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Basic Field Equations APPEND IX Basic Field Equations A.I Introduction In this appendix we list the basic equations of mechani cs. We present these equations in order to exhibit the mathematical expressions of the basic laws of physics (conservation of mass, momentum, and energy) and to show the notations used in this book. How easily the material presented here will be assimilated depends on the background of the reader. For some, a quick glance will be sufficient. For others, some hard study and further reading will be necessary. The author' s book, A First Course in Continuum M echanics (Fung, 1977), is one of the shortest and most con venient references for this purpose. Other larger treati ses are listed at the end of this appendix. The mechanical properties of materials are expressed in constitutive equations. The constitutive equations of biological materials are discussed in the author's book, Biomechanics: Mechanical Properties ofLiving Tissues (Fung, 1981). They are often quoted below without discussion. A.2 Conservation of Mass and Momentum Let us consider the motion of a solid or a fluid. Let XI> X2 'X 3 or X, y, z be rectangular Cartesian coordinates . Let the velocity components along the X-, y-, z-axis directions be denoted by VI'V2' V3 or u, v, IV, respectively. Let p denote pressure, and uij or (ux x ' uYY' Uzz ' Ux }.' uyx ' u zz) be the stress components. Throughout this book the index notation will be used. Unless stated otherwise, all indices will range over 1,2,3. The summation convention will be used : repetition of an index means summation over I, 2, 3. The basic equations of mechanics consist of the equation of continuity 370 A.2 Con servation of Ma ss and Momentum 371 which expresses the law of conservation of mass, the equation of motion which expresses the law of conservation of momentum, the equation of balance of energy, and the constitutive equation and boundary conditions. A general derivation of the conservation equations is given in the author's First Course, (Fung, 1977, Chap. 10). This derivation makes use of the Gauss theorem and the concept of "material" derivatives. The Gauss theorem states that (i = 1,2,3), (A.2 : I) where A is a quantity (a scalar, a vector, or a tensor) continuously differen­ tiable in a convex region V which is bounded by a surface S that consists of a finite number of parts whose outer normals Vi form a continuous vector field. Vi is unit normal vector of S. To discuss the concept of "material" derivative, consider I(t) = Iv A(x, t)dV, (A.2:2) where A (x, t) is a continuously differentiable function ofspatial coordinates x(x t , X 2' x 3) and time t, V is a regular region as defined above, occupied by a given set of material particles. The integration is extended over a given set of particles, so that when time changes and the particles change their position, the spatial region V changes with them . The rate at which I(t) changes with respect to t is defined as the material derivative of I and is denoted by DI/Dt : ~~ = J!~ ~t [Iv,A (x, t + dt)dV - Iv A (x, t)dVJ , (A.2 : 3) where V' is the volume occupied by the same set of particles at time t + dt. It is shown in First Course (Fung, 1977, p. 249) that ~t ~~ Iv AdV = Iv dV + fAvjvjdS = Iv ( ~~ + vj :~ + A ~~ ) d V (A.2 :4) = f (DA + A OVj ) dV, v Dt oXj where DA ( aA) vA vA vA - = - + V 1-+V2 -+V3 -, (A.2: 5) Dt at x e const. ox1 VX2 ox3 is the mat erial derivative ofA ; i.e., the rate at which the quantity A associated with a particle is seen changing as the particle moves about in a velocity field. Vi is the velocity field, ofcourse. 