Basic assumptions and equations in Fluid Mechanics
Basic assumptions -
The The flu fluid id is con continu tinuou ous. s. All All the the fiel fields ds of inte intere rest st like like pressure pressure,, velocity velocity,, density density,, temperature temperature,, etc., are differentiable (i.e. differentiable (i.e. no phase no phase transitions). transitions).
The equations are derived from the basic principles of conservation of mass, mass, momentum,, and energy momentum energy.. For that matter sometimes it is necessary to consider a finite arbitrary volume, called a control volume, volume , over hich these principles can be easily applied. This finite volume is denoted by ! and its bounding surface . The control volume can remain fi"ed in space or can move ith the fluid.
The substantive derivative The derivative of a field ith respect to fi"ed position in space is called the spatial or or Eulerian derivative. Eulerian derivative. The derivative folloing a moving particle is called the substantive or substantive or Lagrangian Lagrangian derivative. The substantive derivative is defined as the operator# D Dt
( ⋅) ≡
∂( ⋅) + ( ⋅ ∇) ( ⋅) v ∂t
$here v is the velocity of the fluid. The first term on the right-hand side of the equation is the ordinary %ulerian derivative (i.e. the derivative on a fi"ed reference frame) hereas the second term represents the changes brought about by the moving fluid. This effect is referred to as advection advection.. The The op operat erato or ∇=
∂ ∂ x
i +
(nab (nabla la)) is is def defin ined ed## ∂
∂ y
j +
∂
∂ z
k
Equation of continuity &onservation of mass is ritten#
∂ ρ + ∇( ) = D ρ + ∇ ⋅ = v 0 ρ v ρ ∂t Dt $here ' is the density, and v is the velocity of the fluid. n the case of an incompressible fluid ' is not a function of time or space the equation is reduced to#
∇ ⋅ v = 0 The General Equation of Fluid Dynamics Applying *+Alembert rinciple to the control volume F + F i
+ F + F = 0
m
p
ν
e have# Dv
ρ
Dt
= −∇ p + ∇T + ρ f
The tensor T is given by#
τ τ τ
τ
τ
xx
xy
xz
yx
τ yy
τ yz
zx
τ zy
τ zz
σ = τ τ
xx
τ xy
τ xz
yx
σ yy
τ yz
zx
τ zy
σ zz
p + 0 0
0
0
p
0
0
p
*epending on the tensor of the surface forces applied on a fluid particle and the σ + σ + σ pressure p = . 3 xx
yy
zz
The closure problem These equations are incomplete. n the case of a perfect fluid components are nil, for e"ample. Those equations used to complete the set are equations of state. For e"ample, the pressure can be function of, notably, density and temperature. ϕ ( p , ρ ,T ) = 0
The variables to be solved for are the velocity components, the fluid density, static pressure, and temperature. The flo is assumed to be differentiable and continuous, alloing these balances to be e"pressed as partial differential equations. The equations can be converted to $ilkinson equations for the secondary variables vorticity and stream function. olution depends on the fluid properties (such as viscosity, specific heats, and thermal conductivity), and on the boundary conditions of the domain of study.
A Newtonian fluid is a fluid that flos like ater - its shear stress is linearly proportional to the velocity gradient in the direction perpendicular to the plane of shear. The constant of proportionality is knon as the viscosity. A simple equation to describe /etonian fluid behaviour is τ = µ
dv dx
here is the shear stress e"erted by the fluid (0drag0) 1 is the fluid viscosity - a constant of proportionality dv dx
is the velocity gradient perpendicular to the direction of shear
n common terms, this means the fluid continues to flo, regardless of the forces acting on it. For e"ample, ater is /etonian, because it continues to e"emplify fluid properties no matter ho fast it is stirred or mi"ed. For a /etonian fluid, the viscosity, by definition, depends only on temperature and pressure (and also the chemical composition of the fluid if the fluid is not a pure substance), not on the forces acting upon it. f the fluid is incompressible and viscosity is constant across the fluid, the equation governing the shear stress, in the &artesian coordinate system, is
τ
ij
∂v = µ ∂v + ∂ x ∂ x i
j
j
i
here ij is the shear stress on the ith face of a fluid element in the jth direction vi is the velocity in the ith direction x j is the jth direction coordinate f a fluid does not obey this relation, it is termed a non-/etonian fluid,
Navier to!es Equations
The motion of a non-turbulent, /etonian fluid is governed by the /avier-tokes equations. For an incompressible fluid results# ! ∂ p
∂ u ∂ u =− + + + + # ρ ∂ x ρ ∂ x ∂ y ∂ z Dt Dv ! ∂ p µ ∂ v ∂ v ∂ v =− + + + + " ρ ∂ y ρ ∂ x ∂ y ∂ z Dt D ! ∂ p µ ∂ ∂ ∂ =− + + + + Z Dt ρ ∂ z ρ ∂ x ∂ y ∂ z Du
µ ∂ u $
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$here % is the molecular viscosity and %&' is called the kinematical viscosity
Euler"s Equations For the case hen the viscosity is 2ero the /avier tokes equations become %uler+s equations# Du
=−
ρ ∂ x
Dt Dv
! ∂ p
=−
Dt D Dt
! ∂ p
ρ ∂ y =−
! ∂ p
ρ ∂ z
(3.34)
− g
Bernoulli"s Equation 5esults from %uler+s equations in the conditions of irrotationality ( u = −∇ ) for an incompresibile fluid
−
p ∂φ ! + ) u + ( + + gz = * ) t ( ∂t $ ρ $
∂ p ! ∂ ∂ − φ + + φ + φ + gz = * ) t ( ∂t ρ $ ∂ x ∂ z $
or#
$
$
(3.36)
For the case of the potential motion the ecuation of continuity becomes# hich implies ∇∇φ =0 , or ∆φ = 0 , that is $aplace"s equation#
∇u = 0
∂ φ ∂ φ ∂ φ + + =0 ∂ x ∂ y ∂ z $
$
$
$
$
$
Two Dimensional tream Function The stream function 7 for a to dimensional flo is defined such that the flo velocity can be e"pressed as# u
= − ∂ψ , = ∂ψ ∂ z ∂ x
$here u and v are the velocities in the x and y directions, respectively. u=
∂ x
∂ z , = ∂t ∂t
This formulation of the stream function satisfies the to dimensional continuity equation# ∂u ∂ x
+
∂ ∂ z
=0
The stream line, or line of constant stream function, and lines of constant velocity potential are perpendicular as can be seen from the fact that their gradients are perpendicular# ∆Φ ⋅ ∆ψ = 0
