- Determine \(\oiint_{A} ho \vec{V} \otimes \vec{V} \cdot \vec{n} d A\) for a \(1 \mathrm{~cm} \times 1 \mathrm{~cm} \times 1 \mathrm{~cm}\) Cartesian element if \(\vec{V}=5 \sqrt{x} \hat{i}
- The flux of a quantity given by the vector, ⃗f = 5xyˆi + 10yˆj + 0ˆk, passes through a cube with dimensions (in x−, y−, and z− directions) of 2 x 1 x 1 (assume the cube “starts” at the
- If the velocity profile of flow in a channel is given by: u = U∞H y, what is the mass flow rate through the channel per length into the page if the height of the channel is H?
- The velocity profile of flow in a pipe in the z−direction (Vz) is given by:Vz = ΔpR2 4μL 1 −r2 R2 !where r is the radial component (in cylindrical coordinates), R in this case is the radius of
- The divergence of the velocity vector in spherical coordinates can be written as:⃗∇· ⃗V = 1 r2 ∂r2Vr ∂r +1 r sin (θ)∂∂θ(Vθ sin (θ)) +1 r sin (θ)∂Vϕ∂ϕwhere Vr, Vθ, and Vϕ
- Find the mass flow rate of a fluid at a constant density of 1.2 kg/m3 passing through a surface whose area is 1 m2 and whose normal is ⃗n =1 √14ˆi+ 2ˆj + 3ˆk. The velocity of the flow is:
- A metal block, whose dimensions in the \(x-, y\)-, and \(z\)-directions are respectively \(L \times H \times W\), is in a steady state with each of its sides all held fixed at various temperatures.
- Scale the one dimensional heat equation with the heat generation term included. Write the non-dimensional heat equation in terms of Fourier number ( \(F o\) ), a non-dimensional time parameter
- Consider a conduction problem similar to the one given in Figure 5.5, except the boundary condition of the right side (at \(x=L\) ) is no longer held at a fixed temperature and is instead in contact
- The book provided the boundary layer equations in scaled form. Rescale the boundary layer equations in dimensional form.
- Scale Cauchy's momentum equation.
- Consider Couette flow between two plates a distance of 2 millimeters apart whose top plate moves with a velocity of \(1 \mathrm{~m} / \mathrm{s}\) and the bottom plate is held fixed. The temperature
- Consider flow past a flat plate. At what point downstream from the leading edge does the boundary layer double in size compared to the size of the boundary layer at \(x=1\) meter? You may assume the
- Evaluate the viscous dissipation term at an \((x, y)\) location of \((1,5)\) meters with a dynamic viscosity of \(10^{-3} \mathrm{~Pa} \cdot \mathrm{s}\) if velocity field is given by: \(\vec{V}=2 y
- Calculate the power (per volume) if the stress tensor is:\[\overrightarrow{\vec{T}}=\left(\begin{array}{ccc} 10 x & 5 y & 6 x \\ 5 y & 2 y & 25 y \\ 6 x & 25 y & 3 z \end{array}\right)\]with a
- Does the follow expression satisfy the Laplace equation?\[T=T_{0}+T_{1} \frac{\sin (\pi x) \sinh (\pi y)}{\sinh (\pi L)}\]You may assume \(T_{0}, T_{1}\), and \(L\) are constants.
- Consider a 2D flow whose velocity vector is given by \(\vec{V}=-10 x \hat{i}+10 y \hat{j}\) \(\mathrm{m} / \mathrm{s}\) and whose temperature is \(T=50 e^{-0.03 t} \sin (2 x) \cosh (5 y)\) kelvin.
- Find the steady state temperature in the middle of a \(10 \mathrm{~cm}\) bar whose sides in the \(y\) - and \(z\)-directions can be ignored and the temperature of the right (at \(x=10 \mathrm{~cm}\)
- Does the following function for temperature satisfy the heat equation when there is no heat generation ( \(\alpha\) is thermal diffusivity):\[T=T_{0}+T_{m} \exp \left(\frac{-2 \pi^{2} \alpha}{L^{2}}
- Consider a system of water with a pinch of salt centered in the middle. If we were to model the "stirring" of the system of water using the advection equation, what would the result be? What would we
- Consider incompressible flow between two parallel plates. The flow is driven by the bottom plate moving to the right with a velocity of 100 \(\mathrm{m} / \mathrm{s}\). The viscosity of the fluid is
- In your own words, what is the definition of a boundary layer?
- Find \(\vec{A} \otimes \vec{B}\) if \(\vec{A}=3 \hat{i}+5 \hat{j}-10 \hat{k}\) and \(\vec{B}=5 \hat{i}-2 \hat{j}-\hat{k}\).
- Consider incompressible flow between two parallel plates a distance, \(H\), apart. If the bottom plate moves with a velocity of \(U_{B}\) and the top is fixed, find an expression of the velocity
- Find an expression for the shear stress on the bottom and top plates from Problem 4.3.
- Consider incompressible flow between two parallel plates a distance, \(H=0.5 \mathrm{~cm}\), apart. The top plate is moving with a velocity to the right of \(1 \mathrm{~m} / \mathrm{s}\). There is
- In your own words, describe the various forms of the Navier-Stokes equations discussed in this book and how they were obtained.
