A Student S Guide To The Navier-Stokes Equations 1st Edition Justin W. Garvin - Solutions

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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 \(\vec{V}=5 \sqrt{x} \hat{i} \mathrm{~m} / \mathrm{s}\) and the density is \(1000 \mathrm{~kg} / \mathrm{m}^{3}\).
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 origin). Find the value of the area integral,A⃗f · ⃗ndA.
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 pipe, L is the length of the pipe, Δp is considered to be a pressure difference between the
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 are the velocity coordinates in the r−, θ−, and ϕ−direction, respectively. Determine if a
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: ⃗V = 2ˆi + 3ˆj + 0ˆk m/s.
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. If \(L\) is much smaller than both \(H\) and \(W\), show, by scaling, that the governing equation for
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 defined as:\[ F o=\frac{\alpha t}{L^{2}} \]where \(\alpha\) is the thermal diffusivity and \(L\) is the
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 with a fluid at temperature \(T_{\infty}\). The boundary condition on the right side is given by a
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 of both plates are held fixed at zero degrees Celsius. The thermal conductivity is \(0.6
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 Reynolds number never reaches \(10^{5}\), which is the transition to turbulence.
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: \(\vec{V}=2 y \cos (2 x) \hat{i}+y^{2} \sin (2 x) \hat{j} \mathrm{~m} / \mathrm{s}\).
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 velocity vector is \(\vec{V}=\hat{i}+2 \hat{j}-2 \hat{k}\) and no body force.
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. The fluid has a thermal conductivity of \(0.6 \frac{\mathrm{W}}{\mathrm{mK}}\), a density of \(1000
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 \(x=10 \mathrm{~cm}\) ) is fixed at 0 degrees Celsius and the flux on the left is \(1 \mathrm{~W} / \mathrm{m}^{2}\). The
Does the following function for temperature satisfy the heat equation when there is no heat generation ( \(\alpha\) is thermal diffusivity):\[T=T_{0}+T_{m} \exp \left(\frac{-2 \pi^{2} \alpha}{L^{2}} t\right) \sin \left(\frac{\pi x}{L}\right) \sin \left(\frac{\pi y}{L}\right)\]You may assume \(L,
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 need to do in order to more accurately model the physical reality of the situation?
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 given by \(0.001 \mathrm{~Pa} \cdot \mathrm{s}\) and the thermal conductivity is \(0.6
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 profile assuming there is no pressure gradient.
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 also a constant pressure gradient that is resisting the flow in the \(x\)-direction with a magnitude
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 \(u=1 \mathrm{~m} / \mathrm{s}\). Plot the final temperature distribution at \(t=1.5 \mathrm{~s}\).
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: \(\vec{V}=\hat{i}+2 \hat{j}\).
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 \(x=2, y=1\), and \(t=1\) ?
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 \(y=0.5 \mathrm{~m}\) at time, \(t=1 \mathrm{~s}\) ?
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 not change in time. Mathematically, this can be written as:\[ \frac{d}{d t} \iiint_{\mathcal{V}(t)}
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 of the material derivative after it has gone 60 meters?
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:\[ \vec{abla} f \equiv \hat{i} \frac{\partial f}{\partial x}+\hat{j} \frac{\partial f}{\partial y}+\hat{k} \frac{\partial f}{\partial z} \] Show that \((\vec{V} \cdot \vec{abla}) f=\vec{V} \cdot(\vec{abla} f)\)
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 normal of the surface is given by: \(\vec{n}=\frac{1}{\sqrt{2}} \hat{i}+\frac{1}{\sqrt{2}} \hat{j}+0
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 the acceleration of a fluid particle with density of \(1.2 \frac{\mathrm{kg}}{\mathrm{m}^{3}}\)
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 at a given \((x, y, z)\) point.
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 \(C=0\) ). Be sure to check if this flow is incompressible or not. Is this flow at a steady state?
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 \(x=1 \mathrm{~m}, y=2 \mathrm{~m}\). Assume a dynamic viscosity of \(10^{-4} \mathrm{~Pa} \cdot
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: \[ \begin{gathered} u=\frac{U}{H} y \\ v=w=0 \end{gathered} \]where \(U\) is a velocity parameter and \(H\) is a length scale parameter.
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 density of water is 1000 kg/m3.
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 channel is 1 m/s. If the flow is at a steady state, what will the velocity be if the rectangular
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 with the air speeding up to 115 m/s and the temperature decreases to 500 K, what is the pressure after
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.

A Student S Guide To The Navier-Stokes Equations 1st Edition Justin W. Garvin - Solutions

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