Chapte 3 Stess, Cauchy s equation and the Navie-Stokes equations 3. The concept of taction/stess Conside the volume of fluid shown in the left half of Fig. 3.. The volume of fluid is subjected to distibuted etenal foces (e.g. shea stesses, pessues etc.. Let F be the esultant foce acting on a small suface element S with oute unit nomal n, then the taction vecto t is defined as: t = F lim S 0 S (3. F n S S F n Figue 3.: Sketch illustating taction and stess. The ight half of Fig. 3. illustates the concept of an (intenal stess t which epesents the taction eeted by one half of the fluid volume onto the othe half acoss a ficticious cut (along a plane with oute unit nomal n though the volume. 3.2 The stess tenso The stess vecto t depends on the spatial position in the body and on the oientation of the plane (chaacteised by its oute unit nomal n along which the volume of fluid is cut: whee τ ij = τ ji is the symmetic stess tenso. t i = τ ij n j, (3.2 On an infinitesimal block of fluid whose faces ae paallel to the aes, the component τ ij of the stess tenso epesents the taction component in the positive i-diection on the face j = const. whose oute nomal points in the positive j-diection (see Fig. 3.2. 6
MATH3500 Viscous Fluid Flow: Stess, Cauchy s equation and the Navie-Stokes equations 7 3 3 τ33 τ3 τ23 τ 3 τ 32 τ τ2 τ 2 τ22 2 τ 22 τ τ 2 τ 2 τ 32 τ 3 τ 23 τ 3 τ 33 2 Figue 3.2: Sketch illustating the components of the stess tenso. 3.3 Eamples fo simple stess states Hydostatic pessue: τ ij = δ ij ; note that t i = τ ij n j = δ ij n j = n i, i.e. the stess on any suface is nomal to the suface and pesses against it (i.e. acts in the diection opposite to the oute nomal vecto which is pecisely what we epect a pue pessue to do; see left half of Fig. 3.3 Pue shea stess: E.g. τ 2 = τ 2 = T 0, τ ij = 0 othewise; see ight half of Fig. 3.3. This sketch also illustates that the symmety of the stess tenso is elated to the balance of moments: If τ 2 wee not equal to τ 2 (i.e. if the tangential stess acting on the vetical faces was not equal to the tangential stess acting on the hoizontal ones then the block would otate about the 3 ais. T 0 2 T 0 T 0 T 0 Figue 3.3: Simple stess states: Hydostatic pessue (left and pue shea stess (ight. 3.4 Cauchy s equation Cauchy s equation is obtained by consideing the equation of motion ( sum of all foces = mass times acceleation of an infinitesimal volume of fluid. Fo a fluid which is subject to a body foce (a foce pe unit mass F i, Cauchy s equation is given by ρa i = ρf i + τ ij j, (3.3 whee ρ is the density of the fluid. a i is the acceleation of the fluid, given by (2.5, theefoe Cauchy s equation can also be witten as ρ Du i Dt = ρf i + τ ij j (3.4
MATH3500 Viscous Fluid Flow: Stess, Cauchy s equation and the Navie-Stokes equations 8 o ( i ρ t + u i k k = ρf i + τ ij j. (3.5 Note that Cauchy s equation is valid fo any continuum (not just fluids! povided its defomation is descibed by an Euleian appoach. 3.5 The constitutive equations fo a Newtonian incompessible fluid In chapte 2 we deived a quantity (the ate of stain tenso ɛ ij which povides a mathematical desciption of the ate of defomation of the fluid. Cauchy s equation povides the equations of motion fo the fluid, povided we know what state of stess (chaacteised by the stess tenso τ ij the fluid is in. The constitutive equations povide the missing link between the ate of defomation and the esulting stesses in the fluid. A lage numbe of pactically impotant fluids (e.g. wate and oil ae incompessible and ehibit a linea elation between the shea ate of stain and the shea stesses. These fluids ae known as Newtonian Fluids and thei constitutive equation is given by o, using the definition of the ate of stain tenso, τ ij = pδ ij + 2µɛ ij, (3.6 τ ij = pδ ij + µ ( i j + j i, (3.7 whee p is the pessue in the fluid and µ is the dynamic viscosity, a quantity that has to be detemined epeimentally. Note that thee ae also many fluids which do not behave as Newtonian fluids and have diffeent constitutive equations (e.g. toothpaste, mayonaise. Not vey imaginatively, these ae often called Non-Newtonian Fluids the behaviou of these fluids is coveed in a diffeent lectue. 3.6 The Navie-Stokes equations fo incompessible Newtonian fluids We inset the constitutive equations fo an incompessible Newtonian fluid into Cauchy s equations and obtain the famous Navie-Stokes equations ( i ρ t + u i k = ρf i p + µ 2 u i, (3.8 k i o symbolically ( ρ + (u u = ρf p + µ 2 u. (3.9 t Dividing the momentum equations by ρ povides an altenative fom whee ν = µ/ρ is the kinematic viscosity. i t + u i k = F i p + ν 2 u i, (3.0 k ρ i 2 j 2 j
MATH3500 Viscous Fluid Flow: Stess, Cauchy s equation and the Navie-Stokes equations 9 In combination with the equation of continuity o symbolically i i = 0 (3. u = 0, (3.2 the thee momentum equations fom a system of fou coupled nonlinea, patial diffeential equations of paabolic type (second ode in space and fist ode in time fo the thee velocity components u i and the pessue p.
