Mass transport equation Cartesian to polar coordinates












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Can someone please advise on how to transform the following equation to polar coordinates?
$$frac{partial rho(x,t)}{partial t}=vfrac{partial left(rho(x,t) L(x)right)}{partial x}+Dfrac{partial}{partial x}left(L(x)frac{partial rho(x,t)}{partial x}right)$$






multivariable-calculus pde polar-coordinates







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  • $begingroup$
    This looks like one dimensional equation (no $y$ and $z$), so no need to transform to polar coordinates
    $endgroup$
    – Andrei
    Dec 12 '18 at 18:55










  • $begingroup$
    What polar coordinates? There is only a one-dimensional coordinate $x$ here
    $endgroup$
    – Federico
    Dec 12 '18 at 18:55










  • $begingroup$
    Maybe you mean $partial_t rho + mathrm{div}(rho V + D nablarho)=0$? As in the en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation
    $endgroup$
    – Federico
    Dec 12 '18 at 18:57












  • $begingroup$
    You can write in several forms, see also here: en.wikipedia.org/wiki/Convection%E2%80%93diffusion_equation
    $endgroup$
    – Federico
    Dec 12 '18 at 18:59










  • $begingroup$
    You just have to use the formula for the divergence in polar coordinates
    $endgroup$
    – Federico
    Dec 12 '18 at 18:59
















0












$begingroup$


Can someone please advise on how to transform the following equation to polar coordinates?
$$frac{partial rho(x,t)}{partial t}=vfrac{partial left(rho(x,t) L(x)right)}{partial x}+Dfrac{partial}{partial x}left(L(x)frac{partial rho(x,t)}{partial x}right)$$






multivariable-calculus pde polar-coordinates







share|cite|improve this question











$endgroup$












  • $begingroup$
    This looks like one dimensional equation (no $y$ and $z$), so no need to transform to polar coordinates
    $endgroup$
    – Andrei
    Dec 12 '18 at 18:55










  • $begingroup$
    What polar coordinates? There is only a one-dimensional coordinate $x$ here
    $endgroup$
    – Federico
    Dec 12 '18 at 18:55










  • $begingroup$
    Maybe you mean $partial_t rho + mathrm{div}(rho V + D nablarho)=0$? As in the en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation
    $endgroup$
    – Federico
    Dec 12 '18 at 18:57












  • $begingroup$
    You can write in several forms, see also here: en.wikipedia.org/wiki/Convection%E2%80%93diffusion_equation
    $endgroup$
    – Federico
    Dec 12 '18 at 18:59










  • $begingroup$
    You just have to use the formula for the divergence in polar coordinates
    $endgroup$
    – Federico
    Dec 12 '18 at 18:59














0












0








0





$begingroup$


Can someone please advise on how to transform the following equation to polar coordinates?
$$frac{partial rho(x,t)}{partial t}=vfrac{partial left(rho(x,t) L(x)right)}{partial x}+Dfrac{partial}{partial x}left(L(x)frac{partial rho(x,t)}{partial x}right)$$






multivariable-calculus pde polar-coordinates







share|cite|improve this question











$endgroup$




Can someone please advise on how to transform the following equation to polar coordinates?
$$frac{partial rho(x,t)}{partial t}=vfrac{partial left(rho(x,t) L(x)right)}{partial x}+Dfrac{partial}{partial x}left(L(x)frac{partial rho(x,t)}{partial x}right)$$






multivariable-calculus pde polar-coordinates




multivariable-calculus pde polar-coordinates






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edited Dec 13 '18 at 10:53









Harry49

6,21331132




6,21331132










asked Dec 12 '18 at 18:52









jarheadjarhead

1308




1308












  • $begingroup$
    This looks like one dimensional equation (no $y$ and $z$), so no need to transform to polar coordinates
    $endgroup$
    – Andrei
    Dec 12 '18 at 18:55










  • $begingroup$
    What polar coordinates? There is only a one-dimensional coordinate $x$ here
    $endgroup$
    – Federico
    Dec 12 '18 at 18:55










  • $begingroup$
    Maybe you mean $partial_t rho + mathrm{div}(rho V + D nablarho)=0$? As in the en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation
    $endgroup$
    – Federico
    Dec 12 '18 at 18:57












  • $begingroup$
    You can write in several forms, see also here: en.wikipedia.org/wiki/Convection%E2%80%93diffusion_equation
    $endgroup$
    – Federico
    Dec 12 '18 at 18:59










  • $begingroup$
    You just have to use the formula for the divergence in polar coordinates
    $endgroup$
    – Federico
    Dec 12 '18 at 18:59


















  • $begingroup$
    This looks like one dimensional equation (no $y$ and $z$), so no need to transform to polar coordinates
    $endgroup$
    – Andrei
    Dec 12 '18 at 18:55










