FLUID MECHANICS FIFTH EDITION

He served as chairman of the Department of Mechanics and Applied Mechanics from 1992 to 1997. The discussion of the wave equation has been placed in the acoustics section of Chapter 15.

COMPANION WEBSITE

And, as a final note, the origins of many of the new exercises are referenced. The current version of this textbook benefited from the comments and suggestions provided by the reviewers of the initial revision proposal and the reviewers of draft versions of several of the chapters.

NOTATION

SYMBOLS )

FLUID MECHANICS

Given that the vast majority of observed mass in the universe exists in a liquid state, that life as we know it is not possible without liquids, and that the atmosphere and oceans that cover this planet are liquids, fluid mechanics has undeniable scientific and practical importance. Thus, some fluency in mathematics, especially multivariable calculus, is useful in the study of fluid mechanics.

UNITS OF MEASUREMENT

SOLIDS, LIQUIDS, AND GASES

If the solid is perfectly elastic, it returns to its preferred shape ABCD when F is withdrawn. Although solids and fluids behave very differently when subjected to shear stresses, they behave similarly under normal compressive stresses.

FIGURE 1.1 Deformation of solid and fluid elements under a constant externally applied shear force

CONTINUUM HYPOTHESIS

Solids can tolerate both normal tensile and compressive stresses, while liquids typically expand or change phase (i.e., boil) when subjected to tensile stresses. Some fluids can tolerate a small amount of tensile stress, the amount depending on the degree of molecular cohesion and the duration of the tensile stress.

MOLECULAR TRANSPORT PHENOMENA

Here is the mass of nitrogen per unit volume rY (sometimes known as the nitrogen concentration density), where is the overall density of the gas mixture. For liquids, shear stress is caused more by the intermolecular cohesive forces than by the thermal motion of the molecules.

FIGURE 1.2 Mass flux J m due to variation in the mass fraction Y(y). Here the mass fraction profile increases with increasing Y, so Fick’s law of diffusion states that the diffusive mass flux that acts to smooth out mass-fraction differences is downward ac

SURFACE TENSION

If are the pressures on the inner and outer sides of the interface, respectively, then a static force balance is given. A more complete discussion of surface tension is presented at the end of Chapter 4 as part of the section on boundary conditions.

FLUID STATICS

The only forces acting on the member are the compressive forces perpendicular to the faces and the weight of the member. Here, the pressure difference between the top and bottom of the element balances the weight of the element.

FIGURE 1.5 Demonstration that p 1 ¼ p 2 ¼ p 3 in a static fluid. Here the vector sum of the three arrows is zero when the volume of the element shrinks to zero.

CLASSICAL THERMODYNAMICS

Internal energy (also thermal energy) is a manifestation of the random molecular movement of the components of the system. In short, e is a thermodynamic property and is a function of the thermodynamic state of the system.

PERFECT GAS

A perfect gas is a gas that satisfies (1.22), even if it is a mixture of different molecular species. From (1.18) it can be shown (Exercise 1.11) that the isentropic flow of a perfect gas with constant specific heat obeys.

STABILITY OF STRATIFIED FLUID MEDIA

Suppose the FIGURE 1.9 Vertical variation of the (a) actual and (b) potential temperature in the atmosphere. This is a reasonable assumption in the lower 70 km of the atmosphere, where the absolute temperature generally remains within 15% of 250 K.

DIMENSIONAL ANALYSIS

The amount of RT/g is thus called the scale height of the atmosphere and provides a reasonable quantitative measure of the thickness of the atmosphere. Of the various formal methods of dimensional analysis, the description here is based on Buckingham's 1914 method.

Select Variables and Parameters

Dimensional analysis is a widely applicable technique for developing scaling laws, interpreting experimental data, and simplifying problems. The dimensional analysis process is presented here as a series of six steps that should be followed one-seven whenever possible.

Create the Dimensional Matrix

Determine the Rank of the Dimensional Matrix

For dimensional matrices, the rank is less than the number of rows only when one of the rows can be obtained by a linear combination of the other rows. A rank of less than 3 is common in statics problems, in which mass or density is really not relevant, but the dimensions of the variables (such as force) involve M.

Determine the Number of Dimensionless Groups

If all possible third-order determinants were zero, we would have concluded that <3 and proceeded with testing the second-order determinants.

Construct the Dimensionless Groups

Each dimensionless array is formed by combining the three iterative parameters, raised to unknown powers, with one of the variables or non-iterative parameters from the list constructed for the first step. The third dimensionless group is obtained by starting with the next unused parameter, 3, to find P3 ¼ 3/d.

