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1 Chapter Flow kinematics Vector and tensor formulae This introductory section presents a brief account of different definitions of ector and tensor analysis that will be used in the following chapters. In the following, we shall consider an orthogonal coordinate system with unit ectors (ē, ē, ē 3 ) at each point. In that coordinate system, any ector ā admits a representation as a function of three scalars a i such that ā = i a iē i. Some common operations between two ectors ā and b include the scalar product ā b = i a ib i and the cross product ā b = ē ē ē 3 a a a 3 b b b 3 = (a b 3 a 3 b )ē +(a 3 b a b 3 )ē +(a b a b )ē 3. (.) A second-order tensor Ā is to be interpreted in the framework of the present course as a linear operator that, when acting oer a ector ā, gies as result another ector b = Ā ā with components b i = j A ija j. The transerse tensor Ā T, with components (Ā T ) ij = (Ā) ji, satisfies ā Ā = Ā T ā. The tensor Ī with components I ii = y I ij = 0 if i j is the so-called identity tensor, which erifies Ī ā = ā for any gien ector ā. Of interest in the following deelopment are the dyadic product of two ectors, ā b, which produces a tensor of components (ā b) ij = a i b j, and the contraction of two tensors, Ā : B which gies as a result a scalar Ā : B = i j A ijb ij. Note that this last result is quite different from the product of the two tensors, Ā B, which is another tensor of components (Ā B)ij = k A ikb kj.also of interest is the cross product of a ector ā = (a,a,a 3 ) and a tensor Ā, which gies a tensor according to ā Ā = 0 a 3 a a 3 0 a a a 0 Ā. (.) Besides cartesian (rectangular) coordinates (x = x, x = y, x 3 = z), two other types of orthogonal curilinear coordinates will be found to be useful: cylindrical coordinates (x = r, x = θ, x 3 = z) and spherical coordinates (x = r, x = θ, x 3 = φ), which are schematically represented in Fig.., along with their associated unit ectors (ē, ē, ē 3 ).

2 0 e z e z e r e θ θ e φ r e θ z e r φ x θ r y e y e x Figure.: Schematic representation of releant coordinate systems. The differential line elements associated for each of the three coordinate systems considered are, respectiely, d x = (dx,dy,dz), d x = (dr,rdθ,dz) and d x = (dr,rdθ,rsinθdφ). The coefficients that premultiply each of the differentials in the aboe expressions are the so-called scale factors (h,h,h 3 ), which reduce to (h =,h =,h 3 = ) for cartesian coordinates, to (h =,h = r,h 3 = ) for cylindrical coordinates, and to (h =,h = r,h 3 = rsinθ) for spherical coordinates. The introduction of these scale factors enables the differential operators to be written in a compact form independent of the coordinate system. Thus, the gradient of a scalar function Φ is in general gien by ( Φ Φ =, Φ, ) Φ, (.3) h x h x h 3 while its laplacian can be expressed in the form [ h h 3 Φ Φ = + h h 3 Φ + ] h h Φ. (.4) h h h 3 x h x x h x h 3 Similarly, the diergence of a ector function ā = a ē +a ē +a 3 ē 3 is [ ā = (h h 3 a )+ (h h 3 a )+ ] (h h a 3 ), (.5) h h h 3 x x while its curl is gien by ā = h ē h ē h 3 ē 3 h h h 3 x x h a h a h 3 a 3 = h h 3 [ ] x (h 3 a 3 ) (h a ) ē + [ ] [ ] h h 3 (h a ) x (h 3 a 3 ) ē + h h x (h a ) x (h a ) ē 3. Also, the gradient of a ector function produces a tensor according to (.6) ( ā) ii = a i + a k h i and ( ā) ij = a j a i h i if i j, (.7) h i x i h i h k x k h i x i h i h j x j k i

