CHAPTER 2

DIFFERENTIAL FORMULATION

OF THE BASIC LAWS

2.1 Introduction

Differential formulation of basic laws:

Conservation of mass

Conservation of momentum

Conservation of energy

2.2 Flow Generation

(i) Forced convection. Motion is driven by mechanical means.

(ii) Free (natural) convection. Motion is driven by natural forces.

2.3 Laminar vs. Turbulent Flow

Laminar flow: no fluctuations in velocity, pressure, temperature, …

Turbulent flow: random fluctuations in velocity, pressure, temperature, …

Transition from laminar to turbulent flow: Determined by the Reynolds number:

Flow over a flat plate: 500,000

Flow through tubes: = 2300

2.4 Conservation of Mass: The Continuity Equation

2.4.1 Cartesian Coordinates

The principle of conservation of mass is applied to an element

(2.1)

Expressing each term in terms of velocity components gives continuity equation

(2.2a)

This equation can be expressed in different forms:

(2.2b)

or

(2.2c)

or

(2.2d)

For constant density (incompressible fluid):

(2.3)

2.4.2 Cylindrical Coordinates

(2.4)

2.4.3 Spherical Coordinates

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(2 .5)

2.5 Conservation of Momentum: The Navier-Stokes Equations of Motion

2.6 2.5.1 Cartesian Coordinates

Application of Newton’s law of motion to the element shown in Fig. 2.5, gives

(a)

Application of (a) in the x-direction, gives

(b)

Each term in (b) is expressed in terms of flow field variables: density, pressure, and velocity components:

Mass of the element:

(c)

Acceleration of the element :

(d)

Substituting (c) and (d) into (b)

(e)

Forces:

(i) Body force

(g)

(ii) Surface force

Summing up all the x-component forces shown in Fig. 2.6 gives

(h)

Combining the above equations

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x-direction: (2.6a)

By analogy:

y-direction: (2.6b)

z-direction: (2.6c)

IMPORTANT

THE NORMAL AND SHEARING STRESSES ARE EXPRESSED IN TERMS OF VELOSICTY AND PRESSURE. THIS IS VALID FOR NEWTOINAN FLUIDS. (See equations 2.7a-2.7f).

THE RESULTING EQUATIONS ARE KNOWN AS THE NAVIER-STOKES EQUAITONS OF MOTION

SPECIAL SIMPLIFIED CASES:

(i) Constant viscosity

(2.9)

(2.9) is valid for: (1) continuum, (2) Newtonian fluid, and (3) constant

viscosity

(ii) Constant viscosity and density

(2.10)

(2.10) is valid for: (1) continuum, (2) Newtonian fluid, (3) constant viscosity and (4) constant density.

The three component of (2.10) are

x: (2.10x)

y- (2.10y)

z- (2.10z)

2.5.2 Cylindrical Coordinates

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The three equations corresponding to (2.10) in cylindrical coordinates are (2.11r), (2.11 ), and (2.11z).

2.5.3 Spherical Coordinates

The three equations corresponding to (2.10) in spherical coordinates are (2.11r), (2.11 ), and (2.11 ).

2.6 Conservation of Energy: The Energy Equation

2.6.1 Formulation

The principle of conservation of energy is applied to an element

The variables u, v, w, p, T, and are used to express each term in (2.14).

Assumptions: (1) continuum, (2) Newtonian fluid, and (3) negligible nuclear, electromagnetic and radiation energy transfer.

Detailed formulation of the terms A, B, C and D is given in Appendix A

The following is the resulting equation

(2.15)

(2.15) is referred to as the energy equation

is the coefficient of thermal expansion, defined as

(2.16)

The dissipation function is associated with energy dissipation due to friction. It is important in high speed flow and for very viscous fluids. In Cartesian coordinates is given by

(2.17)

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2.6.2 Simplified Form of the Energy Equation

Cartesian Coordinates

(i) Incompressible fluid. Equation (2.15) becomes

(2.18)

(ii) Incompressible constant conductivity fluid. Equation (2.18) is simplified further if the conductivity k is assumed constant

(2.19a)

or

(2.19b)

(iii) Ideal gas. (2.15) becomes

(2.22)

or

(2.23)

Cylindrical Coordinates. The corresponding energy equation in cylindrical coordinate is given in (2.24)

Spherical Coordinates. The corresponding energy equation in cylindrical coordinate is given in (2.26)

2.7 Solutions to the Temperature Distribution

The flow field (velocity distribution) is needed for the determination of the temperature distribution.

