1. DIFFERENTIAL EQUATIONS FOR FLUID FLOW Vinay Chandwani (Mtech Struct.)

2. ACCELERATION OF FLUID CARTESIAN VECTOR OF VELOCITY
TOTAL TIME DERIVATIVE OF VELOCITY VECTOR

3. LOCAL VELOCITY COMPONENTS u,v & w respectively
ACCELERATION OF FLUID
Since each scalar component (u,v,w) is a function of four variables (x,y,z,t), we use the chain rule to obtain each scalar time derivative. For example
LOCAL VELOCITY COMPONENTS u,v & w respectively

4. LOCAL ACCELERATION (ZERO IF FLOW IS STEADY)
ACCELERATION OF FLUID
Summing these into a vector, we obtain the total acceleration:
MATERIAL DERIVATIVE
Convective acceleration is defined as the rate of change of velocity due to the change of position of fluid particles in a fluid flow. Local acceleration or Temporal acceleration is defined as the rate of change of velocity with respect to time at a given point in a flow field.
CONVECTIVE ACCELERATION (arises when particle moves through regions of spatially varying velocity, e.g. nozzle or diffuser)
LOCAL ACCELERATION (ZERO IF FLOW IS STEADY)
MATERIAL DERIVATIVE :It describes the time rate of change of some quantity (such as heat or momentum) by following it, while moving with a space and time dependent
velocity field.

5. TIME DERIVATIVE FOR A PATH FOLLOWING THE FLUID MOTION
ACCELERATION OF FLUID
So acceleration a can be represented as material derivative (SUBSTANTIAL DERIVATIVE) of velocity q as
TIME DERIVATIVE FOR A PATH FOLLOWING THE FLUID MOTION

6. EQUATION OF CONTINUITY
All basic differential equations can be derived by considering either an elemental control volume or an elemental system
Let us choose an infinitesimal fixed control volume(dx,dy,dz)
Law of conservation of mass
Rate of decrease of mass in the control volume=the rate of net outflow of the mass through the surface of control volume.
IF ELEMENT IS VERY SMALL

7. EQUATION OF CONTINUITY
The mass-flow term occurs on all six faces, three inlets and three outlets, where all fluid properties are considered to be uniformly varying functions of time & position e.g ρ(x,y,z,t)
if ρu is known on left face, the value of this product on the right face is ρu+(∂(ρu)/∂x)dx
X-direction
Inlet & outlet mass flow
Y-direction
Inlet & outlet mass flow
Z-direction
Inlet & outlet mass flow

8. EQUATION OF CONTINUITY
EQUATION OF MASS CONSERVATION
EQUATION OF CONTINUITY

9. EQUATION OF CONTINUITY
Assumption: density & velocity are continuous functions i.e. the flow may be either steady or unsteady, viscous or frictionless, compressible or incompressible.

10. EQUATION OF CONTINUITY
SUBSTANTIAL DERIVATIVE OF DENSITY or THE TIME DERIVATIVE FOR A PATH FOLLOWING THE FLUID MOTION

11. EQUATION OF CONTINUITY
STEADY COMPRESSIBLE FLOW
ALL PROPERTIES ARE FUNCTIONS OF POSITION ONLY
INCOMPRESSIBLE FLOW
DENSITY CHANGES ARE NEGLIGIBLE

12. EQUATION OF CONTINUITY (CYLINDRICAL COORDINATES)
LET US CONSIDER A POINT P(r,θ,z) IN SPACE. LET δr,δθ & δz BE SMALL INCREMENTS IN THE RADIAL, TANGENTIAL & VERTICAL DIRECTIONS RESPECTIVELY.
LET qr, qθ & qz BE THE COMPONENTS OF THE VELOCITY IN THE DIRECTIONS OF r, θ & z RESPECTIVELY AT POINT P & ρ BE THE MASS DENSITY OF FLUID AT P.

14. EQUATION OF CONTINUITY (CYLINDRICAL COORDINATES)
Mass rate of flow entering the parallelepiped in radial direction through face PQRS
Mass rate of flow leaving the parallelepiped through face P’Q’R’S’
Net gain in mass per unit time in radial direction
Net gain in mass per unit time in vertical direction
Net gain in mass per unit time in tangential direction
TOTAL GAIN IN MASS PER UNIT TIME
TOTAL GAIN IN MASS =RATE OF INCREASE OF MASS WITHIN PARALLELEPIPED

15. EQUATION OF CONTINUITY (CYLINDRICAL COORDINATES)
FLOW IS STEADY
FLOW IS INCOMPRESSIBLE

16. EQUATION OF CONTINUITY (POLAR COORDINATES)
If the flow is two dimensional then polar coordinates (r,θ) are used to describe the flow.
qz =0
STEADY FLOW
INCOMPRESSIBLE FLOW
CONTINUITY EQUATION IN POLAR COORD.

