.. _sphx_glr_api_gallery_fluids_viscous_flow:

Viscous Flow
------------

Viscosity is what potential flow leaves out, and these are the handful of
problems simple enough that adding it back in still yields an exact,
closed-form answer. Couette and Poiseuille flow show how the same viscous
term produces a straight-line profile under a moving wall but a parabola
under a pressure gradient; Stokes drag captures the opposite extreme, where
viscosity so dominates inertia that a settling sphere never notices its own
momentum; the Reynolds number sweep shows that same Stokes-drag physics
recast to make a different point -- that a single dimensionless ratio, not
the sphere's size or the fluid's identity separately, decides the flow
regime; and the Blasius boundary layer shows what happens in between -- a
thin viscous layer clinging to a wall in an otherwise fast, nearly inviscid
stream, thickening as the square root of downstream distance. Watch how
thin that layer stays even at a middling free-stream speed, and how the
wall shear stress it predicts falls off as the flow moves downstream.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="Ludwig Prandtl&#x27;s 1904 boundary-layer theory reduced the full Navier-Stokes equations, in the thin-layer limit next to a wall, to a much simpler boundary-layer approximation; Paul Blasius then found, for steady, incompressible flow over a flat plate with zero streamwise pressure gradient, that the similarity substitution \eta = y\sqrt{U_\infty/(\nu x)}, f(\eta) the dimensionless streamfunction, collapses even that approximation to a single third-order ordinary differential equation:">

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  .. image:: /api/gallery/fluids/viscous_flow/images/thumb/sphx_glr_plot_blasius_boundary_layer_thumb.png
    :alt:

  :doc:`/api/gallery/fluids/viscous_flow/plot_blasius_boundary_layer`

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      <div class="sphx-glr-thumbnail-title">The Blasius laminar boundary layer</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Both flows are steady, unidirectional shear flow between two infinite parallel plates separated by a gap h, for which the full nonlinear Navier-Stokes equations collapse to a single linear ordinary differential equation for u(y):">

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  .. image:: /api/gallery/fluids/viscous_flow/images/thumb/sphx_glr_plot_couette_poiseuille_thumb.png
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  :doc:`/api/gallery/fluids/viscous_flow/plot_couette_poiseuille`

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      <div class="sphx-glr-thumbnail-title">Couette versus Poiseuille flow</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Osborne Reynolds injected a thread of dye into water flowing through a glass pipe and found that a single dimensionless ratio,">

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  .. image:: /api/gallery/fluids/viscous_flow/images/thumb/sphx_glr_plot_reynolds_number_scaling_thumb.png
    :alt:

  :doc:`/api/gallery/fluids/viscous_flow/plot_reynolds_number_scaling`

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      <div class="sphx-glr-thumbnail-title">The Reynolds number: one ratio, regardless of size or fluid</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="In the creeping-flow limit Re = 2R|v|/\nu \ll 1, the nonlinear advection term in the Navier-Stokes equations is negligible next to viscous diffusion, leaving the linear Stokes equations \mu\nabla^2\mathbf{u}=\nabla p -- George Stokes&#x27; 1851 exact solution for uniform flow past a sphere of radius R, giving a drag force">

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  .. image:: /api/gallery/fluids/viscous_flow/images/thumb/sphx_glr_plot_stokes_drag_thumb.png
    :alt:

  :doc:`/api/gallery/fluids/viscous_flow/plot_stokes_drag`

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      <div class="sphx-glr-thumbnail-title">Stokes drag and the terminal velocity of a settling sphere</div>
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    </div>


.. toctree::
   :hidden:

   /api/gallery/fluids/viscous_flow/plot_blasius_boundary_layer
   /api/gallery/fluids/viscous_flow/plot_couette_poiseuille
   /api/gallery/fluids/viscous_flow/plot_reynolds_number_scaling
   /api/gallery/fluids/viscous_flow/plot_stokes_drag