372 Appendix : Basic Field Equations If A in the equation above is the velocity vector with components u, v, w, then DA IDt is the acceleration vector, and oAlot is the transient acceleration, whereas the last three terms in Eq. (A.2 :5) are the convective acceleration. Thus, the components of transient acceleration are au ov ow . at' at' at ' whereas the three components of the convective acceleration are au au ou OV ov ov Ow ow ow u:;- + v:;- + w:;-, u:;- + v:;- + w ~, u ~+v~+w~ . ox uy oz uX uy oz ox uy oz Now, consider the mass contained in a region V: m= LPdV. (A.2:6) Refer to Eq. (A.2:2). Iedntify A with p and 1 with m. The law ofconservation of mass states that Dm]Dt = O. Applying Eq. (A.2:4) we obtain the following equation ofcontinuity, which expresses the law ofconservation of mass : Op + opvj = O. (A.2: 7) at OXj If the fluid density p is a constant, then the material is said to be incompres­ sible. The equation ofcontinuity ofan incompressible fluid is OV. _J =0 or (A.2:8) oXj Next, consider the momentum of particles in V, &>i = L pv;dV. (A.2 :9) Newton's law of motion states that the material rate of change of momentum is equal to the force acting on the body : (A.2: 10) If the body is subjected to surface tractions i; and body force per unit volume Xi' the resultant force acting on the body is ~dS iff:i = f + f XidV. (A.2: 11) s v v Now T; = (JjiVj' Applying Eqs. (6.2: 1), (6.2:4), and (6.2:9)-(6.2: 11), we obtain the Euler's equation ofmotion t», o(Jij p-'= - + Xi' (A.2: 12) Dt oXj A.3 Na vier- Sto kes Eq ua tions for an Incompressible Newto nian Flu id 373 If there is no heat input into the system and no heat source in the region, the equation for balance of energy is the same as balance of mechanical energy and work: it leads to no new independent equation. When these equations are combined with the constitutive equation of the material, we obtain the basic equations of mechanics . We shall illustrate this by several examples in the following sections. A.3 Navier-Stokes Equations foran Incompressible Newtonian Fluid Consider an incompressible, Newtonian viscous fluid. Let us use the same notations for pressure, velocity, and coordinates for position as in the previ­ ous section. Let J1 be the coefficient of viscosity which is a constant for a Newtonian fluid. Then the stress- strain rate relationship is given by aij = -p[)ij + }Ykk[)ij + 2p.f/ij (A.3: I) where [)ij is the Kronecker delta , ~ i j f'ij = (aav + aav ) (A.3 :2) 2 X j Xi is the strain rate tensor, and ..1. and J1 are two material constants. Since the fluid is assumed to be incompressible, the condition of incompressibility au;= 0 or au+ au+ aw= 0 ax; ax ay az (A.3:3) reduces Eq. (A.3: I) to the form (A.3 :4) These are, of course, au ax x = - P + 2J1 ax' au a y y = - P + 2J1 ay, (A.3 :5) aw az z = - p + 2J1 7};" ' «, = J1 (~; + ;~). «; = J1 (~ ; + ~; ). «; = J1 (~: + ~~} Substituting these into Eq. (A.2: 12), we obtain the Navier-Stokes equations u au au au p aa + u + u + w = X - ~ aa + vV 2 u, (A.3:6) I axa y az p x 374 Appendix : Basic Field Equations ov ov ov OV I op 2 - + u- + V- + w- = Y --- + vV' v, ot OX oy OZ p oy (A.3:6) OW OW OW OW I op 2 - + u- + V- + w-=Z -- -+ vV' w ot OX oy OZ p OZ ' where v = /1/p is the kinematic viscosity of the fluid, and V'2 is the Laplacian operator 2 02 02 02 V' = ox 2 + oy2 + OZ 2' (A.3 :7) Equations (A.3: 3) and (A.3 :6) comprise four equations for the four variables u, v, w, and p occurring in an incompressible viscous flow. To solve these equations, we need to specify appropriate boundary conditions. If the fluid is in contact with a solid, the boundary condition is that there is no relative motion between the solid and the fluid. The fluid adheres to the solid, whether the surface is wettable or not. The justification of this condition is discussed at some length in Fung (I977 , p. 268 et seq.). If the fluid is in contact with another fluid, then at the interface the boundary conditions must be consistent with the interfacial surface tension, surface viscosity, and conditions of cavitation or its absence . See Fung (1977, p. 270). A.4 Navier's Equation for an Isotropic Hookean Elastic Solid If the material is an isotropic elastic body obeying Hooke's law, the stress­ strain relationship can be expressed either as (A.4 : 1) or as (A.4 :2) Here eij (i,j = 1,2,3) or Cx x , cYY' ezz , ex y , eyz ' ezx are strain components. )" G, E, v are elastic constants. .Ie and G are called Lame's constants, G is called the shear modulus or the modulus ofrigidity, E is called the Young's modulus or modulus ofelasticity, and v is called the Poisson '.I' ratio. On substituting Eq. (A.4 : 1) into the equation of motion (A.2 : 12), we obtain Dv , 0 oCij P Dt' = 1'-0 c ~~ + 2G-0 + Xi' (A.4 :3) X i X j To proceed further we need to express DvJDt and cij in terms of the displacements of material particles .
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