3 while the diergence of a tensor Ā gies a ector function of components ( Ā) i = h i haij + A ij +A ji h i A jj h j, (.8) h x j j h i h j h j i h j x j h j i h j x i where h = h h h 3. The so-called Gauss formula Σ ( n φ)dσ = V ( φ)dv, (.9) whereφrepresents a scalar, ector or tensor function, V is a domain enclosed by the closed surface Σ, and denotes any of the products defined aboe, will be found to be useful in manipulating the conseration equations. Some preliminary concepts Two different approaches, called respectiely Eulerian and Lagrangian descriptions, can be adopted when describing the elocity field. The Lagrangian approach takes a discrete point of iew by treating the fluid as an ensemble of fluid particles. The flow field is described in terms of the fluid-particle trajectories x = x T (t; x o,t o ), (.0) with each fluid particle identified by its initial position x o at the initial time time t o. The associated elocity and acceleration is computed by straightforward deriation to gie = d x T /dt and ā = d x T /dt. This Lagrangian approximation, which might be of use in specific applications (e.g., for the analysis of the disperse phase in two-phase problems), leads in general to complex conseration laws. For that reason, in the following deriation we shall adopt instead the Eulerian description, which assumes the elocity to be a continuous ector field ( x, t) as a function of the position x and time t. A flow field is said to be uniform when the spatial differences of the different flow properties are strictly zero. Similarly, a fluid flow is said to be steady when it does not change in time. Thus, a uniform elocity field satisfies = (t), whereas whereas a steady elocity field is such that = ( x). At times, the steadiness of a gien flow depends on the obserer. For instance, for a flow oer an aircraft moing with constant elocity, the flow is clearly unsteady for an obserer standing on the ground, whereas for the pilot the flow is effectiely steady. A moing frame of reference in that case would clearly facilitate the analysis of the flow. A stagnation point is defined as a point where the elocity is zero (all three components). For a gien elocity field = ( x,t), the stagnation points are determined by the equation ( x, t) = 0. (.) Clearly, the location of the stagnation points depends on the reference frame selected. Trajectories, path lines, and streak lines In its motion, a fluid particle initially located at x = x o changes its position in time as expressed in (.0). Gien the elocity field ( x, t), the location of the fluid particle at each instant of time is determined by integration of d x = ( x,t) (.) dt

4 with initial condition x = x o at t = t o to gie x = x T (t; x o,t o ). The aboe ector equation can be alternatiely expressed in its three components dt = h dx (x,x,x 3,t) = h dx (x,x,x 3,t) = h 3 dx 3 3 (x,x,x 3,t) (.3) to be integrated subject to x = x o, x = x o and x 3 = x 3o at t = t o. The solution can be expressed in the general form (.0), which reduces to x = x T (t;x o,y o,z o,t o ) y = y T (t;x o,y o,z o,t o ) (.4) z = z T (t;x o,y o,z o,t o ) for cartesian coordinates. The trajectory contains information about the path followed by the particle and also about the rate with which it traels along it. The equations that describe the path lines can be obtained by eliminating the time in (.0). For instance, for cartesian coordinates, once the time has been eliminated in (.4), the following two equations emerge f(x o,y o,z o,t o,x,y,z) = g(x o,y o,z o,t o,x,y,z) = 0 (.5) defining two surfaces whose intersection is the path line followed by the fluid particle initially located at (x o,y o,z o ). Besides path lines, in connection with the particle trajectories it is of interest to introduce the concept of streak lines. At a gien time t = t, consider the fluid particles that hae passed through the point x = x o at an earlier time t o. These particles lie on a cure, that we call streak line. They are useful in experiments for obserational purposes; they can be isualized by injecting dye slowly at some fixed point. Its mathematical description readily follows from ealuating the equations for the particle trajectories (.4) at a fixed time t = t for arying t o. Eliminating t o from the three equations proides the alternatie description F(x o,y o,z o,t,x,y,z) = G(x o,y o,z o,t,x,y,z) = 0 in terms of two intersecting surfaces. Fluid lines, fluid surfaces and fluid olumes Fluid particles initially located along the cure x = x l (λ), (.6) where λ denotes a parameter describing the cure, will continue to form a cure at any subsequent time. The equation describing the eolution of a fluid line is obtained by writing the equation for the trajectories, gien in (.0), for the fluid particles located at the initial time t = t o along the cure defined in (.6) to yield x = x T ( x l (λ),t). (.7) Recall that a cure can be in general expressed in the parametric form x = x l (λ) with λ < λ < λ. For instance, the ertical axis is gien by the expression x = 0,y = 0,z = λ ( < λ < ), while the parametric expression for a circle of radius R contained on the horizontal plane and centered at the origin is x = Rcosλ,y = Rsinλ,z = 0 (0 < λ < π).

