R. von Mises - Mathematical theory of compressible fluid flow, страница 6

If viscosity is admitted, however, thestress vector on a surface element d§> is no longer normal to dS. T h e stresscan be resolved into a normal component σ and tangential or shearingcomponents r. Considering surface elements normal to the coordinate axes(see Fig. 6 ) , there are three normal stresses: σ ,σ , and σ ,χstresses, equal in pairs: rxy=r, ryx=yzusual in the theory of elasticity, rυzyand six shearingζ, and τr=ζχ,τχζ12Here, as is, for example, denotes the componentxyin the ^/-direction of the stress acting on a surface element with outwardand Tnormal in the x-direction.

M o r e explicitly σ , r ,χXZxydenote the com­ponents of the stress vector t acting on the element with outward normal inxthe positive x-direction. Thus, a positive value of σ denotes a tensile stress,χwhile a negative value corresponds to compression. If t , ty , and t denotexzthe stresses acting on surface elements with normals in the χ-, y-, and 2-directions, then equilibrium considerations show that the stress vector t actingnon a surface element with outward normal in the direction η must be givenbyt(1)n= t cos (n, x) + ty cos (n, y) + t cos (n, z ) .xzAlso σ, which is the component of t in the direction n, is given bynσ = t .ή = t(2)nwhere tnx,tnynx, and tnzcos (n, x) + tnycos (n, y) + tnzcos (n, z),are the χ-, y-, and ^-components of t and ή denotesnthe unit vector in the direction n, while the resultant shearing stress is(3)τ =V V-σ .2I n the particular case of an inviscid fluid, t is parallel to ή by hypothesis;nthen in ( 2 ) , σ =t , (3) reduces to τ =nthe x-direction is σ cos (η, χ) =σ = σνσχ0, and the component of (1) incos (n, x), so that σ =σ .

Similarly,χ= σ , as mentioned in Sec. 1.2, where the common value of the nor­ζmal stresses was denoted by —p.In general, however, the three normal stresses σ , σ , and σ are not neces­χsarilyequal, thedifferencesbeinggreatertheυζlargertheshearing3.1VISCOUS STRESSES A N D H Y D R A U L I C25PRESSURE0F I G . 6. Stress vectors and their components acting on surface of rectangular cellin viscous fluid.stresses. In the theory of elasticity this lack of equality causes no difficulty,but in fluid mechanics the equation of state requires the existence of ahydraulic pressure p, equal for all directions. T h e question arises: W h a tcan be used in place of the variable ρ in the equation of state (and in otherthermodynamic equations) when dealing with a viscous fluid?A n answer is suggested by this fact (following from Eqs. (2) and (1) bya simple computation which uses the elementary properties of the directioncosines of orthogonal directions): T h e sum of the normal stresses σ , σ ,and σ is invariant under rotation of the coordinate axes, i.e., if x , y', and z'define a triple of orthogonal directions, and if σ > , σ > , and σ > denote thenormal stresses corresponding to these directions, thenχνfζχ(4)σ > + a > + σ.

= σχyζ+ σχυυζ+ σ .ζFor an inviscid fluid, this invariant quantity has the value — 3p. Thus, itis natural to consider one-third of the sum (4) as the mean or average tensilestress, and its negative as the mean pressure at a given point. I t is usual totake(5)ρ =~\(σ+ a +χyσ)ζas the value of ρ in the thermodynamic equations. Whether or not thisassumption is justified is an essentially experimental question, which cannotbe settled definitely on the basis of the experimental evidence now avail­able.

Equation (5) will be used in this book as a working hypothesis.Once ρ has been defined, the state of stress at any point may be thoughtof as being composed of a hydraulic pressure p, uniform in all directions,and a system of viscous stresses σ , σ' , σ' , r , τ , and τ , whereχ(6)σχ=—ρ + σ ,χσυ=υζxy—ρ + σ' ,νand consequently, using Eq.

( 5 ) ,(7)σ' +χCy+σ'ζ=0.υζσζχζ=—ρ + σ' ,ζ26I.INTRODUCTIONI t is to be noted that the viscous stresses are not additional unknowns,over and above the five unknowns q , q , q , p, and ρ considered thus far.In the theory of viscous fluids the stresses σ' and τ are taken to be givenfunctions of p, p, q, and their derivatives, e.g., σ may be given as propor­tional to dq /dx, etc. T h e exact form of this dependence will not be con­sidered here, although a certain necessary restriction, due to the fact thatviscous forces are "friction forces", will be mentioned in Sec. 3.xyzχx2. Newton's equation for a viscous fluid13T h e general form of Newton's equation, holding for any type of con­tinuum, was given in E q .

(1.1). For the "internal force per unit v o l u m e "there is first the contribution of the pressure p, which is —grad p, exactlyas in the case of an inviscid fluid. I n the rectangular cell of Fig. 6, the viscousforces acting in the x-direction on one set of faces, have the magnitudesσ dy dz,rχdx dz,yxτdx dy,ζχand on the three opposite faces[σ'χ +^dx^j dy dz,[r+yxdy^j dx dz,[r+ ^zxdz^j dx dy.Upon subtracting and dividing through b y the volume dx dy dz, we seethat the x-component of the resultant viscous force, per unit volume, isvx=da. drx—+dx, dr~r —.dzyxzx~~zdyThus, the x-component of Newton's equation is(8)ρ^dt=ρί_ ^ + * £ ί + *!»? +dxdxdyχdzSimilarly, or by cyclic substitution on x, y, ζ in E q . ( 8 ) , the other compo­nents of Newton's equation ared<lvVd, ΰτχν , de'v ,(8)dqdtdpdzzdrdxdrdyxzyzdrzy,dadzzT h e equation can also be given in vector form as(la)αΊΡ=p g~g r a dV+V>where ν is the resultant viscous force per unit volume.

