1What is the physical meaning of the gradient (nabla) when applied to a scalar field like pressure (P)?
It points in the direction of the steepest increase of the scalar field (e.g., pressure) and its magnitude indicates the rate of that increase.
2What does a pressure gradient (nabla P) in a fluid represent?
A force that causes fluid to accelerate from regions of high pressure to regions of low pressure.
3What does a velocity gradient represent in fluid dynamics?
The rate of change of the fluid's velocity with respect to spatial position, indicating shear stress between layers of the fluid.
4What does the divergence (nabla dot v) of a velocity vector field (v) measure?
The net rate of outflow per unit volume at a given point, indicating whether the point is a source (positive divergence) or a sink (negative divergence).
5What condition must the divergence of the velocity field satisfy for an incompressible fluid?
The divergence must be zero everywhere (nabla dot v=0), meaning there are no sources or sinks, and the volume of a fluid parcel remains constant.
6What is a vector field with zero divergence called?
A solenoidal field.
7What does the curl (nabla X v) of a velocity vector field (v) measure?
The microscopic rotation, or "vorticity," of the fluid at a given point. It is a vector pointing along the axis of rotation.
8What does a non-zero curl (nabla X v /= 0) indicate about the fluid flow?
The fluid is rotational, meaning a small paddlewheel placed in the flow would spin.
9What is a flow called if its curl is zero everywhere (nabla X v=0)?
Irrotational flow.
10What is the curl of a gradient (nabla X (nabla f(x, y,z)))?
It is always zero. This means that any conservative vector field (which can be expressed as the gradient of a scalar potential) is irrotational.
11What is the divergence of a curl (nabla dot (nabla X v))?
It is always zero. This is a vector identity which implies that the curl of any vector field is solenoidal.
12What does the fundamental theorem of line integrals state for a conservative vector field?
The line integral from point A to point B is independent of the path taken and depends only on the value of the scalar potential function at the endpoints (f(B)-f(A)).
13What does Green's theorem relate?
The line integral around a simple closed curve to a double integral over the plane region bounded by the curve. It connects the circulation (line integral of tangential components) to the curl (double integral of the normal component of the curl).
14What does the divergence theorem relate?
The outward flux of a vector field through a closed surface to the triple integral of the divergence over the volume enclosed by the surface.
15What does Stokes' theorem relate?
The line integral of a vector field around a closed loop to the surface integral of the curl of the vector field over any surface bounded by that loop.
16What does the vector cross product of two vectors (aXb) measure?
It measures how much the two vectors are perpendicular to each other, producing a third vector that is mutually perpendicular to both.
17What is the fundamental difference between the result of a dot product and a cross product?
The dot product results in a scalar (a number), while the cross product results in a new vector.
18What determines the direction of the vector resulting from a cross product (aXb)?
The right-hand rule. Your thumb points in the direction of the resultant vector when your index finger points in the direction of the first vector (a) and your middle finger points in the direction of the second vector (b).
19When is the magnitude of the cross product maximal and when is it zero?
It is maximal when the two vectors are perpendicular (90 degrees) and zero when they are parallel or anti-parallel (0 degrees) or (180 degrees).
20When is the magnitude of the dot product maximal and when is it zero?
It is maximal when the two vectors are parallel (0 degree) and zero when they are perpendicular (90 degrees).
21How is the cross product relevant to a fluid's rotational motion?
It is the key operation in the curl (nabla X v) of a velocity field, which is a measure of the fluid's vorticity, or local rotation.
22How does the cross product define torque in fluid mechanics?
Torque (tau), or the turning effect of a force, is defined as the cross product of the position vector (r) and the force vector (F): (tau =r X F).
23How is the cross product used to describe the Coriolis effect?
The Coriolis force, which causes objects to be deflected in a rotating reference frame, is calculated using a cross product involving the angular velocity vector and the object's velocity vector.
24How can the cross product tell you if two vectors are parallel?
If the cross product of two non-zero vectors is the zero vector, then the vectors are parallel or anti-parallel.
25nabla applied to scalar field results in?
gradient
26nable applied to a vector results in?
Divergence
27nablaXvector (cross product) results in?
rotation/curl
28what is index notation of FxG (cross product)?
(epsilon)ijk fj gk ei
29What is the equation for nable applied to phi?
The partial of phi with respect to xi * ei
30What's the equation for dot product of nabla and vector F?
partial derivative with respect to xi * fi
31What's the equation for the cross product of nabla and vector F?
epsilonijk * partial derivative with respect to xj * fk * vector ei
32What is a streamline?
An imaginary line in a flow field that is tangent to the velocity vector at every point at a given instant in time.
