** DYNAMICS**

**4.2 Equations of Motion: Navier-Stokes Equations**

and thus, with aid of the continuity equation, ρ

∂q_{i}

∂t +qj

∂q_{i}

∂x_{j}

= ∂τ_{ij}

∂x_{j} +FB_{i} (4.19)

4.2.2 Surface Forces for Differential Control Volume

Figure 4.4 Stresses in thex-direction on a differential control volume in Cartesian coordi- nates. (From Fox, McDonald & Pritchard 2004)

• Surface forces acting on a fluid particle:

dF_{S}_{x} =

∂σ_{xx}

∂x + ∂τ_{yx}

∂y + ∂τ_{zx}

∂z

dx dy dz (4.20)
dF_{S}_{y} =

∂τ_{xy}

∂x + ∂σ_{yy}

∂y + ∂τ_{zx}

∂z

dx dy dz (4.21)
dFS_{z} =

∂τ_{xz}

∂x + ∂τ_{yz}

∂y + ∂σ_{zz}

∂z

dx dy dz (4.22)

These components correspond to ∂τ_{ij}

∂x_{j} dx dy dz.

4.2.3 Stress and Strain Rate in a Newtonian Fluid

• A fluid is defined as a material that cannot be in stationary equilibrium with applied shear stress.

A Newtonian fluid has a resistance to shear deformation that is linearly
proportional to the rate of deformation(i.e., proportional to the gradient of
velocity), while an ideal(perfect) fluid has no resistance to shear deforma-
tion. ^{3}

• Four motions of a fluid particle element are possible: (1) translation, (2)
rotation, (3) squeeze motion (angular defomation), and (4) volumetric
change (linear deformation). ^{4}

Among them, (3) and (4) cause stress in a fluid, while (1) and (2) repre-

Figure 4.5 Motion of a fluid particle. (From Fox, McDonald & Pritchard 2004)

sent only rigid body motion for which there will be no stress developed.

3In solid mechanics, this linearity can be applied to elastic solids that follow Hooke’s law.

4 Movie: Motion of fluid particle (Boundary layer)

./mmfm_movies/1_02032.mov ./mmfm_movies/1_02030.mov

Figure 4.6 Deformation of fluid element in 2-D flows. (From Fox, McDonald &

Pritchard 2004)

• Rigid body motion without deformation: Translation + Rotation

Figure 4.7 Rotational motion of a fluid particle. (From MIT website 2004)

q = dx_{0}

dt + Ω×(x−x_{0}) (4.23)

– Fluid Translation: Acceleration of a Fluid Particle in a Velocity Field
a_{p} = Dq

Dt = ∂q

∂t + q · ∇q (4.24)

– Fluid Rotation: Relationship between rotation and vorticity^{5}
Ω = 1

2∇ × q (4.25)

5 Movie: Motion of fluid particle (Rotation)

./mmfm_movies/tire.mov ./mmfm_movies/3_14035.mov

• Squeeze motion (angular deformation):

Figure 4.8 Squeeze motion of a fluid particle. (From MIT website 2004)

Rate of angular deformation in xy plane = lim

4t→0

∂v

∂x 4x

4x4t+ ∂u

∂y 4y 4y4t

4t

=

∂v

∂x + ∂u

∂y

(4.26) Rate of angular deformation inyz plane =

∂w

∂y + ∂v

∂z

(4.27) Rate of angular deformation inzxplane =

∂u

∂z + ∂w

∂x

(4.28)

• Volumetric change (linear deformation)^{6}
Volume dilation rate = ∂u

∂x + ∂v

∂y + ∂w

∂z = ∇ ·q (4.29) – It vanishes for incompressible fluid.

– This velocity vector is called solenoidal.

• If there were no deformation(also including rigid body motion), then only a static pressure acts normal to the surface of the volume of interest. The stress vector is simply τ = −p n. Thus the stress tensor T = −p I =

−p δ_{ij}. The pressure diagram of a fluid is shown in Figure4.9.

Figure 4.9 Pressure diagram of a fluid.

6 Movie: Motion of fluid particle (Volumetric change) ./mmfm_movies/3_14040.mov

• Combining the above statements leads one to, for a Newtonian incom-
pressible fluid,^{7}

T ≡ τ_{ij} = −pI+µ

∇q + (∇q)^{T}

(4.31) In the tensor notation,

τ_{ij} = −p δ_{ij} +µ

∂q_{i}

∂x_{j} + ∂q_{j}

∂x_{i}

(4.32) In the matrix form,

{τ_{ij}}=

−p 0 0 0 −p 0 0 0 −p

+µ

2∂u

∂x

∂u

∂y + ∂v

∂x

∂u

∂z + ∂w

∂x

∂v

∂x + ∂u

∂y 2∂v

∂y

∂v

∂z + ∂w

∂y

∂w

∂x + ∂u

∂z

∂w

∂y + ∂v

∂z 2∂w

∂z

(4.33)

• The diagonal terms consist of the divergence of the velocity vector and (averaged) static pressures, and hence, for an incompressible fluid, are as- sociated with volumetric changes.

