StreamPlot3D

StreamPlot3D[{vx,vy,vz},{x,xmin,xmax},{y,ymin,ymax},{z,zmin,zmax}]

plots streamlines for the vector field {vx,vy,vz} as functions of x, y and z.

StreamPlot3D[{vx,vy,vz},{x,y,z}reg]

takes the variables {x,y,z} to be in the geometric region reg.

Details and Options

  • StreamPlot3D plots streamlines that show the local direction of the vector field at each point, effectively solving the system of differential equations , , and then plotting .
  • By default, the direction of the vector field is indicated by the paths of the streamlines, and the magnitude is indicated by the color of the streamlines.
  • StreamPlot3D by default shows enough streamlines to achieve a roughly uniform density throughout the plot and shows no background scalar field.
  • StreamPlot3D treats the variables x, y and z as local, effectively using Block.
  • StreamPlot3D has attribute HoldAll and evaluates the vi etc. only after assigning specific numerical values to x, y and z. In some cases, it may be more efficient to use Evaluate to evaluate the vi etc. symbolically first.
  • StreamPlot3D has the same options as Graphics3D, with the following additions and changes:
  • BoxRatios{1,1,1}ratio of height to width
    EvaluationMonitorNoneexpression to evaluate at every function evaluation
    MethodAutomaticmethods to use for the plot
    PerformanceGoal$PerformanceGoalaspects of performance to try to optimize
    PlotLegendsNonelegends to include
    PlotRange{Full,Full,Full}range of x, y, z values to include
    PlotRangePaddingAutomatichow much to pad the range of values
    PlotTheme$PlotThemeoverall theme for the plot
    RegionBoundaryStyleAutomatichow to style plot region boundaries
    RegionFunctionTrue&determine what region to include
    ScalingFunctionsNonehow to scale individual coordinates
    StreamColorFunctionAutomatichow to color streamlines
    StreamColorFunctionScalingTruewhether to scale the argument to StreamColorFunction
    StreamMarkersAutomaticshape to use for streams
    StreamPointsAutomaticthe number or placement of streamlines
    StreamScaleNonehow to scale the sizes of streamlines
    StreamStyleAutomatichow to draw streamlines
    WorkingPrecisionMachinePrecisionprecision to use in internal computations
  • The arguments supplied to functions in RegionFunction and ColorFunction are x,y,z,vx,vy,vz,Norm[{vx,vy,vz}].
  • Possible settings for StreamMarkers include:
  • "Arrow"lines with 2D arrowheads
    "Arrow3D"tubes with 3D arrowheads
    "Line"lines
    "Tube"tubes
    "Ribbon"flat ribbons
    "ArrowRibbon"ribbons with built-in arrowheads
  • With StreamScaleAutomatic and "arrow" stream markers, the streamlines are split into segments to make it easier to see the direction of the streamlines.
  • Possible settings for StreamScale are:
  • Automaticautomatically determine the streamline segments
    Fullshow the streamline as one piece
    Tiny,Small,Medium,Largenamed settings for how long the segments should be
    {len,npts,ratio}use explicit specification of streamline segmentation
  • The length len of streamline segments can be one of the following forms:
  • Automaticautomatically determine the length
    Noneshow the streamline as one piece
    Tiny,Small,Medium,Largeuse named segment lengths
    suse a length s that is a fraction of the graphic size
  • The number of points npts used to draw each segment can be Automatic or a specific number of points.
  • The aspect ratio ratio specifies how wide the cross section of a streamline is relative to the streamline segment.
  • Possible settings for ScalingFunctions include:
  • {sx,sy,sz}scale x, y and z axes
  • Common built-in scaling functions s include:
  • "Log"log scale with automatic tick labeling
    "Log10"base-10 log scale with powers of 10 for ticks
    "SignedLog"log-like scale that includes 0 and negative numbers
    "Reverse"reverse the coordinate direction
    "Infinite"infinite scale

