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:

dx dx dx 1 = 2 = 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.,

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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

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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

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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 (x ) = k = 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

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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:

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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

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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)

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