CONTINUUM FLUID MECHANICS Motion A body is a collection of material particles. The point X is a material point and it is the position of the material particles at time zero. x3 Definition: A one-to-one one-parameter mapping X3 x(X, t ) x = x(X, t) X x X 2 x 2 is called motion. The inverse t =0 x1 X1 X = X(x, t) Figure 1. Schematic of motion. is the inverse motion. X k are referred to as the material coordinates of particle x , and x is the spatial point occupied by X at time t. Theorem: The inverse motion exists if the Jacobian, J of the transformation is nonzero. That is ∂x ∂x J = det = det k ≠ 0 . ∂X ∂X K This is the statement of the fundamental theorem of calculus. Definition: Streamlines are the family of curves tangent to the velocity vector field at time t . Given a velocity vector field v, streamlines are governed by the following equations: dx1 dx 2 dx 3 = = for a given t. v1 v 2 v3 Definition: The streak line of point x 0 at time t is a line, which is made up of material points, which have passed through point x 0 at different times τ ≤ t . Given a motion x i = x i (X, t) and its inverse X K = X K (x, t), it follows that the 0 0 material particle X k will pass through the spatial point x at time τ . i.e., ME639 1 G. Ahmadi 0 0 0 X k = X k (x ,τ). Thus, the equation for the streak line of x 0 at time t is given by 0 0 x i = x i (X (x ,τ), t) for fixed t . Deformation Rate Tensors Suppose a motion x = x(X, t) is given. Definition: Deformation Gradient ∂x k dx k = dx K = x k,K dX K . (1) ∂x K ∂x k x k,K = is referred to as the deformation gradient tensor. ∂X K ∂X K X K,k = is the (inverse) deformation gradient tensor. ∂x k Definition: Deformation Tensors An element of arc in the deformed body is given as ds 2 = dx dx δ . (2) k l kl The distance from the origin in the un-deformed body is given by 2 dS = dX K X Lδ KL (3) Using the deformation gradient, (1) may be restated as ds 2 = x dX x dX δ = C dX X , (4) k,K K l,L L kl KL K L where C = δ x x is the Green deformation tensor. KL kl k,K l,L Similarly, (3) may be rewritten as dS2 = X X δ dx dx = c dx dx , (5) K,k L,l KL k l kl k l where c = X X δ is the Cauchy deformation tensor. kl K,k L,l KL ME639 2 G. Ahmadi Definition: Strain Tensors The change in the square of arc length during the deformation is given by 2 2 ds − dS = (C KL − δ KL )dX K dX L = 2E KLdX K dX L , (6) Here, we introduced the Lagrangian strain tensor 2E KL = CKL − δKL . (7) Similarly, (6) may be restated as ds 2 − dS2 = (δ − c )dx dx = 2e dx dx , (8) kl kl k l kl k l where the Eulerian strain tensor is defined as 2e = δ − c . (9) kl kl kl Partial and Total Time Derivatives Let A be any scalar or tensor quantity. The partial time derivative is defined as ∂A ∂A = . (10) ∂t ∂t x The material derivative (total time derivative) is defined as dA ∂A ∂A ∂A ∂x = = + i . (11) dt ∂t X ∂t ∂x i ∂t x Definition: Velocity ∂x i dx i vi = = = x& i . (12) ∂t x dt Definition: Acceleration dvi ∂vi ∂vi a i = = + v j . (13) dt ∂t ∂x j ME639 3 G. Ahmadi Definition: Path lines The curve in space along which the material point x travels is referred to as