FluidsEulerian ViewCompressibilityFluid ChangeVolume ChangeContinuity EquationStressForce in FluidsStress TensorForce DensityFluidsFluidsEulerian ViewEulerian ViewIn a Lagrangian view each In a Lagrangian view each body is described at each body is described at each point in space.point in space.•Difficult for a fluid with many Difficult for a fluid with many particles.particles.In an Eulerian view the In an Eulerian view the points in space are points in space are described.described.•Bulk properties of density Bulk properties of density and velocityand velocity),(0trr),( tr),( trvCompressibilityCompressibilityA change in pressure on a A change in pressure on a fluid can cause deformation.fluid can cause deformation.Compressibility measures Compressibility measures the relationship between the relationship between volume change and volume change and pressure.pressure.•Usually expressed as a bulk Usually expressed as a bulk modulus modulus BBIdeal liquids are Ideal liquids are incompressible.incompressible.VppVV1VpVBFluid ChangeFluid ChangeA change in a property like A change in a property like pressure depends on the pressure depends on the view.view.In a Lagrangian view the In a Lagrangian view the total time derivative depends total time derivative depends on position and time.on position and time.An Eulerian view is just the An Eulerian view is just the partial derivative with time.partial derivative with time.•Points are fixedPoints are fixed22222xlxlxlkFpvtpdtdpdtdzzpdtdyypdtdxxptpdtdp vtdtdtpdtdpconstrVolume ChangeVolume ChangeConsider a fixed amount of Consider a fixed amount of fluid in a volume fluid in a volume VV..•Cubic, Cartesian geometryCubic, Cartesian geometry•Dimensions Dimensions xx, , yy, , zz..The change in The change in VV is related is related to the divergence.to the divergence.•Incompressible fluids must Incompressible fluids must have no velocity divergencehave no velocity divergencezzvzdtdyyvydtdxxvxdtdzyxzyxzvyvxvVdtdzyxVvVdtdContinuity EquationContinuity EquationA mass element must remain A mass element must remain constant in time.constant in time.•Conservation of massConservation of massCombine with divergence Combine with divergence relationship.relationship.Write in terms of a point in Write in terms of a point in space.space.Vm 0 Vdtdmdtd0VvVdtddtVdVdtd0 vdtd0vvt0)( vtStressStressA stress measures the A stress measures the surface force per unit area.surface force per unit area.•A normal stress acts normal A normal stress acts normal to a surface.to a surface.•A shear stress acts parallel A shear stress acts parallel to a surface.to a surface.A fluid at rest cannot support A fluid at rest cannot support a shear stress.a shear stress.FFAAForce in FluidsForce in FluidsConsider a small prism of Consider a small prism of fluid in a continuous fluid.fluid in a continuous fluid.•Describe the stress Describe the stress PP at any at any point. point. •Normal area vectors Normal area vectors SS form form a triangle.a triangle.The stress function is linear.The stress function is linear.1Sd2Sd21SdSd)(2SdP)(1SdP)(21SdSdP)()( SdPcScdP)()( SdPSdP)()()(2121SdSdPSdPSdPStress TensorStress TensorRepresent the stress Represent the stress function by a tensor.function by a tensor.•Symmetric Symmetric •Specified by 6 componentsSpecified by 6 componentsIf the only stress is pressure If the only stress is pressure the tensor is diagonal.the tensor is diagonal.The total force is found by The total force is found by integration.integration.SdSdPP)(SdpSdSdP 1P)(SSdFPForce DensityForce DensityThe force on a closed The force on a closed volume can be found volume can be found through Gauss’ law.through Gauss’ law.•Use outward unit vectorsUse outward unit vectorsA force density due to stress A force density due to stress can be defined from the can be defined from the tensor.tensor.•Due to differences in stress Due to differences in stress as a function of positionas a function of positionnextSdSnFˆPVdVF