Semester 1: Questions Flashcards
Define unit vector
A vector of unit length in a given direction.
How can plane polar coordinates be written in terms of Cartesian coordinates?
How can Cartesian coordinates be written in terms of plane polar coordinates?
Describe the diagram that represents plane polar coordinates
What are the unit vectors for plane polar coordinates in Cartesian coordinates?
How can cylindrical coordinates be written in terms of Cartesian coordinates?
How can Cartesian coordinates be written in terms of cylindrical coordinates?
Describe the diagram that represents cylindrical coordinates
What are the unit vectors for cylindrical coordinates in Cartesian coordinates?
How can spherical polar coordinates be written in terms of Cartesian coordinates?
How can Cartesian coordinates be written in terms of spherical polar coordinates?
Describe the diagram that represents spherical polar coordinates
What are the unit vectors for spherical polar coordinates in Cartesian coordinates?
What is a field?
A quantity defined at all positions in space.
What is the difference between a scalar and a vector field?
A scalar field only has magnitude whilst a vector field has both direction and magnitude.
Give 3 examples of scalar fields in Physics
- Temperature
- Pressure
- Density
Give 3 examples of vector fields in Physics
- Velocity
- Electric field
- Magnetic field
Define fluid mechanics
Mechanics concerned with the behaviour of liquids and gases at rest or in motion.
Give 3 examples of fields in fluid mechanics
- Pressure (scalar)
- Density (scalar)
- Velocity (vector)
Define electromagnetism
The interaction that occurs between particles with electric charge via electromagnetic fields.
Give 2 examples of fields in electromagnetism
- Electric field (vector)
- Magnetic field (vector)
Define streamline
A line tangential to the velocity vector, known as field lines in electromagnetism.
What is the equation for the dot product?
What is the equation for the cross product?
What is the del operator?
An operator represented by the symbol nabla that denotes the standard derivative of a function as defined in calculus.
What is the equation for the del operator?
What is the equation for the del operator in cylindrical coordinates?
What are the three types of line/path integral?
What is the equation for the del operator in spherical polar coordinates?
What is the grad of a field?
The gradient, found by multiplying a scalar field by the del operator. It results in a vector.
It is a measure of how much a scalar field changes.
What is the div of a field?
The divergence, found by calculating the dot product between the vector and the del operator. It results in a scalar.
It is a measure of how quickly a field diverges at every point in space so for a positive divergence there is more flow out than in and for a negative divergence there is more flow in than out.
What is the curl of a field?
The curl, found by calculating the cross product between the vector and the del operator. It results in a vector.
It is a measure of how much a vector field would induce a rotation about its own axis.
What is an irrotational vector field?
A field with zero curl.
What is the directional derivative?
The rate of change of a scalar function in a given (unit vector) direction.
What are the 5 valid combinations of grad, div, and curl?
What two combinations of grad, div and curl are equal to zero?
What is a vector identity?
The relationship between combinations of operators and fields that always holds, whatever the field is.
What is the equation for the Laplacian operator?
What is a conservative field?
An integral taken where the start and end points coincide at the same position.
The curl of a conservative field is equal to ____.
Zero
What is scalar potential?
The scalar function, F, that describes a conservative vector field when it is multiplied by the del operator (the grad of the scalar function). As the vector field is conservative, ∇F is equal to 0.
What is a line integral?
The integration of a scalar or vector field along a specified path, C, from a point, A, to another point, B. It defines a component of the field along a given length of the path.
What are the elemental changes in path length in all three coordinate systems?
What is a surface integral?
The integration of a scalar or vector field over a surface, S.
What are the three types of surface integral?
What are the elemental changes in area in all three coordinate systems?
What is a volume integral?
The integration of a scalar or vector field over a volume, V.
What are the elemental changes in volume in all three coordinate systems?
Define buoyancy
The force on an object immersed in an incompressible fluid.
Define solid angle
A measure of ‘how far around’ a secondary point is from an initial point in 3D. It is defined based on the angle subtended by a surface area (rather than an angle subtended by a curve) and is measured in steradians.
Define flux
The flow of a fluid through a surface per unit time. It is calculated by integrating the dot product of a vector field and the elemental change in area over the total surface area.
The divergence at a point is the ___ ____ out of a small box placed in a vector field.
Net flux
What is the equation for flux?
What are 3 examples of flux in Physics?
- Electric flux
- Magnetic flux
- Flux of water through a pipe
What is a theorem?
A general proposition that is not self-evident but proved by a chain of reasoning.
Define the divergence theorem
The theorem that relates the flux of a vector, A, through a closed surface to the average divergence, ∇.A, within a volume. It is a mathematical tool often used in Physics to deal with fields.
What is the equation for the divergence theorem?
Where is the divergence theorem used in Physics?
In deriving Maxwell’s equations from Gauss’ Law.
Define Stoke’s theorem
The theorem that relates a line integral of a vector field to the surface integral of the curl of that vector field, sometimes called the ‘curl analogue’ of the divergence theorem. It is true for all fields (that do not enclose a singularity).
