Fluid dynamics is a branch of physics that studies the motion of liquids or gases, which involves analysis and solution of partial differential equations like Euler equations, Navier-Stokes equations, etc.

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Showing Bernoulli function is constant on streamlines

An incompressible inviscid fluid, under the influence of gravity, has the velocity field $$\textbf u = (− \cos(x)\sin(y), \, \, \sin(x)\cos(y), \, \, 0)$$ with the $z$-axis vertically upwards, where $g$ ...
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Using Euler's equation

An incompressible inviscid fluid, under the influence of gravity, has the velocity field $$\textbf u = (− \cos(x)\sin(y), \, \, \sin(x)\cos(y), \, \, 0)$$ with the $z$-axis vertically upwards, where $g$ ...
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51 views

Explain a physical problem by mathematics

I am not very good at physics, so I don't understand how to solve the following problem with vector calculus. A perfect incompressible fluid moves steadily under gravity around the outside of a fixed ...
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Prove Bernoulli Function is Constant on Streamline

I have an incompressible, inviscid fluid, under the influence of gravity, with a velocity potential: $$ \mathbf{u} = (-\cos(x)\sin(y), \sin(x)\cos(y), 0) $$ Using Euler's equations, $$ \mathbf{u} ...
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1answer
53 views

How do I determine if the equation is a conservation law?

We have the PDE $\frac{\partial u}{\partial t}+a(x,y)\frac{\partial u}{\partial x}+b(x,y)\frac{\partial u}{\partial y}=0$. What would be conditions on $a$ and $b$ for the equation to constitute a ...
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Linearizing Euler's continuity equations

I have three Euler equations (for a polytropic gas) and I'd like to linearize them to get 2 other equations. Any help would be appreciated! The 3 initial equations are: 1) ...
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40 views

Derivation with Euler's Equations

I have three equations as follows (for a polytropic gas): 1) $\displaystyle\quad\frac{\partial\rho}{\partial t}+\nabla\cdot\rho \mathbf{u} = 0$ 2) $\displaystyle\quad\rho \left( \frac{\partial ...
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13 views

Calculate volume using flow rate and time difference in a container

So bare with me, I am a computer scientist undergrad taking part in an engineering week and this is a calculation we require for our report. So I'd like you to imagine I am filling a cylindrical ...
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30 views

Physical difference between $\nabla^\bot\cdot u=0$, and $\nabla\cdot u^\bot=0$ and the existence of a scalar potential

If there exists a $2D$ vector field $u=u(x)=(u_1,u_2)$ such that $\nabla\cdot u=0$ is it equivalent to saying following? $$\nabla\cdot u=?(\nabla\cdot u)^\bot=\nabla^\bot\cdot u^\bot=\nabla\times ...
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Given these equations how would I go about finding the velocity potential?

An organ pipe of length $L$ is closed at $x=L$. the pressure at $x=0$ is made to vary according to the law $\rho_1 = \rho_0 \sin(nt)$. I just wanted to know what law $\rho_1 = \rho_0 \sin(nt)$ means, ...
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51 views

Material Derivative of the Gradient of a Scalar Field

Let $f$ be a scalar field that is continuous and does not vary along the flow, that is $D_t(f)=0$ where $D_t=\partial_t+\vec u\cdot\nabla$ where $\vec u$ is the incompressible velocity field (i.e ...
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26 views

A model for constant temperature of water in a container

I put some water in a container with initial temperature $T_0$ in a room, and the room's initial temperature is $T_a$. Now the container is filled to the maximum, so any more water coming in will ...
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13 views

How to Derive Material Derivative in Rotating Frame

I am currently working through "An Introduction to Dynamic Meteorology (5th Ed.)" by Holton and Hakim, and am looking for more detail on material derivatives in rotating reference frames than what ...
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30 views

Suitable change of variable to transform ODE into the standard form of Bessel's equation

I've been stuck on this for a while now. I need to use a suitable change of variable $r \rightarrow \xi$ to transform this ODE: $r^2V'' + rV' - (\frac{i\Omega r^2}{\nu}+1)V = 0$ into the standard ...
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17 views

fluid flow velocity , boundary conditions

can someone please help me with par(c)? $ u = U(1-\frac{a^{3}}{2r^{3}}) cos \theta $ - $ U(1+ \frac{a^{3}}{2r^{3}}) sin \theta $ and the boundary conditions are that r = a, u(r) =0, When sin ...
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10 views

fluid between two fixed walls equation help

I am not to sure, how to answer this question, but is it cause v = 0? how to show u is independent of x?
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Pertubation of Euler equation of inviscid flow in cylindrical coordinates

Guys I tried to derive pertubation of Eulers equation of inviscid flow but I suspect that I made an error somewhere as the derivation was long. Can someone help identify the error. The following is ...
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38 views

Equivalence of the $H^1$ norm and the energy entropy norm

Let $D \subset \mathbb R^2$ be a bounded domain with smooth boundary. Let $\mathbf v: D \to \mathbb R^2$ be a divergence free vector field tangent to the boundary, i.e., $\mbox{ div } \mathbf v = 0$ ...
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1answer
44 views

Negative divergence implies convergent flow?