- Show that the conservation form and the non-conservation form of the Navier-Stokes equations are equivalent. You may use Cartesian coordinates.
- Given an initial temperature distribution of \(T(x)=\sin (x)\), create a table similar to Table 2.1 for times of \(0 \mathrm{~s}, 0.5\mathrm{~s}, 1 \mathrm{~s}\), and \(1.5 \mathrm{~s}\) when \(u=1
- Explain why the characteristic curves for the advection equation are parallel when the velocity is a constant value.
- Given an initial \(x\)-velocity distribution of:\[ u(x, 0)= \begin{cases}1 & x \leq 0.5 \\ 2 x & 0.5
- Find an expression for the material derivative of temperature if the temperature is given by the following equation \(T=e^{-t}(\sin (2 x)+\cos (y))\) and if the velocity vector is:
- For the temperature equation used in Problem 2.4, what is the advective transport term of temperature if the velocity field is given by \(\vec{V}=\) \(2 x y \hat{i}-y^{2} \hat{j}\) when \(x=2, y=1\),
- Given the velocity field vector: \(\vec{V}=y(A \cos (2 t)+B \sin (3 t)) \hat{i}+6 x y t \hat{j}\) \(\mathrm{m} / \mathrm{s}\), what is the acceleration of the fluid at \(x=0.5 \mathrm{~m}\) and
- The Lagrangian and non-conservation form of the continuity equation can be obtained by applying mass conservation to a moving fluid element, which states that the mass of a moving fluid element does
- Suppose the density of a fluid element is given by the expression: \(ho=\) \(e^{-0.005 t}+1\). It travels with a velocity of \(2 \mathrm{~m} / \mathrm{s}\) in the \(x\)-direction. What is the value
- What is the summation force vector on a cube fluid element that is 1 meter by 1 meter by 1 meter in size if the density is \(1 \mathrm{~kg} / \mathrm{m}^{3}\) and \(\vec{V}=2 x \hat{i}+4 y \hat{j} ?\)
- The gradient of a scalar function, \(f\), is defined as:\[ \vec{abla} f \equiv \hat{i} \frac{\partial f}{\partial x}+\hat{j} \frac{\partial f}{\partial y}+\hat{k} \frac{\partial f}{\partial z} \]
- Find the stress vector acting on a surface if the stress tensor is:\[ \overrightarrow{\vec{T}}=\left(\begin{array}{ccc} 5 & -3 & 10 \\ -3 & 2 & 4 \\ 10 & 4 & 7 \end{array}\right) \]and the
- Do you spot any potential issue with the stress tensor below?\[ \overrightarrow{\vec{T}}=\left(\begin{array}{ccc} 5 x & 4 x & 10 z \\ -3 y & 2 x y & 4 x \\ 2 z & 4 y & 7 x y z \end{array}\right)
- A stress tensor (in pascals) of a flow is given by:\[ \overrightarrow{\vec{T}}=\left(\begin{array}{ccc} 5 x & -3 y & 10 z \\ -3 y & 5 y & 4 x \\ 10 z & 4 x & 5 z \end{array}\right) \]What is
- If the velocity field of a given flow is given by:\[ \vec{V}=A \cos (x) \hat{i}+B \sin (y) \hat{j}+C \tan (z) \hat{k} \]Determine an expression for the total force (per volume) acting on the fluid
- For the velocity field given in Problem 3.4, if there are no body forces and if the dynamic viscosity is considered a constant, find an expression for the pressure gradient (given \(A=1, B=1\), and
- Determine the Newtonian stress tensor if the pressure is given by \(10 x\) pascals and a velocity field of \(\vec{V}=2 x \sin (2 y) \hat{i}+x \cos (2 y) \hat{j} \mathrm{~m} / \mathrm{s}\) at location
- What is the pressure, in pascals, a person experiences when they have dived 5 meters below the water surface? Assume atmospheric pressure is \(10^{5}\) pascals.
- Estimate the shear stress (i.e. viscous force) on a surface defined by \(\vec{n}=\hat{i}\) where the velocity distribution is given by: \[ \begin{gathered} u=\frac{U}{H} y \\ v=w=0 \end{gathered}
- Show that \((\vec{V} \cdot \vec{abla}) \vec{V}=\vec{V} \cdot(\vec{abla} \vec{V})\) in two dimensional Cartesian coordinates.
- Please explain, in your own words, why an equation of state is not necessarily needed for an incompressible flow.
- Approximate how long will it take, in minutes, to fill up a bathtub with dimensions of 1.3 m x 0.6 m x 0.4 m if water is coming out of a 40 mm diameter faucet at a speed of 1.2 m/s. Assume the
- An incompressible fluid with density ρ = 1000 kg/m3 travels through a channel with a rectangular cross-section of dimensions 25 mm by 30 mm. The average velocity of the flow in this portion of the
- Air in a pipe with a diameter of 10 cm starts out at a temperature of 700 K and pressure of 4 × 105 Pa. The initial flow velocity is 10 meters per second. If the pipe diameter contracts to 5 cm
- Given the following velocity vector: ⃗V = Cy cos(5x)ˆi + Dsin(5x)y2ˆj, what do C and D need to be in order for this vector field to be considered incompressible flow?
- Sketch the velocity vector given by: ⃗V = − sin(y)ˆi + sin(x)ˆj.