MECH 5810 Module 4: Differential FluidDeformation & Conservation of Mass

D.J. Willis

Department of Mechanical EngineeringUniversity of Massachusetts, Lowell

MECH 5810 Advanced Fluid DynamicsFall 2017

Outline

1 Kinematics and Mathematics

Operator Basics

Aside: Gauss’s Theorem

Differential Conservation of Mass

Aside: Stokes Theorem (Optional in module 4)

Kinematics

We would like to examine differential elements of fluid:

How can we mathematically describe the motion for the fluid?How does a fluid element deform due to the velocity field?Can we simplify the deformation into consistent sub-deformations?

Kinematics: Linear Strain Rate – ε

Similar idea to solid mechanics - normalized change in length

The difference is strain rate – continues to deform

Linear strain rate: represents the normalized rate of change in

length of a fluid element with respect to time:

Strain Rate =1lo

D(l)Dt

=∂u∂x

So, how do we get ∂u∂x ?

Kinematics : Linear Strain Rate

Strain Rate =

(1lo

D(l)Dt

)'(

1lo

∆l∆t

)=

(1lo

(u + ∂u

∂x lo)

∆t − (u) ∆t∆t

)

Strain Rate = εxx =

(1

��lo

(�u + ∂u

∂x��lo)��∆t − (�u)��∆t

��∆t

)=

(∂u∂x

)

εxx, εyy, εzz: Linear strains in x, y, z

Kinematics : Volumetric/Bulk Strain Rate

Volumetric strain rate = Rate of change of the volume per unit

volume

The infinitesimal volume δV = δxδyδz.

Volumetric Strain Rate =1δV

DDt

(δV) =1

δxδyδzDDt

(δxδyδz)

=1δx

DDt

(δx) +1δy

DDt

(δy) +1δz

DDt

(δz)

Simplifying, we get:

Volumetric Strain Rate =∂u∂x

+∂v∂y

+∂w∂z

=∂ui

∂xi︸︷︷︸indicial notation

Which can be represented using the divergence operator:

Volumetric Strain Rate = ∇ ·~u =∂u∂x

+∂v∂y

+∂w∂z

εxx, εyy, εzz: Linear strains in x, y, z

Vector Calculus : Gradient – grad(p) = ∇p

The gradient operator provides the path of steepest descent

Sidenote: Fluids will tend to follow minimum energy paths

grad(p) = ∇(p) =

∂p∂x∂p∂y∂p∂z

www.mathworks.com

Grad-dot-u: volumetric strain rate→incompressible conservation of mass

Outline

1 Kinematics and Mathematics

Operator Basics

Aside: Gauss’s Theorem

Differential Conservation of Mass

Aside: Stokes Theorem (Optional in module 4)

Aside 1: Gauss’s Theorem

Gauss’s Theorem or Divergence Theorem∫∫∫V∇ ·~udV =

∫∫S~u · n̂dS

Or, more generally:∫∫∫V∇ · ~QdV =

∫∫S

~Q · n̂dS

States that the volume integral of the divergence of a quantity (~Q)

is equal to the surface flux of that quantity.

This can be loosely translated as: The divergence of a quantity is

similar to the rate of change of a volume quantity with respect to

time.

Concept/Illustration of Divergence Theorem

Illustration of Divergence Theorem

A conceptual view of the divergence theorem:

Consider a volume V to be made up of many elemental volumes

dV

The divergence is, by definition, the normalized rate of change of

a volume

Hence, if the divergence is positive (volume is increasing), and the

boundary of our volume is fixed, then, there must be outflux of

fluid equal to the rate of change of the volume.