Suppose we have a differentiable vector field $X:\Omega\to\mathbb{R^n}$ defined on an open, bounded and simply connected region subset $\Omega$ of $\mathbb{R^n}$, and its divergence is negative ...
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3answers
35 views

How to understand the equality about $(\mathbf{q}\cdot \mathrm{grad})\mathbf{q} $?

For the relation, $$(\mathbf{q}\cdot \mathrm{grad})\mathbf{q} = (\mathrm{curl}\mathbf{q})\times\mathbf{q}+\frac{1}{2}\mathrm{grad}|\mathbf{q}|^2,$$ is there any physics, geometry, or basic intuitive ...
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How can we describe the diffusion of “things” injected into a fluid?

Let $d\in\left\{2,3\right\}$ and $\Omega_t$ be the bounded set occupied by a fluid at time $t\ge 0$. Let $c\in\Omega_0$ be a particle and $$[0,\infty)\to\mathbb R^d\;,\;\;\;t\mapsto ...
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20 views

Fluid Flow and Uniform Convergence of Taylor Series

I'm reading through a text on fluid flow and Laplace's equation, and it makes a statement that I do not understand and would really like to clarify. Here's the setup: Let $u:A \rightarrow R^2$ a ...
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Using Runge-Kutta 4th order for a system of 5 ODE's

I'm an engineer and not quite familiar with solving systems of differential equations numerically, but I need to write a (fluid dynamics) program which contains a system of 4 implicit ODE's and an ...
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1answer
59 views

How can we describe the evolution of a density “injected” into an incompressible Newtonian fluid?

Let $d\in\left\{2,3\right\}$ and $\Omega\subseteq\mathbb R^d$ be a bounded domain. The evolution up to time $T>0$ of an incompressible Newtonian fluid with uniform density $\rho_0$ and viscosity ...
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80 views

Implementation of a simulation of an incompressible Newtonian fluid with uniform density

Let $d\in\left\{2,3\right\}$ and $\Omega\subseteq\mathbb R^d$ be a bounded domain. I want to simulate an incompressible Newtonian fluid with uniform density $\rho$ and viscosity $\nu$. The evolution ...
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21 views

how does replacing $x$ with $x-h^2t$ where $h=h(x,t)$ transform an equation?

I'm working on modelling a drip running down a wall, which is modelled at $t=0$ by $h=1+e^{-x^2}$ And as $t$ increases, I've found it to be modelled by $h=1+e^{-(x-h^2t)^2}$ Obviously, subbing ...
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14 views

How would the following function behave?

I've found a function describing how the free surface of a fluid moves down a wall, and I know our solution is of the from $f(x-h^2t)$. The question then advised me to choose a shape of the form ...
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1answer
16 views

Show $\delta_{KL}$ is a Cartesian tensor

By using the definition of kronecker delta $\delta_{KL}$, show that $\delta_{KL}$ is a Cartesian tensor, that is $$\delta ' _{MN}=L_{MK}L_{NL} \delta_{KL}$$ under the rotation $X_K=L_{MK}X' _M$. I ...
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15 views

Similarity solutions from boundary layer equations - choosing appropriate powers

I've found a governing equation in terms of a streamfunction $\psi$ to be $\frac {\partial \psi} {\partial y} \frac {\partial^2 \psi} {\partial x \partial y} - \frac {\partial \psi} {\partial x} ...
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How to solve $\nabla(u\cdot u)=\lambda u$

Suppose $C^2(\mathbb R^n)$ vector field $u(x)$, how to solve this pde? $$\nabla(u\cdot u)=\lambda u$$ for some constant $\lambda$. The physical intuition is that: The left hand side of the equation ...
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Transforming an infinitesimal line element, dx, to 1/2(curl(u)/\dx)? What does this mean physically?

Consider transforming an infinitesimal line element,say dx, to 1/2(curl(u)/\dx)? Where curl denoted /\ here, and dx is an infinitesimal 3d vector, and u is the displacement vector --What does this ...
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Treatment of Linear Terms with Truncated Fourier Spectral Method

Consider some PDE which can be decomposed into linear and nonlinear terms, i.e.: $$\frac{\partial u}{\partial t} + \mathcal{L}u + \mathcal{N}(u,u) = 0$$ I am trying to solve this type of problem ...
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1answer
31 views

Show $x_{l,L}X_{L,k}=\delta_{kl}$

In lectures we had $$x_{l,L}=\frac{\partial x_l}{\partial X_L}$$ and $$X_{L,k}=\frac{\partial X_L}{\partial x_k}$$ So this is what I THOUGHT it would be but i was told that can't cancel the partial ...
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35 views

What is a Complex Velocity Potential?

