Question

If we have a perfect gas of mass density \(\rho_{m} \mathrm{~kg} / \mathrm{m}^{3}\), and we assign a velocity \(\mathbf{U} \mathrm{m} / \mathrm{s}\) to each differential element, then the mass flow rate is \(\rho_{m} \mathbf{U} \mathrm{kg} /\left(\mathrm{m}^{2}-\mathrm{s}\right)\). Physical reasoning then leads to the continuity equation, \(\nabla \cdot\left(\rho_{m} \mathbf{U}\right)=-\partial \rho_{m} / \partial t .(a)\) Explain in words the physical interpretation of this equation. (b) Show that \(\oint_{s} \rho_{m} \mathbf{U} \cdot d \mathbf{S}=-d M / d t\), where \(M\) is the total mass of the gas within the constant closed surface \(S\), and explain the physical significance of the equation.

Short answer

Answer: The continuity equation, ∇⋅(ρₘU)=-∂ρₘ/∂t, physically represents the conservation of mass for a perfect gas, taking into account changes in density and mass flow rate. The integral expression, ∮ₛρₘU⋅dS=-dM/dt, states that the net mass flow rate through a constant closed surface S is equal to the negative rate of change of the total mass M inside that surface, highlighting the conservation of mass principle for a perfect gas.

Step-by-step solution

Understanding the terms and given equation

Before diving deep into the physical interpretation, let's understand the terms involved: 1. ρₘ: Mass density of the perfect gas (kg/m³) 2. U: Velocity of each differential element (m/s) 3. ρₘU: Mass flow rate (kg/(m²s)) 4. Continuity equation: ∇⋅(ρₘU)=-∂ρₘ/∂t

Physical interpretation of the continuity equation

The continuity equation, ∇⋅(ρₘU)=-∂ρₘ/∂t, expresses the conservation of mass for a perfect gas. It states that the sum of net mass flowing out of a given volume plus the rate of change of density with time should be zero. This is because the mass entering the volume should either stay within it or leave it, thereby keeping the total mass constant. In simpler terms, the equation ensures that mass is conserved in a system when taking into account changes in density and mass flow rate.

Proving the integral expression

Next, we need to prove that ∮ₛρₘU⋅dS=-dM/dt, where M is the total mass of the gas within the constant closed surface S. To do this, we'll use the divergence theorem. The divergence theorem states that: ∮ₛ(F⋅dS) = ∫∫∫_V(∇⋅F)dV In our case, F = ρₘU. Applying the divergence theorem, we get: ∮ₛ(ρₘU⋅dS) = ∫∫∫_V(∇⋅(ρₘU))dV We know from the continuity equation that ∇⋅(ρₘU)=-∂ρₘ/∂t, so: ∮ₛ(ρₘU⋅dS) = -∫∫∫_V(∂ρₘ/∂t)dV Now, the total mass of the gas within the constant closed surface S can be expressed as: M = ∫∫∫_V ρₘdV Differentiating M with respect to time, we get: dM/dt = ∫∫∫_V (∂ρₘ/∂t) dV Therefore, we have: ∮ₛρₘU⋅dS=-dM/dt

Physical significance of the integral expression

The equation ∮ₛρₘU⋅dS=-dM/dt has a clear physical significance. It states that the net mass flow rate through a constant closed surface S is equal to the negative rate of change of the total mass M inside that surface. In other words, if the mass within the volume is increasing, the net mass flow rate entering the surface is larger than the mass flow rate leaving it. Conversely, if the mass within the volume is decreasing, the net mass flow rate leaving the surface is larger than the mass flow rate entering it. This equation essentially highlights the conservation of mass principle for a perfect gas.

Key concepts

Mass Density

Mass density is an important concept in understanding the behavior of gases and fluids. It is defined as the mass per unit volume and is usually denoted by the symbol \(\rho_m\). For a perfect gas, the mass density is expressed in kilograms per cubic meter (kg/m³). This quantity indicates how much mass is present within a certain volume of gas.
Mass density can change with variations in pressure and temperature, especially for gases. In a perfect gas, as the pressure increases at a constant temperature, the density will also increase. Likewise, if the temperature rises at a constant pressure, the density decreases.
Key points about mass density:

• Higher density means more mass packed into a given volume.
• Mass density plays a crucial role in calculations involving mass flow rate.
• In the continuity equation, changes in mass density are linked to changes in flow behavior.

Perfect Gas

The term 'perfect gas' refers to an idealized model of a gas where the molecules do not interact except through elastic collisions. In this model, gases obey the ideal gas law, which can be expressed as \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is temperature.
A perfect gas helps simplify the analysis of fluid dynamics by ignoring complex molecular interactions and focusing only on the macroscopic properties of pressure, volume, and temperature. This allows us to use straightforward mathematical models, such as the continuity equation, to describe gas behavior under different conditions.
Remembering some essential attributes of a perfect gas can be helpful:

• No intermolecular forces act between gas atoms or molecules.
• Gas molecules behave independently of each other.
• The behavior of the gas is described by simple proportionality relations from the ideal gas law.

Conservation of Mass

Conservation of mass is a fundamental principle in physics and chemistry that asserts that mass is neither created nor destroyed in an isolated system. When applying this concept to fluid dynamics, it means that the amount of mass that enters a control volume must equal the amount of mass that exits that volume, plus any accumulation of mass within the volume over time.
The continuity equation is a direct application of the conservation of mass for fluid flow. It states that any change in mass flow out of a region corresponds to a rate of accumulation of mass inside that region. When written mathematically as \(abla \cdot (\rho_m \mathbf{U}) = -\partial \rho_m / \partial t\), the equation ensures that any imbalance in mass flow is due to changing density over time.
Main takeaways from the conservation of mass principle include:

• It is universally valid in all isolated physical systems.
• The principle helps derive the important equations of fluid dynamics.
• It explains how mass flow is related to changes in density and velocity of a gas.

Divergence Theorem

The divergence theorem is a powerful mathematical tool used in vector calculus to relate the flow of a vector field through a surface to the behavior of the field within the volume enclosed by that surface. It provides the link between the flux of a field across a closed surface and the divergence of the field throughout the volume.
For continuous functions and smooth surfaces, the divergence theorem states that the integral of a vector field's divergence over a volume is equal to the flow of the vector field through the surface that bounds the volume. This is expressed mathematically as
\[\oint_{S} \mathbf{F} \cdot d\mathbf{S} = \int\int\int_{V} (abla \cdot \mathbf{F}) dV\]
In the context of the continuity equation, the divergence theorem is used to express the relation between net mass flow across a surface and the change in mass within the surface.
Key insights into the divergence theorem:

• It connects surface integrals to volume integrals, making complex problems more manageable.
• The theorem is essential for deriving and validating conservation principles like mass conservation.
• Its application helps solve practical engineering problems involving fluid dynamics and electromagnetism.