5 Similarly, fluid particles initially forming a surface x = x s (α,β), (.8) where α and β are parameters will continue to form a surface (fluid surface). The corresponding fluid-surface equation x = x T ( x s (α,β),t), (.9) can be simplified by eliminating the parameters α and β, yielding in general an implicit equation of the form f( x,t) = 0. (.0) If the surface is initially closed, it can be expected to remain closed in its eolution. The finite olume of fluid bounded by a closed fluid surface is called a fluid olume, a useful concept to be used later when deriing the conseration laws goerning the fluid motion. In particular, in iew of the definition of the fluid elocity as the elocity of the center of mass of the fluid particle (see (.4)), it is clear that mass cannot cross a fluid surface, which implies that the mass of a fluid olume remains constant, thereby proiding the first conseration law of fluid mechanics. Streamlines, stream surfaces and stream tubes The streamlines of a flow field at a gien instant of time t = t are lines that are tangent to the local elocity ector at each point. This tangency condition can be expressed in the form h dx (x,x,x 3,t ) = h dx (x,x,x 3,t ) = h 3 dx 3 3 (x,x,x 3,t ) (.) The two integration constants for the aboe pair of differential equations sere to identify a specific streamline. For instance, one could identify a streamline by selecting the location (x o,y o ) at which it crosses the horizontal plane z = 0. If two streamlines intersect, the crossing point is necessarily a stagnation point (why?). There are two additional concepts related to the streamlines that are worth mentioning. The surface formed by all streamlines intersecting a gien line is called a stream surface. If the line selected is closed, the resulting stream surface would form a stream tube. Although Eqs. (.3) and (.) look similar, they represent two ery different physical and mathematical concepts. The time t is an integration ariable in the initial alue problem defined by (.3), whereas it merely plays the role of a parameter in the integration of (.). To find a trajectory, we follow the temporal eolution of the fluid particle. To find a streamline, we freeze the time by looking at the elocity field existing at a specific instant. Path lines and streamlines differ in a general unsteady flow. Howeer, for a steady elocity field = ( x) or, more generally, for a elocity field of the form = f(t) V( x) with f(t) representing an arbitrary function of time, streamlines and path lines can be shown to coincide (show it!). In general, a surface can be defined in terms of a pair of parameters. For instance, a sphere of radiusrcentered at the origin admits the parametric representation x = Rsinαcosβ,y = Rsinαsinβ,z = Rcosα (0 < α < π, 0 < β < π). 3

