Questions tagged [fluid-dynamics]

For questions about fluid dynamics which studies the flows of fluids and involves analysis and solution of partial differential equations like Euler equations, Navier-Stokes equations, etc. Tag with [tag:mathematical-physics] if necessary.

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14 views

Curves of equal speed-fluid dynamics [closed]

What are curves of equal speed, in fluid dynamics? Are they stream lines? I'm doing a basic course on fluids.
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Equation of Motion for a Point Vortex System. Need Help Solving 4 Simultaneous ODEs

Background and Statement of the Problem We consider the problem in $\mathbb{R^2}.$ As the title states, we are interested in the following equation involving the Hamiltonian of the point vortex system,...
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Effect of Green's function on aerodynamic lift and drag

I am trying to determine the effect of inserting a Green's function on the lift and drag exerted on a body inmersed in an inviscid, incompressible and irrotational flow in 2D. One concrete example is ...
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Lift and drag exerted by a vortex on a cylinder [closed]

How can we calculate the unsteady lift and drag exerted on a rigid circular cylinder of radius $a$ produced by a parallel line vortex of circulation $\Gamma$ in the presence of a uniform mean flow ...
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1answer
24 views

Circulation of a flow along a closed curve enclosing the $z$-axis

This is a problem we looked at it class the solution to which I still don't completely understand. We were asked to find the circulation along any closed curve $\Gamma$ enclosing the $z$-axis of the ...
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could you explain why we define generalised riemann invariants as below?

I don't understand why we define generalised riemann invariants in the equation? i try following idea like what we could do in linear case. $$U_t+AU_x =0 $$ $$L^{-1}U_t+\Lambda L^{-1}U_x =0$$ if we ...
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Understanding representation of Cauchy stress tensor for the simplest plane steady flow

Consider the simple problem of a flow between two plates, one at $x_2=0$ and one at $x_2=h$ with the bottom one held stationary and the top plate moving in the $x_1$ direction with velocity $V$. Also, ...
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Well-posedness of non-local equations. Deterministic. Singular. Reference request.

Any references would be much appreciated. I'm looking for some well-posedness results for flow problems of the following type (note my issue is that I'm considering a singular non-local term). Let $\...
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63 views

runge kutta 2 in python

I am trying to solve an equation in fluid mechanics using the runge-kutta 2 method, usually it seems quite doable but in this case its with x y and z and i cant seem to make the code. Here is what i ...
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Fluid Dynamics Rainfall

Imagine a uniform rectangular channel with width w and a flat bottom, through which water flows in the shallow water regime. What are the equations for conservation of mass and x-momentum when a ...
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Linearised Euler equations in rotating frame

The Euler momentum equation for an inviscid fluid of constant density $\rho$ is $\rho\left[\frac{Du}{Dt} + \underbrace{2\Omega\times u}_{\text{Coriolis }} + \underbrace{\Omega\times (\Omega \times x)}...
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Integration in cylindrical coordinate system

Context: I am trying to derive an equation given in a Journal of Fluid Mechanics paper (2.2). It deals with the analysis of an axisymmetric turbulent wake where cylindrical coordinate system has been ...
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54 views

Two time derivatives of kinetic energy of fluid

Suppose $D$ is a smooth domain, $\rho > 0$ is fluid density (constant) and $u \in C^1([0,1];D)$ is the fluid velocity. Let $K(t) = \frac{1}{2}\int_D \rho \vert u \vert^2 dV,~0\le t \le 1,$ be the ...
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Turbulent modelling and the reynold stress term

I have got three questions linked with one and another, which is associated with mathematics and derivations, concerning the field of fluid dynamics simulations and general cfd. Any help and step-by-...
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2answers
59 views

Physical intuition behind no extremum of a function

During many of the courses (my background is fluid dynamics), I have seen that if a function $\phi(x,y)$ is smooth and continuous and satisfies a diffusion/Laplace equation of the form: $$\frac{\...
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Variable mean curvature equation

I have been trying to analyse the shapes that in hydrostatics when surface tension and gravity balance each other. The theory in the absence of gravity is well studied, which leads to constant mean ...
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1answer
25 views

Convective derivative vs Divergence of velocity

What is the physical significance or difference between Convective derivative : $\vec{v} \cdot \nabla $ and the Divergence of velocity $\nabla \cdot \vec{v}$? I have understood the convective ...
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Conformal map onto the upper half of the complex plane.