17. MOMENTUM EQUATION
Sum of the forces acting on the control volume = the rate of increase of momentum of the fluid through the control surface.
Forces
Surface Forces Stresses
Body Forces External forces like gravitation
SURFACE FORCES: If σ is the stress tensor then the force due to stress on the on the elemental area dS is σ.dS
Force due to stresses on the control surface is given by

18. FROM DIVERGENCE THEOREM
MOMENTUM EQUATION
BODY FORCES: If f is the body force per unit mass, then the force on a small element of volume dV can be written as (ρdV)f.
BODY FORCES ON THE CONTROL VOLUME V0
FORCES ON CONTROL VOLUME V0
FROM DIVERGENCE THEOREM

19. RATE OF INCREASE OF MOMENTUM
MOMENTUM EQUATION
RATE OF INCREASE OF MOMENTUM

20. FROM DIVERGENCE THEOREM
MOMENTUM EQUATION
Rate of net outflow of momentum: Flow of momentum is due to mass flowing into & out of the control volume through control surface.
For an elemental area dS, the mass flow rate will be ρq.dS. Therefore momentum flow rate through this elemental area is [ρq.dS]*q
Considering outflow as positive, the net rate of outflow of momentum can be written as:
FROM DIVERGENCE THEOREM

21. MOMENTUM EQUATION

22. Rate of increase of Momentum Rate of Net Outflow of Momentum
MOMENTUM EQUATION
Force = Rate of increase of momentum + Rate of net outflow of momentum
Rate of increase of Momentum
Rate of Net Outflow of Momentum
FORCE

23. IF CONTROL VOLUME IS ASSUMED AS V0
MOMENTUM EQUATION
IF CONTROL VOLUME IS ASSUMED AS V0

24. MOMENTUM EQUATION MOMENTUM EQUATION (CAUCHY’S EQUATION)
CONTINUITY EQUATION
MATERIAL DERIVATIVE
MOMENTUM EQUATION
(CAUCHY’S EQUATION)

25. NAVIER STOKES EQUATIONS
Before going into the details of the Navier-Stokes equations, first, it is necessary to make several assumptions about the fluid. The first one is that the fluid is continuous. It signifies that it does not contain voids formed, for example, by bubbles of dissolved gases, or that it does not consist of an aggregate of mist-like particles. Another necessary assumption is that all the fields of interest like pressure, velocity, density, temperature, etc., are differentiable (i.e. no phase transitions) & fluid is Newtonian.
The equations are derived from the basic principles of conservation of mass, momentum, and energy. For that matter sometimes it is necessary to consider a finite arbitrary volume, called a control volume, over which these principles can be easily applied.

26. NAVIER STOKES EQUATIONS
FOR NEWTONIAN FLUIDS (Derived earlier)

27. NAVIER STOKES EQUATIONS
TAKING EACH TERM SEPARATELY

28. NAVIER STOKES EQUATIONS

29. NAVIER STOKES EQUATIONS
REPRESENTS ACCELERATION OF FORCE PER UNIT MASS

30. NAVIER STOKES EQUATIONS

31. NAVIER STOKES EQUATIONS
PRESSURE FORCE PER UNIT MASS
IN X-DIRECTION
IN Y-DIRECTION
IN Y-DIRECTION

32. NAVIER STOKES EQUATIONS
REPRESENT VISCOUS FORCE PER UNIT MASS

33. NAVIER STOKES EQUATIONS
COMPRESSIBILITY FORCE PER UNIT MASS
THIS BECOMES ZERO FOR INCOMPRESSIBLE FLUIDS

34. NAVIER STOKES EQUATIONS
INCOMPRESSIBLE FLUIDS
NON VISCOUS & INCOMPRESSIBLE FLUIDS
EULER’S EQUATION

35. LIMITATIONS OF NAVIER STOKE’S EQUATIONS
The derivation is based on the assumption that viscous stress is directly proportional to the rate of deformation. This characteristic is limited to Newtonian Fluids. Many common fluids do behave in this manner. But these equations cannot be applied to Non-Newtonian Fluids.
It is set of three non-linear equations. For completely solve this equation mathematically, three more relations are necessary: the equation of continuity, the equation of state of the fluid & the equation giving the shear viscosity as a function of the state of the fluid.

36. EULER’S EQUATION OF MOTION
In fluid dynamics, the Euler equations are a set of equations governing inviscid flow. The equations represent conservation of mass (continuity), momentum, and energy, corresponding to the Navier–Stokes equations with zero viscosity and heat conduction terms.
The Euler equations can be applied to compressible as well as to incompressible flow — using either an appropriate equation of state or assuming that the divergence of the flow velocity field is zero, respectively.