The govening equations in selected coodinate systems Rectangula catesian coodinates The ate of stain tenso whee ɛ ij = ɛ = ɛ zz = w z ɛ yz = w 2 y + v z ɛ ɛ y ɛ z ɛ y ɛ yy ɛ yz ɛ z ɛ zy ɛ zz ɛ yy = v y ɛ y = v 2 + y ɛ z = 2 z + w The voticity ω = cul u = ( w y v z, z w, v. y The Navie Stokes equations The Laplace opeato t + u + v y + w z = P ρ + ν 2 u, v t + u v + v v y + w v z = P ρ y + ν 2 v, w t + uw + v w y + w w z = P ρ z + ν 2 w, div u = + v y + w z = 0. 2 2 2 + 2 y 2 + 2 z 2. 0
MATH3500 Viscous Fluid Flow: The govening equations in selected coodinate systems Cylindical Pola Coodinates Relation to Catesian coodinates: = cos ϕ, y = sin ϕ, z = z Velocity components: The ate of stain tenso u = u, v = u ϕ, w = u z ɛ ij = ɛ ɛ ϕ ɛ z ɛ ϕ ɛ ϕϕ ɛ ϕz ɛ z ɛ zϕ ɛ zz whee ɛ = ɛ zz = w ɛ ϕ = z 2 ɛ ϕz = w 2 + v z ɛ ϕϕ = v + u ɛ z = 2 ( v + z + w The voticity ω = cul u = The Navie Stokes equations ( w v z, z w, (v. t + u + v + w z v2 v t + uv + v v + w v z + uv w t + uw + v w + w w z = P ρ + ν = P ρ + ν = ρ P z + ν 2 w, 2 u u 2 2 v 2, 2 v v 2 + 2 2, div u = (u + v + w z = 0. The Laplace opeato 2 ( + 2 2 2 + 2 z 2.
MATH3500 Viscous Fluid Flow: The govening equations in selected coodinate systems2 Spheical Pola Coodinates Relation to Catesian coodinates: = cos θ, y = sin θ cos ϕ, z = sin θ sin ϕ Velocity components: The ate of stain tenso u = u, v = u θ, w = u ϕ ɛ ij = ɛ ɛ θ ɛ ϕ ɛ θ ɛ θθ ɛ θϕ ɛ ϕ ɛ ϕθ ɛ ϕϕ whee ɛ θϕ = 2 ɛ ϕϕ = sin θ sin θ θ ɛ = w + u + v cot θ ( w sin θ + sin θ v ɛ θθ = v θ + u ɛ θ = 2 ɛ ϕ = 2 ( v + sin θ θ + ( w The voticity ω = cul u = ( v (w sin θ, sin θ θ sin θ (w, (v. θ The Navie Stokes equations t + u + v θ + w sin θ v2 + w 2 v t + uv + v v θ + = P ρ + ν 2 u 2u w v sin θ + uv w2 cot θ = P ρ θ + ν w t + uw + v w θ + w w sin θ + uw The Laplace opeato div u = 2 2 2 ρ sin θ P + ν vw cot θ = 2 w 2 2 v 2 θ 2 v + 2 2 θ 2v cot θ 2 2 2 sin θ v 2 sin 2 θ 2 cos θ 2 sin 2 θ w 2 sin 2 θ + 2 2 sin θ + 2 cos θ 2 sin 2 θ (2 u + w (v sin θ + sin θ θ sin θ = 0. ( 2 + 2 sin θ ( sin θ + θ θ 2 sin 2 θ 2 2. v, w, w,