I was recently introduced to Complex Velocity Potentials in Fluid Mechanics without much explanation. We were told, given a stream function $\psi$ for a flow, and a velocity potential $\phi$, the ...
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1answer
26 views

General method for solving a second order homogenous linear ODE with non constant coefficients?

$$y'' +f(r)y' + g(r)y = 0$$ Is the form of the equation I'm given. I'm using $r$ instead of a more conventional $t$ or $x$ because it pertains to fluid dynamics with $r$ being some radius. I'm just ...
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1answer
30 views

Chain rule applies to Navier-Stokes Equations in polar coordinates

Based on the fact that $p=p(r)$ only, consider the Navier-Stokes equations and show that the $e_\theta$ component reduces to I can get to this if the following is true $\frac {1} {r} \frac ...
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Velocity profile for a fluid with no net volume flux

I've established a relationship between the upper horizontal boundary layer $U$ and pressure gradient $G$ where $\mu$ is dynamic viscosity and h is the height of the pipe the fluid is travelling ...
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1answer
31 views

Average velocity of fluid flow

Washburn equation for fluid flow in horizontal capillary is given by : $\frac{dL}{dt} = \frac{γR}{4µL}$ Find the time dependencies of length of travel, $L(T)$ and average velocity. I can find the ...
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1answer
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Meaning of notation: $x_{l,L}$

The question is show that $$x_{l,L}X_{L,k}=\delta _{kl}$$ but what does that notation even mean?? The topic of this is continuum mechanics.
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1answer
19 views

Stream Function in Sobolev Space

Assume $\Omega$ is a multi-connected bounded domain in $\mathbb{R}^2$, and function $v\in W^{1,2}_{0}(\Omega)$ satisfies: $$\text{div} \; v=0$$ Then there exists a stream function $\psi\in ...
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1answer
103 views

Prove $l_{ip}=\frac{\partial \bar x_i}{\partial x_p}=\frac{\partial x_p}{\partial \bar x_i}$

Let $l$ be a rotation tensor such that $$\bar x_i=l_{ip}x_p$$ where $l_{ip}$ is the direction cosine between the unit vectors in the component directions $x_p$ and $\bar x_i$. Prove that ...
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Identity in Deriving Bernoulli's Equation for Barotropic Flows

I'm trying to understand a derivation of Bernoilli's Equation and I'm having a hard time understanding the math behind this indentity $$ \frac{1}{\rho} \nabla p = \nabla \int ...
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Pressure and Surface Integration

I have a disc of which a pressure is applied across its surface. My equation for pressure, though, is a function of radius, i.e. $r$, only: $$p = p(r) $$ My task is to calculate the pressure across ...
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1answer
32 views

Showing a rotational symmetry of the Navier Stokes / Euler equation

I'm going through Majda's "Vorticity and Incompressible Flow" and am having a hard time verifying what looks like an easy equality; I suspect its due to my very poor understanding/familiarity of ...
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1answer
30 views

Fluids contained in a closed cylinder

While reading a fluids book for my course and doing some problems I came across this question that the book didn't provide an answer for and was wondering if one of you kind folks could help me out ...
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43 views

Application of the Bernoulli Equation

I'm attempting a question on fluid dynamics and I'm using a rearranged form of the Bernoulli Equation. But I can't prove the equation for the velocity of water leaving the tap. It should equal $$v = ...
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Lagrange Multiplier FEM for Navier-Stokes

I'm trying to derive the weak formulation for the Navier-Stokes equations with boundaries imposed via Lagrange Multipliers. This technique is used by Urquiza 2014 for the Stokes equations. It's done ...
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14 views

Evaluating gradiant of vector basis r in Spherical Co-ordinates

If I were to define $\nabla$ in spherical polar co-ordinates: $$ ...
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80 views

Navier-Stokes on concentric cylinders

Consider incompressible fluid flowing between two fixed concentric cylinders of radii a and b with b>a, and length L. Radial distance is measured by r and axial distance along the cylinders is ...
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1answer
105 views

Dot product with Del operator in Cylindrical Coordinates?

It's not hard to derive the equation for the Del operator in cylindrical coordinates from the Del operator in cartesian coordinates. From $$\nabla = \hat{\bf{x}} \frac{\partial}{\partial x} + ...