# 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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0answers
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.
0answers
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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 ...
1answer
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### 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 ...
0answers
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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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1answer
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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 ...
1answer
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### 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 ...
1answer
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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-...
2answers
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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 ...
2answers
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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 ...
2answers
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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 ...
0answers
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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}...
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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1answer
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### 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}) \...
1answer
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### 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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### 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 ...
1answer
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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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### 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 ...