Chapter 6, Isothermal Fluid Mechanics Video Solutions, A First Course in Continuum Mechanics

Problem 1

Consider an ideal fluid with mass density $\rho_{0}$ subject to a uniform, constant body force field per unit mass $\boldsymbol{b}$. Find the pressure field $p(\boldsymbol{x})$ assuming the fluid is at rest and that $p\left(\boldsymbol{x}_{*}\right)=p_{*}$ at some reference point $\boldsymbol{x}_{*}$.

Victor Salazar

Numerade Educator

03:50

Problem 2

Consider an ideal fluid in a region $D$ with mass density $\rho_{0}$ subject to zero body force. Suppose $D$ is the unbounded regionexterior to a fixed, solid obstacle $\Omega$. Show that, if the motion of the fluid is steady and irrotational, then the resultant force exerted by the fluid on the obstacle is
$$
\boldsymbol{r}=\frac{\rho_{0}}{2} \int_{\partial \Omega}|\boldsymbol{v}|^{2} \boldsymbol{n} d A_{\boldsymbol{x}}
$$
where $\boldsymbol{n}$ is the outward unit normal field on $\partial \Omega$.

Ajay Singhal

Numerade Educator

01:18

Problem 3

Let $D=\left\{\boldsymbol{x} \in \mathbb{E}^{3} \mid 0<x_{i}<1\right\}$ and consider the scalar Laplace equation
$$
\Delta^{x} \phi=0, \quad \forall \boldsymbol{x} \in D
$$
together with the boundary condition
$$
\nabla^{x} \phi \cdot \boldsymbol{n}=g, \quad \forall \boldsymbol{x} \in \partial D
$$
where $g$ is a given function and $\boldsymbol{n}$ is the outward unit normal field on $\partial D$.
(a) Find $\phi$ assuming $g=\cos \left(k \pi x_{1}\right) \cos \left(l \pi x_{2}\right)$ on the face with $\boldsymbol{n}=\boldsymbol{e}_{3}$, and $g=0$ elsewhere. Here $k$ and $l$ are arbitrary integers.
(b) Find $\phi$ assuming $g=\sum_{k, l=1}^{\infty} a_{k l} \cos \left(k \pi x_{1}\right) \cos \left(l \pi x_{2}\right)$ on the face with $\boldsymbol{n}=\boldsymbol{e}_{3}$, and $g=0$ elsewhere. Here $a_{k l}$ are constants.
(c) Find $\phi$ assuming $g=0$ on all faces. What does this case say about irrotational motions of an ideal fluid in $D$ ?

Raj Bala

Numerade Educator

01:01

Problem 4

Repeat Exercise 1, but now for an elastic fluid with mass density $\rho$ and constitutive relation $p=\lambda \rho^{\gamma}$ where $\lambda>0$ and $\gamma>1$ are constants. (A relation of this form describes various real gases at moderate conditions, for example atmospheric air.)

Amit Srivastava

Numerade Educator

02:56

Problem 5

Consider a body of elastic fluid, subject to zero body force, undergoing a steady, irrotational motion with spatial velocity field $\boldsymbol{v}=\nabla^{x} \phi$ in a fixed region $D .$ Assume the constitutive relation $p=\frac{1}{2} \rho^{2}$.
(a) Find the associated functions $\gamma(s)$ and $\zeta(s)$ for this fluid.
(b) Show that the velocity potential $\phi$ satisfies the nonlinear equation
$$
c \Delta^{x} \phi=\frac{1}{2}\left|\nabla^{x} \phi\right|^{2} \Delta^{x} \phi+\nabla^{x} \phi \cdot\left(\nabla^{x} \nabla^{x} \phi\right) \nabla^{x} \phi, \quad \forall \boldsymbol{x} \in D
$$
where $c$ is a constant. Here $\nabla^{x} \nabla^{x} \phi$ is the second-order tensor with components $\left[\nabla^{x} \nabla^{x} \phi\right]_{i j}=\phi_{, i j}$