State the Dimensionless Relationship

For the pipe flow example, the group Dpd2r/m2 can be formed from P1=P24, and the group3/Dx can be formed as P3/P2.

Use Physical Reasoning or Additional Knowledge to Simplify the Dimensionless Relationship

Second, when l is large compared to the size of the scatterer, the scattered field amplitude will be produced from the dipole moment induced in the scatterer by the incident field, and this scattered field amplitude will be proportional to V. He deduced it in the 1870s while studying light scattering from small diffusers to understand why the cloudless daytime sky was blue while the sun appeared orange or red at dawn and sunset.

FIGURE 1.10 A right triangle with area a, smallest acute angle b, and hypotenuse C. The dashed line is perpendicular to side C.

EXERCISES

SCALARS, VECTORS, TENSORS, NOTATION

Second-order tensors have one component for each pair of coordinate directions and can therefore have as many as 33¼9 separate components. Once a coordinate system is chosen, the nine components of a second-order tensor can be represented by a 33 matrix or by an italic symbol with two subscripts, such as assijfor the stress tensor.

FIGURE 2.1 Position vector OP and its three Cartesian components (x 1 , x 2 , x 3 ). The three unit vectors for the coordinate directions are e 1 , e 2 , and e 3

ROTATION OF AXES: FORMAL DEFINITION OF A VECTOR

Similarly, any letter can also be used for the free index as long as the same free index is used on both sides of the equation. We can now formally define a Cartesian vector as any quantity that transforms like the position vector under rotation of the coordinate system.

MULTIPLICATION OF MATRICES

Of course, the inner product A, B is defined only if the number of columns of A is equal to the number of rows of B. For example, (2.6) can be written six0i ¼ CTikxk, which is now of the form of (2.9) because the summary index is appended.

SECOND-ORDER TENSORS

In general, tensors can be of any order, and the number of free indices corresponds to the order of the tensor. The nine products formed from the components of the two vectors u and v also transform according to (2.12), and therefore form a second-order tensor.

FIGURE 2.4 Illustration of the stress field at a point via stress components on a cubic volume element

CONTRACTION AND MULTIPLICATION

Common second-order tensors are the stress tensor sijan and the velocity-gradient tensor vui/vxj. In addition, the Kronecker delta and alternating tensors are also frequently used; these are defined and discussed in Section 2.7.

FORCE ON A SURFACE

The areas of the faces of the tetrahedron that are perpendicular to the coordinate axis are dAi. Find the magnitude and direction of the force per unit area for an element whose external normal points ¼ 30 from the direction of flow.

FIGURE 2.6 Determination of the force per unit area on a small area element with a normal vector rotated 30 from the flow direction in a simple unidirectional shear flow parallel to the x 1 -axis.

KRONECKER DELTA AND ALTERNATING TENSOR

From its definition, it is clear that diji is anisotropic tensor in the sense that its components are unchanged by a rotation of the reference frame, that is, d0ij ¼ dij. From this definition, it is clear that an index of 3ijk can be moved two places (either to the right or to the left) without changing its value.

VECTOR, DOT, AND CROSS PRODUCTS

For example, 3ijk¼3jki, where it was shifted two places to the right, and 3ijk¼3kij, where it was shifted two places to the left. The dot product,vis is equal to the sum of the diagonal terms of the tensoruiyj.

GRADIENT, DIVERGENCE, AND CURL

So far we have defined the operations of the gradient of a scalar and the divergence of a vector. The three components of the vectorVucan can be easily found from the right-hand side of (2.24).

FIGURE 2.7 An illustration of the gradient, Vf, of a scalar function f. The curves of constant f and Vf are perpendicular, and the spatial derivative of f in the direction n is given by n,Vf

SYMMETRIC AND ANTISYMMETRIC TENSORS

Thus, the doubly contracted product of a symmetric tensor sm any tensor B is equal to s doubly contracted with the symmetric part of B, and the doubly contracted product of a symmetric tensor and an antisymmetric tensor is zero. The latter result is analogous to the fact that the definite integral over an even (symmetric) interval of the product of a symmetric and an antisymmetric function is zero.

EIGENVALUES AND EIGENVECTORS OF A SYMMETRIC TENSOR

For positive G it will be shown in the next chapter that it represents a linear extension with a velocity G along one principal axis and a linear compression with a velocity along the other; the shear strains are zero in the principal axis coordinate system of the strain rate tensor.

