Revision #1423 → #1766 · back to history
modifiedCauchy momentum equation753c615b4727
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| anchors | [{"section":"General continuum equations","snippet":"can be derived as a particular form of the Cauchy momentum equation"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {\\mathrm {D} \\mathbf {u} }{\\mathrm {D} t}}={\\frac {1}{\\rho }}\\nabla \\cdot {\\boldsymbol {\\sigma }}+\\mathbf {a} .}"}] | — |
modifiedMass continuity equationfd6c5202ea88
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| anchors | [{"section":"General continuum equations","snippet":"we can use the mass continuity equation"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}&{\\frac {\\mathbf {D} m}{\\mathbf {Dt} }}=\\iiint \\limits _{V}\\left({\\frac {\\mathbf {D} \\rho }{\\mathbf {Dt} }}+\\rho (\\nabla \\cdot \\mathbf {u} )\\right)\\,dV\\\\[5pt]&{\\frac {\\mathbf {D} \\rho }{\\mathbf {Dt} }}+\\rho (\\nabla \\cdot \\mathbf {u} )={\\frac {\\partial \\rho }{\\partial t}}+(\\nabla \\rho )\\cdot \\mathbf {u} +\\rho (\\nabla \\cdot \\mathbf {u} )={\\frac {\\partial \\rho }{\\partial t}}+\\nabla \\cdot (\\rho \\mathbf {u} )=0\\end{aligned}}}"}] | — |
addedMaterial derivative1a786e178a6f
addedDeviatoric stress tensorb87562246ffd
addedRate-of-strain tensora38cc5ae79af
modifiedBulk viscosity6b4bb2378ee6
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| anchors | [{"section":"Compressible flow","snippet":"Introducing the bulk viscosity"},{"type":"math_alttext","value":"{\\displaystyle \\zeta \\equiv \\lambda +{\\tfrac {2}{3}}\\mu ,}"}] | — |
modifiedStokes hypothesisea3fcbb0c917
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| anchors | [{"section":"Compressible flow","snippet":"is called as the Stokes hypothesis"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}&\\nabla \\cdot (\\nabla \\cdot \\mathbf {u} )\\mathbf {I} =\\nabla (\\nabla \\cdot \\mathbf {u} ),\\\\&{\\bar {p}}\\equiv p-\\zeta \\,\\nabla \\cdot \\mathbf {u} ,\\end{aligned}}}"}] | — |
modifiedLamb vectordfb36ffc7742
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| anchors | [{"section":"Compressible flow","snippet":"is known as the Lamb vector"},{"type":"math_alttext","value":"{\\displaystyle \\mathbf {u} \\cdot \\nabla \\mathbf {u} =(\\nabla \\times \\mathbf {u} )\\times \\mathbf {u} +{\\tfrac {1}{2}}\\nabla \\mathbf {u} ^{2},}"}] | — |
addedIsochoric (solenoidal) incompressible flowa9d222c02718
modifiedDivergence of deviatoric stress (uniform viscosity)8857024b1aaf
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| anchors | [{"section":"Incompressible flow","snippet":"The divergence of the deviatoric stress in case of uniform viscosity is given by"},{"type":"math_alttext","value":"{\\displaystyle \\nabla \\cdot {\\boldsymbol {\\tau }}=2\\mu \\nabla \\cdot {\\boldsymbol {\\varepsilon }}=\\mu \\nabla \\cdot \\left(\\nabla \\mathbf {u} +\\nabla \\mathbf {u} ^{\\mathsf {T}}\\right)=\\mu \\,\\nabla ^{2}\\mathbf {u} }"}] | — |
modifiedLaminar velocity profile40fa968bf143
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| anchors | [{"section":"Incompressible flow","snippet":"Finally this gives the velocity profile"},{"type":"math_alttext","value":"{\\displaystyle u={\\frac {1}{2\\mu }}{\\frac {\\mathrm {d} P}{\\mathrm {d} x}}\\left(y^{2}-h^{2}\\right)}"}] | — |
modifiedViscosity as momentum diffusion (vector Laplacian)4d1258db38db
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| mathlib.module | Mathlib.Analysis.InnerProductSpace.Laplacian | Mathlib.Analysis.Distribution.DerivNotation |
| note | Mathlib defines the (vector-valued) Laplacian operator, but not its identification with the viscous momentum-diffusion term of Navier–Stokes. | Mathlib defines a general Laplacian operator (and an InnerProductSpace instance), but not its identification with the viscous momentum-diffusion term of Navier–Stokes. |
modifiedNS equation with conservative external field71e801b6fe55