If the stress is notsplit up into the uniform pressure ρ and the viscous stresses, the term3.3— (dp/dx)27W O R K D O N E B Y VISCOUS FORCES. D I S S I P A T I O N+in (8) is to be replaced b y da /dx, etc. T h e reader(da /dx)xxfamiliar with the notation of tensor calculus will notice that E q . ( I a ) mayalso be written as(80Λ=άρpgg a d ρ + grad Σ',-rorp | i = pg +(8")gradS,where Σ denotes the stress tensor and Σ' the tensor of the viscous stresses.14Equation ( I a ) [or (8)] takes the place of (1.1). T h e addition of viscosityhas no effect on the form of the equation of continuity (1 . I I ) or ( 1 .

Ι Γ ) ,and some specifying equation of the form ( l . I I I ) must be added t o makethe system of equations complete.3. Work done by viscous forces. DissipationIn A r t . 2 the energy equation for an inviscid fluid was obtained from thevector equation (1.1) by scalar multiplication by q. If we start from ( I a )instead of (1.1), the result of multiplying by q is the same as before, E q .(2.6), except for additional terms on the right resulting from q » v , namely,/θσ'drdr \yx(drzXdc'xydr \yzy,/ drdrxzda \yzzEach term in E q . (9) can be split into t w o terms, e.g.,d ,daxqx/vτ - \qz<r )^— =dxxdx-tdqσχ—dxxdrq,x—-yxd ,=—dy.(q T )x-yxdqrx—-.yxdydyN e x t , let-w'= ^- (q a'dxxx+ qryxy+ qr )zxz+ ~dy(q r+ qax yxyyqr )+z(10)+and, keeping in mind that rxyη(11)' hd(,> dqy,= ryx-r-dz(qxTzx, etc.> dqz , ^ (dqx .

dqA\dy ^dx)+qryzyyz+q a' )zz28I. I N T R O D U C T I O NThen the equation that results in place of (2.6) can be written asT h e definitions given in (10) and (11) have been chosen with a viewto physical interpretation. I t will be shown (a) that w is the work donerper unit of time and volume against the viscous forces, and ( b ) that 0 is anessentially positive quantity having the dimensions of work per unit ofvolume and time, and may be called the dissipation function,or brieflyrepresenting the rate of conversion of mechanical energy intodissipation,heat, per unit of time and volume.(a) W i t h the above definitions of the components of stress, the viscousforce acting on the left-hand face (with outward normal in the negativex-direction) of the rectangular cell of Fig. 6 has components— σ dy dz,—rχanddy dz,xy—rxzdy dzin the χ-, y-, and z-directions, respectively.

Thus, the work done by thisforce per unit time is— qx(r' dy dz — q rxydy dz — q rxy2xzdy dz.A t the opposite face (with outward normal in the x-direction), the work doneis+ ^\^ σχχ(qzVx)d x j dy dz ++^q ryxy^ -+d x j dy dz(q r )yxy^q Txzz+dx~^ dy dz.(q r )^zxzThus, the net work done at this pair of faces by the viscous forces, per unitof time and volume, isdx{qx^'x)+{q T )yxy+dy^dz(q r );zxzanalogous expressions hold for the remaining pairs of faces, so that thetotal work for all faces is exactly the right-hand side of E q . (10), and, there­fore, —w'.

Then w' is the work done per unit time and volume against theviscous forces. Since w is the work done against hydraulic pressure, the termw + w' in (12) includes all work against surface stresses.15(b) T h e velocity terms in (11) can be easily interpreted. For example,dxx= dq /dxxdt the leftqxis the time rate of deformation in the x-direction, since in timefaceoftherectangularcellundergoesthedt, while for the right face the displacement is [ q +xdisplacement(dq /dx)xdx] dt.3.429E N E R G Y E Q U A T I O N F O R A VISCOUS F L U I DAlso, since σ' is a normal stress due to the viscosity of the fluid, it isχnecessary to assume that σ' is positive (tensile stress) when a positiveχexpansion occurs (dq /dxxsion (dq /dxxpositive) and negative in the case of compres­negative); but the first product in (11), σχdq /dx, is nonxnegative, as are the two following terms. I t is known from the elementsof the theory of elasticity (or the kinematics of a continuum) that the termdxy = %(dq /dy + dq /dx) represents the time rate of shear about the 2-axis.xyT h e symmetric tensor D with components dxx, dxy, etc.

is called the rateof deformation tensor or rate of strain tensor. Again, the nature of viscosityrequires that a rate of shear strain be accompanied by a shearing stress ofthe same, not opposite, sign. Thus, the fourth term in Eq. (11), r (dq /dyxydq /dx),yx+is nonnegative, as are the following terms, and consequently thevalue of θ itself is nonnegative.16T h e above assumptions as to the signs ofthe components of viscous stress may be summarized in the statement thatthe viscous stress tensor Σ', has the nature of a resistance force or friction,i.e., of a force that consumes rather than produces energy.17T h e most usual assumption made in the theory of viscous flow is thateach of the six viscous stresses is a linear function of the rates of deforma­tion with coefficients constant or known functions of ρ and p.

So specific ahypothesis is not adopted here, but the assumption concerning signs isessential, since the contrary case would correspond to a ''friction force''between material elements that would tend to increase their relativevelocity. A world in which friction forces accelerated moving bodies wouldcontradict all our experience.4. The energy equation for a viscous fluidT h e energy equation in mechanical terms has already been given inE q . (12). A s in A r t . 2, the heat input Q' may be introduced by using theFirst L a w of Thermodynamics, which gives the relation between Q' andthe mechanical variables.