6 4 Material deriatie To determine the ariation of any intensie scalar property φ following the fluid flow, such as the density, pressure or temperature, one must account for the fluid motion by considering the infinitesimal displacement of a fluid particle, as illustrated in Fig... At time t the fluid t dx t + dt x x + dx Figure.: Infinitesimal displacement of a fluid particle. particle is located at x, so that the initial alue of the intensie property is gien by φ( x,t). A time increment dt later, the particle has moed to occupy a new position gien by x + d x (d x = dt), and the new alue of φ would be gien by φ( x+d x,t+dt). Neglecting small terms in a Taylor expansion of φ about ( x,t) enables the ariation of φ for the fluid particle to be written in the approximate form dφ = φ( x+d x,t+dt) φ( x,t) = dt φ t Diiding the aboe expression by dt and identifying = d x/dt leads to where the material deriatie operator +d x φ. (.) Dφ Dt = φ + φ, (.3) t D() Dt = () + () (.4) t expresses the ariation in time of any intensie scalar property following the fluid particle. The first term on the right-hand side represents the local temporal ariation, while the second term is the so-called conectie deriatie, related to the motion of the fluid particle. Changes in the properties of a fluid particle may arise because the flow is unsteady and/or because the fluid particle moes through a nonuniform field 3. 3 Note that the concept of material deriatie seres to derie the differential equation to be satisfied by the function f( x,t) corresponding to the fluid surface f( x,t) = 0 (see Eq. (.0)). The fluid surface therefore diides the flow field in two regions where the sign of f( x,t) is different. Since the fluid particles defining the fluid surface maintain a constant alue f( x,t) = 0 in their eolution, the function f( x,t) must satisfy the equation Df Dt = f + f = 0. t

7 5 Acceleration The concept of material deriatie can be extended to determine the time eolution of a ector property. In particular, we shall define the acceleration ector as ā = D Dt = + ( ), (.5) t where the gradient of elocity is a tensor. A conenient way to express (.5) is ā = t + ( /) ( ), (.6) which is alid regaress of the coordinate system selected. In cartesian coordinates, the components of the gradient of elocity are gien by ( ) ij = j / x i, so that each component of the acceleration a i reduces to the material deriatie of the corresponding elocity component according to a i = D i Dt = i t + j i. (.7) x j Note that this last result does not hold when cylindrical or spherical coordinates are used. For the description of certain flow fields, it is at times conenient to employ a non-inertial reference frame, that is, a reference frame that is accelerating and/or rotating with respect to the laboratory frame of reference. In that case, when writing the fluid acceleration one must account for the additional inertial contribution ā s = ā o + d Ω dt x+ Ω ( Ω x)+ Ω, (.8) where ā o and Ω are the acceleration and angular elocity of the non-inertial reference frame. Circulation and orticity The circulation Γ along a cure L is defined as the integral Γ = d l, (.9) L where the ector d l represents the differential line element. Physically, the circulation gies as a measure of the fluid motion along the direction of the cure. If x = x l (λ) (λ < λ < λ ) is the parametric representation of the cure, then d l = d x l dλ, (.30) dλ so that (.9) reduces to Γ = L ( x,t) d l = λ λ ( x l (λ),t) d x l dλ dλ. (.3)

8 6 If one considers a closed cure, then according to Stokes theorem Γ = d l = ( ) ndσ, (.3) L where Σ represents any surface bounded by L and dσ is a surface differential element with normal unit ector n, defined for the surface to hae positie orientation according to the so-called righthand rule; see Fig..3. For the last equation to hold, the elocity field ( x,t) must be continuous and the surface Σ must be completely contained within the fluid domain. The ector ω = is called orticity. n Σ σ Σ Figure.3: Application of Stokes theorem for the computation of the circulation. L The circulation along a closed cure limiting an infinitesimally small surface dσ contained in a plane normal to n reduces to d l = ( ) ndσ. (.33) L Hence, ( ) n = ( ) n turns out to be the circulation per unit surface around a cure normal to n. At a gien point, the circulation will be maximum when the cure is contained in a plane normal to. Irrotational flow and elocity potential The fluid motion is said to be irrotational when the orticity is identically zero eerywhere, i.e., ω = = 0. In ector calculus we hae seen that for any irrotational ector field one can define a scalar function, called potential function, such that = ϕ. (.34) The potential is defined by (.34), except for an integration constant, whose alue, a function of time, can be selected arbitrarily. Clearly, the introduction of the elocity potential will simplify notably the description of irrotational motions, because we are effectiely replacing the computation of a three-component ector field (the elocity ) by that of a scalar field (the potential ϕ). If the circulation along any closed cure is zero, the flow is necessarily irrotational. The demonstration begins by considering an infinitesimally small closed cure around a gien point. According to (.33), the circulation around the cure will be zero, regaress of the orientation n, if