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| anchors | [{"section":"Incompressible flow","snippet":"arriving to the incompressible Navier–Stokes equation with conservative external field"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {\\partial \\mathbf {u} }{\\partial t}}+(\\mathbf {u} \\cdot \\nabla )\\mathbf {u} -\\nu \\,\\nabla ^{2}\\mathbf {u} =-\\nabla h.}"}] | — |
modifiedHelmholtz decomposition of incompressible NS295b696d9613
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| anchors | [{"section":"Incompressible flow","snippet":"The incompressible Navier–Stokes equation is composite, the sum of two orthogonal equations"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}{\\frac {\\partial \\mathbf {u} }{\\partial t}}&=\\Pi ^{S}\\left(-(\\mathbf {u} \\cdot \\nabla )\\mathbf {u} +\\nu \\,\\nabla ^{2}\\mathbf {u} \\right)+\\mathbf {f} ^{S}\\\\\\rho ^{-1}\\,\\nabla p&=\\Pi ^{I}\\left(-(\\mathbf {u} \\cdot \\nabla )\\mathbf {u} +\\nu \\,\\nabla ^{2}\\mathbf {u} \\right)+\\mathbf {f} ^{I}\\end{aligned}}}"}] | — |
modifiedProjection operator in 3Dd6d6badeeb70
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| anchors | [{"section":"Incompressible flow","snippet":"The explicit functional form of the projection operator in 3D is found from the Helmholtz theorem"},{"type":"math_alttext","value":"{\\displaystyle \\Pi ^{S}\\,\\mathbf {F} (\\mathbf {r} )={\\frac {1}{4\\pi }}\\nabla \\times \\int {\\frac {\\nabla ^{\\prime }\\times \\mathbf {F} (\\mathbf {r} ')}{|\\mathbf {r} -\\mathbf {r} '|}}\\,\\mathrm {d} V',\\quad \\Pi ^{I}=1-\\Pi ^{S}}"}] | — |
modifiedWeak/variational form equivalencebb611101fee4
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| anchors | [{"section":"Incompressible flow","snippet":"An equivalent weak or variational form of the equation, proved to produce the same velocity solution as the Navier–Stokes equation"},{"type":"math_alttext","value":"{\\displaystyle \\left(\\mathbf {w} ,{\\frac {\\partial \\mathbf {u} }{\\partial t}}\\right)=-{\\bigl (}\\mathbf {w} ,\\left(\\mathbf {u} \\cdot \\nabla \\right)\\mathbf {u} {\\bigr )}-\\nu \\left(\\nabla \\mathbf {w} :\\nabla \\mathbf {u} \\right)+\\left(\\mathbf {w} ,\\mathbf {f} ^{S}\\right)}"}] | — |
modifiedStrong form of incompressible NS1e497e38a663
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| anchors | [{"section":"Strong form","snippet":"Consider the incompressible Navier–Stokes equations for a Newtonian fluid of constant density"},{"type":"math_alttext","value":"{\\displaystyle \\Omega \\subset \\mathbb {R} ^{d}\\quad (d=2,3)}"},{"type":"math_alttext","value":"{\\displaystyle \\partial \\Omega =\\Gamma _{D}\\cup \\Gamma _{N},}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{cases}\\rho {\\dfrac {\\partial \\mathbf {u} }{\\partial t}}+\\rho (\\mathbf {u} \\cdot \\nabla )\\mathbf {u} -\\nabla \\cdot {\\boldsymbol {\\sigma }}(\\mathbf {u} ,p)=\\mathbf {f} &{\\text{ in }}\\Omega \\times (0,T)\\\\\\nabla \\cdot \\mathbf {u} =0&{\\text{ in }}\\Omega \\times (0,T)\\\\\\mathbf {u} =\\mathbf {g} &{\\text{ on }}\\Gamma _{D}\\times (0,T)\\\\{\\boldsymbol {\\sigma }}(\\mathbf {u} ,p){\\hat {\\mathbf {n} }}=\\mathbf {h} &{\\text{ on }}\\Gamma _{N}\\times (0,T)\\\\\\mathbf {u} (0)=\\mathbf {u} _{0}&{\\text{ in }}\\Omega \\times \\{0\\}\\end{cases}}}"},{"type":"math_alttext","value":"{\\displaystyle {\\boldsymbol {\\sigma }}(\\mathbf {u} ,p)=-p\\mathbf {I} +2\\mu {\\boldsymbol {\\varepsilon }}(\\mathbf {u} ).}"},{"type":"math_alttext","value":"{\\displaystyle {\\boldsymbol {\\varepsilon }}(\\mathbf {u} )={\\tfrac {1}{2}}\\left(\\left(\\nabla \\mathbf {u} \\right)+\\left(\\nabla \\mathbf {u} \\right)^{\\mathsf {T}}\\right).}"},{"type":"math_alttext","value":"{\\displaystyle \\nabla \\cdot \\left(\\nabla \\mathbf {f} \\right)^{\\mathsf {T}}=\\nabla (\\nabla \\cdot \\mathbf {f} )}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\nabla \\cdot {\\boldsymbol {\\sigma }}(\\mathbf {u} ,p)&=\\nabla \\cdot \\left(-p\\mathbf {I} +2\\mu {\\boldsymbol {\\varepsilon }}(\\mathbf {u} )\\right)\\\\&=-\\nabla p+2\\mu \\nabla \\cdot {\\boldsymbol {\\varepsilon }}(\\mathbf {u} )\\\\&=-\\nabla p+2\\mu \\nabla \\cdot \\left[{\\tfrac {1}{2}}\\left(\\left(\\nabla \\mathbf {u} \\right)+\\left(\\nabla \\mathbf {u} \\right)^{\\mathsf {T}}\\right)\\right]\\\\&=-\\nabla p+\\mu \\left(\\Delta \\mathbf {u} +\\nabla \\cdot \\left(\\nabla \\mathbf {u} \\right)^{\\mathsf {T}}\\right)\\\\&=-\\nabla p+\\mu {\\bigl (}\\Delta \\mathbf {u} +\\nabla \\underbrace {(\\nabla \\cdot \\mathbf {u} )} _{=0}{\\bigr )}=-\\nabla p+\\mu \\,\\Delta \\mathbf {u} .