GAUSS’ THEOREM

Consider an elementary rectangular volume centered on x with faces perpendicular to the coordinate axes (see Figure 2.4). A Taylor expansion of Q(x) from the center of the volume to the center of each of these sides produces.

FIGURE 2.9 Illustration of Gauss’ theorem for a volume Venclosed by surface area A. A small volume element, dV , and a small area element, dA, with outward normal n are shown.

STOKES’ THEOREM

This theorem means that the surface integral of the curl of a vector field is equal to the line integral of one along the bounding curve of the surface. In addition, (2.34) can be used to define the curl of a vector through the boundary of the circulation around an infinitesimally small surface as.

FIGURE 2.10 Illustration of Stokes’ theorem for surface A bounded by the closed curve C

COMMA NOTATION

Use the vector integral theorems to prove that V,ðVuÞ ¼ 0for any doubly differentiable vector function regardless of the coordinate system. Use Stokes' theorem to prove that V ðVfÞ ¼ 0 for any one-valued doubly differentiable scalar regardless of the coordinate system.

INTRODUCTION AND COORDINATE SYSTEMS

In addition, higher dimensional flows are sometimes analyzed in one dimension by averaging the properties of the higher dimensional flow over a suitable distance or range (Figure 3.1c and d). Spherical polar coordinates of P (Figure 3.3d) are denoted by (r, q, 4) with the corresponding velocity components (ur, uq, u4).

PARTICLE AND FIELD DESCRIPTIONS OF FLUID MOTION

The particle path or particle trajectory (t;ro,to) specifies the location of the fluid particles at later times. DtFðx,tÞ, (3.4) where the final equality defines D/Dtas the total time derivative in the Eulerian description of the fluid motion.

FIGURE 3.2 Sample flow fields where two spatial coordinates are needed. (a) Steady flow of an ideal incom- incom-pressible fluid past a long stationary circular cylinder with its axis perpendicular to the flow

FLOW LINES, FLUID ACCELERATION, AND GALILEAN TRANSFORMATION

The equation of the path line for the fluid particle launched from ro attois obtained from the fluid velocity by integrating. The advective acceleration term, (u,V)u, is non-zero when fluid particles move between locations where the fluid velocity is different.

FIGURE 3.6 Stream tube geometry for the closed curve C.

STRAIN AND ROTATION RATES

Here A0B0¼ABþBB0AA0, and a positive S11¼vu1/vx1, corresponds to an extension of the fluid element. This deformation becomes particularly simple in a coordinate system that coincides with the principal axes of the strain rate tensor.

FIGURE 3.10 Illustration of positive linear strain rate in the first coordinate direction

KINEMATICS OF SIMPLE PLANE FLOWS

Thus, each fluid element rotates about its own center at the same rate as it rotates about the origin of coordinates. Although the circulation around a circuit that contains the origin in an irrotational vortex flow is not zero, the circulation around a circuit that does not contain the origin is zero.

FIGURE 3.15 Solid-body rotation. The streamlines are circular and fluid elements spin about their own centers at the same rate that they revolve around the origin.

REYNOLDS TRANSPORT THEOREM

The purpose of this effort is to determine the time derivative of the integral of a single value FIGURE 3.17 Graphical illustration of the Liebniz theorem. Determine the equivalent of the first equality in (3.7) for two-dimensional (r,q)-polar coordinates, and then find the equation for the streamline passing through (ro,qo) when u¼(ur,uq)¼(A/) r ,B/r) verifiable constants.

FIGURE 3.18 Geometrical depiction of a control volume V*(t) having a surface A*(t) that moves at a nonuniform velocity b during a small time increment D t

INTRODUCTION

To derive and describe the dimensionless numbers that appear naturally when the equations of motion are put into dimensionless form.

CONSERVATION OF MASS

Gases are compressible, but for flow velocities less than ~100 m/s (that is, for Mach numbers <0.3), the fractional change of absolute pressure in an air stream is small. The general form of the continuity equation(4.7) is typically required when the derivative Dr/Dtis is nonzero due to changes in pressure, temperature, or molecular composition of fluid particles.

STREAM FUNCTIONS

In this situation, zi is one of the three-dimensional flow functions, so we can set c ¼ z, choosing the sign to follow the usual convention. Similarly, for asymmetric three-dimensional flow in cylindrical polar coordinates (Figure 3.3c), all streamlines lie in 4¼ constant planes containing the z-axis, soc¼ 4is one of the flow functions.

FIGURE 4.1 Isometric view of two members from each family of stream surfaces. The solid curves are streamlines and these lie at the intersections of the surfaces

CONSERVATION OF MOMENTUM

However, the control volume laws are written for the forces that apply to the contents of the volume. Let the horizontal velocity of the flow be at the outlet side of the control volume and assume that its profile is uniform.