Khoobchandra Agrawal

Numerade Educator

08:15

Problem 6

Let $c>0$ be constant and consider the wave equation for the density disturbance field in an elastic fluid
$$
\frac{\partial^{2}}{\partial t^{2}} \rho^{(1)}=c^{2} \Delta^{x} \rho^{(1)}
$$
Show that
$$
\rho^{(1)}(\boldsymbol{x}, t)=f(\boldsymbol{k} \cdot \boldsymbol{x}-c t)+g(\boldsymbol{k} \cdot \boldsymbol{x}+c t)
$$
is a solution of the wave equation for any functions $f, g: \mathbb{R} \rightarrow \mathbb{R}$. and any unit vector $\boldsymbol{k}$. Solutions of this form are called plane waves. $f(\boldsymbol{k} \cdot \boldsymbol{x}-c t)$ is a wave profile which moves with speed $c$ in the direction $\boldsymbol{k}$, and $g(\boldsymbol{k} \cdot \boldsymbol{x}+c t)$ is a profile which moves with speed $c$ in the direction $-\boldsymbol{k}$.

Shoukat Ali

Other Schools

04:00

Problem 7

Consider a body of elastic fluid with reference configuration $B$ and equation of state $p=\pi(\rho)$ between the spatial pressure and spatial mass density.
(a) Show that the first Piola-Kirchhoff stress tensor for the fluid can be written in the form
$$
\boldsymbol{P}=\boldsymbol{F} \overline{\boldsymbol{\Sigma}}\left(\boldsymbol{C}, \rho_{0}\right)
$$
where $\overline{\boldsymbol{\Sigma}}\left(\boldsymbol{C}, \rho_{0}\right)$ is a certain function depending on the state function $\pi .$ Here $C$ is the Cauchy-Green strain tensor and $\rho_{0}$ is the reference mass density.
(b) Based on the result in (a) show that an elastic fluid may also be viewed as an isotropic elastic material as considered in Chapter 7, with stress response function depending on the reference mass density.

Chai Santi

Numerade Educator

View

Problem 8

Let $\boldsymbol{v}$ be a smooth vector field satisfying the condition $\nabla^{x} \cdot \boldsymbol{v}=0$ in a regular, bounded region $D$. Assuming $\boldsymbol{v} \cdot \boldsymbol{n}=0$ on $\partial D$ show
$$
\int_{D}\left(\nabla^{x} \boldsymbol{v}\right) \boldsymbol{v} d V_{\boldsymbol{x}}=\mathbf{0}
$$

Victor Salazar

Numerade Educator

01:00

Problem 9

Show that the Cauchy reactive stress $\boldsymbol{S}^{(r)}$ in either an ideal or Newtonian fluid body does no work in any motion compatible with the incompressibility constraint. In particular, show that its contribution to the stress power vanishes, that is
$$
\boldsymbol{S}^{(r)}: \boldsymbol{L}=0, \quad \forall \boldsymbol{x} \in B_{t}, t \geq 0
$$

Manik Pulyani

Numerade Educator

03:18

Problem 10

Consider the Navier-Stokes equations with zero body force, and consider solutions for which the velocity field $\boldsymbol{v}$ is of the form
$$
\boldsymbol{v}(\boldsymbol{x}, t)=v_{1}\left(x_{1}, x_{2}, t\right) \boldsymbol{e}_{1}+v_{2}\left(x_{1}, x_{2}, t\right) \boldsymbol{e}_{2}
$$
Such solutions are called planar. Moreover, consider velocity fields which can be represented in the form
$$
\boldsymbol{v}=\frac{\partial \psi}{\partial x_{2}} \boldsymbol{e}_{1}-\frac{\partial \psi}{\partial x_{1}} \boldsymbol{e}_{2}=: \nabla^{x \perp} \psi
$$
where $\psi\left(x_{1}, x_{2}, t\right)$ is an arbitrary scalar-valued function called a streamfunction. (The notation $\nabla^{x \perp}$ is motivated by the fact that $\nabla^{x \perp} \psi \cdot \nabla^{x} \psi=0$ for any $\psi$. The operator $\nabla^{x \perp}$ is sometimes referred to as a skew gradient.) Here we show that the streamfunction $\psi$ for a planar solution of the Navier-Stokes equations satisfies a relatively simple equation.
(a) Show that the incompressibility constraint (conservation of mass equation) is automatically satisfied for any $\psi$.
(b) Let $\boldsymbol{w}=\nabla^{x} \times \boldsymbol{v}$ denote the vorticity field associated with
$v$. Show that
$$
\boldsymbol{w}=w \boldsymbol{e}_{3} \quad \text { where } \quad w=-\Delta^{x} \psi
$$
(c) Use the balance of linear momentum equation to show that $w$ must satisfy
$$
\frac{\partial w}{\partial t}+\boldsymbol{v} \cdot \nabla^{x} w=\nu \Delta^{x} w
$$Remark: A good approximation to the above equation when $\nu \gg 1$ is
$$
\left(\Delta^{x}\right)^{2} \psi=0
$$
This equation is usually called the biharmonic equation. It is also a good approximation when the flow is steady and gradients of $\psi$ are small, so that a linear approximation is valid.