the path line of the material particle X . The equation for the path line of x is x = x(X, t) for fixed X . (14) If the velocity field is known, then equations dx i = v (x, t) for i = 1,2,3 (15) dt i must be solved for evaluating the path lines. Deformation Rate Tensor Material derivatives of dx k and deformation gradients are given as d d (dx ) = (x dX ) = v dX = v dx (16) dt k dt k,K K k,K K k,l l d ∂ dx k (x ) = = v = v x (17) k,K k,K k,l l,K dt ∂X K dt Theorem: The material derivative of the square of the arc length is given by d (ds 2 ) = 2d dx dx (18) dt kl k l 1 where d = (v + v ) is the Eulerian deformation rate tensor. kl 2 k,l l,k Proof: d d dx dx (ds 2 ) = (δ dx dx ) = δ ( k dx + dx l ) dt dt kl k l kl dt l k dt Now using (16), we find ME639 4 G. Ahmadi d (ds 2 ) = δ ()v dx dx + v dx dx = v dx dx + v dx dx dt kl k,m m l l,m m k k,m m k l,m m l . = (v + v )dx dx = 2d dx dx k,l l,k k l kl k l Relationships between Deformation Rate Tensors and Deformation Strain Tensors 2 From (4), recall that ds = C KL dX K dX L . Taking the material derivative, we find d (ds 2 ) = C& dX dX = C& X X dx dx (19) dt KL K L KL K,k L,l k l Equation (18) implies that d (ds 2 ) = 2d dx dx = 2d dx x dX dX (20) dt kl k l kl k,K l,L K L From (7), it follows that C& KL = 2E& KL . (21) From (19)-(21), we find C& = 2E& = 2d x x (22) KL KL kl k,K l,L and 2d = C& X X = 2E& X X (23) kl KL K,k L,l KL K,k L,l Equations (22) and (23) show the relationship between the deformation rate tensor d kl and the material derivative of Green's deformation tensor and Lagrangian strain tensor. Rivlin-Ericksen Tensors Definition: The Rivlin-Ericksen tensor of order n is defined as d n (ds 2 ) = A (n)dx dx . (24) dt n kl k l Clearly, A (1) = 2d . (25) kl kl The Rivlin-Ericksen tensor satisfies the following recurrence relationship: ME639 5 G. Ahmadi d A (n+1) = A (n) + A (n) v + A (n) v (26) kl dt kl km m,l lm m,k For example, A (2) = 2d& + 2d v + 2d v (27) kl kl km m,l lm m,k The Rivlin-Ericksen tensors are important tensors for certain viscoelastic materials. Lemma: The material derivative of the Jacobian is given by d J = Jv . (28) dt k,k Time Rate of Change of Volume Element Noting that dv = JdV , (29) It follows that d dJ dv = dV = v JdV , dt dt k,k or d dv = v dv . (30) dt k,k Reynolds Transport Theorem The material derivative of an integral taken over a material volume is given as d ∂f ∫∫∫fdv = ∫∫∫ dv + ∫∫fv ⋅ds . (31) dt v v ∂t s Proof: d d df dJ ∫∫∫fdv = ∫∫∫f JdV = ∫∫∫( J + f )dV dt v dt V V dt dt ME639 6 G. Ahmadi Using (30), we find d df fdv = ( + v f )JdV = (f& + v f )dv ∫∫∫ ∫∫∫ k,k ∫∫∫ k,k dt v V dt v ∂f ∂f Noting that f& = + v k , we find ∂t ∂x k d ⎛ ∂f ∂ ⎞ fdv = ⎜ + (v f )⎟dv ∫∫∫ ∫∫∫⎜ k ⎟ dt v v ⎝ ∂t ∂x k ⎠ Using the divergence theorem (25) follows. Spin and Vorticity Spin tensor is defined as: 1 ω = (v − v ) (32) kl 2 k,l l,k Vorticity vector is defined as: ζ i = εijk ωkj = εijk v k, j (33) Angular velocity vector is defined as 1 ω = ζ . (34) i 2 i In vector notation, we have 1 ζ = ∇ × v , ω = ∇ × v (35) 2 Note that T (,∇v) = d + ω (v i, j = d ij + ωij ) (36) ME639 7 G. Ahmadi .