# Mathematical Models for FLUID MECHANICS

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Mathematical Models for FLUID MECHANICS. P M V Subbarao Associate Professor Mechanical Engineering Department IIT Delhi. Convert Ideas into A Precise Blue Print before feeling the same. A path line is the trace of the path followed by a selected fluid particle.
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Mathematical Models for FLUID MECHANICSP M V SubbaraoAssociate ProfessorMechanical Engineering DepartmentIIT DelhiConvert Ideas into A Precise Blue Print before feeling the same....A path line is the trace of the path followed by a selected fluid particleFew things to know about streamlines
• At all points the direction of the streamline is the direction of the fluid velocity: this is how they are defined.
• Close to the wall the velocity is parallel to the wall so the streamline is also parallel to the wall.
• It is also important to recognize that the position of streamlines can change with time - this is the case in unsteady flow.
• In steady flow, the position of streamlines does not change
• Because the fluid is moving in the same direction as the streamlines, fluid can not cross a streamline.
• Streamlines can not cross each other.
• If they were to cross this would indicate two different velocities at the same point.
• This is not physically possible.
• The above point implies that any particles of fluid starting on one streamline will stay on that same streamline throughout the fluid.
• A useful technique in fluid flow analysis is to consider only a part of the total fluid in isolation from the rest. This can be done by imagining a tubular surface formed by streamlines along which the fluid flows. This tubular surface is known as a streamtube. A StreamtubeA two dimensional version of the streamtubeThe "walls" of a streamtube are made of streamlines. As we have seen above, fluid cannot flow across a streamline, so fluid cannot cross a streamtube wall. The streamtube can often be viewed as a solid walled pipe. A streamtube is not a pipe - it differs in unsteady flow as the walls will move with time. And it differs because the "wall" is moving with the fluid Fluid Kinematics
• The acceleration of a fluid particle is the rate of change of its velocity.
• In the Lagrangian approach the velocity of a fluid particle is a function of time only since we have described its motion in terms of its position vector.
• In the Eulerian approach the velocity is a function of both space and time; consequently,x,y,z are f(t) since we must follow the total derivative approach in evaluating du/dt.Similarly for ay & az,In vector notation this can be written conciselyxConservation laws can be applied to an infinitesimal element or cube, or may be integrated over a large control volume.Basic Control-Volume ApproachControl Volume
• In fluid mechanics we are usually interested in a region of space, i.e, control volume and not particular systems.
• Therefore, we need to transform GDE’s from a system to a control volume.
• This is accomplished through the use of Reynolds Transport Theorem.
• Actually derived in thermodynamics for CV forms of continuity and 1st and 2nd laws.
• Flowing Fluid Through A CV
• A typical control volume for flow in an funnel-shaped pipe is bounded by the pipe wall and the broken lines.
• At time t0, all the fluid (control mass) is inside the control volume.
• The fluid that was in the control volume at time t0 will be seen at time t0 +dt as:          .The control volume at time t0 +dt      .The control mass at time t0 +dt      .The differences between the fluid (control mass) and the control volume at time t0 +dt      .III
• Consider a system and a control volume (C.V.) as follows:
• the system occupies region I and C.V. (region II) at time t0.
• Fluid particles of region – I are trying to enter C.V. (II) at time t0.
• IIIII
• the same system occupies regions (II+III) at t0 + dt
• Fluid particles of I will enter CV-II in a time dt.
• Few more fluid particles which belong to CV – II at t0 will occupy III at time t0 + dt.
• IIIIIAt time t0+dtIIIAt time t0The control volume may move as time passes.III has left CV at time t0+dtI is trying to enter CV at time t0Reynolds' Transport Theorem
• Consider a fluid scalar property b which is the amount of this property per unit mass of fluid.
• For example, b might be a thermodynamic property, such as the internal energy unit mass, or the electric charge per unit mass of fluid.
• The laws of physics are expressed as applying to a fixed mass of material.
• But most of the real devices are control volumes.
• The total amount of the property b inside the material volume M , designated by B, may be found by integrating the property per unit volume, M ,over the material volume :
• Conservation of B
• total rate of change of any extensive property B of a system(C.M.) occupying a control volume C.V. at time t is equal to the sum of
• a) the temporal rate of change of B within the C.V.
• b) the net flux of B through the control surface C.S. that surrounds the C.V.
• The change of property B of system (C.M.) during Dt is
• add and subtractThe above mentioned change has occurred over a time dt, thereforeTime averaged change in BCMisFor and infinitesimal time duration
• The rate of change of property B of the system.
