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The circulation Γ of a vector field V around a closed curve C is the line integral: [3] [4] =. In a conservative vector field this integral evaluates to zero for every closed curve. That means that a line integral between any two points in the field is independent of the path taken.
Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to ...
so the flow velocity components in relation to the stream function must be =, =. Notice that the stream function is linear in the velocity. Consequently if two incompressible flow fields are superimposed, then the stream function of the resultant flow field is the algebraic sum of the stream functions of the two original fields.
A scalar function whose contour lines define the streamlines is known as the stream function. Dye line may refer either to a streakline: dye released gradually from a fixed location during time; or it may refer to a timeline: a line of dye applied instantaneously at a certain moment in time, and observed at a later instant.
A schematic diagram of the Blasius flow profile. The streamwise velocity component () / is shown, as a function of the similarity variable .. Using scaling arguments, Ludwig Prandtl [1] argued that about half of the terms in the Navier-Stokes equations are negligible in boundary layer flows (except in a small region near the leading edge of the plate).
Since the velocity of the object is the derivative of the position graph, the area under the line in the velocity vs. time graph is the displacement of the object. (Velocity is on the y-axis and time on the x-axis. Multiplying the velocity by the time, the time cancels out, and only displacement remains.)
In order to satisfy the continuity equation, the streamlines get bunched closer and the velocity increases as we get closer to the source. As with source flow, the velocity at all points equidistant from the sink is equal. Fig 3 – Streamlines and potential lines for sink flow. The velocity of the flow around the sink can be given by –
There are two main descriptions of motion: dynamics and kinematics.Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.