I am currently enrolled in a basic complex analysis course. Teacher likes to ask questions as he explains (and grades our answers). Last class, we were examinating an example from Complex Analysis by ...
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Derivation of 2D wave equation using Navier Stokes incompressible motion equation

I have been searching for an answer to this question- could anyone please show me how to derive 2d wave equation using the Navier Stokes Equation for incompressible flow?
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Options to model fluid problem (Navier-Stokes)

I have read that there are many considerations for modeling turbulent fluids (eg LES or RAS models). But to model a fluid it is not enough to solve the Navier-Stokes equations using finite elements, ...
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1answer
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First order PDE from continuity equation

Given the following equation $$\frac{\partial \rho}{\partial t}=-\vec{\nabla}\cdot \left(\rho\vec{v}\right)$$ with $\vec{v}(\vec{r})=x\hat{e}_1-y\hat{e}_2$ we get $$\frac{\partial \rho}{\partial t}=-\...
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Matlab Problem: Badly scaled matrix, very small condition number RCOND when using Chebyshev discretization

I am using MATLAB for the following problem. I have the following problem statement: LHS * q = RHS * f. This can be rewritten to q = H *f with H = LHS\RHS; Hereby q and f are vectors, LHS and RHS ...
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Vena contracta effect, why can't streamlines change direction abruptly?

I am curious about the common explanation for the vena contracta effect that occurs as a flow moves around a sharp corner, or within a free jet of liquid issuing from a nozzle. The explanation goes ...
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Streamline functions for one-component velocity

I would like to plot the streamline function for the two-dimensional flow with only one non-zero velocity component ($v_x, v_y=0$). I have seen the comparable question, which has been asked here, but ...
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How can we justify this proof of local existence of a solution to Burgers' equation using the method of characteristics?

I want to solve \begin{align}&\forall(t,x)\in\Omega:\left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}\right)(t,x)=0;\tag5\\&\forall x\in\mathbb R:u(0,x)=u_0(x)\tag6,\end{align}...
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1answer
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Movement of a partially filled oil tanker

I came across the following question A partially filled oil tanker is being carried on a truck moving with constant-horizontal acceleration. What will happen to the level of oil? which had the ...
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Reducing the Navier Stokes Equation in the X- Direction

The Navier Stokes equation reduces to In the x- direction to u∂u∂x+v∂u∂y=−1ρ∂p∂x+ν(∂2u∂y2) (1) I am unsure how you dervive from the general form (2) to get equation (1), Could someone show me ...
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1D Unsteady Constant Area Duct Flow Problem

For 1D constant area isentropic flow(assume inviscid and adiabatic). Given the following relations: $$\rho = \frac{\partial \phi}{\partial x} $$ $$\rho u = -\frac{\partial \phi}{\partial t} $$ where $...
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1answer
36 views

Stokes flow for a falling sphere

I am following this document on Stokes flow. It is stated that "if we have a falling sphere, doubling the velocity will double $\sigma_{ij} (= -p\delta_{ij} + 2\mu e_{ij})$", but I am ...
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Cast my flow field into Hamiltonian system?

I have derived a flow field for a singularity located inside a circular domain. My boundary conditions and the geometry of the domain are such that it is far easier to derive a boundary correction ...
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1answer
109 views

Dam break problem for shallow water equations

This problem concerns the shallow water equations $\partial_tu+u\partial_xu+g\partial_xh=0$ $\partial_th+u\partial_xh+h\partial_xu=0$. where $g$ is a constant. Water of depth $h_0$ is at rest for $x &...
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1answer
78 views

Sphere falling in viscous fluid boundary condition

A sphere, radius $a$, falls at constant velocity $U_0$ through an incompressible viscous fluid. I am told to assume that \begin{equation} \mathbf{u} = U_0\mathbf{e}_z + \tilde{\mathbf{u}}(\mathbf{x}) \...
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1answer
50 views

Is $\bf{u} \times\bf{(\nabla \times u)}$ $=0$ in the contex for deriving Bernoulli’s theorem

Is $\bf{u} \times\bf{(\nabla \times u)}$ $=0$ I am trying to derive the to Bernoulli’s theorem for a a steady, inviscid, homogeneous, incompressible fluid. Using mass conservation: Using Euler's ...
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How do I derive a time-dependent velocity vector field and position vector field?

