reference 1 is an example of a textbook devoted entirely to numerical analysis offluid motion.

Chapter 4Differential Relations

for Fluid Flow

230 Chapter 4 Differential Relations for Fluid Flow

In Sec. 1.7 we established the cartesian vector form of a velocity field that varies inspace and time:

V(r, t) 5 iu(x, y, z, t) 1 jy(x, y, z, t) 1 kw(x, y, z, t) (1.4)This is the most important variable in fluid mechanics: Knowledge of the velocity vectorfield is nearly equivalent to solving a fluid flow problem. Our coordinates are fixed in space,and we observe the fluid as it passes byas if we had scribed a set of coordinate lineson a glass window in a wind tunnel. This is the eulerian frame of reference, as opposedto the lagrangian frame, which follows the moving position of individual particles.

To write Newtons second law for an infinitesimal fluid system, we need to cal-culate the acceleration vector field a of the flow. Thus we compute the total timederivative of the velocity vector:

Since each scalar component (u, y, w) is a function of the four variables (x, y, z, t),we use the chain rule to obtain each scalar time derivative. For example,

But, by definition, dx/dt is the local velocity component u, and dy/dt 5 y, and dz/dt5 w. The total time derivative of u may thus be written as follows, with exactly sim-ilar expressions for the time derivatives of y and w:

(4.1)

Summing these into a vector, we obtain the total acceleration:

(4.2)Local Convective

The term V/t is called the local acceleration, which vanishes if the flow is steadythat is, independent of time. The three terms in parentheses are called the convectiveacceleration, which arises when the particle moves through regions of spatially vary-ing velocity, as in a nozzle or diffuser. Flows that are nominally steady may havelarge accelerations due to the convective terms.

Note our use of the compact dot product involving V and the gradient operator =:

where

The total time derivativesometimes called the substantial or material derivativeconcept may be applied to any variable, such as the pressure:

= 5 i

x1 j

y1 k

zu

x1 y

y1 w

z5 V ? =

a 5dVdt

5Vt

1 au Vx

1 y Vy

1 w Vzb 5 V

t1 (V ? =)V

az 5 dwdt

5 w

t 1 u

w

x 1 y

w

y 1 w

w

z 5

w

t 1 (V ? =) w

ay 5 dydt

5 y

t 1 u

y

x 1 y

y

y 1 w

y

z 5

y

t 1 (V ? =) y

ax 5 dudt

5 u

t 1 u

u

x 1 y

u

y 1 w

u

z 5

u

t 1 (V ? =) u

du(x, y, z, t)dt

5u

t1

u

x

dxdt

1u

y

dydt

1u

z

dzdt

a 5dVdt

5 i dudt

1 j dydt

1 k dwdt

4.1 The Acceleration Field of a Fluid

(4.3)

Wherever convective effects occur in the basic laws involving mass, momentum, orenergy, the basic differential equations become nonlinear and are usually more com-plicated than flows that do not involve convective changes.

We emphasize that this total time derivative follows a particle of fixed identity,making it convenient for expressing laws of particle mechanics in the eulerian fluidfield description. The operator d/dt is sometimes assigned a special symbol such asD/Dt as a further reminder that it contains four terms and follows a fixed particle.

As another reminder of the special nature of d/dt, some writers give it the namesubstantial or material derivative.

EXAMPLE 4.1

Given the eulerian velocity vector field

V 5 3ti 1 xzj 1 ty2kfind the total acceleration of a particle.

Solution

Assumptions: Given three known unsteady velocity components, u 5 3t, y 5 xz, andw 5 ty2.

Approach: Carry out all the required derivatives with respect to (x, y, z, t), substituteinto the total acceleration vector, Eq. (4.2), and collect terms.

Solution step 1: First work out the local acceleration V/t:

Solution step 2: In a similar manner, the convective acceleration terms, from Eq. (4.2),are

Solution step 3: Combine all four terms above into the single total or substantialderivative:

Ans.

Comments: Assuming that V is valid everywhere as given, this total acceleration vectordV/dt applies to all positions and times within the flow field.

5 3i 1 (3tz 1 txy2)j 1 (y2 1 2txyz)k

dVdt

5Vt

1 u Vx

1 y Vy

1 w Vz

5 (3i 1 y2k) 1 3tzj 1 2txyzk 1 txy2j

w Vz

5 (ty2) z

(3ti 1 xzj 1 ty2k) 5 (ty2)(0i 1 xj 1 0k) 5 txy2 j

y Vy

5 (xz) y

(3ti 1 xzj 1 ty2k) 5 (xz)(0i 1 0j 1 2tyk) 5 2txyz k

u Vx

5 (3t) x

(3ti 1 xzj 1 ty2k) 5 (3t)(0i 1 zj 1 0k) 5 3tz j

Vt

5 i u

t1 j y

t1 k

w

t5 i

t (3t) 1 j

t (xz) 1 k

t (ty2) 5 3i 1 0j 1 y2 k

dpdt

5pt

1 u px

1 y py

1 w pz

5pt

1 (V ? =)p

4.1 The Acceleration Field of a Fluid 231

Conservation of mass, often called the continuity relation, states that the fluid masscannot change. We apply this concept to a very small region. All the basic differen-tial equations can be derived by considering either an elemental control volume or anelemental system. We choose an infinitesimal fixed control volume (dx, dy, dz), as inFig. 4.1, and use our basic control volume relations from Chap. 3. The flow througheach side of the element is approximately one-dimensional, and so the appropriate massconservation relation to use here is

(3.22)

The element is so small that the volume integral simply reduces to a differential term:

The mass flow terms occur on all six faces, three inlets and three outlets. We make useof the field or continuum concept from Chap. 1, where all fluid properties are consid-ered to be uniformly varying functions of time and position, such as r 5 r(x, y, z, t).Thus, if T is the temperature on the left face of the element in Fig. 4.1, the right face willhave a slightly different temperature For mass conservation, if ru isknown on the left face, the value of this product on the right face is

Figure 4.1 shows only the mass flows on the x or left and right faces. The flowson the y (bottom and top) and the z (back and front) faces have been omitted to avoidcluttering up the drawing. We can list all these six flows as follows:

Face Inlet mass flow Outlet mass flow

x ru dy dz

y ry dx dz

z rw dx dy c rw 1 z

(rw) dz d dx dycry 1

y (ry) dy d dx dz

cru 1 x

(ru) dx d dy dz

ru 1 (ru/x) dx.T 1 (T/x) dx.

#CV

r

t d9