2Eulerian ViewIn the Lagrangian view each body is described at each point in space.Difficult for a fluid with many particlesIn the Eulerian view the points in space are described.Bulk properties of density and velocity
3Streamlines A streamline follows the tangents to fluid velocity. Lagrangian viewDashed lines at leftStream tube follows an areaA streakline (blue) shows the current position of a particle starting at a fixed point.A pathline (red) tracks an individual particle.Wikimedia image
4Fluid Change A change in a property like pressure depends on the view. In the Lagrangian view the total time derivative depends on position and time.The Eulerian view uses just the partial derivative with time.Points in space are fixed
5Jacobian TensorA general coordinate transformation can be expressed as a tensor.Partial derivatives between two systemsJacobian NN real matrixInverse for nonsingular JacobiansCartesian coordinate transformations have an additional symmetry.Not generally true for other transformations
6Volume ElementAn infinitessimal volume element is defined by coordinates.dV = dx1dx2dx3Transform a volume element from other coordinates.components from the transformationThe Jacobian determinant is the ratio of the volume elements.x3x2x1
7Compressibility A change in pressure on a fluid can cause deformation. Compressibility measures the relationship between volume change and pressure.Usually expressed as a bulk modulus BIdeal liquids are incompressible.Vp
8Volume Change Consider a fixed amount of fluid in a volume dV. Cubic, Cartesian geometryDimensions dx, dy, dzThe change in dV is related to the divergence.Incompressible fluids - no velocity divergence
9Balance EquationsThe equation of motion for an arbitrary density in a volume is a balance equation.Current J through the sides of the volumeSource s inside the volumeAdditional balance equations describe conservation of mass, momentum and energy.No sources for conserved quantities
10Mass Conservation A mass element must remain constant in time. Conservation of massCombine with divergence relationship.Write in terms of a point in space.
11Pressure ForceEach volume element in a fluid is subject to force due to pressure.Assume a rectangular boxPressure force density is the gradient of pressuredVdzdypdx
12Equation of MotionA fluid element may be subject to an external force.Write as a force densityAssume uniform over small element.The equation of motion uses pressure and external force.Write form as force densityUse stress tensor instead of pressure forceThis is Cauchy’s equation.
13Euler’s Equation Divide by the density. Motion in units of force density per unit mass.The time derivative can be expanded to give a partial differential equation.Pressure or stress tensorThis is Euler’s equation of motion for a fluid.
14Momentum Conservation The momentum is found for a small volume.Euler equation with force densityMass is constantMomentum is not generally constant.Effect of pressureThe total momentum change is found by integration.Gauss’ law
15Energy Conservation The kinetic energy is related to the momentum. Right side is energy densitySome change in energy is related to pressure and volume.Total time derivativeVolume change related to velocity divergence
16Work Supplied The work supplied by expansion depends on pressure. Potential energy associated with change in volumeThis potential energy change goes into the energy conservation equation.
17Bernoulli’s Equation Gravity is an external force. Gradient of potentialNo time dependenceThe result is Bernoulli’s equation.Steady flow no time changeIntegrate to a constant
18Strain Rate TensorRate of strain measures the amount of deformation in response to a stress.Forms symmetric tensorBased on the velocity gradient
19Stress and StrainThere is a general relation between stress and strainConstants a, b include viscosityAn incompressible fluid has no velocity divergence.
20Navier-Stokes Equation The stress and strain relations can be combined with the equation of motion.Reduces to Euler for no viscosity.
21Bernoulli RederivedMake assumptions about flow to approximate fluid motion.IncompressibleInviscidIrrotationalForce from gravityApply to Navier-StokesThe result is Bernoulli’s equation.