  • $begingroup$
    What polar coordinates? There is only a one-dimensional coordinate $x$ here
    $endgroup$
    – Federico
    Dec 12 '18 at 18:55










  • $begingroup$
    Maybe you mean $partial_t rho + mathrm{div}(rho V + D nablarho)=0$? As in the en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation
    $endgroup$
    – Federico
    Dec 12 '18 at 18:57












  • $begingroup$
    You can write in several forms, see also here: en.wikipedia.org/wiki/Convection%E2%80%93diffusion_equation
    $endgroup$
    – Federico
    Dec 12 '18 at 18:59










  • $begingroup$
    You just have to use the formula for the divergence in polar coordinates
    $endgroup$
    – Federico
    Dec 12 '18 at 18:59
















$begingroup$
This looks like one dimensional equation (no $y$ and $z$), so no need to transform to polar coordinates
$endgroup$
– Andrei
Dec 12 '18 at 18:55




$begingroup$
This looks like one dimensional equation (no $y$ and $z$), so no need to transform to polar coordinates
$endgroup$
– Andrei
Dec 12 '18 at 18:55












$begingroup$
What polar coordinates? There is only a one-dimensional coordinate $x$ here
$endgroup$
– Federico
Dec 12 '18 at 18:55




$begingroup$
What polar coordinates? There is only a one-dimensional coordinate $x$ here
$endgroup$
– Federico
Dec 12 '18 at 18:55












$begingroup$
Maybe you mean $partial_t rho + mathrm{div}(rho V + D nablarho)=0$? As in the en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation
$endgroup$
– Federico
Dec 12 '18 at 18:57






$begingroup$
Maybe you mean $partial_t rho + mathrm{div}(rho V + D nablarho)=0$? As in the en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation
$endgroup$
– Federico
Dec 12 '18 at 18:57














$begingroup$
You can write in several forms, see also here: en.wikipedia.org/wiki/Convection%E2%80%93diffusion_equation
$endgroup$
– Federico
Dec 12 '18 at 18:59




$begingroup$
You can write in several forms, see also here: en.wikipedia.org/wiki/Convection%E2%80%93diffusion_equation
$endgroup$
– Federico
Dec 12 '18 at 18:59












$begingroup$
You just have to use the formula for the divergence in polar coordinates
$endgroup$
– Federico
Dec 12 '18 at 18:59




$begingroup$
You just have to use the formula for the divergence in polar coordinates
$endgroup$
– Federico
Dec 12 '18 at 18:59










1 Answer
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$begingroup$

This is a convection-diffusion equation
$$
frac{partial rho}{partial t} = nablacdot(underbrace{D, {boldsymbol nabla}rho - rho{boldsymbol v}}_{-boldsymbol f}) ,
$$
where $rho$ is a species concentration (in mass transfer), $D$ is the diffusivity, ${boldsymbol v} = v_r {boldsymbol e}_r + v_theta {boldsymbol e}_theta$ is the velocity field, and ${boldsymbol f} = f_r {boldsymbol e}_r + f_theta {boldsymbol e}_theta$ is the flux. The differential operators write
$$
{boldsymbol nabla}rho = partial_rrho, {boldsymbol e}_r + frac{partial_thetarho}{r}, {boldsymbol e}_theta,
qquadtext{and}qquad
nablacdot {boldsymbol f} = frac{partial_r (r f_r)}{r} + frac{partial_theta f_theta}{r}
$$
with the flux components $f_r = rho v_r - Dpartial_rrho$ and $f_theta = rho v_theta - D partial_thetarho / r$.
If $D$ and $v$ do not depend on space (as seems to be the case here), we have
$$
frac{partial rho}{partial t} = frac{D partial_r (r partial_rrho) - v_r partial_r(rrho)}{r} + frac{D/r, partial_{thetatheta} rho - v_theta partial_thetarho }{r} .
$$






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    1 Answer
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    1












    $begingroup$

    This is a convection-diffusion equation
    $$
    frac{partial rho}{partial t} = nablacdot(underbrace{D, {boldsymbol nabla}rho - rho{boldsymbol v}}_{-boldsymbol f}) ,
    $$
    where $rho$ is a species concentration (in mass transfer), $D$ is the diffusivity, ${boldsymbol v} = v_r {boldsymbol e}_r + v_theta {boldsymbol e}_theta$ is the velocity field, and ${boldsymbol f} = f_r {boldsymbol e}_r + f_theta {boldsymbol e}_theta$ is the flux. The differential operators write
    $$
    {boldsymbol nabla}rho = partial_rrho, {boldsymbol e}_r + frac{partial_thetarho}{r}, {boldsymbol e}_theta,
    qquadtext{and}qquad
    nablacdot {boldsymbol f} = frac{partial_r (r f_r)}{r} + frac{partial_theta f_theta}{r}
    $$
    with the flux components $f_r = rho v_r - Dpartial_rrho$ and $f_theta = rho v_theta - D partial_thetarho / r$.
    If $D$ and $v$ do not depend on space (as seems to be the case here), we have
    $$
    frac{partial rho}{partial t} = frac{D partial_r (r partial_rrho) - v_r partial_r(rrho)}{r} + frac{D/r, partial_{thetatheta} rho - v_theta partial_thetarho }{r} .
    $$