33What visualization technique shows the actual path traced by a single, specific fluid particle over a period of time?
A pathline.
34What visualization technique is formed by connecting all the fluid particles that have passed through a single, fixed point over a period of time?
A streakline.
35Which flow visualization technique is represented by a single, fixed-in-time snapshot of the flow field?
A streamline.
36In steady flow, how do streamlines, pathlines, and streaklines relate to each other?
They are all identical.
37What happens to streamlines when the fluid velocity increases?
They get closer together.
38How are pathlines different from streamlines in unsteady flow?
In unsteady flow, pathlines and streamlines are different. Pathlines are a time history of a single particle, while streamlines are an instantaneous snapshot of the flow direction.
39Why are streaklines often used in laboratory flow visualization?
They are the easiest to create experimentally by injecting dye or smoke from a fixed point.
40What happens at a point where streamlines, pathlines, and streaklines are all identical?
The flow is steady at that point.
41In unsteady flow, if you released a dye from a fixed point, which flow line would you be visualizing?
A streakline.
42What is vorticity in fluid dynamics?
Vorticity is a measure of the local spinning motion or angular velocity of a fluid element and is defined as the curl of the velocity field.
43What is the mathematical definition of vorticity?
Vorticity is defined as the curl of the velocity field:
ω⃗=∇×u⃗
44Is vorticity a scalar or a vector?
Vorticity is a vector quantity.
45What does the magnitude of the vorticity vector represent?
The magnitude represents how quickly a fluid element is rotating.
46What does the direction of the vorticity vector indicate?
The direction indicates the axis of rotation, following the right-hand rule.
47Why is vorticity considered a local property?
Because it is defined at every point in the flow and can vary from one location to another.
48What distinguishes vorticity from simple circular motion?
Vorticity describes rotation of a fluid particle about its own axis, not just motion along a curved path.
49What is an example of flow with zero vorticity but circular motion?
An ideal irrotational vortex, where particles move in circles but local vorticity is zero except at the center.
50How is vorticity related to vortices?
Vorticity is the defining characteristic of vortices and describes their rotational strength.
51What causes high vorticity in boundary layers?
Steep velocity gradients due to the no-slip condition near solid surfaces create high vorticity.
52How does vorticity behave in turbulent flows?
In turbulence, vorticity forms chaotic eddies that transfer energy from large to small scales.
53What role does vorticity play in lift and drag on an airplane wing?
Vorticity is shed from the wingtips, forming vortices that are linked to lift generation and induced drag.
54How do meteorologists use vorticity?
They use it to track and predict the motion of rotating weather systems like cyclones and anticyclones.
55In which direction does the vorticity vector point in a tornado?
Upward, along the central axis of the vortex.
56What condition at a wall causes vorticity in fluid flow?
The no-slip condition, which results in shear and steep velocity gradients near the wall.
57What does the strain rate tensor describe in fluid dynamics?
It describes the rate at which a fluid element is deforming at a specific point in the flow.
58How does the strain rate tensor differ from vorticity?
Vorticity measures local rotation, while the strain rate tensor quantifies local stretching, compressing, and shearing.
59What three motions can a fluid element undergo in flow?
Translate, rotate, and deform.
60Which of the three motions is measured by the strain rate tensor?
Deformation of the fluid element.
61From which tensor is the strain rate tensor derived?
It is derived from the velocity gradient tensor.
62What part of the velocity gradient tensor forms the strain rate tensor?
The symmetric part.
63What does the antisymmetric part of the velocity gradient tensor represent?
It represents the vorticity tensor, associated with rigid-body rotation.
64What type of matrix is the strain rate tensor?
It is a symmetric 3x3 matrix.
65What do the diagonal components of the strain rate tensor represent?
They represent linear strain rates—stretching or compressing along each axis.
66What does a positive diagonal component of the strain rate tensor indicate?
Stretching along that axis.
67What does a negative diagonal component of the strain rate tensor indicate?
Compressing along that axis.
68What does the trace of the strain rate tensor represent?
It represents the divergence of the velocity field, indicating the rate of volume change.
69What do the off-diagonal components of the strain rate tensor represent?
They represent shear strain rates—how angles between fluid elements change.
70 What does
ϵxy and ϵyx
represent in the strain rate tensor?
Shear strain rate in the x-y plane.
71Why is the strain rate tensor important for viscous fluids?
Because viscous stresses are directly proportional to the strain rate tensor in Newtonian fluids.
72What causes viscous stresses in a fluid?
Deformation of the fluid, as measured by the strain rate tensor.
73How does the strain rate tensor explain pressure drop in pipe flow?
High shear strain rates near the wall cause viscous stresses, which result in a pressure drop along the pipe.