BMEN90036 Biofluid Mechanics Assignment 2
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Assignment 2
Biofluid Mechanics (BMEN90036)
Question 1 [20 marks]
As we have seen in class, a hydraulic jump is a region of flow where the flow depth increases rapidly (see figure b elow). H ydraulic j umps a re i ncredibly t urbulent r egions o f t he fl ow, wh ere a large amount of energy is dissipated.
You are studying the hydraulic jump in a steady river flow where the downstream flow depth (h2) is 2 times the upstream flow depth (h1). Use the conservation laws of fluid mechanics to determine the flow rate in this river as a function of only the river width (b) and the upstream depth (h1 ). Hint: use the integral forms of the conservation laws covered in class, and consider the surfaces across which momentum flows a nd w hat forces a re a cting o n t hose surfaces.
Hydraulic&Jump
h2 > hl
l
Flow
Question 2: Part A [5 marks]
When small particles move through a fluid, the Reynolds number is typically very small (Re << 1). Such flows are called creeping flows. The drag force (FD ) on an object in creeping flow is a function only of its speed U , its characteristic length scale L and the dynamic fluid viscosity, µ .
D
U
Consider pollen particles falling at terminal velocity through air. At terminal velocity, the drag force on the object is equal and opposite to its weight. If the length scale of a pollen particle is tripled, what happens to its terminal velocity?
Question 2: Part B [15 marks]
Consider a unique fluid, in which its density (ρ) is dependent on its depth from the free surface by the following relationship.
ρz = ρ0 z3
Where ρz represents the instantaneous density at a given depth z (m) and ρ0 is the initial density at the surface of the fluid. A silver ball bearing (ρAg = 10,500 kg/m3) of diameter 1 mm is placed into the fluid. You can assume that the ball bearing begins initially from rest and its velocity and drag-force at any instance in time is given by.
vk+1 = vk + at
Fd = ρz v C2d A
Cd = 0.45 +
Where a is the acceleration of the bearing, A is the cross-sectional area of the ball-bearing and k is the iteration in the loop. In this instance you can neglect the buoyancy forces present on the ball-bearing and the fluid has an initial density equal to that of water ( i.e ρ0 = 1000 kg/m3 ) at the free surface and a constant viscosity of µ = 0.001 Pas.
(a) Derive an expression for the force balance on the ball bearing.
(b) What is the maximum velocity the ball bearing experiences? Using MATLAB, plot the
velocity of the bearing and the penetration depth as a function of time.
(c) After the ball bearing begins to slow down, at what depth and time is the ball bearing traveling at 0.1 m/s? What is the density at this location?
It may help to use Eulers method to track the position of the ball bearing as a function of time. Remember that Eulers method states that the new location of a particle is simply the old location plus the velocity vector times a small time step. Formally this can be written as,
xk+1 = xk + u-∆t
We haven’t specifically provided you with a time step for this problem. You should chose an appro- priate time step that results in convergence of your output result.
Question 3 [20 marks]
A pair of opposite signed (but equal strength) potential vortices are held fixed in space in a uniform flow from right to left. The counter-clockwise potential vortex is located at (0, l),and the clockwise vortex is located at (0, -l).
(a) Write out the streamfunction ψ for this flowfield
(b) Determine the components of the velocity in the x and y directions
(c) Hence show that the flow pattern and stagnation point location depends on the quantity Γ/πlU2 , and that for,
< 1
The stagnation points are located on the y axis at,
y = 士 ′ ╱l2 - 、
(d) Use MATLAB to sketch the resulting flow pattern and velocity vector field. Assume that Γ = 1, l = 1 and U2 = 1. Hint: remember you can use the quiverInLogScale function found on the mathworks website, to more easily visualize the vector field.
Question 4 [20 marks]
U
Figure 1: Diagram of the problem statement. Note that the velocity profile i n t he fi gure ma y or may not be representative of the solution you obtain
Consider steady laminar flow of a viscous incompressible fluid between two parallel plates, with spacing a. By assuming developed flow, t he z a nd y c omponents o f t he fl uid ve locity ar e ze ro (i.e v = w = 0). The top wall is stationary, and the fluid motion is driven by the motion of the bottom wall at constant velocity u and by the pressure gradient in the x-direction dp/dx.
Starting from the full incompressible continuity equation and the x-momentum equations, derive an expression of the fluid velocity u(y). The effects of gravity can be neglected.
Question 5 [20 marks]
Carbon dioxide is often used in biomedical engineering labs for many applications, one of which is cell culturing. UltraSuperCell Services忿is a large commercial laboratory, and requires large amounts of carbon dioxide. They have determined that it is much more efficient to pump this carbon dioxide from a neighbouring facility located 1000 m away via a long pipe with an internal diameter of 12 cm and a fanning friction factor of f = 0.0005. The flow is isothermal at 25C and carbon dioxide has the chemical formula CO2, where carbon has a molecular weight of 12g/mol and Oxygen has a molecular weight of 16 g/mol. The neighboring facility stores the carbon dioxide at a pressure 521 kPa and the lab requires the pressure to be 200 kPa.
(a) what is the mass flow rate of the carbon dioxide?
(b) What is the pressure one-third the way down the pipe? What would be the pressure if you were pumping an incompressible fluid? Are the pressures the same? Why or why not?
(c) An operator at the neighbouring facility proudly tells your that he thinks he can triple the flow-rate through the piping system by tripling the pressure from 521 kPa to 1563 kPa. Do you think that this is possible? Why or why not?
(d) What is the maximum initial pressure that you would recommend in order to maximize the flow-rate of carbon dioxide? What is the sonic velocity of the gas at this condition? At what percentage of the speed of sound is the gas flowing?
2022-10-07