What is the equation for Stoke’s theorem?
Stoke’s theorem is true for _________ fields
Conservative
Why is Stoke’s theorem true for conservative fields?
For a conservative field, the left-hand and right-hand sides of the equation are both equal to 0.
Where is Stoke’s theorem used in Physics?
- Writing Ampere’s law in differential form
- Vorticity in fluids
Define vorticity
The tendency for rotation of a fluid element about its own axis, given by the curl of the fluid velocity vector.
What is the equation for vorticity?
ζ = vorticity
v = fluid velocity
What are vortex lines?
Lines of constant vorticity (constant curl). These describe the angular motion in a fluid just as streamlines describe the translational motion.
Define shearing strain
The angular deformation of a fluid element.
Define circulation
An integral quantity related to vorticity. The integral traverses anticlockwise by convention (as with line integrals in general).
What is the equation for circulation?
Γ = circulation
v = fluid velocity
What is the equation that relates vorticity and circulation?
What theorem relates vorticity and circulation?
Stoke’s theorem
What is a fluid?
A substance that deforms continuously when acted on by a shearing stress (i.e. a force tangential to a surface) of any magnitude. This includes both liquids and gases that are homogeneous, incompressible, and non-viscous (so not including metals or materials like tar, putty, or toothpaste).
What are the three types of fluid?
- Inviscid fluid
- Incompressible fluid
- Ideal fluid
What is an inviscid fluid?
A fluid that is non-viscous. Very few fluids are truly inviscid and those that are are called superfluids.
What is an incompressible fluid?
A fluid with constant density, such as air or water.
What is an ideal fluid?
A fluid that is both inviscid and incompressible.
Define flow
A fluid that moves, so has a non-zero vector velocity field. General flows are either pressure-induced or shear-induced.
What are the four types of flow?
- Steady flow
- Unsteady flow
- Uniform flow
- Non-uniform flow
What is steady flow?
A flow that occurs if the velocity vector at a given point does not vary with time (∂v/∂t = 0).
What is unsteady flow?
A flow that occurs if the velocity vector at a given point does vary with time.
What is uniform flow?
A flow that has the same velocity vector at all positions (within the defined boundary of the system being considered). This velocity vector may change with time.
What is non-uniform flow?
A flow that has different velocity vectors across the space within the defined boundary of the system being considered. This velocity vector can change with time.
Unsteady effects within a system can either be _________ or ________.
Periodic
Random
Define lamina flow
Flows which can be determined.
Define turbulent flow
Flows which cannot be determined.
Uniform fields are irrotational so they are _________.
Conservative
What is a source?
Where a fluid enters a system. Positive charges are labelled as sources by convention.
What is the equation for the fluid velocity of a line source in 2D?
v = fluid velocity
m = mass
What is the equation for the fluid velocity of a 3D source?
v = fluid velocity
m = mass
What is a sink?
Where a fluid leaves a system. Negative charges are labelled as sinks by convention.
Why can’t Stoke’s theorem be used when a source/sink is enclosed?
Because they are an example of a singularity where flow is undefined at r = 0 or ρ = 0.
What is a vortex?
Fluids that revolve around the axis line, formed when streamlines are circular.
What are the three types of vortex?
- Forced vortex
- Free vortex
- Rankine vortex
What is the equation for the velocity of a forced vortex?
v = velocity
ρ = distance from origin
ω = angular speed
What is the vorticity of a forced vortex?
Vorticity = 2x rotation vector
What is the circulation of a forced vortex?
What is the equation for the velocity of a free vortex?
v = velocity
k = constant
ρ = distance from origin
What is the equation for the angular momentum of a free vortex?
L = angular momentum (= Iω)
I = moment of inertia
ω = angular speed
m = mass
ρ = distance from origin
v = velocity
What is the vorticity of a free vortex?
0
What is the circulation of a free vortex?
Non-zero
Why isn’t Stoke’s theorem valid for a free vortex?
The vortex encloses a singularity at its centre.
Why can’t real vortices be free or forced?
- Forced vortices become unphysical at large ρ as v(ρ) tends to ρ.
- Free vortices become unphysical at small ρ because v(ρ) tends to 1/ρ.
What is the Rankine vortex model?
A model of vortices in real fluids that combines free and forced vortices. The velocity profile is approximated as a forced vortex at the centre where ρ is small and as a free vortex on the outskirts where ρ is large.
Describe the shape of the graph for a Rankine vortex
What is the continuity equation for an incompressible fluid flowing between two tubes?
ρ = density
v = fluid velocity
A = area of flow tube
What is the calculus version of the continuity equation?
ρ = density
t = time
v = fluid velocity
What are the three ways that the calculus version of the continuity equation can be written?
What is the continuity equation for a steady flow?
What is the continuity equation for a steady, incompressible flow?
What is the equation for the steady flow of an incompressible fluid in two dimensions (the x-y plane)?
u = velocity in x direction
v = velocity in y direction
What are the equations for the two velocity components of a stream function?