\\end{aligned}}}"},{"type":"math_alttext","value":"{\\displaystyle {\\boldsymbol {\\sigma }}(\\mathbf {u} ,p){\\hat {\\mathbf {n} }}={\\bigl (}-p\\mathbf {I} +2\\mu {\\boldsymbol {\\varepsilon }}(\\mathbf {u} ){\\bigr )}{\\hat {\\mathbf {n} }}=-p{\\hat {\\mathbf {n} }}+\\mu {\\frac {\\partial {\\boldsymbol {u}}}{\\partial {\\hat {\\mathbf {n} }}}}.}"}] | — |
modifiedWeak form of NS equationsa6e4ddfafd15
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| anchors | [{"section":"Weak form","snippet":"In order to find the weak form of the Navier–Stokes equations"},{"type":"math_alttext","value":"{\\displaystyle \\rho {\\frac {\\partial \\mathbf {u} }{\\partial t}}-\\mu \\Delta \\mathbf {u} +\\rho (\\mathbf {u} \\cdot \\nabla )\\mathbf {u} +\\nabla p=\\mathbf {f} }"},{"type":"math_alttext","value":"{\\displaystyle \\int \\limits _{\\Omega }\\rho {\\frac {\\partial \\mathbf {u} }{\\partial t}}\\cdot \\mathbf {v} -\\int \\limits _{\\Omega }\\mu \\Delta \\mathbf {u} \\cdot \\mathbf {v} +\\int \\limits _{\\Omega }\\rho (\\mathbf {u} \\cdot \\nabla )\\mathbf {u} \\cdot \\mathbf {v} +\\int \\limits _{\\Omega }\\nabla p\\cdot \\mathbf {v} =\\int \\limits _{\\Omega }\\mathbf {f} \\cdot \\mathbf {v} }"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}-\\int \\limits _{\\Omega }\\mu \\Delta \\mathbf {u} \\cdot \\mathbf {v} &=\\int \\limits _{\\Omega }\\mu \\nabla \\mathbf {u} \\cdot \\nabla \\mathbf {v} -\\int \\limits _{\\partial \\Omega }\\mu {\\frac {\\partial \\mathbf {u} }{\\partial {\\hat {\\mathbf {n} }}}}\\cdot \\mathbf {v} \\\\\\int \\limits _{\\Omega }\\nabla p\\cdot \\mathbf {v} &=-\\int \\limits _{\\Omega }p\\nabla \\cdot \\mathbf {v} +\\int \\limits _{\\partial \\Omega }p\\mathbf {v} \\cdot {\\hat {\\mathbf {n} }}\\end{aligned}}}"}] | — |
modifiedDiscrete form of governing equationd9168a2f1c84
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| anchors | [{"section":"Discrete velocity","snippet":"the discrete form of the governing equation is"},{"type":"math_alttext","value":"{\\displaystyle \\left(\\mathbf {w} _{i},{\\frac {\\partial \\mathbf {u} _{j}}{\\partial t}}\\right)=-{\\bigl (}\\mathbf {w} _{i},\\left(\\mathbf {u} \\cdot \\nabla \\right)\\mathbf {u} _{j}{\\bigr )}-\\nu \\left(\\nabla \\mathbf {w} _{i}:\\nabla \\mathbf {u} _{j}\\right)+\\left(\\mathbf {w} _{i},\\mathbf {f} ^{S}\\right).}"}] | — |
modifiedOrthogonality of gradient and curl in 2Da717fea56b6c
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| anchors | [{"section":"Discrete velocity","snippet":"the gradient and curl of a scalar are clearly orthogonal"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\nabla \\varphi &=\\left({\\frac {\\partial \\varphi }{\\partial x}},\\,{\\frac {\\partial \\varphi }{\\partial y}}\\right)^{\\mathsf {T}},\\\\[5pt]\\nabla \\times \\varphi &=\\left({\\frac {\\partial \\varphi }{\\partial y}},\\,-{\\frac {\\partial \\varphi }{\\partial x}}\\right)^{\\mathsf {T}}.\\end{aligned}}}"}] | — |
modifiedDiscrete weak pressure gradient equation8eb993235ac5
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| anchors | [{"section":"Pressure recovery","snippet":"The discrete weak equation for the pressure gradient is"},{"type":"math_alttext","value":"{\\displaystyle (\\mathbf {g} _{i},\\nabla p)=-{\\bigl (}\\mathbf {g} _{i},\\left(\\mathbf {u} \\cdot \\nabla \\right)\\mathbf {u} _{j}{\\bigr )}-\\nu \\left(\\nabla \\mathbf {g} _{i}:\\nabla \\mathbf {u} _{j}\\right)+\\left(\\mathbf {g} _{i},\\mathbf {f} ^{I}\\right)}"}] | — |
addedCoriolis accelerationefd2280ee325
addedCentrifugal accelerationf3cd6455f3b5
modifiedIncompressible fluidc7a3f428b63b