9 7 = 0. The opposite is not necessarily true, that is, een though = 0 eerywhere in the flow field, the circulation around a closed cure might not be zero. This is so because the alidity of (.3) is restricted to surfaces Σ that are completely contained in the fluid domain. Therefore, for an irrotational field, one can claim that the circulation around a gien closed cure is zero if and only if we can find a surface bounded by the cure that is completely contained in the flow field, that is, when the cure is continuously reducible to a point (the fluid domain is simply connected). This feature of irrotational motion arises in the study of flows oer airfoils, releant to the design of wings, compressor and turbine blades, etc. In the two-dimensional flow field that appears, the circulation around a closed cure circling the airfoil - which is non-reducible to a point - is non-zero, and turns out to be linearly proportional to the lift force proided by the airfoil. Conectie flux Mass, momentum and energy are transported by the moing fluid. To quantify in general this transport rate, we introduce φ( x, t) to represent a gien fluid property expressed per unit olume, with particular cases of interest being the mass, momentum and energy per unit olume (φ = ρ, φ = ρ and φ = ρ(e + /)). With this definition, the total amount of a gien quantity existing within a gien olume V is gien by V φdv (e.g., the total amount of mass contained in a olume is V ρdv ). Let us now consider the fixed surface Σ o shown in the figure. The fluid will cross the surface in its motion, transporting across mass, momentum and energy. The olume of fluid crossing in a n dt dσ Σ o Figure.4: Conectie flux. time dt through the differential surface element dσ oriented perpendicular to n is that contained in the parallelepiped of base dσ and side dt shown in the figure, which can be computed as ndσdt. The amount of mass, momentum and energy crossing with the fluid is therefore gien by φ ndσdt, and the total conectie flux across the surface (mass, momentum or energy that crosses the surface per unit time) is obtained by adding the contribution of the different surface

10 8 elements and diiding the result by the unit time dt to gie Σ o φ ndσ. (.35) Note that, with φ =, the aboe integral seres to compute the olume of fluid that crosses Σ o per unit time, the olume flux Q = ndσ. (.36) Σ o When φ =, we find inside the integral (.35) the so-called momentum flux tensor ρ. If Σ o is a closed surface (and φ is a continuous function), Gauss formula (.9) enables the conectie flux to be written in the form φ ndσ = (φ )dv, (.37) Σ o V o where V o is the olume bounded by Σ o. If we now consider an infinitesimally small olume, it then follows that (φ ) is the rate at which mass, momentum or energy abandons the unit olume due to the fluid outflow. Similarly, is the amount of olume that abandons the unit olume per unit time (the expansion rate). Note that, for a perfect liquid, whose density is strictly constant, the condition that the specific olume cannot change requires that the condition = 0 (.38) be satisfied at each point. The concept of conectie flux can be extended to a moing surface Σ c (t) whose points are moing with elocity c ( x c,t) by replacing the fluid elocity with the relatie elocity c when computing the flux to gie φ( c ) ndσ. (.39) The stream function Σ c(t) The description of flows that are either planar or axisymmetric can be simplified when the elocity field is solenoidal (i.e., satisfies = 0), as occurs in constant-density fluids (see the discussion leading to (.63)). To present the deelopment, we restrict our attention to the case of a planar flow described in a cartesian x y coordinate system. Extensions to steady gas flow, for which the continuity equation becomes (ρ ) = 0, and also to polar coordinates and axisymmetric flows can be performed, but will not be considered further in this introductory section. For a planar flow = 0 reduces to x x + y y = 0. (.40) To sole the problem, it is conenient to introduce a scalar function ψ, called stream function defined from the two equations x = ψ y and y = ψ x, (.4)