What is the equation for the slope of a streamline?
Where are streamlines found in a steady, incompressible flow?
Where the stream function, Ψ(x, y), is constant. This is tangential to the velocity field.
What is the Laplace equation?
v = fluid velocity
F = scalar function
The Laplace equation is used for ___________ flow by combining the equation for the velocity field (v = ∇F) and the continuity equation for ___________, ___________ flow (∇.v = 0).
Irrotational
Homogeneous
Incompressible
What is the uniqueness theorem (in terms of the Laplace equation?
If there is a solution to the Laplace equation that satisfies a set of boundary conditions then it is the only solution for those conditions.
The Laplace equation is a ________ _________ ____________.
Linear differential equation
What type of equation can the principle of superposition be used for?
Linear differential equations: the linear combination of different solutions is also a solution.
What is a Rankine half-body?
A feature of fluid flow where the boundary between uniform fluid flow and a source resembles a solid object; the uniform flow cannot enter this boundary and the source flow cannot leave it.
Where is the Rankine half-body found in physics?
Fluid dynamics
What is the method of images?
A method of determining the potential due to a source in the presence of a hard boundary by replacing the hard boundary with an appropriately chosen source (such that the boundary conditions are still obeyed). These fictitious sources are known as images.
Where is the method of images used in physics?
Electrostatics
What is the Euler equation?
The expression of Newton’s laws of motion for a fluid, but neglecting any viscous effects and considering gravity and pressure acting on the fluid.
What is the equation for the acceleration of a particle?
a = acceleration
v = velocity
t = time
What is the equation for acceleration in terms of the del operator?
a = acceleration
v = velocity
t = time
What are the two key forces that act on a small mass of fluid?
- Gravity
- Pressure
What is the equation for a small change in the force acting on a small mass of fluid due to gravity?
f = force
M = mass
g = acceleration due to gravity
What is the equation for a small change in the force acting on a small mass of fluid due to pressure?
f = force
p = pressure
S = area
How can the Euler equation be derived?
By using the divergence theorem to define the total force acting due to changes in pressure and applying that to an infinitesimal element of fluid mass. This can then be combined with the equation for the force due to gravity for an infinitesimal element of fluid mass then rearranged to give Euler’s equation.
What is the formula for the Euler equation?
ρ = density
v = velocity
t = time
g = acceleration due to gravity
p = pressure
What is the Bernoulli equation?
An equation derived from Euler’s equation that considers components along a streamline for an ideal (incompressible and inviscid) fluid in steady flow. It is essentially a statement about the conservation of energy for ideal fluids.
What is the formula for the Bernoulli equation?
ρ = density
v = velocity
g = acceleration due to gravity
p = pressure
How can the Bernoulli equation be used to measure the speed of a fluid?
By measuring the difference in pressure between a stagnation point (where speed = 0) and a given point on the streamline then substituting all of the values into the Bernoulli equation.
What is a Venturi tube?
A tube that has two wide ends and a narrow centre which causes pressure changes so that the fluid speed can be measured.
Define dynamical viscosity
The internal friction between different layers of fluid moving at different velocities (which form a gradient of velocity/distance). The internal friction gives rise to shear stresses in the fluid (force/area). It is generally given the symbol µ and has the units kg/m/s.
What is the equation for dynamical viscosity
µ = dynamical viscosity
F = force
A = area
v = velocity
d = distance
What is Reynolds number?
A dimensionless quantity, Re, that is a measurement of the ratio between the magnitude of inertial forces to viscous forces.
What is the equation for the Reynolds number?
Re = Reynolds number
ρ = density of fluid
v = mean velocity of fluid relative to surface
L = characteristic linear dimension (e.g. pipe radius)
µ = dynamic viscosity
Inviscid flow has __ dynamic viscosity so the Reynolds number is _________.
No
Infinite
The ____________ the Reynolds number, the greater the viscosity.
Smaller
What is the Navier-Stokes equation?
A statement of conservation of momentum that incorporates viscous effects into the equation of motion by modifying Newton’s laws for an element of fluid.
What are the viscous effects considered in the Navier-Stokes equation?
- Normal stress force
- Shearing stress force
What is the formula for the Navier-Stokes equation in vector form?
ρ = density of fluid
v = velocity
t = time
µ = dynamic viscosity
f = body forces per unit volume
What situations have analytical solutions to the Navier-Stokes equation?
- Drag-induced flow
- Pressure-induced flow
What is a Couette viscometer?
A device used to measure (shear) viscosity.
What boundary conditions are required to solve the Navier-Stokes equation for drag-induced flow?
- Steady flow
- Incompressible flow
- Flow is only in x-direction
- Not under external pressure
What boundary conditions are required to solve the Navier-Stokes equation for pressure-induced flow?
- Steady flow
- Incompressible flow
- Flow is only in x-direction (in the same direction as two parallel plates)
- Infinite plates
- No-slip boundary condition is applied
