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In classical field theories, the Lagrangian specification of the flow field is a way of looking at fluid motion where the observer follows an individual fluid parcel as it moves through space and time. [1] [2] Plotting the position of an individual parcel through time gives the pathline of the parcel. This can be visualized as sitting in a boat ...
In many engineering applications the local flow velocity vector field is not known in every point and the only accessible velocity is the bulk velocity or average flow velocity ¯ (with the usual dimension of length per time), defined as the quotient between the volume flow rate ˙ (with dimension of cubed length per time) and the cross sectional area (with dimension of square length):
The two-dimensional stream function is based on the following assumptions: The space domain is three-dimensional. The flow field can be described as two-dimensional plane flow, with velocity vector = [(,,) (,,)].
An example of a solenoidal vector field, (,) = (,) In vector calculus a solenoidal vector field (also known as an incompressible vector field , a divergence-free vector field , or a transverse vector field ) is a vector field v with divergence zero at all points in the field: ∇ ⋅ v = 0. {\displaystyle \nabla \cdot \mathbf {v} =0.}
The divergence of a vector field is often illustrated using the simple example of the velocity field of a fluid, a liquid or gas. A moving gas has a velocity, a speed and direction at each point, which can be represented by a vector, so the velocity of the gas forms a vector field. If a gas is heated, it will expand.
A vector field V defined on an open set S is called a gradient field or a conservative field if there exists a real-valued function (a scalar field) f on S such that = = (,,, …,). The associated flow is called the gradient flow , and is used in the method of gradient descent .
For constant velocity the position at time t will be = +, where x 0 is the position at time t = 0. Velocity is the time derivative of position. Its dimensions are length/time. Acceleration a of a point is vector which is the time derivative of velocity.
In steady flow (when the velocity vector-field does not change with time), the streamlines, pathlines, and streaklines coincide. This is because when a particle on a streamline reaches a point, a 0 {\displaystyle a_{0}} , further on that streamline the equations governing the flow will send it in a certain direction x → {\displaystyle {\vec ...