Examples

open allclose all

Basic Examples  (4)

Plot streamlines through a vector field in 3D:

Use tubes to show the streamlines:

Include a legend for the vector field magnitudes:

Plot streamlines over an arbitrary region:

Scope  (12)

Sampling  (3)

Specify the density of seed points for the streamlines:

Specify specific seed points for the streamlines:

Plot streamlines over a specified region:

Presentation  (9)

Streamlines are drawn as lines by default:

Use 3D tubes for the streamlines:

Use flat ribbons:

Use "arrow" versions of the stream markers to indicate the direction of flow along the streamlines:

Arrows on tubes:

Ribbons are turned into arrows by tapering the heads and notching the tails of the streamlines:

Use a single color for the streamlines:

Use a named color gradient for the streamlines:

Include a legend for the field magnitude:

Use StreamScale to split streamlines into multiple shorter line segments:

Increase the number of points in each segment and increase the marker aspect ratio:

Use a theme:

Use a log scale for the x axis:

Reverse the y scale so it increases toward the bottom:

Options  (42)

BoxRatios  (2)

By default, BoxRatios is set to Automatic:

Make the box twice as long in the x direction:

PlotLegends  (3)

No legends are included by default:

Include a legend that indicates the vector field norm:

Specify the location of the legend:

PlotTheme  (1)

Specify a theme:

RegionBoundaryStyle  (4)

Show the region defined by a RegionFunction:

Use None to avoid showing the boundary:

Specify the color of the region boundary:

The boundaries of full rectangular regions are not shown:

RegionFunction  (4)

Plot streamlines in a ball:

Plot streams only where the field magnitude exceeds a given threshold:

Region functions depend, in general, on seven arguments:

Use RegionBoundaryStyleNone to avoid showing the boundary:

ScalingFunctions  (1)

Use a log scale for the x axis:

Reverse the y scale so it increases toward the bottom:

StreamColorFunction  (4)

Color the streams by their norm:

Use any named color gradient from ColorData:

Color the streamlines according to their x value:

Use StreamColorFunctionScalingFalse to get unscaled values:

StreamColorFunctionScaling  (2)

By default, scaled values are used:

Use StreamColorFunctionScalingFalse to get unscaled values:

StreamMarkers  (5)

By default, lines are used:

Draw the streamlines as tubes:

Draw them as flat ribbons:

"Arrow" stream markers automatically break the streamlines into shorter segments:

Use 3D arrowheads on tubes:

Use directional ribbons:

Make segmented markers continuous:

Break continuous markers into segments:

StreamPoints  (4)

Use automatically determined stream points to seed the curves:

Specify a maximum number of streamlines:

Give specific seed points for the streams:

Use coarsely spaced streamlines:

Use more finely spaced streamlines:

StreamScale  (9)

Segmented markers have default lengths, numbers of points and aspect ratios:

Modify the lengths of the segments:

Specify the number of sample points in each segment:

Modify the aspect ratios for the stream markers:

Make segmented markers continuous:

Break continuous markers into segments:

The aspect ratio controls the thickness of ribbons and tubes:

Increase the width of the ribbons and tubes:

The aspect ratio controls the sizes of arrowheads:

Control the number of points in each segment:

StreamStyle  (3)

Change the appearance of the streamlines:

StreamColorFunction takes precedence over StreamStyle:

Use StreamColorFunctionNone to specify a streamline color with StreamStyle:

Applications  (10)

Basic Applications  (1)

Consider a vector differential equation where f(x)=If[TemplateBox[{x}, Norm]<=1,x,{1,0,0}] is defined piecewise.

