Differential Analysis

1
Differential Analysis

• Control volume methods useful but sometimes
  limited whenever we require a detailed knowledge
  of pressure and velocity variations.
• Control volume treated problems with black-box
  approach No detailed knowledge was required

Accumulation or depletion
Loss through outlet
Addition through inlet
CV

• Knowledge of flow details are sometimes needed
• Examples Pressure and shear stress along the
  surface of an airplane wing, determination of
  velocity profiles inside non-circular conduits,
  flow of polymer inside a die during extrusion

2
Differential Analysis of Fluid Flow

• Application of the fundamental laws to an
  infinitesimal control volume (differential
  element)
• Principles same as for Control Volume analysis
  Conservation of mass, momentum and energy
• We apply these principles on an element of
  dimensions Dx, Dy, Dz

Element of volume Dx Dy Dz
CV
3
Differential Continuity Equation
z
Vz
Vx
Vy
x
y
Recall mass balance (3.1)
All mass flow rates in
All mass flow rates out
Rate of accumulation
-

4
Differential Continuity Equation
x-direction

• Rate of mass flow in

• Rate of mass flow out

y-direction

• Rate of mass flow in

• Rate of mass flow out

z-direction

• Rate of mass flow in

• Rate of mass flow out

5
Differential Continuity Equation
Rate of mass accumulation
Inserting into equation 3.1, dividing by DxDyDz
and taking the limit as the cube shrinks to a
point
Continuity Equation
(15.1)
6
Differential Continuity Equation

• Simplifications
• For steady-state conditions

(15.2)

• For incompressible fluids

(15.3)
7
Differential Continuity Equation
In cylindrical coordinates
(15.4)

• For steady state, incompressible flow

(15.5)
8
Differential Momentum Balance
Rate of accumulation of momentum
Sum of forces acting on system
Rate of momentum out
Rate of momentum in
-

(15.6)

• Estimation of net rate of momentum out of element

z
x
y

• Estimation of forces acting on the element

9
Reminder Definition of stress

• Stress force per unit area (F/A)
• Normal stress acts perpendicular to the surface
  (Fnormal force).

F
F
F
F
A
A
Tensile causes elongation
Compressive causes shrinkage

• Shear stress acts tangentially to the surface
  (Ftangential force).

F
A
F
10
Forces acting on a differential element
Consider a force, DF, acting on a surface element
DA
DFs
DFn
DA
DF1
DF2
Normal stress
Shear stress
11
Forces acting on a differential element (3-D)
P
szz
z
tzx
tzy
txz
x
y
tyz
P
sxx
P
txy
tyx
syy

• The first subscript indicates the direction of
  the normal to the plane on which the stress acts.
• The second subscript indicates the direction of
  the stress.

12
Differential Momentum Balance

• Estimation of forces acting on the element

z
x
y
13
Equations of Motion

• x-component of momentum equation

(15.7a)

• y-component of momentum equation

(15.7b)

• z-component of momentum equation

(15.7c)
14
Stress-Deformation Newtons Law
For one-dimensional flow (say flow between two
flat plates)
y
x
15
Stress Deformation relationship

• In general the stresses are linearly related to
  the rates of deformation
• (shear stress) (viscosity)x(rate of shear
  strain)
• In Cartesian coordinates

(15.8)
16
Stress Deformation relationship

• In Cylindrical coordinates

(15.9)
17
Navier-Stokes Equations

• Taking into account the stress-deformation
  relationships (Eqs. 15.8, 15.9) and making the
  following assumptions
• The fluid has constant density
• The flow is laminar throughout
• The fluid is Newtonian
• we obtain the Navier-Stokes Equations

18
Navier-Stokes Equations
(15.10a)

• x-component

• y-component

(15.10b)

• z-component

(15.10c)
19
Navier-Stokes Equations
In cylindrical (polar) coordinates
(15.11a)

• r-component

(15.11b)

• ?-component

20
Navier-Stokes Equations
(15.11c)

• z-component

21
Solution Procedure

• Make reasonable simplifying assumptions (i.e.
  steady state, incompressible flow, coordinate
  direction of flow)
• Write down continuity and momentum (or
  Navier-Stokes) equations and simplify them
  according to the assumptions of Step 1.
• Integrate the simplified equations.
• Invoke boundary conditions in order to evaluate
  integration constants obtained in Step 3.
• No-slip condition
• Continuity of velocity
• Continuity of shear stress
• Solve for pressure and velocity. Derive shear
  stress distributions if desired. Apply numerical
  values.