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| anchors | [{"section":"Continuity equation for incompressible fluid","snippet":"is constant is called incompressible"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}{\\frac {\\mathbf {D} m}{\\mathbf {Dt} }}&={\\iiint \\limits _{V}}\\left({{\\frac {\\mathbf {D} \\rho }{\\mathbf {Dt} }}+\\rho (\\nabla \\cdot \\mathbf {u} )}\\right)dV\\\\{\\frac {\\mathbf {D} \\rho }{\\mathbf {Dt} }}+\\rho (\\nabla \\cdot {\\mathbf {u} })&={\\frac {\\partial \\rho }{\\partial t}}+({\\nabla \\rho })\\cdot {\\mathbf {u} }+{\\rho }(\\nabla \\cdot \\mathbf {u} )={\\frac {\\partial \\rho }{\\partial t}}+\\nabla \\cdot ({\\rho \\mathbf {u} })=0\\end{aligned}}}"},{"type":"math_alttext","value":"{\\displaystyle \\rho (\\nabla {\\cdot }{\\mathbf {u} })=0}"},{"type":"math_alttext","value":"{\\displaystyle (\\nabla {\\cdot {\\mathbf {u} }})=0}"},{"type":"math_alttext","value":"{\\displaystyle \\nabla ^{2}\\mathbf {u} =-{\\bigl (}\\nabla \\times (\\nabla \\times \\mathbf {u} ){\\bigr )}=-(\\nabla \\times {\\boldsymbol {\\omega }})}"}] | — |
modifiedContinuity reduces to divergence-free conditione822dc0a4057
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| anchors | [{"section":"Continuity equation for incompressible fluid","snippet":"the continuity equation for an incompressible fluid reduces further to"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}{\\frac {\\mathbf {D} m}{\\mathbf {Dt} }}&={\\iiint \\limits _{V}}\\left({{\\frac {\\mathbf {D} \\rho }{\\mathbf {Dt} }}+\\rho (\\nabla \\cdot \\mathbf {u} )}\\right)dV\\\\{\\frac {\\mathbf {D} \\rho }{\\mathbf {Dt} }}+\\rho (\\nabla \\cdot {\\mathbf {u} })&={\\frac {\\partial \\rho }{\\partial t}}+({\\nabla \\rho })\\cdot {\\mathbf {u} }+{\\rho }(\\nabla \\cdot \\mathbf {u} )={\\frac {\\partial \\rho }{\\partial t}}+\\nabla \\cdot ({\\rho \\mathbf {u} })=0\\end{aligned}}}"},{"type":"math_alttext","value":"{\\displaystyle \\rho (\\nabla {\\cdot }{\\mathbf {u} })=0}"},{"type":"math_alttext","value":"{\\displaystyle (\\nabla {\\cdot {\\mathbf {u} }})=0}"},{"type":"math_alttext","value":"{\\displaystyle \\nabla ^{2}\\mathbf {u} =-{\\bigl (}\\nabla \\times (\\nabla \\times \\mathbf {u} ){\\bigr )}=-(\\nabla \\times {\\boldsymbol {\\omega }})}"}] | — |
addedVorticity66199c11fac3
modifiedCurl eliminates pressure681bcd0ff043
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| anchors | [{"section":"Stream function for incompressible 2D fluid","snippet":"Taking the curl of the incompressible Navier–Stokes equation results in the elimination of pressure"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\rho \\left({\\frac {\\partial u_{x}}{\\partial t}}+u_{x}{\\frac {\\partial u_{x}}{\\partial x}}+u_{y}{\\frac {\\partial u_{x}}{\\partial y}}\\right)&=-{\\frac {\\partial p}{\\partial x}}+\\mu \\left({\\frac {\\partial ^{2}u_{x}}{\\partial x^{2}}}+{\\frac {\\partial ^{2}u_{x}}{\\partial y^{2}}}\\right)+\\rho g_{x}\\\\\\rho \\left({\\frac {\\partial u_{y}}{\\partial t}}+u_{x}{\\frac {\\partial u_{y}}{\\partial x}}+u_{y}{\\frac {\\partial u_{y}}{\\partial y}}\\right)&=-{\\frac {\\partial p}{\\partial y}}+\\mu \\left({\\frac {\\partial ^{2}u_{y}}{\\partial x^{2}}}+{\\frac {\\partial ^{2}u_{y}}{\\partial y^{2}}}\\right)+\\rho g_{y}.\\end{aligned}}}"}] | — |
modifiedStream function (2D)6164080abb6c
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| anchors | [{"section":"Stream function for incompressible 2D fluid","snippet":"results in mass continuity being unconditionally satisfied"},{"type":"math_alttext","value":"{\\displaystyle u_{x}={\\frac {\\partial \\psi }{\\partial y}};\\quad u_{y}=-{\\frac {\\partial \\psi }{\\partial x}}}"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {\\partial }{\\partial t}}\\left(\\nabla ^{2}\\psi \\right)+{\\frac {\\partial \\psi }{\\partial y}}{\\frac {\\partial }{\\partial x}}\\left(\\nabla ^{2}\\psi \\right)-{\\frac {\\partial \\psi }{\\partial x}}{\\frac {\\partial }{\\partial y}}\\left(\\nabla ^{2}\\psi \\right)=\\nu \\nabla ^{4}\\psi }"}] | — |
modified2D biharmonic operator9b175e5dd8d0
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| anchors | [{"section":"Stream function for incompressible 2D fluid","snippet":"is the 2D biharmonic operator"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {\\partial }{\\partial t}}\\left(\\nabla ^{2}\\psi \\right)+{\\frac {\\partial \\left(\\psi ,\\nabla ^{2}\\psi \\right)}{\\partial (y,x)}}=\\nu \\nabla ^{4}\\psi .}"}] | — |