Visualize solutions of using seed points for the streamlines that are inside the unit sphere:

Fluid Flow  (3)

Consider Stokes flow for a point force of the form , where is a constant vector and is a Dirac delta function. For example, a force pointing down:

Define the fluid velocity vector , the pressure and the viscosity :

Confirm that the equations for Stokes flow are satisfied so that and :

Plot streamlines for the flow:

Visualize Stokes flow around a unit sphere. Define the fluid velocity vector , the pressure , the viscosity and the far-field fluid speed :

Confirm that the equations for Stokes flow are satisfied so that and :

Plot streamlines for the flow:

Plot the pressure for the flow:

Specify the NavierStokes equations for a fluid through a pipe with a bulge:

Specify the geometry for the flow:

Specify the boundary conditions for flow from left to right:

Solve for the flow velocities and pressure:

Specify seed points for the streamlines:

Plot the streamlines for the flow:

Electrical Systems  (1)

Visualize electric field lines for a dipole:

The streamlines appear to have uniform color because of the extremely rapid change in the vector field norm near the point charges at . Bounding the magnitude of the vector field norm with a region function makes the colors visible:

Add spheres to indicate the positive (red) and negative (black) point charges:

Arrows can be used to provide more information, but the colors change because arrow markers are colored by the field magnitude at the tip of the arrow:

Use a custom StreamColorFunction to exert more control over the colors:

Miscellaneous  (5)

Lorenz attractor:

Use ribbons or arrow ribbons to visualize the torsion of a twisted cubic:

Visualize solutions of differential equations on manifolds:

Visualize streamlines for Poiseuille flow. The fluid speed is fastest along the central axis:

Visualize solutions of Euler's equations for a rotating rigid body:

Properties & Relations  (9)

Use VectorPlot3D to visualize a field with discrete arrows:

Use ListStreamPlot3D or ListVectorPlot3D to generate plots based on data:

Use StreamPlot to plot streamlines of 2D vector fields:

Use VectorPlot to plot with vectors instead of streamlines:

Generate plots based on data:

Use StreamDensityPlot to add a density plot of the scalar field:

Use VectorDensityPlot to plot with arrows instead of streamlines:

Generate plots based on data:

Use LineIntegralConvolutionPlot to plot the line integral convolution of a vector field:

Use VectorDisplacementPlot to visualize the deformation of a region associated with a displacement vector field:

Use ListVectorDisplacementPlot to visualize the same deformation based on data:

Plot vectors along surfaces with SliceVectorPlot3D:

Use VectorDisplacementPlot3D to visualize the deformation of a 3D region associated with a displacement vector field:

Use ListVectorDisplacementPlot3D to visualize the same deformation based on data:

Use ComplexVectorPlot or ComplexStreamPlot to visualize a complex function of a complex variable as a vector field or with streamlines:

Use GeoVectorPlot to plot vectors on a map:

Use GeoStreamPlot to plot streamlines instead of vectors:

Possible Issues  (3)

Tube StreamMarkers can be distorted by the BoxRatios:

Carefully adjusting the BoxRatios can eliminate the tube distortion:

The colors of "Arrow" and "Arrow3D" stream markers are determined at the tip of the arrow, which can result in inconsistent colors for long arrows:

Wolfram Research (2021), StreamPlot3D, Wolfram Language function, https://reference.wolfram.com/language/ref/StreamPlot3D.html (updated 2022).

Text

Wolfram Research (2021), StreamPlot3D, Wolfram Language function, https://reference.wolfram.com/language/ref/StreamPlot3D.html (updated 2022).

CMS

Wolfram Language. 2021. "StreamPlot3D." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/StreamPlot3D.html.

APA

Wolfram Language. (2021). StreamPlot3D. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/StreamPlot3D.html

BibTeX

@misc{reference.wolfram_2022_streamplot3d, author="Wolfram Research", title="{StreamPlot3D}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/StreamPlot3D.html}", note=[Accessed: 29-September-2022 ]}

BibLaTeX

@online{reference.wolfram_2022_streamplot3d, organization={Wolfram Research}, title={StreamPlot3D}, year={2022}, url={https://reference.wolfram.com/language/ref/StreamPlot3D.html}, note=[Accessed: 29-September-2022 ]}