Bernoullis Equation

1、Newtons Second Law:Body Forces: Gravity, Magnetic Fields, etc.Surface Forces: Pressure and Shear StressesF are the forces acting on the fluid particle. m is the mass of a fluid particle, and a is the acceleration of the fluid particle.If a flow is inviscid, it has zero viscosity.Inviscid Flow:Possib

2、le Forces, F : Body Forces and Surface ForcesConservation of Linear Momentum :Let P = linear momentum,Applying the differential form,Then,Body Forces:Surface Forces:Normal Stress:Shear Stress:Then the total forces:Looking at the various sides of the differential element, the shear and normal stresse

3、s are shown for an x-face.In components:Material derivative for aForce Terms(1)For x-component,Divide by(x direction)Viscosity effects for a Newtonian FluidNormal Stresses:Shear Stresses:(1) Navier-Stokes EqnViscous Flows: Navier-Stokes EquationsFrench Mathematician, L. M. H. Navier (1758-1836) and

4、English Mathematician Sir G. G. Stokes (1819-1903) formulated the Navier-Stokes Equations by including viscous effects in the equations of motion. L. M. H. Navier (1758-1836)Sir G. G. Stokes (1819-1903)Local AccelerationAdvective Acceleration(non-linear terms)Pressure termWeight termViscous termsTer

5、ms in the x-direction:Inviscid Flow : An inviscid flow is a flow in which viscosity effects or shearing effects become negligible.The equations of motion for this type of flow then becomes the following:Eulers Equations ;In vector notation Eulers Equation:Inviscid Flow: Eulers EquationsLeonhard Eule

6、r(1707 1783)Famous Swiss mathematician who pioneered work on the relationship between pressure and flow.There is no general method of solving these equations for an analytical solution.The Eulers equation, for special situations can lead to some useful information about inviscid flow fields. Invisci

7、d Flow: Bernoulli EquationFrom the Euler Equation,First, assume steady state:Select, the vertical direction as “up”, opposite gravity:Use the vector identity:Now, rewriting the Euler Equation:Rearrange:Now, take the dot product with the differential length ds along a streamline:ds and V are parralle

8、l, is perpendicular to V, and thus to ds.We note, Now, combining the terms:Integrate:Then,1) Inviscid flow2) Steady flow3) Incompressible flow4) Along a streamlinepdBernoullis Equation:Daniel Bernoulli(1700-1782)Swiss mathematician, son of Johann Bernoulli, who showed that as the velocity of a fluid

9、 increases, the pressure decreases, a statement known as the Bernoulli principle. He won the annual prize of the French Academy ten times for work on vibrating strings, ocean tides, and the kinetic theory of gases. For one of these victories, he was ejected from his jealous fathers house, as his fat

10、her had also submitted an entry for the prize. His kinetic theory proposed that the properties of a gas could be explained by the motions of its particles. Bernoulli Equation := H (total head) = constantPressureHeadVelocityHeadPotentialHead* LStatic PressureVelocity(Dynamic) PressureHydrostatic Pres

11、sure F/L2 (work)FL/L3=Total pressure = constant = (energy grade line) p. 212 4-18 = (hydraulic grade line)= Em =Total energy( ) = constantStatic energyKineticenergyPotential energyModified Bernoulli Equation : real fluid(:head loss)(:pressure loss)(:energy loss)Application of Bernoulli Equation: Fre

12、e JetsBernoulli Equation for along a streamline between any two points:Free Jets:Following the streamline between (1) and (2):0 gage0 gage0h0 Torricellis Equation (1643):Velocity at (5):()()Cv=coefficient of velocityQ()=AVFlow Rate Measurement through OrificeFlowrate Measurements in Pipes :Using continuity equation:So, if we measure the pressure difference between (1) and (2) we have the flow rate.Z1=Z2 VenturiOrificeA2A1A0P1P2Then, Qactual= AVactual Follow a Streamline from point 1 to 2,