The velocity vector field can be described as follows: $$\vec{V} = u(x,y,z,t)\hat{i} + v(x,y,z,t)\hat{j} + w(x,y,z,t)\hat{k}$$ What I understand from this is that for ALL particles at some random ...
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Numerical scheme for unsteady 2D Navier-Stokes with variable density

I'm working on a project in wich I am supposed to simulate the 2D Navier-Stokes problem for a fluid which density may vary. In that setting, the equations are: $$\left\{\begin{array}{l} \frac{\partial ...
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2answers
82 views

How to derive Bernoulli's theorem for an elastic fluid from Lamb's equation

Here I state Lamb's version of the Euler's equation for an elastic fluid: $$ \partial_t \vec v + \vec \Omega \times \vec v + \vec \nabla \left( \frac{\|\vec v\|^2}{2} + \frac{p}{\rho} + \psi + \phi_b\...
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General solution for sound propagation in a semi-infinite pipe

I need to find the velocity potential $\Phi$ defined by $\vec{u}=\nabla\Phi$ in the domain $D=\{(x,y,z) : x^2+y^2\leq R^2, z\geq0\}$. We are considering sound propagation so $\Phi$ satisfies the wave ...
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1answer
63 views

Poiseuille flow: can I avoid the simplification that $u=u(y)$?

Let's consider a viscous, incompressible and irrotational fluid that flows in a canal in two dimensions. It is reasonable to assume that the velocity of the fluid has the form $\vec v = u(x,y,t) \hat ...
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A flow in three dimensions has velocity field $u=(cx,cy,-2cz)$ where $c$ is a positive constant. Find and sketch the streamlines through $(1,1,1)$

The velocity field is: $$u=(cx,cy,-2cz)$$ where $c$ is positive constant. Using the parametric equation: $\frac{dx}{u}=\frac{dy}{v}=\frac{dz}{w}=ds$] $\frac{dx}{ds}=u=cx$ --> $x=x_0 e^{cs}$ $\frac{...
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1answer
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When is a vector field the gradient of the pressure

In Frank White's book Fluid mechanics problem 4.27 (8th ed.) a 2D-velocity field is given as $$ \vec{v} = (2xy,-y^2) $$ Using Euler's equation (assuming stationary, incompressible flow and no ...
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$\log r$ term zero in solution to Laplace's equation in $2$D polars for flow around aerofoil?

The general form of the velocity potential $\phi$ in the separation of variables solution to Laplace's equation in $2$D polars is $\phi = A_0\log r + B_0\theta + \sum_{n=1}^{\infty} [A_n r^n\cos(n\...
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1answer
76 views

Riemann problem for Burgers' equation with both shock waves and rarefaction waves.

Given the inviscid Burgers' equation with piecewise initial data $$ u_t + u u_x = 0,\qquad u(x,0) = \left\lbrace \begin{aligned} &0 && \text{if } x<-1 \\ &2 && \text{...
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76 views

Where is the grad trace operator ($\nabla \operatorname{trace} A$ for matrix field $A$) used?

I recently stumbled upon the linear partial differential operator $$\mathcal{L}A\colon=\nabla \mathrm{tr} A\quad\text{for matrix-valued fields } A\colon\mathbb{R}^d\rightarrow\mathbb{R}^{d\times d}.$$ ...
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38 views

Material Derivative of a Line Integral?

I was wondering if there was an identity concerning the material derivative of a line integral, e.g., the material derivative of the zonal barotropic wind $u_{\tau}$ (which itself is the vertical-...
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Aubin-Lions Lemma

Let $\alpha \ge \frac{1}{2}$ and $d=2$ or $d=3$. If we consider the sequence $w^n$ satisfying \begin{align*} w^{n}\overset{*}{\rightharpoonup} w \quad\text{in}\quad L^{\infty}(0,T,B_{2,1}^{1+\frac{d}{...
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Lagrangian (particle) approach in 1D partial differential equations

I know that for Euler or Navier-Stokes equation (2D or 3D) there is possibility of Lagrange aprroach. I suppose that it means we take trajectories of particles in the fluid, instead of looking on ...
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1answer
34 views

Fréchet derivative of scalarfield operator $ f \mapsto \nabla f \cdot \sigma $?

Let $\sigma \in \mathfrak{X}(\Omega)$ be a fixed smooth vector field on an open domain $\Omega \subset \mathbb{R}^d$. Consider the operator $T\colon C^\infty(\Omega) \to C^\infty(\Omega)$ on the space ...
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Clarification of vector calculus for fluid mechanics

I have been reading on fluid dynamics and am currently deriving the Navier-Stockes Equations for a Generalized Newtonian Fluid. One of the expression I came across is the following: $\eta (\nabla \vec{...
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37 views

Show that a potential flow must satisfy Euler's Equations

I am trying to show that a potential flow must satisfy the incompressible Euler's equation: \begin{align} \rho \partial_t v + \rho (v \cdot \nabla ) v = - \nabla p \end{align} As the flow is steady we ...
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Potential energy of a compressed fluid given compressiblity and ratio of pressured and unpressured volume

Considering an idealized fluid with a certain compressibility (and the assumption that the compression behaves spring-like; i.e. hookes law), how can I compute the potential energy stored in the ...

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