modifiedStream function equation via Jacobian6627f22ec06d
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| anchors | [{"section":"Stream function for incompressible 2D fluid","snippet":"We can also express this compactly using the Jacobian determinant"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {\\partial }{\\partial t}}\\left(\\nabla ^{2}\\psi \\right)+{\\frac {\\partial \\left(\\psi ,\\nabla ^{2}\\psi \\right)}{\\partial (y,x)}}=\\nu \\nabla ^{4}\\psi .}"}] | — |
addedReynolds number67af34097e73
modifiedParallel flow between plates03c09e006676
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| anchors | [{"section":"Parallel flow","snippet":"Assume steady, parallel, one-dimensional, non-convective pressure-driven flow between parallel plates"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {\\mathrm {d} ^{2}u}{\\mathrm {d} y^{2}}}=-1;\\quad u(0)=u(1)=0.}"}] | — |
modifiedRadial flow between platesc3c0b5012eb1
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| anchors | [{"section":"Radial flow","snippet":"the radial flow between parallel plates"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {\\mathrm {d} ^{2}f}{\\mathrm {d} z^{2}}}+Rf^{2}=-1;\\quad f(-1)=f(1)=0.}"}] | — |
modified2D polar exact solution75e09b486ae6
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| anchors | [{"section":"Exact solutions of the Navier–Stokes equations","snippet":"flow in polar coordinates ( r , φ )"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}u_{r}&={\\frac {A}{r}},\\\\u_{\\varphi }&=B\\left({\\frac {1}{r}}-r^{{\\frac {A}{\\nu }}+1}\\right),\\\\p&=-{\\frac {A^{2}+B^{2}}{2r^{2}}}-{\\frac {2B^{2}\\nu r^{\\frac {A}{\\nu }}}{A}}+{\\frac {B^{2}r^{\\left({\\frac {2A}{\\nu }}+2\\right)}}{{\\frac {2A}{\\nu }}+2}}\\end{aligned}}}"}] | — |
modifiedCartesian zero-viscosity solutionf7195786f859
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| anchors | [{"section":"Exact solutions of the Navier–Stokes equations","snippet":"In Cartesian coordinates, when the viscosity is zero"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\mathbf {v} (x,y)&={\\frac {1}{x^{2}+y^{2}}}{\\begin{pmatrix}Ax+By\\\\Ay-Bx\\end{pmatrix}},\\\\p(x,y)&=-{\\frac {A^{2}+B^{2}}{2\\left(x^{2}+y^{2}\\right)}}\\end{aligned}}}"}] | — |
modified3D radial Cartesian solutionce618651be35
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|---|
| anchors | [{"section":"Exact solutions of the Navier–Stokes equations","snippet":"radial flow in Cartesian coordinates ( x , y , z )"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\mathbf {v} (x,y,z)&={\\frac {A}{x^{2}+y^{2}+z^{2}}}{\\begin{pmatrix}x\\\\y\\\\z\\end{pmatrix}},\\\\p(x,y,z)&=-{\\frac {A^{2}}{2\\left(x^{2}+y^{2}+z^{2}\\right)}}.\\end{aligned}}}"}] | — |
modifiedHopf fibration vortex solution2bb490350d83
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| anchors | [{"section":"A three-dimensional steady-state vortex solution","snippet":"comes from considering the flow along the lines of a Hopf fibration"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\rho (x,y,z)&={\\frac {3B}{r^{2}+x^{2}+y^{2}+z^{2}}}\\\\p(x,y,z)&={\\frac {-A^{2}B}{\\left(r^{2}+x^{2}+y^{2}+z^{2}\\right)^{3}}}\\\\\\mathbf {u} (x,y,z)&={\\frac {A}{\\left(r^{2}+x^{2}+y^{2}+z^{2}\\right)^{2}}}{\\begin{pmatrix}2(-ry+xz)\\\\2(rx+yz)\\\\r^{2}-x^{2}-y^{2}+z^{2}\\end{pmatrix}}\\\\g&=0\\\\\\mu &=0\\end{aligned}}}"}] | — |
modifiedAlternate pressure/density choicee28331dd08b0
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|---|
| anchors | [{"section":"A three-dimensional steady-state vortex solution","snippet":"Another choice of pressure and density with the same velocity vector above"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\rho (x,y,z)&={\\frac {20B\\left(x^{2}+y^{2}\\right)}{\\left(r^{2}+x^{2}+y^{2}+z^{2}\\right)^{3}}}\\\\p(x,y,z)&={\\frac {-A^{2}B}{\\left(r^{2}+x^{2}+y^{2}+z^{2}\\right)^{4}}}+{\\frac {-4A^{2}B\\left(x^{2}+y^{2}\\right)}{\\left(r^{2}+x^{2}+y^{2}+z^{2}\\right)^{5}}}.\\end{aligned}}}"}] | — |
modifiedSolutions for polynomial densitya457dc81854f