IMPORTANT:

Table 2.1 shows that for constant density and viscosity, continuity and momentum (four equations) give the solution to u, v, w, and p. Thus for this condition the flow field and temperature fields are uncoupled (smallest rectangle).

For compressible fluid the density is an added variable. Energy equation and the equation of state provide the fifth and sixth required equations. For this

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case the velocity and temperature fields are coupled and thus must be solved simultaneously (largest rectangle in Table 2.1).

2.8 The Boussinesq Approximation

Fluid motion in free convection is driven by buoyancy forces.

Gravity and density change due to temperature change give rise to buoyancy.

According to Table 2.1, continuity, momentum, energy and equation of state must be solved simultaneously for the 6 unknowns: u. v, w, p, T and

Starting with the definition of coefficient of thermal expansion , defined as

(2.16)

or

(f)

This result gives (2.28)

Based on the above approximation, the momentum equation becomes

(2.29)

2.9 Boundary Conditions

(1) No-slip condition. At the wall,

(2.30a)

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or

(2.30b)

(2) Free stream condition. Far away from an object ( )

(2.31)

Similarly, uniform temperature far away from an object is expressed as

(2.32)

(3) Surface thermal conditions. Two common surface thermal conditions are used in the analysis of convection problems. They are:

(i) Specified temperature. At the wall:

(2.33)

(ii) Specified heat flux. Heated or cooled surface:

(2.34)

2.10 Non-dimensional Form of the Governing Equations: Dynamic and Thermal Similarity Parameters

Express the governing equations in dimensionless form to:

(1) identify the governing parameters

(2) plan experiments

(3) guide in the presentation of experimental results and theoretical solutions

Dimensional form:

Independent variables: x, y, z and t

Unknown variables are: and These variables depend on the four independent variables. In addition various quantities affect the solutions. They are

, V , , , and

Fluid properties k, , , and

Geometry

2.10.1 Dimensionless Variables

Dependent and independent variables are made dimensionless as follows:

, , ,

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(2.35)

, , ,

Using (2.35) the governing equations are rewritten in dimensionless form.

2.10.2 Dimensionless Form of Continuity

(2.37)

No parameters appear in (2.37)

2.10.3 Dimensionless Form of the Navier-Stokes Equations of Motion

(2.38)

Constant (characteristic) quantities combine into two governing parameters:

, Reynolds number (viscous effect) (2.39)

, Grashof number (free convection effect) (2.40)

2.10.4 Dimensionless Form of the Energy Equation

Consider two cases:

(i) Incompressible, constant conductivity

(2.41a)

Constant (characteristic) quantities combine into two additional governing parameters:

, Prandtl number (heat transfer effect) (2.42)

, Eckert number (dissipation effect – high speed, large viscosity) (2.43)

(ii) Ideal gas, constant conductivity and viscosity

(2.41b)

No new parameters appear.

2.10.5 Significance of the Governing Parameters

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Dimensionless temperature solution: (2.45)

NOTE:

Six quantities: , , V , and five properties k, , , and , are replaced by four dimensionless parameters: Re, Pr, Gr and Ec.

Special case: negligible free convection and dissipation: Two governing parameters:

(2.46)

Geometrically similar bodies have the same solution when the parameters are the same.

Experiments and correlation of data are expressed in terms of parameters rather than dimensional quantities.

Numerical solutions are expressed in terms of parameters rather than dimensional quantities.

2.10.6 Heat Transfer Coefficient: The Nusselt Number

Local Nusselt number: (2.51)

Special case: negligible buoyancy and dissipation: (2.52)

Free convection, negligible dissipation

(2. 53)

For the average Nusselt number, is eliminated in the above.

2.9 Scale Analysis

A procedure used to obtain order of magnitude estimates without solving governing equations.

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