FIGURE 4.2 Momentum and mass balance for flow past long bar of constant cross section placed perpendicular to the flow

CONSTITUTIVE EQUATION FOR A NEWTONIAN FLUID

The relationship(s) between stress and strain rate of the fluid or the constitutive equation provide much of the required reduction. We can therefore take the average of the diagonal terms of sand and define an average pressure (as opposed to the thermodynamic pressure p) as.

NAVIER-STOKES MOMENTUM EQUATION

For example, shear stress can be an online function of the local strain rate, which is the case for many liquid plastics that undergo shear thinning; their viscosity decreases with increasing strain rate. If viscous effects are negligible, which is usually the case outside the boundaries of the flow field, (4.39) further simplifies Euler's equation.

NONINERTIAL FRAME OF REFERENCE

The last new acceleration term in (4.45), the centrifugal acceleration, depends strongly on the rotational speed and the distance of the fluid particle from the axis of rotation. In the first two momentum equations, the terms in [] brackets are the result of rotation of the coordinate system.

FIGURE 4.7 Geometry showing the relationship between U, the rotational velocity vector of O 0 1 0 2 0 3 0 , and the first coordinate unit vector e 0 1 in O 0 1 0 2 0 3 0

CONSERVATION OF ENERGY

It is the product of the viscous stress acting on a fluid element and the strain rate of a fluid element, and represents the viscous work put into fluid element deformation. The final energy equation manipulation is to express in terms of the other dependent field variables.

SPECIAL FORMS OF THE EQUATIONS

The flow is irrotational at all points outside the thin viscous layer near the surface of the body. Applying Bernoulli's equation (4.19) to steady flow at constant density between a point on the free surface in the reservoir and a point at C gives.

FIGURE 4.12 Lawn sprinkler.

BOUNDARY CONDITIONS

When is>0, it and the curvature of the fluid interface determine the pressure difference across the interface. Calculation of the shape of the free surface of a liquid adhering to an infinite vertical plane wall.

FIGURE 4.19 The curved surface shown is tangent to the x-y plane at the origin of coordinates

DIMENSIONLESS FORMS OF THE EQUATIONS AND DYNAMIC SIMILARITY

It is clear that the boundary conditions in terms of the dimensionless variables (4.100) are independent of l, U and U. Also show that the velocity of the fluid at the plane of the propeller has the average value U ¼(U1 þU2)/2 is.

INTRODUCTION

An vortex line is a curve in a fluid that is everywhere tangent to the local vorticity vector. When the axis of rotation is parallel to the (downward) gravitational acceleration, the surfaces of constant pressure in the fluid are paraboloids of revolution.

FIGURE 5.1 Analogy between stream tubes and vortex tubes. The lateral sides of stream and vortex tubes are locally tangent to the flow’s velocity and vorticity fields, respectively

KELVIN’S CIRCULATION THEOREM

For a barotropic fluid, the first term on the right-hand side of (5.10) is zero because it is a closed contour, and another one valued at each point in space. Similarly, the second integral on the right-hand side of (5.10) is zero since Fis is also single-valued at every point in space.

FIGURE 5.5 Schematic drawings of two fluids with differing density that are initially stationary and separated within a rectangular container

HELMHOLTZ’S VORTEX THEOREMS

Thus, the vortex tubes move with the liquid, which we can also get from the vorticity field equation. Applying this result to an infinitely thin vortex tube gives Helmholtz's vortex theorem that vortex lines move with the fluid.

VORTICITY EQUATION IN A NONROTATING FRAME

The term V2 represents the rate of change caused by the diffusion of eddy current in the same way that nV2u represents acceleration caused by the diffusion of momentum. The term (u,V) represents the rate of change of eddy current caused by the stretching and tilting of vorticity lines.

VELOCITY INDUCED BY A VORTEX FILAMENT

Note that pressure and gravity terms do not appear in (5.13) since these forces act through the center of mass of an element and therefore do not generate any torque. However, the vorticity is the curl of the velocity, so both the advective part of the Du/Dt term and the (u,V)u term represent nonlinearities.

LAW OF BIOT AND SAVART

VORTICITY EQUATION IN A ROTATING FRAME

Equation (5.25) is another form of the Navier-Stokes momentum equation, so we obtain the vorticity equation of the rotating reference system by taking its twist. The second term on the right-hand side is the rate of vorticity generation due to the baroclinicity of the flow, as discussed in section 5.2.