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|---|
| anchors | [{"section":"A three-dimensional steady-state vortex solution","snippet":"there are simple solutions for any polynomial function f where the density is"},{"type":"math_alttext","value":"{\\displaystyle \\rho (x,y,z)={\\frac {1}{r^{2}+x^{2}+y^{2}+z^{2}}}f\\left({\\frac {x^{2}+y^{2}}{\\left(r^{2}+x^{2}+y^{2}+z^{2}\\right)^{2}}}\\right).}"}] | — |
modifiedPeriodic 3D viscous solutions1c5e34f4998d
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|---|
| anchors | [{"section":"Viscous three-dimensional periodic solutions","snippet":"Two examples of periodic fully-three-dimensional viscous solutions are described in"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}u_{x}&={\\frac {4{\\sqrt {2}}}{3{\\sqrt {3}}}}\\,U_{0}\\left[\\,\\sin \\left(kx-{\\frac {\\pi }{3}}\\right)\\cos \\left(ky+{\\frac {\\pi }{3}}\\right)\\sin \\left(kz+{\\frac {\\pi }{2}}\\right)-\\cos \\left(kz-{\\frac {\\pi }{3}}\\right)\\sin \\left(kx+{\\frac {\\pi }{3}}\\right)\\sin \\left(ky+{\\frac {\\pi }{2}}\\right)\\,\\right]e^{-3\\nu k^{2}t}\\\\u_{y}&={\\frac {4{\\sqrt {2}}}{3{\\sqrt {3}}}}\\,U_{0}\\left[\\,\\sin \\left(ky-{\\frac {\\pi }{3}}\\right)\\cos \\left(kz+{\\frac {\\pi }{3}}\\right)\\sin \\left(kx+{\\frac {\\pi }{2}}\\right)-\\cos \\left(kx-{\\frac {\\pi }{3}}\\right)\\sin \\left(ky+{\\frac {\\pi }{3}}\\right)\\sin \\left(kz+{\\frac {\\pi }{2}}\\right)\\,\\right]e^{-3\\nu k^{2}t}\\\\u_{z}&={\\frac {4{\\sqrt {2}}}{3{\\sqrt {3}}}}\\,U_{0}\\left[\\,\\sin \\left(kz-{\\frac {\\pi }{3}}\\right)\\cos \\left(kx+{\\frac {\\pi }{3}}\\right)\\sin \\left(ky+{\\frac {\\pi }{2}}\\right)-\\cos \\left(ky-{\\frac {\\pi }{3}}\\right)\\sin \\left(kz+{\\frac {\\pi }{3}}\\right)\\sin \\left(kx+{\\frac {\\pi }{2}}\\right)\\,\\right]e^{-3\\nu k^{2}t}\\end{aligned}}}"}] | — |
modifiedPositive-helicity solutioncc59f1549067
| Field | From #1423 | To #1766 |
|---|
| anchors | [{"section":"Viscous three-dimensional periodic solutions","snippet":"The solution with positive helicity is given by"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}u_{x}&={\\frac {4{\\sqrt {2}}}{3{\\sqrt {3}}}}\\,U_{0}\\left[\\,\\sin \\left(kx-{\\frac {\\pi }{3}}\\right)\\cos \\left(ky+{\\frac {\\pi }{3}}\\right)\\sin \\left(kz+{\\frac {\\pi }{2}}\\right)-\\cos \\left(kz-{\\frac {\\pi }{3}}\\right)\\sin \\left(kx+{\\frac {\\pi }{3}}\\right)\\sin \\left(ky+{\\frac {\\pi }{2}}\\right)\\,\\right]e^{-3\\nu k^{2}t}\\\\u_{y}&={\\frac {4{\\sqrt {2}}}{3{\\sqrt {3}}}}\\,U_{0}\\left[\\,\\sin \\left(ky-{\\frac {\\pi }{3}}\\right)\\cos \\left(kz+{\\frac {\\pi }{3}}\\right)\\sin \\left(kx+{\\frac {\\pi }{2}}\\right)-\\cos \\left(kx-{\\frac {\\pi }{3}}\\right)\\sin \\left(ky+{\\frac {\\pi }{3}}\\right)\\sin \\left(kz+{\\frac {\\pi }{2}}\\right)\\,\\right]e^{-3\\nu k^{2}t}\\\\u_{z}&={\\frac {4{\\sqrt {2}}}{3{\\sqrt {3}}}}\\,U_{0}\\left[\\,\\sin \\left(kz-{\\frac {\\pi }{3}}\\right)\\cos \\left(kx+{\\frac {\\pi }{3}}\\right)\\sin \\left(ky+{\\frac {\\pi }{2}}\\right)-\\cos \\left(ky-{\\frac {\\pi }{3}}\\right)\\sin \\left(kz+{\\frac {\\pi }{3}}\\right)\\sin \\left(kx+{\\frac {\\pi }{2}}\\right)\\,\\right]e^{-3\\nu k^{2}t}\\end{aligned}}}"}] | — |
addedBeltrami flow60590a452554
modifiedNS in Cartesian coordinatesb63b5610eb4f
| Field | From #1423 | To #1766 |
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| anchors | [{"section":"Cartesian coordinates","snippet":"we may write the vector equation explicitly"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}x:\\ &\\rho \\left({\\partial _{t}u_{x}}+u_{x}\\,{\\partial _{x}u_{x}}+u_{y}\\,{\\partial _{y}u_{x}}+u_{z}\\,{\\partial _{z}u_{x}}\\right)\\\\&\\quad =-\\partial _{x}p+\\mu \\left({\\partial _{x}^{2}u_{x}}+{\\partial _{y}^{2}u_{x}}+{\\partial _{z}^{2}u_{x}}\\right)+{\\frac {1}{3}}\\mu \\ \\partial _{x}\\left({\\partial _{x}u_{x}}+{\\partial _{y}u_{y}}+{\\partial _{z}u_{z}}\\right)+\\rho g_{x}\\\\\\end{aligned}}}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}y:\\ &\\rho \\left({\\partial _{t}u_{y}}+u_{x}{\\partial _{x}u_{y}}+u_{y}{\\partial _{y}u_{y}}+u_{z}{\\partial _{z}u_{y}}\\right)\\\\&\\quad =-{\\partial _{y}p}+\\mu \\left({\\partial _{x}^{2}u_{y}}+{\\partial _{y}^{2}u_{y}}+{\\partial _{z}^{2}u_{y}}\\right)+{\\frac {1}{3}}\\mu \\ \\partial _{y}\\left({\\partial _{x}u_{x}}+{\\partial _{y}u_{y}}+{\\partial _{z}u_{z}}\\right)+\\rho g_{y}\\\\\\end{aligned}}}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}z:\\ &\\rho \\left({\\partial _{t}u_{z}}+u_{x}{\\partial _{x}u_{z}}+u_{y}{\\partial _{y}u_{z}}+u_{z}{\\partial _{z}u_{z}}\\right)\\\\&\\quad =-{\\partial _{z}p}+\\mu \\left({\\partial _{x}^{2}u_{z}}+{\\partial _{y}^{2}u_{z}}+{\\partial _{z}^{2}u_{z}}\\right)+{\\frac {1}{3}}\\mu \\ \\partial _{z}\\left({\\partial _{x}u_{x}}+{\\partial _{y}u_{y}}+{\\partial _{z}u_{z}}\\right)+\\rho g_{z}.\\end{aligned}}}"}] | — |
modifiedCartesian continuity (incompressible)9cd9bc8d908c
| Field | From #1423 | To #1766 |
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| anchors | [{"section":"Cartesian coordinates","snippet":"The continuity equation is reduced to"},{"type":"math_alttext","value":"{\\displaystyle \\partial _{x}u_{x}+\\partial _{y}u_{y}+\\partial _{z}u_{z}=0.}"}] | — |
modifiedNS in cylindrical coordinatesa14b7f704bb7
| Field | From #1423 | To #1766 |
|---|
| anchors | [{"section":"Cylindrical coordinates","snippet":"A change of variables on the Cartesian equations will yield"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}r:\\ &\\rho \\left({\\partial _{t}u_{r}}+u_{r}{\\partial _{r}u_{r}}+{\\frac {u_{\\varphi }}{r}}{\\partial _{\\varphi }u_{r}}+u_{z}{\\partial _{z}u_{r}}-{\\frac {u_{\\varphi }^{2}}{r}}\\right)\\\\&\\quad =-{\\partial _{r}p}\\\\&\\qquad +\\mu \\left({\\frac {1}{r}}\\partial _{r}\\left(r{\\partial _{r}u_{r}}\\right)+{\\frac {1}{r^{2}}}{\\partial _{\\varphi }^{2}u_{r}}+{\\partial _{z}^{2}u_{r}}-{\\frac {u_{r}}{r^{2}}}-{\\frac {2}{r^{2}}}{\\partial _{\\varphi }u_{\\varphi }}\\right)\\\\&\\qquad +{\\frac {1}{3}}\\mu \\partial _{r}\\left({\\frac {1}{r}}{\\partial _{r}\\left(ru_{r}\\right)}+{\\frac {1}{r}}{\\partial _{\\varphi }u_{\\varphi }}+{\\partial _{z}u_{z}}\\right)\\\\&\\qquad +\\rho g_{r}\\\\[8px]\\end{aligned}}}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\varphi :\\ &\\rho \\left({\\partial _{t}u_{\\varphi }}+u_{r}{\\partial _{r}u_{\\varphi }}+{\\frac {u_{\\varphi }}{r}}{\\partial _{\\varphi }u_{\\varphi }}+u_{z}{\\partial _{z}u_{\\varphi }}+{\\frac {u_{r}u_{\\varphi }}{r}}\\right)\\\\&\\quad =-{\\frac {1}{r}}{\\partial _{\\varphi }p}\\\\&\\qquad +\\mu \\left({\\frac {1}{r}}\\ \\partial _{r}\\left(r{\\partial _{r}u_{\\varphi }}\\right)+{\\frac {1}{r^{2}}}{\\partial _{\\varphi }^{2}u_{\\varphi }}+{\\partial _{z}^{2}u_{\\varphi }}-{\\frac {u_{\\varphi }}{r^{2}}}+{\\frac {2}{r^{2}}}{\\partial _{\\varphi }u_{r}}\\right)\\\\&\\qquad +{\\frac {1}{3}}\\mu {\\frac {1}{r}}\\partial _{\\varphi }\\left({\\frac {1}{r}}{\\partial _{r}\\left(ru_{r}\\right)}+{\\frac {1}{r}}{\\partial _{\\varphi }u_{\\varphi }}+{\\partial _{z}u_{z}}\\right)\\\\&\\qquad +\\rho g_{\\varphi }\\\\[8px]\\end{aligned}}}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}z:\\ &\\rho \\left({\\partial _{t}u_{z}}+u_{r}{\\partial _{r}u_{z}}+{\\frac {u_{\\varphi }}{r}}{\\partial _{\\varphi }u_{z}}+u_{z}{\\partial _{z}u_{z}}\\right)\\\\&\\quad =-{\\partial _{z}p}\\\\&\\qquad +\\mu \\left({\\frac {1}{r}}\\partial _{r}\\left(r{\\partial _{r}u_{z}}\\right)+{\\frac {1}{r^{2}}}{\\partial _{\\varphi }^{2}u_{z}}+{\\partial _{z}^{2}u_{z}}\\right)\\\\&\\qquad +{\\frac {1}{3}}\\mu \\partial _{z}\\left({\\frac {1}{r}}{\\partial _{r}\\left(ru_{r}\\right)}+{\\frac {1}{r}}{\\partial _{\\varphi }u_{\\varphi }}+{\\partial _{z}u_{z}}\\right)\\\\&\\qquad +\\rho g_{z}.\\end{aligned}}}"}] | — |
modifiedAxisymmetric flowd3df8b91f565
| Field | From #1423 | To #1766 |
|---|
| anchors | [{"section":"Cylindrical coordinates","snippet":"A very common case is axisymmetric flow with the assumption of no tangential velocity"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\rho \\left({\\partial _{t}u_{r}}+u_{r}{\\partial _{r}u_{r}}+u_{z}{\\partial _{z}u_{r}}\\right)&=-{\\partial _{r}p}+\\mu \\left({\\frac {1}{r}}\\partial _{r}\\left(r{\\partial _{r}u_{r}}\\right)+{\\partial _{z}^{2}u_{r}}-{\\frac {u_{r}}{r^{2}}}\\right)+\\rho g_{r}\\\\\\rho \\left({\\partial _{t}u_{z}}+u_{r}{\\partial _{r}u_{z}}+u_{z}{\\partial _{z}u_{z}}\\right)&=-{\\partial _{z}p}+\\mu \\left({\\frac {1}{r}}\\partial _{r}\\left(r{\\partial _{r}u_{z}}\\right)+{\\partial _{z}^{2}u_{z}}\\right)+\\rho g_{z}\\\\{\\frac {1}{r}}\\partial _{r}\\left(ru_{r}\\right)+{\\partial _{z}u_{z}}&=0.\\end{aligned}}}"}] | — |
