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

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

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

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.

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

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

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

EQUATION OF CONTINUITY EQUATION OF MASS CONSERVATION EQUATION OF CONTINUITY

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.

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

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

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.

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

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

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.

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

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

RATE OF INCREASE OF MOMENTUM MOMENTUM EQUATION RATE OF INCREASE OF MOMENTUM

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

MOMENTUM EQUATION

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

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

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

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.

NAVIER STOKES EQUATIONS FOR NEWTONIAN FLUIDS (Derived earlier)

NAVIER STOKES EQUATIONS TAKING EACH TERM SEPARATELY

NAVIER STOKES EQUATIONS

NAVIER STOKES EQUATIONS REPRESENTS ACCELERATION OF FORCE PER UNIT MASS

NAVIER STOKES EQUATIONS

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

NAVIER STOKES EQUATIONS REPRESENT VISCOUS FORCE PER UNIT MASS

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

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

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.

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.

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.

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

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

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.

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

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

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

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

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

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

BERNOULLI’S EQUATION (ALTERNATE PROOF)

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

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

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

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

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

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

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.

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

PRINCIPAL STRESSES