    share|cite|improve this answer











    $endgroup$


















      1












      $begingroup$

      This is a convection-diffusion equation
      $$
      frac{partial rho}{partial t} = nablacdot(underbrace{D, {boldsymbol nabla}rho - rho{boldsymbol v}}_{-boldsymbol f}) ,
      $$
      where $rho$ is a species concentration (in mass transfer), $D$ is the diffusivity, ${boldsymbol v} = v_r {boldsymbol e}_r + v_theta {boldsymbol e}_theta$ is the velocity field, and ${boldsymbol f} = f_r {boldsymbol e}_r + f_theta {boldsymbol e}_theta$ is the flux. The differential operators write
      $$
      {boldsymbol nabla}rho = partial_rrho, {boldsymbol e}_r + frac{partial_thetarho}{r}, {boldsymbol e}_theta,
      qquadtext{and}qquad
      nablacdot {boldsymbol f} = frac{partial_r (r f_r)}{r} + frac{partial_theta f_theta}{r}
      $$
      with the flux components $f_r = rho v_r - Dpartial_rrho$ and $f_theta = rho v_theta - D partial_thetarho / r$.
      If $D$ and $v$ do not depend on space (as seems to be the case here), we have
      $$
      frac{partial rho}{partial t} = frac{D partial_r (r partial_rrho) - v_r partial_r(rrho)}{r} + frac{D/r, partial_{thetatheta} rho - v_theta partial_thetarho }{r} .
      $$






      share|cite|improve this answer











      $endgroup$
















        1












        1








        1





        $begingroup$

        This is a convection-diffusion equation
        $$
        frac{partial rho}{partial t} = nablacdot(underbrace{D, {boldsymbol nabla}rho - rho{boldsymbol v}}_{-boldsymbol f}) ,
        $$
        where $rho$ is a species concentration (in mass transfer), $D$ is the diffusivity, ${boldsymbol v} = v_r {boldsymbol e}_r + v_theta {boldsymbol e}_theta$ is the velocity field, and ${boldsymbol f} = f_r {boldsymbol e}_r + f_theta {boldsymbol e}_theta$ is the flux. The differential operators write
        $$
        {boldsymbol nabla}rho = partial_rrho, {boldsymbol e}_r + frac{partial_thetarho}{r}, {boldsymbol e}_theta,
        qquadtext{and}qquad
        nablacdot {boldsymbol f} = frac{partial_r (r f_r)}{r} + frac{partial_theta f_theta}{r}
        $$
        with the flux components $f_r = rho v_r - Dpartial_rrho$ and $f_theta = rho v_theta - D partial_thetarho / r$.
        If $D$ and $v$ do not depend on space (as seems to be the case here), we have
        $$
        frac{partial rho}{partial t} = frac{D partial_r (r partial_rrho) - v_r partial_r(rrho)}{r} + frac{D/r, partial_{thetatheta} rho - v_theta partial_thetarho }{r} .
        $$






        share|cite|improve this answer











        $endgroup$



        This is a convection-diffusion equation
        $$
        frac{partial rho}{partial t} = nablacdot(underbrace{D, {boldsymbol nabla}rho - rho{boldsymbol v}}_{-boldsymbol f}) ,
        $$
        where $rho$ is a species concentration (in mass transfer), $D$ is the diffusivity, ${boldsymbol v} = v_r {boldsymbol e}_r + v_theta {boldsymbol e}_theta$ is the velocity field, and ${boldsymbol f} = f_r {boldsymbol e}_r + f_theta {boldsymbol e}_theta$ is the flux. The differential operators write
        $$
        {boldsymbol nabla}rho = partial_rrho, {boldsymbol e}_r + frac{partial_thetarho}{r}, {boldsymbol e}_theta,
        qquadtext{and}qquad
        nablacdot {boldsymbol f} = frac{partial_r (r f_r)}{r} + frac{partial_theta f_theta}{r}
        $$
        with the flux components $f_r = rho v_r - Dpartial_rrho$ and $f_theta = rho v_theta - D partial_thetarho / r$.
        If $D$ and $v$ do not depend on space (as seems to be the case here), we have
        $$
        frac{partial rho}{partial t} = frac{D partial_r (r partial_rrho) - v_r partial_r(rrho)}{r} + frac{D/r, partial_{thetatheta} rho - v_theta partial_thetarho }{r} .
        $$







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 13 '18 at 11:14

























        answered Dec 13 '18 at 10:53









        Harry49Harry49

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        6,21331132






























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