Mathematical Theory of Non-Newtonian Fluid 1. Derivation of the Incompressible Fluid Dynamics 2. Existence of Non-Newtonian Flow and its Dynamics 3. Existence in the Domain with Boundary Hyeong Ohk Bae Ajou Unversity,
Drivation of the Incompressible Fluid Equations 1. Introduction 2. Fluids 3. Navier Stokes Equations 4. History for Navier Stokes Equations 5. Non Newtonian Fluids
2 Introduction Fluids: liquids and gases A solid can resist an applied shear force and remain at rest, while a fluid cannot. Newtonian Fluid : water, air Non Newtonian Fluid: grease, polymer, biological fluid Moving Plate u 0 y t Fluid x Fixed Plate Velocity distribution between two parallel plate
3 Properties of Fluids: 1. Kinematic property linear velocity, angular velocity, vorticity, accerlation, and strain rate 2. Transport property viscosity, termal conductivity, mass diffusivity 3. Thermodynamic property pressure, thermal conductivity, temperature, enthalpy, entropy, specific heat 4. Other miscellaneous property surface tension, vapor pressure
4 Kinematic Property: Lagrangean description : the scheme of following the trajectories of indivisual particles Eulerian description : the scheme describing the flow at fixed point as a function of time Eulerian velocity vector fields u(x, t) = u(x 1, x 2, x 3, t) = (u 1 (x 1, x 2, x 3, t), u 2 (x 1, x 2, x 3, t), u 3 (x 1, x 2, x 3, t)) Three fundamental law of mechanics : Conservation of mass, momentum, energy in Lagrangean sense
[Relationship of Lagrangean and Eulerian Coordinates] Let Q be any property Since dq = Q x 1 dx 1 + Q x 2 dx 2 + Q x 3 dx 3 + Q t dt. we have dx 1 = u 1 dt, dx 2 = u 2 dt, dx 3 = u 3 dt, dq dt = Q t + u Q Q Q 1 + u 2 + u 3 x 1 x 2 x 3 = Q + (u )Q t DQ Dt = dq dt material derivative (u )Q convective derivative Q t local derivative 5
6 [Reynolds Transport Theorem] For a control volume, D Dt Q(t)dV V (t) { [ 1 = lim δt 0 δt V Q(t + δt)dv Q(t)dV (t+δt) V (t) = lim δt 0 = lim = = δt 0 { [ 1 Q(t + δt)dv δt V (t+δt) V (t) V Q(t + δt)u nds + (t) V (t) Q(t)u nds + V (t) [ (Qu) + Q t V (t) ] dv. Q t dv V (t) } + Q t dv V (t) } Q(t) dv t
7 Motion of Fluid particle: translation u(x, t) rotation 2( 1 ui x u ) j j x i deformation 1. dilation : expansion or reduction of volume u x, v y, 2. shear strain : distortion the average decrease of the angle between two lines which are initially perpendicular in the unstrained state ( 1 ui + u ) j 2 x j x i w z
8 Notice that u i,j = 1 2 (u i,j + u j,i ) + 1 2 (u i,j u j,i ). Means : each velocity derivative can be resolved into a strain rate plus an angular velocity Rotation :
9 Distortion of a moving fluid element: y dy+ v y dydt udt u y dydt v x dxdt Time t + dt dx+ u x dxdt dy vdt dx Time t x
10 Transport property : Viscosity : the property of a fluid which relates applied stress to the resulting straing rate Moving Plate u 0 y t Fluid x Fixed Plate By the viscosity property of a fluid, if the upper plate move with speed V, then the fluid on the plate move with the same speed V. Since there is no movement on fixed plate, the fluid on the ground doee not move.
11 [Shear strain = ɛ ij ] = 1 ( u 2 y + v ) = 1 du x 2 dy = τ xy In general, shear stress τ ij is a function of the strain rate ɛ ij : τ ij = f(ɛ ij ) If the relation is linear, it is Newtonian; τ ij = µ u y. The constant µ is the viscosity. If not, it is non-newtonian.
12 τ xy = Kɛ r xy τ xy : shear stress, u y : rate of strain µ : viscosity
τ plastic non Newtonian (shear thinning) Newtonian non Newtonian (shear thickning) ɛ
13 Navier-Stokes equation 1. Conservation of Mass : Equation of Continuity 0 = D Dt ρdv V [ ρ = V t + ] (ρu k ) dv x k Reynold s transport theorem ρ: density Therefore, 0 = ρ t + div(ρu) = Dρ Dt + ρdivu. If div u = 0, then the fluid is called incompressible.
14 2. Conservation of Momentum (F=ma): Du Dt = u t + (u )u = D ρudv = [total force] Dt V = [body force] + [surface (exterior) force] = = ρf idv + τ ijn j ds V S ( ρf i + τ ) ij dv = [divergence theorem] V x j ρ Du i Dt = ρf i + τ ij x j
15 Body forace F i = g = external fields such as gravity or electromagnetic potential Surface force divτ = external stress on the sides of the particle
16 Stresses y τ 22 τ 23 τ 21 z τ 33 τ 32 τ 31 τ 13 τ 12 τ 11 x τ ij : stress in the j direction on a face normal to the i axis
17 [Relation of force and deformation] fluid at rest : τ ij = ρδ ij p = τ xx = τ yy = τ zz : normal stress = pressure fluid in motion: Newtonian fluid: linear relation of τ ij and deformation rate ( ui τ ij = pδ ij + µ + u ) j x j x i + δ ij λdivu Newtonian fluid: non-linear relation of τ ij and deformation rate
18 Navier-Stokes Equations: Incompressible and Newtonian Fluid = ρ Du Dt = ρg p + µ 2 u du i dt ν u i + (u )u i + i p = f i, u = 0 Leray, 1934 Hopf, 1951 Existence of Weak Solutions 2D N-S: Uniquness and Regularity u L (0, ; H 1 ) L 2 (0, ; H 2 )
3D N-S: Short Time Regularity u L (0, T ; H 1 ) L 2 (0, T ; H 2 ) 3D N-S: Small Data, or Large Viscosity = Global Regularity 3D N-S: Regularity and Uniqueness : qquad Open Problem
19 Regularity Serrin, 1962 L r,r, 3 r + 2 r < 1. Caffarelli,Kohn & Nirenberg, 1982 Constantin & Fefferman Choe, 1996 : Boundary L Regularity
20 Regularity Theorem for Navier-Stokes Equations[Bae and Choe, 1996, 2007] Suppose that (u, p) is a weak solution. Let u = (u 1, u 2, u 3 ). If u 1, u 2 L, loc (Q), then Therefore, u is smooth. u 3 L, loc (Q).
21 Our Plan Use u 3 dx dt C Show sup u 3 dx C Show loc u u 2 dx dt C Show u L 5,5 Use Struwe s Regularity: If u L 5,5 Then u regular
22 Existence and Regularity Theorem for Non-Newtonian Fluids[Bae and Choe, 1997] du i dt ν 1 u i ν 2 e ij r ɛ ij x j + (u )u i + i p = f i, u = 0 ɛ ij = 1 2( ui j + u ) j x i Then thereis a weak solution for r > 1. For r > 2/5 the solution is regular.