Chai Santi

Numerade Educator

05:35

Problem 11

Consider a Newtonian fluid subject to zero body force occupying the unbounded region of space
$$
D=\left\{\boldsymbol{x} \in \mathbb{E}^{3} \mid \quad 0<x_{2}<h, \quad-\infty<x_{1}, x_{3}<\infty\right\}
$$
Here we find steady solutions of the Navier-Stokes equations under various different boundary conditions assuming the velocity field has the simple, planar form
$$
\boldsymbol{v}(\boldsymbol{x})=v_{1}\left(x_{1}, x_{2}\right) \boldsymbol{e}_{1}
$$
(a) Show that, when written in components, the balance of linear momentum and conservation of mass equations reduce to
$$
\mu \frac{\partial^{2} v_{1}}{\partial x_{2}^{2}}=\frac{\partial p}{\partial x_{1}}, \quad \frac{\partial p}{\partial x_{2}}=0, \quad \frac{\partial p}{\partial x_{3}}=0, \quad \frac{\partial v_{1}}{\partial x_{1}}=0
$$
Moreover, show that the general solution of these equations is
$$
v_{1}=\frac{\alpha}{2 \mu} x_{2}^{2}+\beta x_{2}+\gamma, \quad p=\alpha x_{1}+\delta
$$
where $\alpha, \beta, \gamma, \delta$ are arbitrary constants.
(b) Find the velocity and pressure fields assuming the no-slip boundary condition $\boldsymbol{v}=\mathbf{0}$ at $x_{2}=0$ and $\boldsymbol{v}=\vartheta \boldsymbol{e}_{1}$ at $x_{2}=$ $h$. This solution describes a fluid between two infinite parallel plates. One plate is at $x_{2}=0$ and is stationary, the other is at $x_{2}=h$ and is moving with speed $\vartheta$ in the $x_{1}$-direction. Further assumptions on the pressure are needed to fix a unique solution.
(c) Find the velocity and pressure fields assuming the no-slip boundary condition $\boldsymbol{v}=\mathbf{0}$ at $x_{2}=0$ and assuming a traction boundary condition of the form $\boldsymbol{S n}=a \boldsymbol{e}_{1}+b \boldsymbol{e}_{2}$ at $x_{2}=h$. Here $S$ is the Cauchy stress, $n$ is the outward unit normal on $\partial D$ and $a, b$ are given constants. This solution describes a film of fluid of constant thickness on a flat plate. The plate is at $x_{2}=0$ and the free surface of the film is at $x_{2}=h$.

Khoobchandra Agrawal

Numerade Educator

03:26

Problem 12

Consider a steady velocity field of the form $\boldsymbol{u}(\boldsymbol{x})=\boldsymbol{A} \nabla^{x} \psi(\boldsymbol{x})$, where $\psi$ is a scalar function and $\boldsymbol{A}$ is a second-order tensor defined in a given frame by
$$
\psi(\boldsymbol{x})=\sin \left(x_{1}\right) \sin \left(x_{2}\right), \quad[\boldsymbol{A}]=\left(\begin{array}{rrr}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 1
\end{array}\right)
$$(a) Show that $\left(\nabla^{x} \boldsymbol{u}\right) \boldsymbol{u}=-\nabla^{x} \phi$, where
$$
\phi(\boldsymbol{x})=\frac{1}{4} \cos \left(2 x_{1}\right)+\frac{1}{4} \cos \left(2 x_{2}\right)
$$
(b) Show that, for an appropriate choice of pressure field $q(\boldsymbol{x})$, the pair $(\boldsymbol{u}, q)$ is a steady solution of the Euler equations $(6.1)$ with zero body force.
(c) Let $\boldsymbol{v}(\boldsymbol{x}, t)=e^{-\lambda t} \boldsymbol{u}(\boldsymbol{x}) .$ Show that, for an appropriate choice of pressure field $p(\boldsymbol{x}, t)$ and constant $\lambda$, the pair $(\boldsymbol{v}, p)$ is a nonsteady solution of the Navier-Stokes equations (6.39) with zero body force.