11 9 which determine ψ upon integration, except for an arbitrary additie constant. Straightforward substitution of (.4) into (.40) reeals that the elocity field deriing from a stream function satisfies automatically the condition of zero expansion rate. An important property of the stream function is that the isolines ψ = constant are streamlines, as can be seen by noticing that along an isoline dψ = ψ ψ dx+ x y dy = ydx+ x dy = 0. (.4) The last of these equations can be written as dx x = dy y, (.43) corresponding to the planar counterpart of (.), thereby demonstrating that isolines and streamlines coincide. Another useful property of ψ is that the difference ψ ψ between the alue of the stream function corresponding to two different streamlines equals the olumetric flux (.36) for the planar stream tube defined by the two streamlines. Gien a cure with end points on each one ψ n ψ n Figure.5: Volumetric flux between two streamlines. of the streamlines, as shown in the figure, the alue of ψ ψ is gien by the integral ψ ψ = dψ = ( y dx+ x dy), (.44) as can be seen from (.4). Bearing in mind that (dy, dx) = n, where n is the unit ector normal to the integration cure, the preious equation can finally be rewritten in the form ψ ψ = n, (.45) where the integral on the right-hand side is the olumetric flux between the two stream surfaces considered (per unit length perpendicular to the plane) Relatie motion near a point As preiously mentioned, the force exerted by a portion of fluid on an adjacent portion of fluid is proportional to the rate at which the fluid is being deformed. To determine that rate, let us

12 0 consider the motion of a differential element of fluid line d x, whose ends are initially located at x and x+d x. The elocities of these two ends differ by a small amount d, which can be expressed in the first approximation as d = d x (.46) in terms of the gradient of elocity. After an infinitesimally small time dt, the two ends of the fluid line element hae moed to occupy new positions gien by x+ dt and x+d x+( +d )dt, respectiely. Clearly, besides a translation dt, the fluid element has undergone a distortion, with d x becoming d x+d dt, as indicated in Fig..6. d t + d d t d d t d d d t d d t r d x d t Figure.6: Motion of a differential element of fluid line. The geometrical character of d dt = d x dt is best inestigated by decomposing the elocity gradient into parts that are symmetric and anti-symmetric, according to = ( + T )+ ( T ) = Td + Tr. (.47) where the symmetric tensor Td is called the rate-of-strain tensor and the anti-symmetric tensor T r is called the rotation tensor. In cartesian coordinates, the two tensors Td and Tr can be computed according to T d = ( x ) x + x ( 3 x + ) ( x + 3 x x 3 x + ) x + 3 x + 3 (.48) and T r = 0 x x x x 3 x 0 3 x ( 3 ) x 3 x 0 (.49)

13 This last expression can be rewritten in the form T r = 0 ω 3 ω ω 3 0 ω ω ω 0, (.50) in terms of the three components ω, ω and ω 3 of the orticity ector ω =. Introducing (.47) into (.46) yields d = d x Td +d x Tr = d d +d r. (.5) The contribution of the anti-symmetric tensor Tr corresponds to a rotation of the fluid element d x with angular elocity ( )/, as can be seen by writing d r in the form d r = d x Tr = ( ) d x = ω d x. (.5) Thus, if Td were identically zero, the motion of the fluid would be exactly that of a rigid body, that is, exclusiely translation and rotation. On the other hand, the symmetry of the rate-of-strain tensor Td enables d d = d x Td to be written in the form d d = Td nds, (.53) where the differential fluid element has been expressed as d x = nds in terms of its differential length ds and the unit ector n. In general, the strain rate is not aligned with n, implying that the element d x undergoes both extension and shear strain. The magnitude of the extension rate is gien by the scalar product n Td nds, so that n Td n represents the extension rate per unit length along the direction defined by n. On the other hand, the contribution of d d to the shear deformation is obtained by appropriately subtracting from the total strain rate the extension rate according to [ Td n ( n Td n) n]ds. There are three priileged directions of space, called principal directions of strain, along which the strain reduces to an extension, without any shear strain (the resulting ector d d is parallel to n). These principal directions n, n and n 3, and their corresponding strain rates, λ, λ and λ 3, are computed by soling the eigenalue problem T d n = λ n. (.54) In particular, the characteristic equation that determines λ i, Td λī = 0 (.55) follows from imposing the existence of nontriial solutions. The symmetry of the tensor Td guarantees the existence of three real roots for this equation, associated with three mutually perpendicular principal directions. Obiously, in the local reference frame defined by the three directions n i the associated rate-of-strain tensor becomes diagonal, with diagonal components λ, λ and λ 3.

