An inhomogeneous fluid

3.2.2 An inhomogeneous fluid

An inhomogeneous fluid is one whose physical properties vary with position 5 . Let ρ(#x) and K(#x)

be the density and bulk modulus of such a fluid. Let P (#x) represent a pressure fluctuation from the ambient pressure P 0 in the fluid. Thus the total fluid pressure is P (#x) + P 0 . The bulk modulus is a material property defined by the relation

(3.9) The quantity θ is the volume strain or dilatation and is a measure of local fractional volume change.

P (#x, t) = −K(#x)θ(#x, t).

This relation says that the pressure fluctuations are directly proportional to induced volume changes. θ is related to particle velocity, #ν, though ∂ t θ=# ∇ · #ν. The minus sign in equation (3.9) is needed because an increase in pressure causes a decrease in volume and vice-versa. This is an example of a constitutive relation that defines how stress relates to strain.

We assume that the fluid is at rest except for those motions induced by the pressure disturbance P (#x, t). As with the vibrating string, the motion of the fluid is defined by Newton’s second law

d#ν(#x, t)

−# ∇P (#x, t) = ρ(#x)

dt

This says that spatially changing pressure causes local forces in the fluid that give rise to local particle accelerations. The minus sign is also necessary here because the local forces caused by the

5 In this context this is just a fancy way of saying force per unit length. It is unfortunate that inhomogeneous is used in at least two very different contexts in this theory. Here the word refers to physical parameters that vary with position. In the previous section the word referred to a pde with a source

term and that usage has nothing to do with this.

70 CHAPTER 3. WAVE PROPAGATION pressure gradient point in the direction opposite to the gradient. That is, the gradient points from

low to high pressure but the fluid will move from high to low pressure. The interpretation of the d dt ν requires some care. Generally in fluid dynamics, such time deriva-

tives consist of two terms as dT dt = ∂T ∂t + (#ν · # ∇)T where T denotes any scalar fluid property such as temperature. The first term describes local temporal changes (e.g. local heating) while the second describes changes due to new material being transported to the analysis location by fluid motion. For this reason, the second term is known as the convective term. The assumption that the fluid is at rest except for the motions caused by the passing pressure wave allows the convective term to be neglected. If the term is included, a nonlinear wave equation will result. Thus, equation (3.10) is rewritten approximately as

∂#ν(#x, t)

−# ∇P (#x, t) = ρ(#x)

∂t

A wave equation for pressure can be derived by taking the divergence of (3.11),

∂#ν(#x, t)

∂# ∇ · #ν(#x, t)

−# ∇·# ∇P (#x, t) = # ∇ρ(#x) ·

and substituting equation (3.11) into the first term on the right side and ∂ t of equation (3.9) into the second. This gives

∇ 2 ρ(#x) ∂ P (#x, t) − # P (#x, t) ∇ ln ρ(#x) · # ∇P (#x, t) −

=0 (3.13) where ∇ 2 =# ∇·# ∇ and 1 ρ ∇ρ = # # ∇ln ρ have been used. If the second term were absent in equation

K(#x)

∂t 2

(3.13) then we would have a classic scalar wave equation for pressure. This occurs exactly if ρ(#x) is constant. In many other cases when ρ(#x) varies slowly the term will be negligible. Neglecting the

∇ln ρ term then gives #

2 1 ∂ 2 ∇ P (#x, t) P (#x, t) −

(3.14) where v =

v 2 ∂t 2

ρ . Equation (3.14) is a scalar wave equation that approximately models the propa- gation of pressure waves through an inhomogeneous fluid. Though a number of assumptions have been made in deriving this result, it still has quite general application. Also, it is possible to use

a solution of (3.14) to develop an approximate solution to the more complex Equation (3.13). Let P (#x, t) be a solution to equation (3.14), then it can be used to approximately express the # ˆ ∇ ln ρ · # ∇P term in (3.13) to give

2 1 ∂ 2 P (#x, t)

(3.15) The term on the right does not depend upon the unknown P (#x, t) but rather plays the role of a

∇ P (#x, t) −

=# ∇ ln ρ(#x) · # ∇ˆ P (#x, t).

v 2 (#x)

∂t 2

source term in scalar wave equation. Thus the effect of strong density inhomogeneity introduces an approximate source term whose strength is proportional to the logarithmic gradient of density. This process can be repeated indefinitely with the solution from iteration n-1 being used to compute the effective source term for iteration n.

Exercise 3.2.2. Show that The volume dilatation, θ also satisfies a scalar wave equation. Exercise 3.2.3. Consider the 1-D inhomogeneous fluid. Write down the wave equation that ap-

71 proximatelymodels pressure waves in a long narrow cylinder.

3.3. FINITE DIFFERENCE MODELLING WITH THE ACOUSTIC WAVE EQUATION