modifiedNS in spherical coordinates35c9a2be074d
| Field | From #1423 | To #1766 |
|---|
| anchors | [{"section":"Spherical coordinates","snippet":"momentum equations are"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}r:\\ &\\rho \\left({\\partial _{t}u_{r}}+u_{r}{\\partial _{r}u_{r}}+{\\frac {u_{\\varphi }}{r\\sin \\theta }}{\\partial _{\\varphi }u_{r}}+{\\frac {u_{\\theta }}{r}}{\\partial _{\\theta }u_{r}}-{\\frac {u_{\\varphi }^{2}+u_{\\theta }^{2}}{r}}\\right)\\\\&\\quad =-{\\partial _{r}p}\\\\&\\qquad +\\mu \\left({\\frac {1}{r^{2}}}\\partial _{r}\\left(r^{2}{\\partial _{r}u_{r}}\\right)+{\\frac {1}{r^{2}\\sin ^{2}\\theta }}{\\partial _{\\varphi }^{2}u_{r}}+{\\frac {1}{r^{2}\\sin \\theta }}\\partial _{\\theta }\\left(\\sin \\theta {\\partial _{\\theta }u_{r}}\\right)-2{\\frac {u_{r}+{\\partial _{\\theta }u_{\\theta }}+u_{\\theta }\\cot \\theta }{r^{2}}}-{\\frac {2}{r^{2}\\sin \\theta }}{\\partial _{\\varphi }u_{\\varphi }}\\right)\\\\&\\qquad +{\\frac {1}{3}}\\mu \\partial _{r}\\left({\\frac {1}{r^{2}}}\\partial _{r}\\left(r^{2}u_{r}\\right)+{\\frac {1}{r\\sin \\theta }}\\partial _{\\theta }\\left(u_{\\theta }\\sin \\theta \\right)+{\\frac {1}{r\\sin \\theta }}{\\partial _{\\varphi }u_{\\varphi }}\\right)\\\\&\\qquad +\\rho g_{r}\\\\[8px]\\end{aligned}}}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\varphi :\\ &\\rho \\left({\\partial _{t}u_{\\varphi }}+u_{r}{\\partial _{r}u_{\\varphi }}+{\\frac {u_{\\varphi }}{r\\sin \\theta }}{\\partial _{\\varphi }u_{\\varphi }}+{\\frac {u_{\\theta }}{r}}{\\partial _{\\theta }u_{\\varphi }}+{\\frac {u_{r}u_{\\varphi }+u_{\\varphi }u_{\\theta }\\cot \\theta }{r}}\\right)\\\\&\\quad =-{\\frac {1}{r\\sin \\theta }}{\\partial _{\\varphi }p}\\\\&\\qquad +\\mu \\left({\\frac {1}{r^{2}}}\\partial _{r}\\left(r^{2}{\\partial _{r}u_{\\varphi }}\\right)+{\\frac {1}{r^{2}\\sin ^{2}\\theta }}{\\partial _{\\varphi }^{2}u_{\\varphi }}+{\\frac {1}{r^{2}\\sin \\theta }}\\partial _{\\theta }\\left(\\sin \\theta {\\partial _{\\theta }u_{\\varphi }}\\right)+{\\frac {2\\sin \\theta {\\partial _{\\varphi }u_{r}}+2\\cos \\theta {\\partial _{\\varphi }u_{\\theta }}-u_{\\varphi }}{r^{2}\\sin ^{2}\\theta }}\\right)\\\\&\\qquad +{\\frac {1}{3}}\\mu {\\frac {1}{r\\sin \\theta }}\\partial _{\\varphi }\\left({\\frac {1}{r^{2}}}\\partial _{r}\\left(r^{2}u_{r}\\right)+{\\frac {1}{r\\sin \\theta }}\\partial _{\\theta }\\left(u_{\\theta }\\sin \\theta \\right)+{\\frac {1}{r\\sin \\theta }}{\\partial _{\\varphi }u_{\\varphi }}\\right)\\\\&\\qquad +\\rho g_{\\varphi }\\\\[8px]\\end{aligned}}}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\theta :\\ &\\rho \\left({\\partial _{t}u_{\\theta }}+u_{r}{\\partial _{r}u_{\\theta }}+{\\frac {u_{\\varphi }}{r\\sin \\theta }}{\\partial _{\\varphi }u_{\\theta }}+{\\frac {u_{\\theta }}{r}}{\\partial _{\\theta }u_{\\theta }}+{\\frac {u_{r}u_{\\theta }-u_{\\varphi }^{2}\\cot \\theta }{r}}\\right)\\\\&\\quad =-{\\frac {1}{r}}{\\partial _{\\theta }p}\\\\&\\qquad +\\mu \\left({\\frac {1}{r^{2}}}\\partial _{r}\\left(r^{2}{\\partial _{r}u_{\\theta }}\\right)+{\\frac {1}{r^{2}\\sin ^{2}\\theta }}{\\partial _{\\varphi }^{2}u_{\\theta }}+{\\frac {1}{r^{2}\\sin \\theta }}\\partial _{\\theta }\\left(\\sin \\theta {\\partial _{\\theta }u_{\\theta }}\\right)+{\\frac {2}{r^{2}}}{\\partial _{\\theta }u_{r}}-{\\frac {u_{\\theta }+2\\cos \\theta {\\partial _{\\varphi }u_{\\varphi }}}{r^{2}\\sin ^{2}\\theta }}\\right)\\\\&\\qquad +{\\frac {1}{3}}\\mu {\\frac {1}{r}}\\partial _{\\theta }\\left({\\frac {1}{r^{2}}}\\partial _{r}\\left(r^{2}u_{r}\\right)+{\\frac {1}{r\\sin \\theta }}\\partial _{\\theta }\\left(u_{\\theta }\\sin \\theta \\right)+{\\frac {1}{r\\sin \\theta }}{\\partial _{\\varphi }u_{\\varphi }}\\right)\\\\&\\qquad +\\rho g_{\\theta }.\\end{aligned}}}"}] | — |