# Speed of sound in non-newtonian fluids

Recently I saw some videos of non-newtonian (shear-thickening) fluids like corn starch mixed with water(sometimes known as oobleck), where the fluid is placed on top of a speaker cone and starts to create odd shapes as it becomes thicker in some places. Now I wondered what would happen in the following situation:

Suppose a room with reflective walls is fully filled with such a fluid. On one side of the room a sound is produced by a speaker. On the other side of the room is a microphone. How will a sound wave propagate through such a substance? Does the fluid increase its viscosity before, during or after a wave moves through it? What will the speed-of-sound be in such a case? I've read that the speed of sound is much faster in solids than in liquids, but how is that for this kind of substance?

• Nonlinear behavior of oobleck is associated with shear stresses. Sound propagation is normal stress, so (I think) not much would be noticeable. – user23660 Oct 25 '13 at 3:56
• @simplelikeanegg shows below that the velocity of sound waves in a fluid doesn't depend upon viscosity. But as aside, seismic waves traveling through rock are affected by non-linear elastic moduli. – Mark Rovetta Feb 1 '14 at 4:08

I will give you mathematical background that explains where sound waves, as well as shear, and other waves come from in continuum approximations and why viscosity has no influence on such waves: In Eulerian form, the vector equations of motion for a fluid or solid continuum can be written as $$\frac{\partial \mathbf{q}}{\partial t} + \frac{\partial\mathbf{F}(\mathbf{q})}{\partial \mathbf{x}} = \vec{\nabla} \cdot \underline{\mathbf{S}}$$ where $\mathbf{q}$ is a vector of size $n$ and $\underline{\mathbf{S}} = \underline{\mathbf{S}}(\frac{\partial\mathbf{q}}{\partial\mathbf{x}})$ is a matrix of size $n \times 3$. For fluids you would typically have $\mathbf{q}=(\rho,\rho u,\rho v,\rho E)$ and solids $\mathbf{q}=(\rho,\rho u,\rho v,\rho E, \epsilon_x,\epsilon_y)$. The term $\mathbf{F}(\mathbf{q})$ represents conservative flux. It is important to note that $\mathbf{F}$ only depends on $\mathbf{q}$ directly and not on any of its spatial derivatives. Alternatively, $\underline{\mathbf{S}}$ depends only on spatial derivatives of $\mathbf{q}$. While this equation is a very complicated non-linear partial differential equation, for small disturbances it is entirely analogous to the simple advection-diffusion equation $$\frac{\partial q}{\partial t} + \lambda\frac{\partial q}{\partial x} = \kappa\frac{\partial^2 q}{\partial x^2}$$ with wave speed $\lambda$ and diffusion coefficient $\kappa$. To put the equations of motion for solids and fluids into a similar form simply apply the chain rule to yield $$\frac{\partial \mathbf{q}}{\partial t} + \frac{\partial\mathbf{F}(\mathbf{q})}{\partial \mathbf{q}}\frac{\partial \mathbf{q}}{\partial\mathbf{x}} = \frac{\partial \underline{\mathbf{S}}}{\partial(\partial\mathbf{q}/\partial \mathbf{x})}\vec{\nabla} \frac{\partial\mathbf{q}}{\partial\mathbf{x}}$$ Comparing this to the simple linear advection-diffusion equation we can extract all of the state dependent wave speeds from the Jacobian $\mathbf{\underline{J}}=\frac{\partial\mathbf{F}(\mathbf{q})}{\partial \mathbf{q}}$ by noting that if $\mathbf{F}(\mathbf{q})$ were constant then a transformation of variables according to it's eigenvectors will produce $n$ independent advection-diffusion equations with wave speeds corresponding to the eigenvalues of the Jacobian. Likewise, the right hand side corresponds strictly to diffusion like processes.