22
Types of Flow encountered in Problems

• Couette flow (or drag flow) A moving surface
  drags adjacent fluid along with it and thereby
  imparts a motion to the rest of the fluid
• Poiseuille flow (or pressure driven flow) The
  applied pressure difference causes fluid motion
  between stationary surfaces

23
Example1 Drag flow between two parallel plates

• Consider two flat parallel plates separated by a
  distance b as shown in the figure. The top plate
  moves in the x-direction at a constant speed V,
  while the bottom plate remains stationary. The
  fluid between the plates is assumed
  incompressible. As the top plate moves the fluid
  is dragged along. This type of flow is often
  referred as Couette flow. It has important
  applications in lubrication applications (such as
  rotating journal bearings) and instruments for
  measurement of viscosity.
• Prove that the velocity profile for this type of
  flow is linear. What is the volumetric flow rate?

24
Sample Worksheet

• Step 1 State assumptions
• - Steady-state (all derivatives with respect to
  time 0), incompressible flow (r const.).
• - Decide on coordinate system, determine
  direction of flow, identify non-zero velocity
  components.
•  - Inspect for any other reasonable assumptions.
•  
• Step 2 Write down continuity (chose from
  15.1-15.5) and Navier-Stokes equations (chose
  from 15.10 or 15.11) for the appropriate
  coordinate system and direction of flow.
• Then simplify them, according to assumptions of
  Step 1.
• Step 3 Integrate the simplified Navier-Stokes
  equation.

25
Sample Worksheet

• Step 4 Identify appropriate boundary conditions.
  Use them to determine the integration constants
  obtained above.
• Step 5 Obtain velocity profile.
• Step 6 (If needed) Obtain volumetric flow rate
  by integrating
• For flow in channels (Wwidth)
•  
• - For flow through circular cross-sections
• Step 7 (If needed) Obtain shear stress
  distributions, chosing the appropriate
  stress-deformation relationship, from eqs (15.8)
  or (15.9) and simplifying it.

26
Example 2 Pressure driven flow between parallel
plates

• The figure below shows a fluid of viscosity m
  that flows in the x direction between two
  rectangular plates, whose width is very large in
  the z direction when compared to their separation
  in the y direction. Such a situation could occur
  in a die when a polymer is being extruded at the
  exit into a sheet, which is subsequently cooled
  and solidified. We will determine the
  relationship between the flow rate and the
  pressure drop between the inlet and exit,
  together with several other quantities of
  interest.

27
Example 2 Pressure driven flow between parallel
plates

• Now solve the following problem
• A highly viscous fluid having a viscosity of 950
  Pa.s and density of 780 kg/m3 is flowing through
  a rectangular (flat) die having length of 25 cm,
  width of 1.75 m and gap of 1.8 mm. The pressure
  drop for this flow is 55.6 MPa. What is the mass
  flow rate? How much is the shear stress at the
  wall?

28
Summary of some useful results

• Steady pressure driven, laminar flow between
  fixed parallel plates

W
L
Velocity Profile
where
Volumetric flow rate
29
Summary of some useful results

• Steady, laminar, Drag (Couette) flow between
  parallel plates

Velocity profile
Volumetric flow rate
30
Summary of some useful results

• Steady, pressure driven, laminar flow in circular
  tubes

Velocity Profile
where
Volumetric flow rate
31
Summary of some useful results

• Steady, Pressure driven, Axial, Laminar flow in
  an Annulus

r
Vz
z
Vz