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NIU PHYS 600 - Navier-Stokes

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Navier-StokesPressure ForceEquation of MotionEuler’s EquationViscosityStrain Rate TensorStress and StrainNavier-Stokes EquationNavier-StokesNavier-StokesPressure ForcePressure ForceEach volume element in a Each volume element in a fluid is subject to force due fluid is subject to force due to pressure.to pressure.•Assume a rectangular boxAssume a rectangular box•Pressure force density is the Pressure force density is the gradient of pressuregradient of pressureVpVxpzyxxpFxVpFVzpzVypyVxpxFˆˆˆxyzEquation of MotionEquation of MotionA fluid element may be A fluid element may be subject to an external force.subject to an external force.•Write as a force densityWrite as a force density•Assume uniform over small Assume uniform over small element.element.The equation of motion uses The equation of motion uses pressure and external force.pressure and external force.•Write form as force densityWrite form as force density•Use stress tensor instead of Use stress tensor instead of pressure forcepressure forceThis is Cauchy’s equation.This is Cauchy’s equation.22222xlxlxlkFVfF FdtvdmVpVfdtvdVfpdtvdfdtvd PEuler’s EquationEuler’s EquationDivide by the density.Divide by the density.•Motion in units of force Motion in units of force density per unit mass.density per unit mass.The time derivative can be The time derivative can be expanded to give a partial expanded to give a partial differential equation.differential equation.•Pressure or stress tensorPressure or stress tensorThis is Euler’s equation of This is Euler’s equation of motion for a fluid.motion for a fluid.fpdtvd1 fpvvtv 1 fvvtvP1ViscosityViscosityA static fluid cannot support A static fluid cannot support a shear.a shear.A moving fluid with viscosity A moving fluid with viscosity can have shear.can have shear.•Dynamic viscosity Dynamic viscosity •Kinematic viscosity Kinematic viscosity yvxFdydvSFxxStrain Rate TensorStrain Rate TensorRate of strain measures the Rate of strain measures the amount of deformation in amount of deformation in response to a stress.response to a stress.•Forms symmetric tensorForms symmetric tensor•Based on the velocity Based on the velocity gradientgradientzvyvzvxvzvyvzvyvxvyvxvzvxvyvxvzzyzxzyyyxzxyxx212121212121EStress and StrainStress and StrainThere is a general relation There is a general relation between stress and strainbetween stress and strain•Constants Constants a, ba, b include include viscosityviscosityAn incompressible fluid has An incompressible fluid has no velocity divergence.no velocity divergence.1EP ba 2avpb321EP vp3221EP p2Navier-Stokes EquationNavier-Stokes EquationThe stress and strain The stress and strain relations can be combined relations can be combined with the equation of motion.with the equation of motion.Reduces to Euler for no Reduces to Euler for no viscosity.viscosity.next


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NIU PHYS 600 - Navier-Stokes

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