fluid mechanics

What is a fluid

A substance which cannot resist a shear force without motion and doesn’t return to its original state when stress is removed

Define stress

Defined as tau which = force/ Area. It has the same dimensions as pressure.

Define strain

Strain is defined as e which equals the change in X(y) / change in y (partial)

X(y) represents change in x dim

What is the small displacement strain approximation

e = tan (theta) = theta

Theta is the angle between the y axis and the changed y dimension of the cube

What is the relation of tau and e in a solid

Tau = G e where G is the shear modulus

Compare removing stress from a cube of solid to a cube of fluid

If stress is removed from a solid. It either stays in that configuration or returns to original position (depending on deformation)

For a fluid the angle theta will increase as time increases sunce fluids have ndiferent bonding therefore no restoring force

Relate stress and strain in a fluid

Tau = mu (change in strain/change in time)

Mu represents the viscosity

What is a lagrangian fluid element

AN LFE moves with the fluid such that there is no flow in or out of the element

U = d/dt (r-r_0) = (dx/dt)i+(dy/dt)j+(dz/dt)z

What is an eulerian fluid element

An eulerian fluid element is fixed in space and the mass of the element may change

u= u(x,y,z,t)

What is a streamline

A 3D curve which at a fixed time t at each location is tangential to the local vector of u

How can we solve to find the streamline equations

let dr be along a streamline. So clearly it is parallel to u hence dr=d(lambda)u where d(lambda) is a scalar constant

Hence dx=d(lambda)u_x (same for y and z) then equate d(lambdas)

When is flow laminar

Flow is smooth typically the same shape as the surface, for parallel plates flow is also parallel

When is flow viscous

Viscosity is important

When is flow steady

u(r,t)=u( r ) and not dependent on time

What is the viscous force

Fluid element with top and bottom area A = dx dz

Then the viscous force will be F_v = A(mu)(du_x/dy)

= (function above evaluated at y+dy)-(function evaluated at y)

Note here dx etc is delta where obvious

As dxdydz→0 F_v =mu (d²u_x/dy²) dxdydz

What is the pressure force

F_v= -F_p = -dp/dx deltaxdeltaydeltaz

From here we can cancel out all the delta terms and equate everything to -Q a constant as both sides are dependent on different things.

Solve the pressure force, viscous force equation for the two fixed plate 2d scenario

Mu (d²u_x/dy²)=dp/dx = -Q

u_x(y) = -Q/(2mu) y² +Ay + B

A and B are found from BCs u_x(a)=u_x(-a)=0

Q= -dp/dx = ( P_1-P_2)/L

what is poiselle flow

Cylindrical steady, symmetric, viscous, laminar flow through a pipe

What is the average molecular speed

U_mol is the flow due to random thermal motion

What is the kinematic viscosity

v(accent) = mu/rho

Rho is the fluid mass density

What is the Reynolds number

Re = (rho_0)(L_0)(u_0)/mu

Rho is the typical mass density

L is the characteristic length

u is the typical flow speed

If the Reynolds number is small then the flow is giscous dominated

What is the continuity equation for steady flow

If the fluid is incompressible then div(u)=0

What is the strouhal number

St=D/Pv (P here is the period)

What is the vorticity

w = grad (u) and if it = 0 the flow is irrotational

What is kelvins circulation theorem

The circulation around a closed loop moving with the fluid is constant for inviscid flow.

DK/dt = d/dt (integral u•dl)=0

What is div u in cylindrical co odds

1/r d_r(ru_r) + 1/r d_theta(u_theta) + d_z(u_z)

What is grad u in cylindrical

1/r outside grad and theta top term has r and theta u has r multi

Gauss theorem

Integral A•ds= integral div(A)dV

Stokes theorem

Integral (A•dl) = integral grad(A)•dS

How to show irrot-/→not rot

U=A/r theta hat

Uniform potential flow

U = vi = grad phi

Phi = ux+ const

Horizontal streamlines

What is the boundary layer thickness

Delta = d/Re^1/2

Explain Magnus effect

The force per unit length on a 2D body in relative motion with a fluid with virticity and circulation is L=rho uxk

State navier stokes equation

P du/dt + rho (div(u))u = -grad(rho)…

Give examples of simple 2D potential flows

State the continuity equation

drho/dt + rho div(u)= 0

Action equation

A = integral L dt

Electrostatic potential energy

V = 1/ 4 pi Epsilon_0 (q1q2/r)

Generalised Hamiltonian

H = q* dL/dq* -L

Posselle flow

V_z ( r ) = Q/ 4 mu (a² -r²)

Adianatic pressure equation thingy

d/dt (P/ rho^r) =0

Drag force

F = b rho v² A/2