• The off-diagonal terms are associated with the squeeze-like motion.

• The proportionality constant µ is called the viscosity coefficient of the fluid, and the second term is called the viscous stress tensor.

Most of the common fluids (water, air, oil, etc.) are Newtonian fluids

7Stress Tensor for Compressible Fluids

T =−pI+µ[∇q+ (∇q)^{T}] +λ(∇ ·q)I (4.30)
whereµis the coefficient of dynamic viscosity andλis the ‘second’ coefficient of viscosity (the bulk elasticity).

(“Linear”fluids): ^{8}

τ_{yx} ∝ du

dy =⇒τ_{yx} = µdu

dy (4.35)

Figure 4.10 Stress and strain for simple shear flow of a Newtonian fluid. (From New- man 1977)

4.2.4 Navier-Stokes Equations for Incompressible Newtonian Fluids

• Divergence of stress tensor in conservation of momentum for a Newtonian incompressible fluid

∂τij

∂x_{j} = −∂p

∂x_{i} + µ ∂

∂x_{j}

∂qi

∂x_{j} + ∂qj

∂x_{i}

= −∂p

∂x_{i} + µ

∂^{2}qi

∂x_{j}∂x_{j}

(4.36)

8Some special fluids (e.g., most biological fluids, toothpaste, some paints, etc.) are non-Newtonian Fluids (“Non-linear”fluids), for which we might approximate the fluid viscosity as, in the locally linearized sense:

τyx=k

du

dy
^{n}

=⇒τyx=k

du dy

n−1du dy

=µ^{∗}du

dy (4.34)

• For incompressible Newtonian fluids, the corresponding differential form becomes the so-called Navier-Stokes equations:

ρ ∂q

∂t +q · ∇q

= −∇p+ F_{B} +µ∇^{2}q (4.37)
In the tensor form,

ρ
∂q_{i}

∂t +q_{j}∂q_{i}

∂xj

= −∂p

∂xi

+F_{B}_{i} +µ

∂^{2}q_{i}

∂xj∂xj

(4.38)

• Navier-Stokes equations in Cartesian coordinates:

∂u

∂t +u∂u

∂x +v∂u

∂y +w∂u

∂z = −1 ρ

∂p

∂x +ν∇^{2}u+ 1

ρF_{B}_{x} (4.39)

∂v

∂t +u∂v

∂x +v∂v

∂y +w∂v

∂z = −1 ρ

∂p

∂y +ν∇^{2}v + 1

ρF_{B}_{y} (4.40)

∂w

∂t +u∂w

∂x +v∂w

∂y + w∂w

∂z = −1 ρ

∂p

∂z +ν∇^{2}w+ 1

ρF_{B}_{z} (4.41)

4.2.4.1 Alternate forms of the convective and diffusion terms

Alternate forms of the non-linear convective and the viscous term of the Navier-
Stokes equations: ^{9}

(1) Alternate form of the convective term, q · ∇q

(a) Divergence form: ∇ ·(q q) = q · ∇q + q(∇ ·q)

(b) Advective/convective form: q · ∇q = ^{1}_{2}∇q^{2} −q ×(∇ ×q)
(c) Rotational form: ω×q

(d) Skew-symmetric (transpose of a matrix equals minus the matrix):

1 2

∇ ·(q q) +q · ∇q

= q · ∇q + ^{1}_{2}q(∇ ·q)
(2) Alternate form of the viscous term, ∇^{2}q

9Gresho, P. M. (1991), “Incompressible fluid dynamics: some fundamental formulation issues”,Annual Review of Fluid Mechanics, vol. 23, pp. 413–453.

