StreamDensityPlot

StreamDensityPlot[{{v_x,v_y},s},{x,x_min,x_max},{y,y_min,y_max}]

generates a stream plot of the vector field {v_x,v_y} as a function of x and y, superimposed on a background density plot of the scalar field s.

StreamDensityPlot[{v_x,v_y},{x,x_min,x_max},{y,y_min,y_max}]

takes the scalar field to be the norm of the vector field.

StreamDensityPlot[{{v_x,v_y},{w_x,w_y},…,s},{x,x_min,x_max},{y,y_min,y_max}]

generates plots of several vector fields.

StreamDensityPlot[…,{x,y}∈reg]

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

Details and Options

StreamDensityPlot plots streamlines that show the local direction of the vector field at every point.
StreamDensityPlot by default shows enough streamlines to achieve a roughly uniform density throughout the plot.
StreamDensityPlot does not show streamlines at any positions for which the v_i etc. do not evaluate to real numbers.
StreamDensityPlot treats the variables x and y as local, effectively using Block.
StreamDensityPlot has attribute HoldAll, and evaluates the v_i etc. only after assigning specific numerical values to x and y.
In some cases it may be more efficient to use Evaluate to evaluate the v_i etc. symbolically before specific numerical values are assigned to x and y.
StreamDensityPlot has the same options as Graphics, with the following additions and changes:

AspectRatio	1	ratio of height to width
BoundaryStyle	None	how to draw RegionFunction boundaries
BoxRatios	Automatic	effective 3D box ratios for simulated lighting
ColorFunction	Automatic	how to color background densities
ColorFunctionScaling	True	whether to scale arguments to ColorFunction
EvaluationMonitor	None	expression to evaluate at every function evaluation
Frame	True	whether to draw a frame around the plot
FrameTicks	Automatic	frame tick marks
LightingAngle	None	effective angle for simulated lighting
MaxRecursion	Automatic	the maximum number of recursive subdivisions allowed for the scalar field
Mesh	None	how many mesh lines to draw in the background
MeshFunctions	{#5&}	how to determine the placement of mesh lines
MeshShading	None	how to shade regions between mesh lines
MeshStyle	Automatic	the style of mesh lines
Method	Automatic	methods to use for the plot
PerformanceGoal	$PerformanceGoal	aspects of performance to try to optimize
PlotLegends	None	legends to include
PlotRange	{Full,Full}	range of x, y values to include
PlotRangePadding	Automatic	how much to pad the range of values
PlotTheme	$PlotTheme	overall theme for the plot
RegionFunction	True&	determine what region to include
StreamColorFunction	None	how to color streamlines
StreamColorFunctionScaling	True	whether to scale the argument to StreamColorFunction
StreamPoints	Automatic	determine number, placement, and closeness of streamlines
StreamScale	Automatic	determine sizes and segmenting of individual streamlines
StreamStyle	Automatic	how to draw streamlines
WorkingPrecision	MachinePrecision	precision to use in internal computations

The arguments supplied to functions in MeshFunctions, RegionFunction, ColorFunction, and StreamColorFunction are x, y, v_x, v_y, s.
The default setting MeshFunctions->{#5&} draws mesh lines for the scalar field s.