Remark: The results in (b) and (c) are known as the TaylorGreen solutions. They are simple examples of exact solutions to the Euler and Navier-Stokes equations for bodies occupying all of space.

Stanley Enemuo

Numerade Educator

08:26

Problem 13

Let $D$ be a regular, bounded region. Assume the eigenvalue problem
$$
\left\{\begin{aligned}
-\Delta^{x} \boldsymbol{\eta} &=\lambda \boldsymbol{\eta}, \quad \forall \boldsymbol{x} \in D \\
\boldsymbol{\eta} &=\mathbf{0}, \quad \forall \boldsymbol{x} \in \partial D
\end{aligned}\right.
$$
has an infinite set of smooth eigenfunctions $\left\{\boldsymbol{\eta}_{i}\right\}_{i=1}^{\infty}$ and corresponding positive eigenvalues $\left\{\lambda_{i}\right\}_{i=1}^{\infty}$, ordered so that
$$
0<\lambda_{1} \leq \lambda_{2} \leq \cdots
$$
Since the eigenvalue problem is self-adjoint, the eigenfunctions may be chosen so that
$$
\int_{D} \boldsymbol{\eta}_{i} \cdot \boldsymbol{\eta}_{j} d V_{\boldsymbol{x}}=\delta_{i j}, \quad i, j=1,2, \ldots
$$
Prove the Poincaré Inequality (Result $6.10)$ assuming that every smooth vector field $\boldsymbol{w}$ on $D$ can be represented by an eigenfunction expansion
$$
\boldsymbol{w}=\sum_{i=1}^{\infty} \alpha_{i} \boldsymbol{\eta}_{i}
$$
where $\left\{\alpha_{i}\right\}_{i=1}^{\infty}$ are constants depending on $\boldsymbol{w}$.

Hafiz Shahzaib

Numerade Educator

04:03

Problem 14

Let $D$ be a given region. For any two smooth vector fields $\boldsymbol{v}, \boldsymbol{b}$ and any real number $s>0$ show that
$$
\int_{D}|\boldsymbol{b} \cdot \boldsymbol{v}| d V_{\boldsymbol{x}} \leq \frac{s^{2}}{2} \int_{D}|\boldsymbol{b}|^{2} d V_{\boldsymbol{x}}+\frac{1}{2 s^{2}} \int_{D}|\boldsymbol{v}|^{2} d V_{\boldsymbol{x}}
$$

Chai Santi

Numerade Educator

07:53

Problem 15

Let $K(t)$ denote the kinetic energy of a Newtonian fluid body with mass density $\rho_{0}$ and viscosity $\mu$ occupying a fixed region $D$. Here we generalize the estimate in Result $6.11$ to the case of a steady, non-conservative body force field per unit mass $\boldsymbol{b}$.
(a) Use the result of Exercise 14 to show
$$
\begin{aligned}
&\frac{d}{d t} K(t)+\mu \int_{D} \nabla^{x} \boldsymbol{v}: \nabla^{x} \boldsymbol{v} d V_{\boldsymbol{x}} \\
&\leq \frac{\alpha^{2}}{2} \int_{D} \rho_{0}|\boldsymbol{b}|^{2} d V_{\boldsymbol{x}}+\frac{1}{\alpha^{2}} K(t)
\end{aligned}
$$
where $\boldsymbol{v}$ is the spatial velocity field and $\alpha>0$ is an arbitrary constant.
(b) Use part (a) and the Poincaré inequality to show
$$
K(t) \leq K_{0} e^{-\mu t / \lambda \rho_{0}}+\frac{\lambda \rho_{0} g}{\mu}\left[1-e^{-\mu t / \lambda \rho_{0}}\right], \quad \forall t \geq 0
$$
Here $\lambda>0$ is the constant from the Poincaré inequality, $g \geq 0$ is a constant defined by $g=\frac{\lambda \rho_{0}}{2 \mu} \int_{D} \rho_{0}|\boldsymbol{b}|^{2} d V_{\boldsymbol{x}}$ and $K_{0}$ is the initial kinetic energy.
(c) Show that
$$
K(t) \leq \max \left(K_{0}, \frac{\lambda \rho_{0} g}{\mu}\right), \quad \forall t \geq 0
$$

Mahipal Kumawat

Numerade Educator