(a) Stress-divergence form: ∇ ·

(∇q) + (∇q)^{T}

= ∇^{2}q +∇(∇ ·q)
(b) Div-curl form: ∇^{2}q = ∇(∇ ·q)− ∇ ×(∇ × q)

(c) Curl form: −∇ ×(∇ ×q) = −∇ ×ω

4.2.5 Navier-Stokes Equations in a Moving Frame

4.2.5.1 Kinematic description

• Let q(x, y, z, t) describe the flow field in a moving coordinate system
that is in motion relative to a space fixed system x^{0}, y^{0}, z^{0}. In general, the

Figure 4.11 Moving coordinate system.

relations between coordinates, velocities, and accelerations are those of Eqs. (A.174–A.177);

x^{0} = x+R (4.42)

q^{0} = q+ ˙R+ Ω×x (4.43)

d^{0}q

dt = dq

dt + 2 Ω×q + ˙Ω×x+ Ω×(Ω×x) + ¨R (4.44)

• Thus the left-hand side of Eq. (4.44) must be augmented by addition of four new terms, in general, in order to constitute a differential equation for q(x, y, z, t).

• On the other hand, space derivatives such as grad, div, and curl are un- affected in form by the transformation of axes. The only change in such terms therefore arise from the process of carrying out these operations on quantities, such asq, which have additional terms. There are no such terms in the scalar quantity p; hence Eq. (4.44) are altered only by addition of the four new left-hand terms mentioned.

• Before writing down the new equation of motion, let us consider the equa-
tion of continuity. Consider the term∇·^{0}q, and let∇·denote the divergence
operator in the moving system:

∇^{0}·q^{0} = ∇ ·q^{0} = ∇ ·

q + ˙R+ Ω×x

= ∇ ·q (4.45) because the divergences of the last two terms are zero (see Eq. (A.73)).

• It is important to note that the vorticity is not the same in the two systems.

Using the primes as before, we see that, according to Eq. (A.74),

∇^{0}×q^{0} = ∇ ×q^{0} = ∇ ×

q + ˙R+ Ω×x

= ∇ ×q + 2 Ω (4.46) We find the physical meaning of this result. An important case is one of flow that is irrotational but is viewed in a rotating coordinate system; it appears to be rotational. The case of a rotating propeller frame is typical.

4.2.5.2 Representation of velocity field

• The total velocity, observed in the inertial frame fixed in space is made up of two parts:

q^{0}

T(x^{0}, t) = q^{0}

o(x^{0}) +u^{0}(x^{0}, t), (4.47)
whereq^{0}

o(x^{0}) is an onset velocity field that satisfies the continuity equation
itself (a steady potential flow when measured in the inertial frame) and
u^{0}(x^{0}, t) is the disturbance velocity component to be determined herein.

• In the moving frame the total velocity is the sum of the frame velocity (q

F) and the fluid velocity (q) measured by an observer in the moving frame:

qT(x, t) =q

F(x, t) +q(x, t), (4.48) wherexand q are measured relative to the moving frame.

• Equating the two different expressions of Eqs. (4.47) and (4.48), the rela- tive velocity (q) is expressed by

q(x, t) = q

o(x, t)−q

F(x, t) +u(x, t) (4.49)

4.2.5.3 Governing equations

• The relations between coordinates, velocities, and acceleration are, again,

x^{0} = x+R (4.50)

q^{0} = q + Ω×x+ ˙R (4.51)

a^{0} ≡ d^{02}x^{0}
dt^{2}

= a+ 2 Ω×q+ dΩ

dt ×x+ Ω ×(Ω×x) + ¨R (4.52)

• Space derivatives such as grad, div, and curl are unaffected:

∇^{0} ·q^{0} = ∇ ·q^{0} = ∇ ·n

q + Ω×x+ ˙R o

= ∇ ·q (4.53)

• Continuity equation and Navier-Stokes equation:

Dρ

Dt +ρ∇ ·q = 0 (4.54)

and Dq

Dt + 2 Ω×q+ ˙Ω×x+ Ω×(Ω×x) + ¨R = −1

ρ∇p+ 1

ρF_{B} +ν∇^{2}q
(4.55)

4.2.6 Boundary Conditions

• Kinematic (Velocity Continuity) Condition ^{10}
– Fluid velocity equals the velocity of the body.

– Both the normal and tangential velocity components of the fluid and the boundary to be equal.

– Note: For inviscid flow, only the normal component must be consid- ered.

• Dynamic (Stress Continuity) Condition ^{11}

– Stress on one side of the interface between the fluid and the boundary equals the stress on the other side.

– Both the normal and tangential stress components of the fluid and the boundary to be equal.

– Note: The surface tension is included, if its effect is significant.

10 Movie: Kinematic Condition (Solid-Liqud, Liquid-Gas, Liquid-Liquid, Sloshing Stratified Liquds) ./mmfm_movies/438.mov ./mmfm_movies/439.mov

./mmfm_movies/436.mov ./mmfm_movies/76.mov

11 Movie: Dynamic Condition (Sloshing Stratified Liquds) ./mmfm_movies/454.mov