Outline

1 Kinematics and Mathematics

Operator Basics

Aside: Gauss’s Theorem

Differential Conservation of Mass

Aside: Stokes Theorem (Optional in module 4)

Differential conservation of mass

Consider a small fluid element dV = dxdydz fixed in space

Differential conservation of mass

Apply the integral conservation of mass equation to this small

volume:

0 =ddt

∫∫∫CV(t)

ρdV +

∫∫CS(t)

ρurndA

Applying divergence theorem to the second integral

0 =ddt

∫∫∫CV(t)

ρdV +

∫∫∫CV(t)

∇ · (ρ~u)dV︸ ︷︷ ︸∫∫CS(t) ρ~urndA

Differential conservation of mass

Since the differential control volume is fixed in space, we can bring

the ddt term inside the integral:

0 =

∫∫∫CV(t)

[∂

∂tρ+∇ · (ρ~u)

]dV

We also know that conservation of mass must apply at all points in

the fluid, as a result, we know that the integrand must be zero

everywhere (not just at select points):

0 =∂ (ρ)

∂t+∇ · (ρ~u)

This is the differential form of the conservation of mass statement

and it must hold true at all locations in a flowfield.

Differential conservation of mass

Means that (1) Local change in density w.r.t. time must be

balanced by the (2) divergence of the product of the density and

velocity:∂ρ

∂t︸︷︷︸1

+∇ · (ρ~u)︸ ︷︷ ︸2

= 0

Differential conservation of mass

For a constant density flow, ρ = const., this conservation of mass

statement becomes:

∇ · (~u) = 0

Does this make sense? The divergence of a constant density fluid

is zero.

Differential conservation of mass

Let’s try some examples...

Outline

1 Kinematics and Mathematics

Operator Basics

Aside: Gauss’s Theorem

Differential Conservation of Mass

Aside: Stokes Theorem (Optional in module 4)

Kinematics : Shear Strain Rate

Shear strain rate: Rate of decrease of the angle formed by two

mutually perpendicular lines on the element

The shear strain rate value depends on the orientation of the line

pair

εij =dα+ dβ

dt

Kinematics : Shear Strain RateLet’s look at dα first:

dα = atan(OA

) ' OA

' ∆llo

'∂u∂y · lo · dt

lo

' ∂u∂y

dt

Kinematics : Shear Strain Rate

So:dαdt

=∂u∂y

Similarly, it can be shown that:

dβdt

=∂v∂x

εxy =dα+ dβ

dt=∂u∂y

+∂v∂x

Kinematics : Strain Rate Tensor

εij =12

(∂ui

∂xj+∂uj

∂xi

)

∂u∂x

12

(∂u∂y + ∂v

∂x

)12

(∂u∂z + ∂w

∂x

)12

(∂v∂x + ∂u

∂y

)∂v∂y

12

(∂v∂z + ∂w

∂y

)12

(∂w∂x + ∂u

∂z

) 12

(∂w∂y + ∂v

∂z

)∂w∂z

The diagonal is the normal strain rate terms (εii)

The off-diagonals are half the shear strain rate terms

εij = εji

Kinematics : Vorticity and Curl - curl(~u) = ∇× ~u

Rotation or angular velocity is an important component of

kinematics as we shall see

We define vorticity to be twice the angular velocity of the fluid

element

Kinematics : Vorticity and Curl - curl(~u) = ∇× ~u

The curl of a vector indicates whether there is rotationality in the vector

field.

curl(~u) = ∇×~u =

∂∂x∂∂y∂∂z

×

u

v

w

=

∂w∂y −

∂v∂z

∂u∂z −

∂w∂x

∂v∂x −

∂u∂y

=

ωx

ωy

ωz

= ~ω

Pure rotation of an element:

Kinematics : Summary of Possible Fluid Motions

Vorticity, Rotation Rate

Shear Strain Rate

Linear Strain Rate

Volumetric Strain Rate

Aside 2: Stoke’s Theorem

∫∫A∇×~udA =

∮l~u · ~dl

If we put to use our latest kinematics knowledge, ω = ∇×~u, then:∫∫Vω︸︷︷︸∇×~u

dS =

∮l~u · ~dl = Γ︸︷︷︸

Circulation

Essentially a statement of the circulation theorem

The ~dl is an infinitesimal length of the boundary

The∮

represents the path integral