• Conservation of Mass
• Let b=1, the B = mass of the system, m.
• The rate of change of mass in a control mass should be zero.Conservation of Momentum
• Let b=V, the B = momentum of the system, mV.
• The rate of change of momentum for a control mass should be equalto resultant external force.Conservation of Energy
• Let b=e, the B = Energy of the system, mV.
• The rate of change of energy of a control mass should be equalto difference of work and heat transfers.Applications of Momentum AnalysisThis is a vector equation and will have three components in x, y and zDirections.X – component of momentum equation:X – component of momentum equation:Y – component of momentum equation:Z – component of momentum equation:For a fluid, which is static or moving with uniform velocity, the Resultant forces in all directions should be individually equal to zero.X – component of momentum equation:Y – component of momentum equation:Z – component of momentum equation:For a fluid, which is static or moving with uniform velocity, the Resultant forces in all directions should be individually equal to zero.X – component of momentum equation:Y – component of momentum equation:Z – component of momentum equation:For a fluid, which is static or moving with uniform velocity, the Resultant forces in all directions should be individually equal to zero.Vector equation for momentum:Vector momentum equation per unit volume:Body force per unit volume:Gravitational force: Electrostatic PrecipitatorsElectric body force: Lorentz force densityThe total electrical force acting on a group of free charges (charged ash particles) . Supporting an applied volumetric charge density.Where= Volumetric charge density= Local electric field= Local Magnetic flux density field= Current density Electric Body Force
• This is also called electrical force density.
• This represents the body force density on a ponderable medium.
• The Coulomb force on the ions becomes an electrical body force on gaseous medium.
• This ion-drag effect on the fluid is called as electrohydrodynamic body force.
• 0Ideal Fluids….Pressure Variation in Flowing Fluids
• For fluids in motion, the pressure variation is no longer hydrostatic and is determined from and is determined from application of Newton’s 2nd Law to a fluid element.
• Various Forces in A Flow field
• For fluids in motion, various forces are important:
• Inertia Force per unit volume :
• Body Force:
• Hydrostatic Surface Force:
• Viscous Surface Force:
• Relative magnitudes of Inertial Forces and Viscous Surface Force are very important in design of basic fluid devices.
• Comparison of Magnitudes of Inertia Force and Viscous Force
• Internal vs. External Flows
• Internal flows = completely wall bounded;
• Both viscous and Inertial Forces are important.
• External flows = unbounded; i.e., at some distance from body or wall flow is uniform.
• External Flow exhibits flow-field regions such that both inviscid and viscous analysis can be used depending on the body shape.
• Ideal or Inviscid FlowsEuler’s Momentum EquationX – Momentum Equation:Euler’s Equation for One Dimensional FlowDefine an exclusive direction along theaxis of the pipe and corresponding unit direction vectorAlong a path of zero acceleration the pressure variation is hydrostaticPressure Variation Due to AccelerationFor steady flow along l – direction (stream line)Integration of above equation yieldsMomentum Transfer in A Pump
• Shaft power Disc Power Fluid Power.
• Flow Machines & Non Flow Machines.
• Compressible fluids & Incompressible Fluids.
• Rotary Machines & Reciprocating Machines.
• Flow inPump
• Rotate a cylinder containing fluid at constant speed.
• Supply continuously fluid from bottom.
• See What happens?
• Any More Ideas?
• Momentum PrincipleP M V SubbaraoAssociate ProfessorMechanical Engineering DepartmentIIT DelhiA primary basis for the design of flow devices ..Momentum EquationApplications of of the Momentum EquationInitial Setup and Signs
• 1. Jet deflected by a plate or a vane
• 2. Flow through a nozzle
• 3. Forces on bends
• 4. Problems involving non-uniform velocity distribution
• 5. Motion of a rocket
• 6. Force on rectangular sluice gate
• 7. Water hammer
• Navier-Stokes EquationsDifferential form of momentum equationX-component:Y-component:z-component:Applications of Momentum EquationGeneration of Motive Power Through Newton’s Second LawConsider a jet of gas/steam/water turned through an angleJet Deflected by a Plate or BladeCV and CS arefor jet so that Fxand Fy are blade reactions forces on fluid.Steady 2 Dimensional FlowX-component:Y-component:Continuity equation: Steady 2 Dimensional Invisicid FlowX-component:Y-component:Continuity equation: Inlet conditions : u = U & v = 0 Pure Impulse Blade Pressure remains constant along the entire jet.X-component:Y-component:Continuity equation:
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