14 x dt ( x x ) dt x ( x dt + x ) dt x dt x dt Figure.7: Deformation and rotation of an infinitesimally small fluid element of square shape. Deformation of a square fluid element Let us now consider the eolution of a square fluid element. The two sides, of initial length, eole after a time dt from d l to d l+d l dt, so that and d l = (,0) d l = (0,) T d = x x x x x x x x = (, ) x x = (, ) x x (.56) (.57) The obseration of Fig..7 helps to reeal the physical meaning of each of the terms appearing in the tensors x x + x x + x x (.58) and T r = 0 x x x x 0 (.59) Recalling that the decomposition d l dt = d d dt + d r dt applies, with d d = d l Td and

15 3 d r = d l Tr, one obtains for the horizontal side (,0) [ d d =, ( + )] [ and d r = 0, ( )], (.60) x x x x x whereas for the ertical side [ d d = ( x + x ), ] x [ and d r = ( ) ],0. (.6) x x According to the figure, the elements along the diagonal of the rate-of-strain tensor represent the rate of extension per unit length in the directions of the three axes. Besides, the sides hae undergone rotations of angles dt( / x ) and dt( / x ), so that the aerage angular elocity for the square element in its plane is gien by ( / x / x ). Therefore, the off-diagonal elements of the anti-symmetric tensor Tr represent the three components of the angular elocity of the fluid element. On the other hand, the angle formed by the fluid-element sides, initially perpendicular, hae decreased an amount ( / x + / x )dt, indicating that ( / x + / x ) is the shear strain rate (the rate at which the angle formed by the directions and is decreasing). In other words, the off-diagonal components in Td correspond to half of the shear strain rate perpendicular to the three axis of the reference frame. Deformation of a cubic fluid element d t 3 d t 3 d t 3 3 d t d t d t 3 d t d t d t 3 Figure.8: Deformation and rotation of a cubic fluid element of infinitesimally small size. The aboe figure considers the eolution of an infinitesimally small fluid element of cubic shape, with the simplified notation i j = j / x i adopted for conenience. The discussion gien aboe for the square shape also applies here. In this case, the distortion of the fluid element associated with the straining motion produces a ariation of olume from its initial alue 3. After a differential time dt, the new olume is gien by 3 + dt dt 3 dt dt + dt 3 dt 3 dt 3 dt dt dt( ), (.6)

16 4 after small terms of order dt and dt 3 are neglected. The preious equation reeals that the elocity diergence = represents physically the rate at which the olume of the fluid element is changing per unit olume. This quantity, equal to the trace of the tensors and Td, is called rate of expansion or dilatation rate. For a perfect liquid, for which the olume of the fluid element does not change since the density is constant, the elocity field must satisfy = 0, (.63) a result that was anticipated earlier. Clearly, for the dilatation rate to be zero, the linear extension rates, and 3 3 cannot all three hae the same sign. In other words, for the olume of an incompressible fluid particle to remain constant, a positie extension rate appears in one or two directions, with a compensating compression rate appearing in the other one (or two) directions to gie a zero net expansion rate.