37. EULER’S EQUATION OF MOTION
Let a closed surface S enclosing the fluid (non-viscous) be moving with the fluid, so that S contains the same fluid particles at any time.
Now take a point P inside S. Let ρ be the density of the fluid at P & δV be the elementary volume enclosing P, q being the velocity of the fluid at P.

38. EULER’S EQUATION OF MOTION
Since the mass ρδV of the element remains unchanged, we have the momentum M of the volume V in S given by
RATE OF CHANGE OF MOMENTUM
MASS REMAINS UNCHANGED

39. EULER’S EQUATION OF MOTION
TOTAL FORCE ON LIQUID IN VOLUME V
P IS THE PRESSURE AT A POINT IN SURFACE dS, TOTAL FORCE
FROM GAUSS’S THEOREM
RATE OF CHANGE OF MOMENTUM = TOTAL FORCE ACTING ON THE MASS

40. EULER’S EQUATION OF MOTION
COMPLETE INTEGRAL
EULER’S EQUATION OF MOTION
Though the equations appear to be very complex, they are actually simplifications of the more general Navier-Stokes equations of fluid dynamics. The Euler equations neglect the effects of the viscosity of the fluid which are included in the Navier-Stokes equations. A solution of the Euler equations is therefore only an approximation to a real fluids problem.

41. BERNOULLI’S EQUATION MATERIAL DERIVATIVE EULER’S EQUATION
BODY FORCE Fg
For steady flow

42. BERNOULLI’S EQUATION h is elevation potential
Negative, because acting in downward direction
BERNOULLI’S POLYNOMIAL

43. BERNOULLI’S EQUATION
To evaluate the rate of change of Bernoulli's polynomial along a streamline, it has to multiplied with a unit vector s tangent to the streamline.
Vector s is parallel to q and normal to
Therefore
Bernoulli’s constant, independent of time. It is conserved along stream lines

44. BERNOULLI’S EQUATION BERNOULLI’S EQUATION Static head Pressure head
Velocity head

45. BERNOULLI’S EQUATION (ALTERNATE PROOF)
NAVIER STOKES EQUATION IN x-DIRECTION
NAVIER STOKES EQUATION IN x-DIRECTION
STATIONARY INVISCID INCOMPRESSIBLE FLOW

46. BERNOULLI’S EQUATION (ALTERNATE PROOF)
MULTIPLY THROUGHOUT BY dx
FOR MOTION ALONG STREAMLINES

47. BERNOULLI’S EQUATION (ALTERNATE PROOF)

48. BERNOULLI’S EQUATION (ALTERNATE PROOF)
For incompressible flow the density is constant, so we integrate these infinitesimal change between the two points on the streamline
BERNOULLI’S EQUATION

49. Differential form of conservation of K.E
ENERGY CONSERVATION
KINETIC ENERGY PER UNIT MASS = q2/2
Body force per unit mass
Navier Stokes Equation for incompressible viscous flow
Viscous force per unit mass
Multiply throughout by q
Differential form of conservation of K.E
Rate at which pressure, gravity & viscous forces are increasing
Rate of increase of K.E

50. ENERGY CONSERVATION (ALTERNATE PROOF)
WE HAVE 5 VARIABLES ρ,u,v,w & p. ONLY 3 EQUATIONS OF NAVIER STOKES & ONE EQUATION OF CONTINUITY ARE AVAILABLE. IF THE EFFECTS OF COMPRESSIBILITY ARE NEGLECTED WE ARE LEFT WITH 4 UNKNOWNS & 4 EQUATIONS.
From first law of thermodynamics rate of change of internal energy (∆E ) is equal to the sum of rate of change in heat energy (∆Q) & work done (∆W)
Rate of work done per unit volume

51. ENERGY CONSERVATION (ALTERNATE PROOF)
Relation between stress & strain
Rate of Work done per unit volume
Viscous dissipation function, heat generated due to frictional forces

52. ENERGY CONSERVATION (ALTERNATE PROOF)
General equation for Newtonian fluid under very general conditions of unsteady, compressible, viscous, heat conducting flow.

53. ENERGY CONSERVATION (ALTERNATE PROOF)
HEAT CONDUCTION EQUATION
For fluids at rest or having negligible velocity, dissipation & convective terms become negligible

54. PRINCIPAL STRESSES
At every point in a stressed body there are at least three planes, called principal planes, with normal vectors n , called principal directions, where the corresponding stress vector is perpendicular to the plane, i.e., parallel or in the same direction as the normal vector, and where there are no shear stresses . The three stresses normal to these principal planes are called principal stresses.

55. COMPONENTS OF PRINCIPAL STRESSES λ ALONG THE COORDINATE AXIS
COMPONENTS OF PRINCIPAL STRESSES DIRECTED ALONG THE NORMAL TO THE PRINCIPAL PLANE
COMPONENTS OF PRINCIPAL STRESSES λ ALONG THE COORDINATE AXIS

56. PRINCIPAL STRESSES